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Guided Waves in Layered Media

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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.

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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.

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Symmetric Slab Waveguides The following equation describes the index profile of a symmetric dielectric waveguide:

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Symmetric Slab Waveguides The propagation of monochromatic radiation along the z axis. Maxwell’s equation can be written in the form

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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

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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.

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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.

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Guided TE Modes The electric field amplitude of the guided TE modes can be written in the form

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Guided TE Modes The mode function is taken as

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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.

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Guided TE Modes The propagation constants of the TE modes can be found from a numerical or graphical solution.

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Guided TE Modes

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Guided TM Modes The field amplitudes are written

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Guided TM Modes The wavefunction H(x) is

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Guided TM Modes The continuity of Hy and Ez at the two (x=±(1/2)d) interface leads to the eigenvalue equation

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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

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Asymmetric Slab Waveguides Typical field distributions corresponding to different values of β

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Guided TE Modes The field component E y of the TE mode can be written as The function Em(x) assumes the following forms in each of the three regions

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Guided TE Modes By imposing the continuity requirements, we get or

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Guided TE Modes The normalization condition is given by Or equivalently,

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Guided TM Modes The field components are

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Guided TM Modes The wavefunction is

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Guided TM Modes

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We define the parameter At long wavelengths, such that No confined mode exists in the waveguide.

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Guided TM Modes As the wavelength decreases such that One solution exists to the mode condition

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Guided TM Modes As the wavelength decreases further such that Two solutions exist to the mode condition

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Guided TM Modes The mth satifies

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Surface Plasmons Confined propagation of electromagnetic radiation can also exist at the interface between two semi- infinite 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.

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Surface Plasmons A typical example is the interface between air and silver where n1^2 = 1 and n3^2 = -16.40- i0.54 at λ = 6328( 艾 ).

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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.

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Surface Plasmons For TM waves, the mode funcion H y (x) can be written as The mode condition can be obtained by insisting on the continuity of E z at the interface x = 0 of from Eq.(11.2-11)

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Surface Plasmons The propagation constant β is given by A confined propagation mode must have a real propagation constant, since <0,

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Surface Plasmons The attenuation constants p and q are given

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Surface Plasmons The electric field components are given

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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.

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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

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Surface Plasmons In terms of the dielectric constants The propagation and attenuation constants can be written as

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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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