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 theguiding layer must be greaterthan those of the surroundmedia.

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

• n2 is greater than n1 and n3, assuming n1<n3<n2

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 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.

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 ).

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