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
• 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 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 n1^2 = 1 and n3^2 = -16.40i0.54 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