Transverse mode
Transverse modes occur in radio waves and microwaves confined to a waveguide, and also in light waves in an optical fiber and in a laser's optical resonator.
[1] Transverse modes occur because of boundary conditions imposed on the wave by the waveguide.
For example, a radio wave in a hollow metal waveguide must have zero tangential electric field amplitude at the walls of the waveguide, so the transverse pattern of the electric field of waves is restricted to those that fit between the walls.
The allowed modes can be found by solving Maxwell's equations for the boundary conditions of a given waveguide.
Unguided electromagnetic waves in free space, or in a bulk isotropic dielectric, can be described as a superposition of plane waves; these can be described as TEM modes as defined below.
However in any sort of waveguide where boundary conditions are imposed by a physical structure, a wave of a particular frequency can be described in terms of a transverse mode (or superposition of such modes).
In coaxial cable energy is normally transported in the fundamental TEM mode.
The TEM mode is also usually assumed for most other electrical conductor line formats as well.
This is mostly an accurate assumption, but a major exception is microstrip which has a significant longitudinal component to the propagated wave due to the inhomogeneity at the boundary of the dielectric substrate below the conductor and the air above it.
In an optical fiber or other dielectric waveguide, modes are generally of the hybrid type.
In circular waveguides, circular modes exist and here m is the number of full-wave patterns along the circumference and n is the number of half-wave patterns along the diameter.
are the refractive indices of the core and cladding, respectively.
[4] Decomposition of field distributions into modes is useful because a large number of field amplitudes readings can be simplified into a much smaller number of mode amplitudes.
Because these modes change over time according to a simple set of rules, it is also possible to anticipate future behavior of the field distribution.
These simplifications of complex field distributions ease the signal processing requirements of fiber-optic communication systems.
[5] The modes in typical low refractive index contrast fibers are usually referred to as LP (linear polarization) modes, which refers to a scalar approximation for the field solution, treating it as if it contains only one transverse field component.
[6] In a laser with cylindrical symmetry, the transverse mode patterns are described by a combination of a Gaussian beam profile with a Laguerre polynomial.
The intensity at a point (r,φ) (in polar coordinates) from the centre of the mode is given by:
where ρ = 2r2/w2, Llp is the associated Laguerre polynomial of order p and index l, and w is the spot size of the mode corresponding to the Gaussian beam radius.
It is the fundamental transverse mode of the laser resonator and has the same form as a Gaussian beam.
The pattern has a single lobe, and has a constant phase across the mode.
Modes with increasing p show concentric rings of intensity, and modes with increasing l show angularly distributed lobes.
The overall size of the mode is determined by the Gaussian beam radius w, and this may increase or decrease with the propagation of the beam, however the modes preserve their general shape during propagation.
In many lasers, the symmetry of the optical resonator is restricted by polarizing elements such as Brewster's angle windows.
In these lasers, transverse modes with rectangular symmetry are formed.
These modes are designated TEMmn with m and n being the horizontal and vertical orders of the pattern.
The electric field pattern at a point (x,y,z) for a beam propagating along the z-axis is given by[7]
are the waist, spot size, radius of curvature, and Gouy phase shift as given for a Gaussian beam;
The phase of each lobe of a TEMmn is offset by π radians with respect to its horizontal or vertical neighbours.
The overall intensity profile of a laser's output may be made up from the superposition of any of the allowed transverse modes of the laser's cavity, though often it is desirable to operate only on the fundamental mode.
