A
transverse mode of a beam of
electromagnetic radiationElectromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...
is a particular electromagnetic field pattern of radiation measured in a plane perpendicular (i.e., transverse) to the propagation direction of the beam. Transverse modes occur in
radioRadio is the transmission of signals through free space by modulation of electromagnetic waves with frequencies below those of visible light. Electromagnetic radiation travels by means of oscillating electromagnetic fields that pass through the air and the vacuum of space...
waves and
microwaveMicrowaves, a subset of radio waves, have wavelengths ranging from as long as one meter to as short as one millimeter, or equivalently, with frequencies between 300 MHz and 300 GHz. This broad definition includes both UHF and EHF , and various sources use different boundaries...
s confined to a
waveguideA waveguide is a structure which guides waves, such as electromagnetic waves or sound waves. There are different types of waveguides for each type of wave...
, and also in
lightLight or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...
waves in an
optical fiberAn optical fiber is a flexible, transparent fiber made of a pure glass not much wider than a human hair. It functions as a waveguide, or "light pipe", to transmit light between the two ends of the fiber. The field of applied science and engineering concerned with the design and application of...
and in a
laserA laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...
's optical resonator.
Transverse modes occur because of boundary conditions imposed on the wave by the waveguide. For example, a radio wave in a hollow metal
waveguideA waveguide is a structure which guides waves, such as electromagnetic waves or sound waves. There are different types of waveguides for each type of wave...
must have zero tangential
electric fieldIn physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
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. For this reason, the modes supported by a waveguide are
quantizedIn physics, quantization is the process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as "quantum mechanics". It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the...
. The allowed modes can be found by solving
Maxwell's equationsMaxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
for the boundary conditions of a given waveguide.
Types of modes
Transverse modes are classified into different types:
- TE modes (Transverse Electric) no electric field in the direction of propagation.
- TM modes (Transverse Magnetic) no magnetic field in the direction of propagation.
- TEM modes (Transverse ElectroMagnetic) neither electric nor magnetic field in the direction of propagation.
- Hybrid modes nonzero electric and magnetic fields in the direction of propagation.
Some authors use an alternate notation;
- H modes have a magnetic field component in the direction of propagation. H modes are equivalent to TE modes.
- E modes have an electric field component in the direction of propagations. E modes are equivalent to TM modes.
In rectangular waveguide, rectangular mode numbers are designated by two suffix numbers attached to the mode type, such as TE
mn, where
m is the number of half-wavelengths across the width of the waveguide and
n is the number of half-wavelengths across the height of the waveguide. In circular waveguide circular modes exist and here
m is the number of half-wavelengths along a half-circumference and
n is the number of half-wavelengths along a radius.
Modes of hollow metallic waveguides filled with a homogeneous, isotropic material fall into the first two categories. Otherwise, except in cases of special symmetry, modes are generally of hybrid type. For example, light travelling in an optical fiber or other dielectric waveguide forms hybrid-type modes. The fiber modes are usually referred to as
LP (linear polarization) modes, which refers to a
scalarIn physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations . This is in contrast to a vector...
approximation for the field solution, treating it as if it contains only one transverse field component (this is accurate because of the low refractive index contrast in typical fibers), the transverse electromagnetic (TEM) type. A planar Fabry-Perot resonator or etalon can also exhibit linearly polarized TEM modes. A resonator employing curved mirrors cannot support a TEM mode. Such modes are almost always hybrid except for the special cases of the electric field polarized either radially or azimuthally. The former case corresponds to a TM mode, the latter to a TE mode. Equivalently, linear polarization and TEM modes are mutually exclusive for a Gaussian beam or any other beam with curved wavefronts.
Laser modes
In a laser with cylindrical symmetry, the transverse mode patterns are described by a combination of a
Gaussian beamIn optics, a Gaussian beam is a beam of electromagnetic radiation whose transverse electric field and intensity distributions are well approximated by Gaussian functions. Many lasers emit beams that approximate a Gaussian profile, in which case the laser is said to be operating on the fundamental...
profile with a
Laguerre polynomialIn mathematics, the Laguerre polynomials, named after Edmond Laguerre ,are the canonical solutions of Laguerre's equation:x\,y + \,y' + n\,y = 0\,which is a second-order linear differential equation....
. The modes are denoted TEM
pl where
p and
l are integers labeling the radial and angular mode orders, respectively. The intensity at a point
r,φ (in polar coordinates) from the centre of the mode is given by:
where ρ = 2
r2/
w2, and
Lpl is the associate Laguerre polynomial of order
p and index
l.
w is the spot size of the mode corresponding to the Gaussian beam radius.
With
p=
l=0, the TEM
00 mode is the lowest order, or 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
phasePhase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...
across the mode. Modes with increasing
p show concentric rings of intensity, and modes with increasing
l show angularly distributed lobes. In general there are 2
l(
p+1) spots in the mode pattern (except for
l=0). The TEM
0i* mode, the so-called
doughnut mode, is a special case consisting of a superposition of two TEM
0i modes (i=1,2,3), rotated 360°/4i with respect to one another.
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. Higher order modes are relatively larger compared to the TEM
00 mode, and thus the fundamental Gaussian mode of a laser may be selected by placing an appropriately sized aperture in the laser cavity.
In many lasers, the symmetry of the optical resonator is restricted by
polarizing elementsA polarizer is an optical filter that passes light of a specific polarization and blocks waves of other polarizations. It can convert a beam of light of undefined or mixed polarization into a beam with well-defined polarization. The common types of polarizers are linear polarizers and circular...
such as
Brewster's angleBrewster's angle is an angle of incidence at which light with a particular polarization is perfectly transmitted through a transparent dielectric surface, with no reflection. When unpolarized light is incident at this angle, the light that is reflected from the surface is therefore perfectly...
windows. In these lasers, transverse modes with rectangular symmetry are formed. These modes are designated TEM
mn with
m and
n being the horizontal and vertical orders of the pattern. The intensity at point
x,
y is given by:
where H
m(
x) is the
mth order
Hermite polynomialIn mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; in numerical analysis as Gaussian quadrature; and in physics, where...
.
The TEM
00 mode corresponds to exactly the same fundamental mode as in the cylindrical geometry. Modes with increasing
m and
n show lobes appearing in the horizontal and vertical directions, with in general (
m+1)(
n+1) lobes present in the pattern. As before, higher-order modes have a larger spatial extent than the 00 mode.
The
phasePhase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...
of each lobe of a TEM
mn is offset by π radians with respect to its horizontal or vertical neighbours. This is equivalent to the polarization of each lobe being flipped in direction.
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.
Modes in an optical fiber
The number of modes in an optical fiber distinguishes
multi-mode optical fiberMulti-mode optical fiber is a type of optical fiber mostly used for communication over short distances, such as within a building or on a campus...
from
single-mode optical fiberIn fiber-optic communication, a single-mode optical fiber is an optical fiber designed to carry only a single ray of light . Modes are the possible solutions of the Helmholtz equation for waves, which is obtained by combining Maxwell's equations and the boundary conditions...
. To determine the number of modes in a step-index fiber, the
V number needs to be determined:
where
is the
wavenumberIn the physical sciences, the wavenumber is a property of a wave, its spatial frequency, that is proportional to the reciprocal of the wavelength. It is also the magnitude of the wave vector...
,
is the fiber's core radius, and
and
are the
refractive indicesIn optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....
of the core and
claddingCladding is one or more layers of material of lower refractive index, in intimate contact with a core material of higher refractive index. The cladding causes light to be confined to the core of the fiber by total internal reflection at the boundary between the two. Light propagation in the...
, respectively. Fiber with a V-parameter of less than 2.405 only supports the fundamental mode (a hybrid mode), and is therefore a single-mode fiber whereas fiber with a higher V-parameter has multiple modes.
http://eeclass.stanford.edu/ee247/
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 processingSignal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
requirements of fiber-optic communication systems.
http://www.rp-photonics.com/modes.html
External links