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A transverse mode of
electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) li ...
is a particular electromagnetic field pattern of the radiation in the plane perpendicular (i.e., transverse) to the radiation's propagation direction. Transverse modes occur in
radio Radio is the technology of signaling and communicating using radio waves. Radio waves are electromagnetic waves of frequency between 30 hertz (Hz) and 300 gigahertz (GHz). They are generated by an electronic device called a transmit ...
waves and
microwave Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ra ...
s confined to a
waveguide A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave intensities de ...
, and also in
light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 tera ...
waves in an
optical fiber An optical fiber, or optical fibre in Commonwealth English, is a flexible, transparent fiber made by drawing glass ( silica) or plastic to a diameter slightly thicker than that of a human hair. Optical fibers are used most often as a mea ...
and in a
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fir ...
's
optical resonator An optical cavity, resonating cavity or optical resonator is an arrangement of mirrors or other optical elements that forms a cavity resonator for light waves. Optical cavities are a major component of lasers, surrounding the gain medium and provi ...
. Transverse modes occur because of
boundary condition In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
s 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. For this reason, the modes supported by a waveguide are quantized. The allowed modes can be found by solving
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
for the boundary conditions of a given waveguide.


Types of modes

Unguided electromagnetic waves in free space, or in a bulk
isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
dielectric In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mate ...
, can be described as a superposition of
plane wave In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, ...
s; these can be described as TEM modes as defined below. However in any sort of
waveguide A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave intensities de ...
where boundary conditions are imposed by a physical structure, a wave of a particular frequency can be described in terms of a transverse
mode Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Arts and entertainment * '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine * ''Mode'' magazine, a fictional fashion magazine which is ...
(or superposition of such modes). These modes generally follow different
propagation constant The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a c ...
s. When two or more modes have an identical propagation constant along the waveguide, then there is more than one modal decomposition possible in order to describe a wave with that propagation constant (for instance, a non-central
Gaussian Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
laser mode can be equivalently described as a superposition of Hermite-Gaussian modes or
Laguerre-Gaussian modes In optics, a Gaussian beam is a beam of electromagnetic radiation with high monochromaticity whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This ...
which are described below).


Waveguides

Modes in waveguides can be classified as follows: ; Transverse electromagnetic (TEM) modes: Neither electric nor magnetic field in the direction of propagation. ; Transverse electric (TE) modes: No electric field in the direction of propagation. These are sometimes called ''H modes'' because there is only a magnetic field along the direction of propagation (''H'' is the conventional symbol for magnetic field). ; Transverse magnetic (TM) modes: No magnetic field in the direction of propagation. These are sometimes called ''E modes'' because there is only an electric field along the direction of propagation. ; Hybrid modes: Non-zero electric and magnetic fields in the direction of propagation. ''See also ''. Hollow metallic waveguides filled with a homogeneous, isotropic material (usually air) support TE and TM modes but not the TEM mode. In
coaxial cable Coaxial cable, or coax (pronounced ) is a type of electrical cable consisting of an inner conductor surrounded by a concentric conducting shield, with the two separated by a dielectric ( insulating material); many coaxial cables also have a p ...
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 Microstrip is a type of electrical transmission line which can be fabricated with any technology where a conductor is separated from a ground plane by a dielectric layer known as the substrate. Microstrip lines are used to convey microwave-frequ ...
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 rectangular waveguides, rectangular mode numbers are designated by two suffix numbers attached to the mode type, such as TE''mn'' or TM''mn'', where ''m'' is the number of half-wave patterns across the width of the waveguide and ''n'' is the number of half-wave patterns across the height of the waveguide. 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.


Optical fibers

The number of modes in an optical fiber distinguishes
multi-mode optical fiber Multi-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. Multi-mode links can be used for data rates up to 100 Gbit/s. Multi-mode fiber has a fairly large ...
from
single-mode optical fiber In fiber-optic communication, a single-mode optical fiber (SMF), also known as fundamental- or mono-mode, is an optical fiber designed to carry only a single mode of light - the transverse mode. Modes are the possible solutions of the Helmholtz ...
. To determine the number of modes in a step-index fiber, the V number needs to be determined: V = k_0 a \sqrt where k_0 is the
wavenumber In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
, a is the fiber's core radius, and n_1 and n_2 are the
refractive indices In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
of the core and
cladding Cladding is an outer layer of material covering another. It may refer to the following: *Cladding (boiler), the layer of insulation and outer wrapping around a boiler shell *Cladding (construction), materials applied to the exterior of buildings ...
, 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. 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 Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
requirements of fiber-optic communication systems. The modes in typical low refractive index contrast fibers are usually referred to as ''LP'' (linear polarization) modes, which refers to a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
approximation for the field solution, treating it as if it contains only one transverse field component.K. Okamoto, ''Fundamentals of Optical Waveguides'', pp. 71–79, Elsevier Academic Press, 2006, .


Lasers

In a laser with cylindrical symmetry, the transverse mode patterns are described by a combination of a
Gaussian beam In optics, a Gaussian beam is a beam of electromagnetic radiation with high monochromaticity whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity (irradiance) profile. Thi ...
profile with a
Laguerre polynomial In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation: xy'' + (1 - x)y' + ny = 0 which is a second-order linear differential equation. This equation has nonsingular solutions only ...
. The modes are denoted where and are integers labeling the radial and angular mode orders, respectively. The intensity at a point (in
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
) from the centre of the mode is given by: I_ (\rho, \varphi) = I_0 \rho^l \left _p^l (\rho)\right2 \cos^2 (l\varphi) e^ where , is the associated
Laguerre polynomial In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation: xy'' + (1 - x)y' + ny = 0 which is a second-order linear differential equation. This equation has nonsingular solutions only ...
of order and index , and is the spot size of the mode corresponding to the Gaussian beam radius. With , the TEM00 mode is the lowest order. 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 Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform * Phase space, a mathematic ...
across the mode. Modes with increasing show concentric rings of intensity, and modes with increasing show angularly distributed lobes. In general there are spots in the mode pattern (except for ). The mode, the so-called ''doughnut mode'', is a special case consisting of a superposition of two modes (), rotated with respect to one another. The overall size of the mode is determined by the Gaussian beam radius , 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 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 elements such as
Brewster's angle Brewster's angle (also known as the polarization 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 ...
windows. In these lasers, transverse modes with rectangular symmetry are formed. These modes are designated with and being the horizontal and vertical orders of the pattern. The electric field pattern at a point for a beam propagating along the z-axis is given by E_(x, y, z) = E_0 \frac H_m\left(\frac\right) H_n\left(\frac\right)\exp\left (x^2 + y^2) \left(\frac + \frac\right) - jkz - j(m + n + 1)\zeta\right/math> where w_0, w(z), R(z), and \zeta(z) are the waist, spot size, radius of curvature, and
Gouy phase shift In optics, a Gaussian beam is a Light beam, beam of electromagnetic radiation with high Monochrome, monochromaticity whose Envelope (waves), amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussi ...
as given for a
Gaussian beam In optics, a Gaussian beam is a beam of electromagnetic radiation with high monochromaticity whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity (irradiance) profile. Thi ...
; E_0 is a normalization constant; and H_k is the -th physicist's
Hermite polynomial In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well as ...
. The corresponding intensity pattern is I_(x, y, z) = I_0 \left( \frac \right)^2 \left H_m \left( \frac \right) \exp \left( \frac \right) \right2 \left H_n \left( \frac \right) \exp \left( \frac \right) \right2 The TEM00 mode corresponds to exactly the same fundamental mode as in the cylindrical geometry. Modes with increasing and show lobes appearing in the horizontal and vertical directions, with in general lobes present in the pattern. As before, higher-order modes have a larger spatial extent than the 00 mode. The
phase Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform * Phase space, a mathematic ...
of each lobe of a is offset by 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.


See also

*
Normal mode A normal mode of a dynamical system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies. ...
*
Longitudinal mode A longitudinal mode of a resonant cavity is a particular standing wave pattern formed by waves confined in the cavity. The longitudinal modes correspond to the wavelengths of the wave which are reinforced by constructive interference after man ...
*
Laser beam profiler A laser beam profiler captures, displays, and records the spatial intensity profile of a laser beam at a particular plane transverse to the beam propagation path. Since there are many types of lasers — ultraviolet, visible, infrared, contin ...
*
Spatial filter A spatial filter is an optical device which uses the principles of Fourier optics to alter the structure of a beam of light or other electromagnetic radiation, typically coherent laser light. Spatial filtering is commonly used to "clean up" the o ...
*
Transverse wave In physics, a transverse wave is a wave whose oscillations are perpendicular to the direction of the wave's advance. This is in contrast to a longitudinal wave which travels in the direction of its oscillations. Water waves are an example of t ...


References


External links


Detailed descriptions of laser modes
{{DEFAULTSORT:Transverse Mode Wave mechanics Electromagnetic radiation Laser science