A
transverse mode of a beam of
electromagnetic radiationElectromagnetic radiation is a ubiquitous phenomenon that takes the form of self-propagating waves in a vacuum or in matter. It consists of electric and magnetic field components which oscillate in phase perpendicular to each other and perpendicular to the direction of energy propagation...
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 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 are electromagnetic waves with wavelengths ranging from as long as one meter to as short as one millimeter, or equivalently, with frequencies between 300MHz and 300 GHz. This is an extremely broad definition including 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 waveguide for each type of wave...
, and also in
lightLight is electromagnetic radiation, particularly radiation of a wavelength that is visible to the human eye ....
waves in an optical fibre and in a
laserA laser is a device that emits light through a process called stimulated emission. Laser light is usually spatially coherent, which means that the light either is emitted in a narrow, low-divergence beam, or can be converted into one with the help of optical components such as lenses...
'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 waveguide for each type of wave...
must have zero tangential
electric fieldIn physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. This electric field exerts a force on other electrically charged objects...
amplitude at the walls of the waveguide, and so the transverse pattern of the electric field of waves is restricted to those which 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 four partial differential equations that relate the electric and magnetic fields to their sources, charge density and current density. These equations can be combined to show that light is an electromagnetic wave...
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) no electric nor magnetic field in the direction of propagation.
- Hybrid modes nonzero electric and magnetic fields in the direction of propagation.
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 . -Physical quantity:...
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 described by Gaussian functions. Many lasers emit beams with a Gaussian profile, in which case the laser is said to be operating on the fundamental transverse mode, or...
profile with a
Laguerre polynomialIn mathematics, the Laguerre polynomials, named after Edmond Laguerre ,are the canonical solutions of Laguerre's equation: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
phaseThe phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0. Phase is a frequency domain or Fourier transform domain concept, and as such, can be readily understood in terms of simple harmonic...
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
polarizingPolarization is a property of waves that describes the orientation of their oscillations. This article primarily covers the polarization of electromagnetic waves such as light, although other types of wave also exhibit polarization....
elements such as
Brewster's angleBrewster's angle is an angle of incidence at which light with a particular polarization is perfectly transmitted through a surface, with no reflection...
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; and in physics, where they give rise to the eigenstates of the quantum...
.
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
phaseThe phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0. Phase is a frequency domain or Fourier transform domain concept, and as such, can be readily understood in terms of simple harmonic...
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 shorter 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 . This ray of light often contains a variety of different wavelengths...
. To determine the number of modes in a step-index fiber, the
V numberNormalized frequency can refer to:* Normalized frequency * Normalized frequency , also known as V number...
needs to be determined: where is the
wavenumberWavenumber in most physical sciences is a wave property inversely related to wavelength, having SI units of reciprocal meters . Wavenumber is the spatial analog of frequency, that is, it is the measurement of the number of wavelengths per unit distance, or more commonly times that, or the number...
, is the fiber's core radius, and and are the
refractive indicesThe refractive index of a medium is a measure of how much the speed of light is reduced inside the medium. For example, typical soda-lime glass has a refractive index close to 1.5, which means that in glass, light travels at 1 / 1.5 = 2/3 the speed of light in a vacuum...
of the core and cladding, 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 electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time to perform useful operations on those signals...
requirements of fiber-optic communication systems.
http://www.rp-photonics.com/modes.html
External links