Longitudinal mode
Encyclopedia
For the longitudinal mode of conduction of electric currents, see Common mode
A longitudinal mode of a resonant cavity is a particular standing wave
pattern formed by wave
s confined in the cavity. The longitudinal modes correspond to the wavelength
s of the wave which are reinforced by constructive interference after many reflections from the cavity's reflecting surfaces. All other wavelengths are suppressed by destructive interference.
A longitudinal mode pattern has its nodes
located axially along the length of the cavity. Transverse mode
s, with nodes located perpendicular to the axis of the cavity, may also exist.
wavelengths produced by a laser
. In the simplest case, the laser's optical cavity
is formed by two opposed plane (flat) mirror
s surrounding the gain medium (a plane-parallel or Fabry–Pérot cavity). The allowed modes of the cavity are those where the mirror separation distance L is equal to an exact multiple of half the wavelength, λ:
where q is an integer known as the mode order.
In practice, the separation distance of the mirrors L is usually much greater than the wavelength of light λ, so the relevant values of q are large (around 105 to 106). The frequency separation between any two adjacent modes, q and q+1 , in a material that is transparent at the laser wavelength, are given (for an empty linear resonator of length L) by Δν:
where c is the speed of light and n is the refractive index of the material (note: n=1 in air).
), the values of L used are the optical path length
s for each element. The frequency spacing of longitudinal modes in the cavity is then given by:
where ni is the refractive index of the i'th element of length Li.
More generally, the longitudinal modes may be found for any type of wave in a cavity by solving the relevant wave equation
with the appropriate boundary conditions.
Both transverse
and longitudinal wave
s may have longitudinal modes when confined to a cavity.
The analysis of longitudinal modes is especially important in lasers with single transversal mode, for example, in single-mode fiber laser
s. The number of longitudinal modes of such a laser can be estimated as ratio of the spectral width of gain to the spectral separation of longitudinal modes.
of lasers. Such addition allows one to both scale-up the output power of a single-transverse-mode laser and reduce number of longitudinal modes; because the system chooses automatically only the modes which are common for all the combined lasers. The reduction of the number of longitudinal modes determines the limits of the coherent addition
. The ability to coherently add one additional laser is exhausted when one longitudinal mode, common for the combined lasers, lies within the spectral width of the gain; a subsequent addition will lead to loss of efficiency of the coherent combination and will not increase the power per longitudinal mode of such a laser.
Common mode
Common mode is a term in engineering with at least two independent meanings.*Of electrical signals,*Common-mode rejection ratio, the ratio of rejection of common mode signals to differential signals...
Standing wave
In physics, a standing wave – also known as a stationary wave – is a wave that remains in a constant position.This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling...
pattern formed by wave
Wave
In physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...
s confined in the cavity. The longitudinal modes correspond to the wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
s of the wave which are reinforced by constructive interference after many reflections from the cavity's reflecting surfaces. All other wavelengths are suppressed by destructive interference.
A longitudinal mode pattern has its nodes
Node (physics)
A node is a point along a standing wave where the wave has minimal amplitude. For instance, in a vibrating guitar string, the ends of the string are nodes. By changing the position of the end node through frets, the guitarist changes the effective length of the vibrating string and thereby the...
located axially along the length of the cavity. Transverse mode
Transverse mode
A transverse mode of a beam of electromagnetic radiation is a particular electromagnetic field pattern of radiation measured in a plane perpendicular to the propagation direction of the beam...
s, with nodes located perpendicular to the axis of the cavity, may also exist.
Simple cavity
A common example of longitudinal modes are the lightLight
Light 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...
wavelengths produced by a laser
Laser
A 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...
. In the simplest case, the laser's optical cavity
Optical cavity
An optical cavity or optical resonator is an arrangement of mirrors that forms a standing wave cavity resonator for light waves. Optical cavities are a major component of lasers, surrounding the gain medium and providing feedback of the laser light. They are also used in optical parametric...
is formed by two opposed plane (flat) mirror
Mirror
A mirror is an object that reflects light or sound in a way that preserves much of its original quality prior to its contact with the mirror. Some mirrors also filter out some wavelengths, while preserving other wavelengths in the reflection...
s surrounding the gain medium (a plane-parallel or Fabry–Pérot cavity). The allowed modes of the cavity are those where the mirror separation distance L is equal to an exact multiple of half the wavelength, λ:
where q is an integer known as the mode order.
In practice, the separation distance of the mirrors L is usually much greater than the wavelength of light λ, so the relevant values of q are large (around 105 to 106). The frequency separation between any two adjacent modes, q and q+1 , in a material that is transparent at the laser wavelength, are given (for an empty linear resonator of length L) by Δν:
where c is the speed of light and n is the refractive index of the material (note: n=1 in air).
Composite cavity
If the cavity is non-empty (i.e. contains one or more elements with different values of refractive indexRefractive index
In 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....
), the values of L used are the optical path length
Optical path length
In optics, optical path length or optical distance is the product of the geometric length of the path light follows through the system, and the index of refraction of the medium through which it propagates. A difference in optical path length between two paths is often called the optical path...
s for each element. The frequency spacing of longitudinal modes in the cavity is then given by:
where ni is the refractive index of the i'th element of length Li.
More generally, the longitudinal modes may be found for any type of wave in a cavity by solving the relevant wave equation
Wave equation
The wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...
with the appropriate boundary conditions.
Both transverse
Transverse wave
A transverse wave is a moving wave that consists of oscillations occurring perpendicular to the direction of energy transfer...
and longitudinal wave
Longitudinal wave
Longitudinal waves, as known as "l-waves", are waves that have the same direction of vibration as their direction of travel, which means that the movement of the medium is in the same direction as or the opposite direction to the motion of the wave. Mechanical longitudinal waves have been also...
s may have longitudinal modes when confined to a cavity.
The analysis of longitudinal modes is especially important in lasers with single transversal mode, for example, in single-mode fiber laser
Fiber laser
A fiber laser or fibre laser is a laser in which the active gain medium is an optical fiber doped with rare-earth elements such as erbium, ytterbium, neodymium, dysprosium, praseodymium, and thulium. They are related to doped fiber amplifiers, which provide light amplification without lasing...
s. The number of longitudinal modes of such a laser can be estimated as ratio of the spectral width of gain to the spectral separation of longitudinal modes.
Power per longitudinal mode
For lasers with single transversal mode, the power per one longitudinal mode can be significantly increased by the coherent additionCoherent addition
Coherent addition of lasersis one of methods of the power scaling. It allows a to increase the output power and brightness of single-transversal mode laser.Usually, the term coherent addition applies to fiber lasers...
of lasers. Such addition allows one to both scale-up the output power of a single-transverse-mode laser and reduce number of longitudinal modes; because the system chooses automatically only the modes which are common for all the combined lasers. The reduction of the number of longitudinal modes determines the limits of the coherent addition
Coherent addition
Coherent addition of lasersis one of methods of the power scaling. It allows a to increase the output power and brightness of single-transversal mode laser.Usually, the term coherent addition applies to fiber lasers...
. The ability to coherently add one additional laser is exhausted when one longitudinal mode, common for the combined lasers, lies within the spectral width of the gain; a subsequent addition will lead to loss of efficiency of the coherent combination and will not increase the power per longitudinal mode of such a laser.