A transverse mode of a beam of electromagnetic radiation is a particular intensity pattern of radiation measured in a plane perpendicular (i.e. transverse) to the propagation direction of the beam. Transverse modes occur in radio waves and microwaves confined to a waveguide, and also in light waves in an optical fibre and in a laser's optical resonator. Jump to: navigation, search Electromagnetic radiation can be conceptualized as a self propagating transverse oscillating wave of electric and magnetic fields. ... Jump to: navigation, search In physics, intensity is a measure of the time-averaged energy flux. ... Jump to: navigation, search This page is about the radiation; for the appliance, see microwave oven. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Jump to: navigation, search Prism splitting light Light is electromagnetic radiation with a wavelength that is visible to the eye (visible light) or, in a technical or scientific setting, electromagnetic radiation of any wavelength. ... Fiber Optic strands An optical fiber in American English or fibre in British English is a transparent thin fiber for transmitting light. ... Jump to: navigation, search The range of sizes in which lasers exist is immense, extending from microscopic diode lasers (top) to football field sized neodymium glass lasers (bottom) used for inertial confinement fusion. ... This page meets Wikipedias criteria for speedy deletion. ...

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 electric field amplitude parallel to 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 quantized. The allowed modes can be found by solving Maxwell's equations for the boundary conditions of a given waveguide. In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ... Jump to: navigation, search In physics, an electric field or E-field is an effect produced by an electric charge that exerts a force on charged objects in its vicinity. ... In physics, quantization is a widely observed fact about nature. ... Jump to: navigation, search Maxwells equations are the set of four equations, attributed to James Clerk Maxwell (written by Oliver Heaviside), that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ...

Transverse modes are classified into different types:

• TE modes (Transverse Electric) have no electric field in the direction of propagation.
• TM modes (Transverse Magnetic) have no magnetic field in the direction of propagation.
• TEM modes (Transverse ElectroMagnetic) have no electric or magnetic field in the direction of propagation.
• Hybrid modes are those which have both electric and magnetic field components in the direction of propagation.

Modes of hollow metallic waveguides filled with a homogeneous, isotropic material fall into the first three 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 scalar 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). Scalar is a concept that has meaning in mathematics, physics, and computing. ...

A plane light wave propagating through free space is of the transverse electromagnetic (TEM) type and these are also the type of modes generated in a laser's optical resonator.

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 modes are denoted TEM_pl where p and l are integers labelling 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: In optics, a Gaussian beam, named in honor of Carl Friedrich Gauss (rhymes with house), is a beam of light whose electric field intensity distribution is a Gaussian function. ... In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 - 1886), are a polynomial sequence defined by These polynomials are orthogonal to each other with respect to the inner product given by Also, for each n, Ln(x) is a solution of Laguerres equation which is a second-order... This article describes some of the common coordinate systems that appear in elementary mathematics. ...

,

where ρ = 2r²/w², and L_p^l 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 TEM₀₀ 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 phase 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 2l(p+1) spots in the mode pattern (except for l=0). The TEM_01* mode, the so-called doughnut mode, is a special case consisting of a superposition of two TEM₀₁ modes, rotated 90° with respect to one another. Jump to: navigation, search Image File history File links Laguerre-gaussian. ... Waves with the same phase Waves with different phases The phase of a wave relates the position of a feature, typically a peak or a trough of the waveform, to that same feature in another part of the waveform (or, which amounts to the same, on a second waveform). ...

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 TEM₀₀ 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 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: This article treats polarization in electrodynamics. ... An illustration of the polarisation of light which is incident on an interface at Brewsters angle. ...

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where H_m(x) is the mth order Hermite polynomial. In mathematics, the Hermite polynomials, named in honor of Charles Hermite (pronounced air MEET), are a polynomial sequence defined either by (the probabilists Hermite polynomials), or sometimes by (the physicists Hermite polynomials). These two definitions are not exactly equivalent; either is a trivial rescaling of the other. ...

The TEM₀₀ 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. Jump to: navigation, search Image File history File links Hermite-gaussian. ...

The phase 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. Waves with the same phase Waves with different phases The phase of a wave relates the position of a feature, typically a peak or a trough of the waveform, to that same feature in another part of the waveform (or, which amounts to the same, on a second waveform). ...

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

• Longitudinal mode