Transverse mode

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A transverse mode of a beam of electromagnetic radiation 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 radio waves and microwaves confined to a waveguide, and also in light waves in an optical fiber and in a laser's optical resonator.^[1]

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 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 for the boundary conditions of a given waveguide.

Types of modes

Transverse modes are classified into different types:

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.
Transverse electromagnetic (TEM) modes: neither electric nor magnetic field in the direction of propagation.
Hybrid modes: non-zero electric and magnetic fields in the direction of propagation.

In rectangular waveguides, 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 waveguides, 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.^[2]

Modes of hollow metallic waveguides filled with a homogeneous, isotropic material are TE or TM. 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, the transverse electromagnetic (TEM) type.^[3] Optical resonators exhibit linearly polarized TEM modes.

Laser modes

Cylindrical transverse mode patterns TEM(pl)

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 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:

$I_{{pl}}(\rho ,\varphi )=I_{0}\rho ^{l}\left[L_{p}^{l}(\rho )\right]^{2}\cos ^{2}(l\varphi )e^{{-\rho }}$

where ρ = 2r²/w², and L_p^l is the associated 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_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 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 electric field pattern at a point (x,y,z) for a beam propagating along the z-axis is given by^[4]

$E_{{mn}}(x,y,z)=E_{0}{\frac {w_{0}}{w}}H_{m}\left({\frac {{\sqrt {2}}x}{w}}\right)H_{n}\left({\frac {{\sqrt {2}}y}{w}}\right)\exp \left[-(x^{2}+y^{2})\left({\frac {1}{w^{2}}}+{\frac {jk}{2R}}\right)-jkz-j(m+n+1)\zeta (z)\right]$

where $w_{0}$ , $w$ , $R$ , and $\zeta (z)$ are the waist, spot size, radius of curvature, and Gouy phase shift as given for a Gaussian beam; $E_{0}$ is a normalization constant; and $H_{k}$ is the k^th physicist's Hermite polynomial. The corresponding intensity pattern is

$I_{{mn}}(x,y,z)=I_{0}\left[H_{m}\left({\frac {{\sqrt {2}}x}{w}}\right)\exp \left({\frac {-x^{2}}{w^{2}}}\right)\right]^{2}\left[H_{n}\left({\frac {{\sqrt {2}}y}{w}}\right)\exp \left({\frac {-y^{2}}{w^{2}}}\right)\right]^{2}$

Rectangular transverse mode patterns TEM(mn)

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.

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.

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 fiber from single-mode optical fiber. To determine the number of modes in a step-index fiber, the V number needs to be determined: $V=k_{0}a{\sqrt {n_{1}^{2}-n_{2}^{2}}}$ where $k_{0}$ is the wavenumber, $a$ is the fiber's core radius, and $n_{1}$ and $n_{2}$ are the refractive indices 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.^[5]

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 requirements of fiber-optic communication systems.^[6]

References

↑ "Transverse electromagnetic mode"
↑ F. R. Connor, Wave Transmission, pp.52-53, London: Edward Arnold 1971 ISBN 0-7131-3278-7.
↑ K. Okamoto, Fundamentals of Optical Waveguides, pp. 71–79, Elsevier Academic Press, 2006, ISBN 0-12-525096-7.
↑ Svelto, O. (2010). Principles of Lasers (5th ed.). p. 158.
↑ http://eeclass.stanford.edu/ee247/
↑ http://www.rp-photonics.com/modes.html

External links

Detailed descriptions of laser modes

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