Editing Transverse mode (section)

==Types of modes==
Unguided electromagnetic waves in free space, or in a bulk [[isotropic]] [[dielectric]], can be described as a superposition of [[plane wave]]s; these can be described as TEM modes as defined below.

However in any sort of [[waveguide]] where [[boundary conditions]] are imposed by a physical structure, a wave of a particular frequency can be described in terms of a transverse [[Normal mode|mode]] (or superposition of such modes). These modes generally follow different [[propagation constant]]s. When two or more modes have an identical propagation constant along the waveguide, then there is more than one [[Normal mode|modal decomposition]] possible in order to describe a wave with that propagation constant (for instance, a non-central [[Gaussian beam|Gaussian]] laser mode can be equivalently described as a superposition of [[Gaussian beam#Hermite-Gaussian modes|Hermite-Gaussian modes]] or [[Gaussian beam#Laguerre-Gaussian modes|Laguerre-Gaussian modes]] which are described below).

===Waveguides===
[[File:Selected modes.svg|thumb|Field patterns of some commonly used waveguide modes]]
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 {{section link|Planar transmission line|Modes}}''.

====Conductor-based transmission lines====
In [[coaxial cable]] 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]] 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.
Inhomogeneity also occurs at connectors or bends in a coaxial cable. Non-TEM modes created by connectors are usually negligible unless the signal has a high enough frequency. This is referred to as maximum '''extraneous-mode-free operation'''<ref>{{cite journal |last1=Eisenhart |first1=R. L. |title=A novel wideband TM01-to-TE11 mode converter |journal=998 IEEE MTT-S International Microwave Symposium Digest (Cat. No.98CH36192) |date=1998 |volume=q |pages=249–252}}</ref> or simply '''mode-free operation'''<ref>{{cite web |last1=Hannon |first1=Michael J. |last2=Malloy |first2=Pat |title=Application Guide to RF Coaxial Connectors and Cables |url=https://www.globalspec.com/ARRFMicrowaveInstrumentation/ref/appnote51B.pdf |access-date=26 February 2025}}</ref><ref>{{cite web |last1=hobbs |title=What does "mode-free" mean, in the context of coaxial connectors? |url=https://electronics.stackexchange.com/q/575353 |website=Electrical Engineering Stack Exchange |access-date=26 February 2025 |date=14 July 2021}}</ref> frequency of the connector.

In an optical fiber or other dielectric waveguide, modes are generally of the hybrid type.

====Waveguides====
Hollow metallic waveguides filled with a homogeneous, isotropic material (usually air) support TE and TM modes but not the TEM mode.
In rectangular waveguides, rectangular mode numbers are designated by two suffix numbers attached to the mode type, such as TE<sub>''mn''</sub> or TM<sub>''mn''</sub>, 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.<ref>F. R. Connor, ''Wave Transmission'', pp.52-53, London: Edward Arnold 1971 {{ISBN|0-7131-3278-7}}.</ref><ref>U.S. Navy-Marine Corps Military Auxiliary Radio System (MARS), NAVMARCORMARS Operator Course, [https://web.archive.org/web/20150412011516/http://www.navymars.org/national/training/nmo_courses/nmo1/module11/14183_ch1.pdf Chapter 1, Waveguide Theory and Application], Figure 1-38.—Various modes of operation for rectangular and circular waveguides.</ref>

====Optical fibers====
{{see also|Equilibrium mode distribution|Mode volume|Cladding mode}}

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 [[Normalized frequency (fiber optics)|V number]] needs to be determined: <math display="inline">V = k_0 a \sqrt{n_1^2 - n_2^2}</math> where <math>k_0</math> is the [[wavenumber]], <math>a</math> is the fiber's core radius, and <math>n_1</math> and <math>n_2</math> are the [[Refractive index|refractive indices]] of the core and [[Cladding (fiber optics)|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.<ref>{{cite web |publisher=Stanford University |work=EE 247: Introduction to Optical Fiber Communications, Lecture Notes |date=Sep 21, 2006 |first=Joseph M. |last=Kahn |title=Lecture 3: Wave Optics Description of Optical Fibers |page=8 |archive-url=https://web.archive.org/web/20070614013059/http://eeclass.stanford.edu/cgi-bin/handouts.cgi?s=&t=1181774370&cc=ee247&action=handout_download&handout_id=ID112717331418248&viewfile=lecture_3.pdf |url=http://eeclass.stanford.edu/cgi-bin/handouts.cgi?s=&t=1181774370&cc=ee247&action=handout_download&handout_id=ID112717331418248&viewfile=lecture_3.pdf |archive-date=June 14, 2007 |access-date=27 Jan 2015}}</ref>

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 [[Optical communications#Optical fiber communication|fiber-optic communication]] systems.<ref>{{cite encyclopedia |title=Modes |url=https://www.rp-photonics.com/modes.html |encyclopedia=Encyclopedia of Laser Physics and Technology |publisher=RP Photonics |first=Rüdiger |last=Paschotta |access-date=Jan 26, 2015}}</ref>

The modes in typical low refractive index contrast fibers are usually referred to as ''LP'' (linear polarization) modes, which refers to a [[scalar (physics)|scalar]] approximation for the field solution, treating it as if it contains only one transverse field component.<ref>K. Okamoto, ''Fundamentals of Optical Waveguides'', pp. 71–79, Elsevier Academic Press, 2006, {{ISBN|0-12-525096-7}}.</ref>

===Lasers===
[[File:Laguerre-gaussian.png|thumb|right|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 polynomials|Laguerre polynomial]]. The modes are denoted {{math|TEM<sub>''pl''</sub>}} where {{mvar|p}} and {{mvar|l}} are integers labeling the radial and angular mode orders, respectively. The intensity at a point {{math|(''r'',''φ'')}} (in [[Coordinates (elementary mathematics)#Circular coordinates|polar coordinates]]) from the centre of the mode is given by:
<math display="block">I_{pl} (\rho, \varphi) = I_0 \rho^l \left[L_p^l (\rho)\right]^2 \cos^2 (l\varphi) e^{-\rho}</math>
where {{math|1=''ρ'' = 2''r''<sup>2</sup>/''w''<sup>2</sup>}}, {{math|''L{{su|b=p|p=l}}''}} is the associated [[Laguerre polynomial]] of order {{mvar|p}} and index {{mvar|l}}, and {{mvar|w}} is the spot size of the mode corresponding to the Gaussian beam radius.
[[File:Tem p 2 l 1 plot.png|thumb|upright|Cylindrical transverse mode with ''p''=2, ''l''=1]]
With {{math|1=''p'' = ''l'' = 0}}, the TEM<sub>00</sub> 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 (waves)|phase]] across the mode. Modes with increasing {{mvar|p}} show concentric rings of intensity, and modes with increasing {{mvar|l}} show angularly distributed lobes. In general there are {{math|2''l''(''p''+1)}} spots in the mode pattern (except for {{math|1=''l'' = 0}}). The {{math|TEM<sub>0''i''*</sub>}} mode, the so-called ''doughnut mode'', is a special case consisting of a superposition of two {{math|TEM<sub>0''i''</sub>}} modes ({{math|1=''i'' = 1, 2, 3}}), rotated {{math|360°/4''i''}} with respect to one another.

The overall size of the mode is determined by the Gaussian beam radius {{mvar|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 {{math|TEM<sub>00</sub>}} 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 [[polarizer|polarizing elements]] such as [[Brewster's angle]] windows. In these lasers, transverse modes with rectangular symmetry are formed. These modes are designated {{math|TEM<sub>''mn''</sub>}} with {{mvar|m}} and {{mvar|n}} being the horizontal and vertical orders of the pattern. The electric field pattern at a point {{math|(''x'',''y'',''z'')}} for a beam propagating along the z-axis is given by<ref name="svelto">{{cite book| author=Svelto, O.|title=Principles of Lasers |edition=5th|year=2010|page=158}}</ref>
<math display="block">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\right]</math>
where <math>w_0</math>, <math>w(z)</math>, <math>R(z)</math>, and <math>\zeta(z)</math> are the waist, spot size, radius of curvature, and [[Gouy phase shift]] as given for a [[Gaussian beam]]; <math>E_0</math> is a normalization constant; and <math>H_k</math> is the {{mvar|k}}-th physicist's [[Hermite polynomials|Hermite polynomial]]. The corresponding intensity pattern is
<math display="block">I_{mn}(x, y, z) = I_0 \left( \frac{w_0}{w} \right)^2 \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</math>

[[File:Hermite-gaussian.png|thumb|right|Rectangular transverse mode patterns TEM(mn)]]

The TEM<sub>00</sub> mode corresponds to exactly the same fundamental mode as in the cylindrical geometry. Modes with increasing {{mvar|m}} and {{mvar|n}} show lobes appearing in the horizontal and vertical directions, with in general {{math|(''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 (waves)|phase]] of each lobe of a {{math|TEM<sub>''mn''</sub>}} is offset by {{math|''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.