# Transverse mode

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A transverse mode 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]

In physics, electromagnetic radiation refers to the waves of the electromagnetic field, propagating (radiating) through space, carrying electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) light, ultraviolet, X-rays, and gamma rays.

Radio is the technology of using radio waves to carry information, such as sound and images, by systematically modulating properties of electromagnetic energy waves transmitted through space, such as their amplitude, frequency, phase, or pulse width. When radio waves strike an electrical conductor, the oscillating fields induce an alternating current in the conductor. The information in the waves can be extracted and transformed back into its original form.

Microwaves are a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter; with frequencies between 300 MHz (1 m) and 300 GHz (1 mm). Different sources define different frequency ranges as microwaves; the above broad definition includes both UHF and EHF bands. A more common definition in radio engineering is the range between 1 and 100 GHz. In all cases, microwaves include the entire SHF band at minimum. Frequencies in the microwave range are often referred to by their IEEE radar band designations: S, C, X, Ku, K, or Ka band, or by similar NATO or EU designations.

## Contents

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.

An electric field surrounds an electric charge, and exerts force on other charges in the field, attracting or repelling them. Electric field is sometimes abbreviated as E-field. Mathematically the electric field is a vector field that associates to each point in space the force per unit of charge exerted on an infinitesimal positive test charge at rest at that point. The SI unit for electric field strength is volt per meter (V/m). Newtons per coulomb (N/C) is also used as a unit of electric field strengh. Electric fields are created by electric charges, or by time-varying magnetic fields. Electric fields are important in many areas of physics, and are exploited practically in electrical technology. On an atomic scale, the electric field is responsible for the attractive force between the atomic nucleus and electrons that holds atoms together, and the forces between atoms that cause chemical bonding. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces of nature.

In physics, quantization is the process of transition from a classical understanding of physical phenomena to a newer understanding known as quantum mechanics. This is a generalization of the procedure for building quantum mechanics from classical mechanics. One also speaks of field quantization, as in the "quantization of the electromagnetic field", where one refers to photons as field "quanta". This procedure is basic to theories of particle physics, nuclear physics, condensed matter physics, and quantum optics.

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. One important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at the speed of light. Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum from radio waves to γ-rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who between 1861 and 1862 published an early form of the equations that included the Lorentz force law. He also first used the equations to propose that light is an electromagnetic phenomenon.

## Types of modes

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

A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material as they do in an electrical conductor but only slightly shift from their average equilibrium positions causing dielectric polarization. Because of dielectric polarization, positive charges are displaced in the direction of the field and negative charges shift in the opposite direction. This creates an internal electric field that reduces the overall field within the dielectric itself. If a dielectric is composed of weakly bonded molecules, those molecules not only become polarized, but also reorient so that their symmetry axes align to the field.

In the physics of wave propagation, a plane wave is a wave whose wavefronts are infinite parallel planes. Mathematically a plane wave takes the form

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 mode (or superposition of such modes). These modes generally follow different propagation constants. When two or more modes have an identical propagation constant along the waveguide, then there is more than one modal decomposition possible in order to describe a wave with that propagation constant (for instance, a non-central Gaussian laser mode can be equivalently described as a superposition of Hermite-Gaussian modes or Laguerre-Gaussian modes which are described below).

A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting expansion to one dimension or two. There is a similar effect in water waves constrained within a canal, or guns that have barrels which restrict hot gas expansion to maximize energy transfer to their bullets. Without the physical constraint of a waveguide, wave amplitudes decrease according to the inverse square law as they expand into three dimensional space.

A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at the fixed frequencies. These fixed frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies. A physical object, such as a building, bridge, or molecule, has a set of normal modes and their natural frequencies that depend on its structure, materials and boundary conditions. When relating to music, normal modes of vibrating instruments are called "harmonics" or "overtones".

The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the change per unit length, but it is otherwise dimensionless. In the context of two-port networks and their cascades, propagation constant measures the change undergone by the source quantity as it propagates from one port to the next.

### Waveguide modes

Modes in waveguides can be further 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.

Hollow metallic waveguides filled with a homogeneous, isotropic material (usually air) support TE and TM modes but not the TEM mode. 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. In an optical fiber or other dielectric waveguide, modes are generally of the hybrid type.

Coaxial cable, or coax, is a type of electrical cable that has an inner conductor surrounded by a tubular insulating layer, surrounded by a tubular conducting shield. Many coaxial cables also have an insulating outer sheath or jacket. The term coaxial comes from the inner conductor and the outer shield sharing a geometric axis. Coaxial cable was invented by English engineer and mathematician Oliver Heaviside, who patented the design in 1880.

Microstrip is a type of electrical transmission line which can be fabricated using printed circuit board technology, and is used to convey microwave-frequency signals. It consists of a conducting strip separated from a ground plane by a dielectric layer known as the substrate. Microwave components such as antennas, couplers, filters, power dividers etc. can be formed from microstrip, with the entire device existing as the pattern of metallization on the substrate. Microstrip is thus much less expensive than traditional waveguide technology, as well as being far lighter and more compact. Microstrip was developed by ITT laboratories as a competitor to stripline.

In rectangular waveguides, rectangular mode numbers are designated by two suffix numbers attached to the mode type, such as TEmn or TMmn, 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. [2] [3]

## Laser modes

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 TEMpl 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:

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

where ρ = 2r2/w2, Ll
p
is the associated Laguerre polynomial of order p and index l, and 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. 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 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 TEM0i* mode, the so-called doughnut mode, is a special case consisting of a superposition of two TEM0i 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 TEMmn 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]

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

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

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

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 TEMmn 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: ${\displaystyle V=k_{0}a{\sqrt {n_{1}^{2}-n_{2}^{2}}}}$ where ${\displaystyle k_{0}}$ is the wavenumber, ${\displaystyle a}$ is the fiber's core radius, and ${\displaystyle n_{1}}$ and ${\displaystyle n_{2}}$ 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]

The modes in typical low refractive index contrast fibers 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. [7]

## Related Research Articles

Diffraction refers to various phenomena that occur when a wave encounters an obstacle or a slit. It is defined as the bending of waves around the corners of an obstacle or aperture into the region of geometrical shadow of the obstacle. In classical physics, the diffraction phenomenon is described as the interference of waves according to the Huygens–Fresnel principle that treats each point in the wave-front as a collection of individual spherical wavelets. These characteristic behaviors are exhibited when a wave encounters an obstacle or a slit that is comparable in size to its wavelength. Similar effects occur when a light wave travels through a medium with a varying refractive index, or when a sound wave travels through a medium with varying acoustic impedance. Diffraction has an impact on the acoustic space. Diffraction occurs with all waves, including sound waves, water waves, and electromagnetic waves such as visible light, X-rays and radio waves.

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In optics, a Gaussian beam is a beam of monochromatic electromagnetic radiation whose transverse magnetic and electric field amplitude profiles are given by the Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This fundamental (or TEM00) transverse gaussian mode describes the intended output of most (but not all) lasers, as such a beam can be focused into the most concentrated spot. When such a beam is refocused by a lens, the transverse phase dependence is altered; this results in a different Gaussian beam. The electric and magnetic field amplitude profiles along any such circular Gaussian beam (for a given wavelength and polarization) are determined by a single parameter: the so-called waist w0. At any position z relative to the waist (focus) along a beam having a specified w0, the field amplitudes and phases are thereby determined as detailed below.

Polarization is a property applying to transverse waves that specifies the geometrical orientation of the oscillations. In a transverse wave, the direction of the oscillation is perpendicular to the direction of motion of the wave. A simple example of a polarized transverse wave is vibrations traveling along a taut string (see image); for example, in a musical instrument like a guitar string. Depending on how the string is plucked, the vibrations can be in a vertical direction, horizontal direction, or at any angle perpendicular to the string. In contrast, in longitudinal waves, such as sound waves in a liquid or gas, the displacement of the particles in the oscillation is always in the direction of propagation, so these waves do not exhibit polarization. Transverse waves that exhibit polarization include electromagnetic waves such as light and radio waves, gravitational waves, and transverse sound waves in solids. In some types of transverse waves, the wave displacement is limited to a single direction, so these also do not exhibit polarization; for example, in surface waves in liquids, the wave displacement of the particles is always in a vertical plane.

The wave impedance of an electromagnetic wave is the ratio of the transverse components of the electric and magnetic fields. For a transverse-electric-magnetic (TEM) plane wave traveling through a homogeneous medium, the wave impedance is everywhere equal to the intrinsic impedance of the medium. In particular, for a plane wave travelling through empty space, the wave impedance is equal to the impedance of free space. The symbol Z is used to represent it and it is expressed in units of ohms. The symbol η (eta) may be used instead of Z for wave impedance to avoid confusion with electrical impedance.

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Eigenmode expansion (EME) is a computational electrodynamics modelling technique. It is also referred to as the mode matching technique or the bidirectional eigenmode propagation method. Eigenmode expansion is a linear frequency-domain method.

The model usually designated as Lugiato–Lefever equation (LLE) was formulated in 1987 by Luigi Lugiato and Renè Lefever as a paradigm for spontaneous pattern formation in nonlinear optical systems. The patterns originate from the interaction of a coherent field, that is injected into a resonant optical cavity, with a Kerr medium that fills the cavity.

## References

1. F. R. Connor, Wave Transmission, pp.52-53, London: Edward Arnold 1971 ISBN   0-7131-3278-7.
2. U.S. Navy-Marine Corps Military Auxiliary Radio System (MARS), NAVMARCORMARS Operator Course, Chapter 1, Waveguide Theory and Application, Figure 1-38.—Various modes of operation for rectangular and circular waveguides.
3. Svelto, O. (2010). Principles of Lasers (5th ed.). p. 158.
4. Kahn, Joseph M. (Sep 21, 2006). "Lecture 3: Wave Optics Description of Optical Fibers" (PDF). EE 247: Introduction to Optical Fiber Communications, Lecture Notes. Stanford University. p. 8. Archived from the original (PDF) on Jun 14, 2007. Retrieved 27 Jan 2015.
5. Paschotta, Rüdiger. "Modes". Encyclopedia of Laser Physics and Technology. RP Photonics. Retrieved Jan 26, 2015.
6. K. Okamoto, Fundamentals of Optical Waveguides, pp. 71–79, Elsevier Academic Press, 2006, ISBN   0-12-525096-7.