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optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
, a Gaussian beam is a
beam Beam may refer to: Streams of particles or energy *Light beam, or beam of light, a directional projection of light energy **Laser beam *Particle beam, a stream of charged or neutral particles **Charged particle beam, a spatially localized grou ...
of
electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic field, electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, inf ...
with high monochromaticity whose
amplitude envelope A synthesizer (also spelled synthesiser) is an electronic musical instrument that generates audio signals. Synthesizers typically create sounds by generating waveforms through methods including subtractive synthesis, additive synthesis an ...
in the transverse plane is given by a
Gaussian function In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form f(x) = \exp (-x^2) and with parametric extension f(x) = a \exp\left( -\frac \right) for arbitrary real constants , and non-zero . It is n ...
; this also implies a Gaussian intensity (irradiance) profile. This fundamental (or TEM00)
transverse Transverse may refer to: *Transverse engine, an engine in which the crankshaft is oriented side-to-side relative to the wheels of the vehicle *Transverse flute, a flute that is held horizontally * Transverse force (or ''Euler force''), the tangen ...
Gaussian mode describes the intended output of most (but not all)
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fir ...
s, as such a beam can be focused into the most concentrated spot. When such a beam is refocused by a
lens A lens is a transmissive optical device which focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements''), ...
, the transverse ''phase'' dependence is altered; this results in a ''different'' Gaussian beam. The
electric Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...
and
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
amplitude profiles along any such circular Gaussian beam (for a given
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
and polarization) are determined by a single parameter: the so-called
waist The waist is the part of the abdomen between the rib cage and hips. On people with slim bodies, the waist is the narrowest part of the torso. ''Waistline'' refers to the horizontal line where the waist is narrowest, or to the general appearan ...
. At any position relative to the waist (focus) along a beam having a specified , the field amplitudes and phases are thereby determinedSvelto, pp. 153–5. as detailed below. The equations below assume a beam with a circular cross-section at all values of ; this can be seen by noting that a single transverse dimension, , appears. Beams with
elliptical Elliptical may mean: * having the shape of an ellipse, or more broadly, any oval shape ** in botany, having an elliptic leaf shape ** of aircraft wings, having an elliptical planform * characterised by ellipsis (the omission of words), or by conc ...
cross-sections, or with waists at different positions in for the two transverse dimensions ( astigmatic beams) can also be described as Gaussian beams, but with distinct values of and of the location for the two transverse dimensions and . Arbitrary solutions of the
paraxial Helmholtz equation In mathematics, the eigenvalue problem for the Laplace operator is known as the Hermann von Helmholtz, Helmholtz equation. It corresponds to the linear partial differential equation \nabla^2 f = -k^2 f, where is the Laplace operator (or "Laplaci ...
can be expressed as combinations of Hermite–Gaussian modes (whose amplitude profiles are separable in and using Cartesian coordinates) or similarly as combinations of Laguerre–Gaussian modes (whose amplitude profiles are separable in and using
cylindrical coordinates A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference di ...
).Siegman, p. 642.probably first considered by Goubau and Schwering (1961). At any point along the beam these modes include the same Gaussian factor as the fundamental Gaussian mode multiplying the additional geometrical factors for the specified mode. However different modes propagate with a different
Gouy phase In optics, a Gaussian beam is a beam of electromagnetic radiation with high monochromaticity whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This ...
which is why the net transverse profile due to a superposition of modes evolves in , whereas the propagation of any ''single'' Hermite–Gaussian (or Laguerre–Gaussian) mode retains the same form along a beam. Although there are other possible modal decompositions, these families of solutions are the most useful for problems involving compact beams, that is, where the optical power is rather closely confined along an axis. Even when a laser is ''not'' operating in the fundamental Gaussian mode, its power will generally be found among the lowest-order modes using these decompositions, as the spatial extent of higher order modes will tend to exceed the bounds of a laser's
resonator A resonator is a device or system that exhibits resonance or resonant behavior. That is, it naturally oscillates with greater amplitude at some frequencies, called resonant frequencies, than at other frequencies. The oscillations in a resonator ...
(cavity). "Gaussian beam" normally implies radiation confined to the fundamental (TEM00) Gaussian mode.


Mathematical form

The Gaussian beam is a transverse electromagnetic (TEM) mode.Svelto, p. 158. The mathematical expression for the electric field amplitude is a solution to the
paraxial Helmholtz equation In mathematics, the eigenvalue problem for the Laplace operator is known as the Hermann von Helmholtz, Helmholtz equation. It corresponds to the linear partial differential equation \nabla^2 f = -k^2 f, where is the Laplace operator (or "Laplaci ...
. Assuming polarization in the direction and propagation in the direction, the electric field in
phasor In physics and engineering, a phasor (a portmanteau of phase vector) is a complex number representing a sinusoidal function whose amplitude (''A''), angular frequency (''ω''), and initial phase (''θ'') are time-invariant. It is related to ...
(complex) notation is given by: = E_0 \, \hat \, \frac \exp \left( \frac\right ) \exp \left(\! -i \left(kz +k \frac - \psi(z) \right) \!\right) where * is the radial distance from the center axis of the beam, * is the axial distance from the beam's focus (or "waist"), * is the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
, * is the
wave number In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
(in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s per meter) for a free-space wavelength , and is the index of refraction of the medium in which the beam propagates, *, the electric field amplitude (and phase) at the origin (, ), * is the radius at which the field amplitudes fall to of their axial values (i.e., where the intensity values fall to of their axial values), at the plane along the beam, * is the waist radius, * is the
radius of curvature In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius o ...
of the beam's
wavefront In physics, the wavefront of a time-varying ''wave field'' is the set (locus) of all points having the same ''phase''. The term is generally meaningful only for fields that, at each point, vary sinusoidally in time with a single temporal freque ...
s at , and * is the
Gouy phase In optics, a Gaussian beam is a beam of electromagnetic radiation with high monochromaticity whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This ...
at , an extra phase term beyond that attributable to the
phase velocity The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
of light. There is also an understood time dependence multiplying such
phasor In physics and engineering, a phasor (a portmanteau of phase vector) is a complex number representing a sinusoidal function whose amplitude (''A''), angular frequency (''ω''), and initial phase (''θ'') are time-invariant. It is related to ...
quantities; the actual field at a point in time and space is given by the
real part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
of that complex quantity. This time factor involves an arbitrary
sign convention In physics, a sign convention is a choice of the physical significance of signs (plus or minus) for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly describ ...
, as discussed at . Since this solution relies on the paraxial approximation, it is not accurate for very strongly diverging beams. The above form is valid in most practical cases, where . The corresponding intensity (or
irradiance In radiometry, irradiance is the radiant flux ''received'' by a ''surface'' per unit area. The SI unit of irradiance is the watt per square metre (W⋅m−2). The CGS unit erg per square centimetre per second (erg⋅cm−2⋅s−1) is often used ...
) distribution is given by I(r,z) = = I_0 \left( \frac \right)^2 \exp \left( \frac\right), where the constant is the
wave impedance The wave impedance of an electromagnetic wave is the ratio of the transverse components of the electric and magnetic fields (the transverse components being those at right angles to the direction of propagation). For a transverse-electric-magnetic ...
of the medium in which the beam is propagating. For free space, ≈ 377 Ω. is the intensity at the center of the beam at its waist. If is the total
power Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may a ...
of the beam, I_0 = .


Evolving beam width

At a position along the beam (measured from the focus), the spot size parameter is given by a hyperbolic relation: w(z) = w_0 \, \sqrt, where z_\mathrm = \frac is called the
Rayleigh range In optics and especially laser science, the Rayleigh length or Rayleigh range, z_\mathrm, is the distance along the propagation direction of a light beam, beam from the beam waist, waist to the place where the area of the Cross_section_(geometry ...
as further discussed below, and n is the refractive index of the medium. The radius of the beam , at any position along the beam, is related to the
full width at half maximum In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve mea ...
(FWHM) of the intensity distribution at that position according to: w(z)=.


Wavefront curvature

The curvature of the wavefronts is largest at the Rayleigh distance, , on either side of the waist, crossing zero at the waist itself. Beyond the Rayleigh distance, , it again decreases in magnitude, approaching zero as . The curvature is often expressed in terms of its reciprocal, , the ''
radius of curvature In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius o ...
''; for a fundamental Gaussian beam the curvature at position is given by: \frac = \frac , so the radius of curvature is R(z) = z \left \right Being the reciprocal of the curvature, the radius of curvature reverses sign and is infinite at the beam waist where the curvature goes through zero.


Gouy phase

The '' Gouy phase'' is a phase advance gradually acquired by a beam around the focal region. At position the Gouy phase of a fundamental Gaussian beam is given by \psi(z) = \arctan \left( \frac \right). The Gouy phase results in an increase in the apparent wavelength near the waist (). Thus the phase velocity in that region formally exceeds the speed of light. That paradoxical behavior must be understood as a near-field phenomenon where the departure from the phase velocity of light (as would apply exactly to a plane wave) is very small except in the case of a beam with large
numerical aperture In optics, the numerical aperture (NA) of an optical system is a dimensionless number that characterizes the range of angles over which the system can accept or emit light. By incorporating index of refraction in its definition, NA has the proper ...
, in which case the wavefronts' curvature (see previous section) changes substantially over the distance of a single wavelength. In all cases the
wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
is satisfied at every position. The sign of the Gouy phase depends on the sign convention chosen for the electric field phasor. With dependence, the Gouy phase changes from to , while with dependence it changes from to along the axis. For a fundamental Gaussian beam, the Gouy phase results in a net phase discrepancy with respect to the speed of light amounting to radians (thus a phase reversal) as one moves from the far field on one side of the waist to the far field on the other side. This phase variation is not observable in most experiments. It is, however, of theoretical importance and takes on a greater range for higher-order Gaussian modes.


Elliptical and astigmatic beams

Many laser beams have an elliptical cross-section. Also common are beams with waist positions which are different for the two transverse dimensions, called astigmatic beams. These beams can be dealt with using the above two evolution equations, but with distinct values of each parameter for and and distinct definitions of the point. The Gouy phase is a single value calculated correctly by summing the contribution from each dimension, with a Gouy phase within the range contributed by each dimension. An elliptical beam will invert its ellipticity ratio as it propagates from the far field to the waist. The dimension which was the larger far from the waist, will be the smaller near the waist.


Beam parameters

The geometric dependence of the fields of a Gaussian beam are governed by the light's wavelength (''in'' the dielectric medium, if not free space) and the following beam parameters, all of which are connected as detailed in the following sections.


Beam waist

The shape of a Gaussian beam of a given wavelength is governed solely by one parameter, the ''beam waist'' . This is a measure of the beam size at the point of its focus ( in the above equations) where the beam width (as defined above) is the smallest (and likewise where the intensity on-axis () is the largest). From this parameter the other parameters describing the beam geometry are determined. This includes the
Rayleigh range In optics and especially laser science, the Rayleigh length or Rayleigh range, z_\mathrm, is the distance along the propagation direction of a light beam, beam from the beam waist, waist to the place where the area of the Cross_section_(geometry ...
and asymptotic beam divergence , as detailed below.


Rayleigh range and confocal parameter

The ''Rayleigh distance'' or ''Rayleigh range'' is determined given a Gaussian beam's waist size: z_\mathrm = \frac. Here is the wavelength of the light, is the index of refraction. At a distance from the waist equal to the Rayleigh range , the width of the beam is larger than it is at the focus where , the beam waist. That also implies that the on-axis () intensity there is one half of the peak intensity (at ). That point along the beam also happens to be where the wavefront curvature () is greatest. The distance between the two points is called the ''confocal parameter'' or ''depth of focus'' of the beam.


Beam divergence

Although the tails of a Gaussian function never actually reach zero, for the purposes of the following discussion the "edge" of a beam is considered to be the radius where . That is where the intensity has dropped to of its on-axis value. Now, for the parameter increases linearly with . This means that far from the waist, the beam "edge" (in the above sense) is cone-shaped. The angle between that cone (whose ) and the beam axis () defines the ''divergence'' of the beam: \theta = \lim_ \arctan\left(\frac\right). In the paraxial case, as we have been considering, (in radians) is then approximately \theta = \frac where is the refractive index of the medium the beam propagates through, and is the free-space wavelength. The total angular spread of the diverging beam, or ''apex angle'' of the above-described cone, is then given by \Theta = 2 \theta\, . That cone then contains 86% of the Gaussian beam's total power. Because the divergence is inversely proportional to the spot size, for a given wavelength , a Gaussian beam that is focused to a small spot diverges rapidly as it propagates away from the focus. Conversely, to ''minimize'' the divergence of a laser beam in the far field (and increase its peak intensity at large distances) it must have a large cross-section () at the waist (and thus a large diameter where it is launched, since is never less than ). This relationship between beam width and divergence is a fundamental characteristic of
diffraction Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ...
, and of the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
which describes
Fraunhofer diffraction In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when plane waves are incident on a diffracting object, and the diffraction pattern is viewed at a sufficiently long distance (a distance satisfying Fraunhofer ...
. A beam with any specified amplitude profile also obeys this inverse relationship, but the fundamental Gaussian mode is a special case where the product of beam size at focus and far-field divergence is smaller than for any other case. Since the Gaussian beam model uses the paraxial approximation, it fails when wavefronts are tilted by more than about 30° from the axis of the beam. From the above expression for divergence, this means the Gaussian beam model is only accurate for beams with waists larger than about . Laser beam quality is quantified by the
beam parameter product In laser science, the beam parameter product (BPP) is the product of a laser beam's divergence angle (half-angle) and the radius of the beam at its narrowest point (the beam waist). The BPP quantifies the quality of a laser beam, and how well it can ...
(BPP). For a Gaussian beam, the BPP is the product of the beam's divergence and waist size . The BPP of a real beam is obtained by measuring the beam's minimum diameter and far-field divergence, and taking their product. The ratio of the BPP of the real beam to that of an ideal Gaussian beam at the same wavelength is known as ("
M squared In laser science, the parameter M2, also known as the beam propagation ratio or beam quality factor is a measure of laser beam quality. It represents the degree of variation of a beam from an ideal Gaussian beam. It is calculated from the ratio of t ...
"). The for a Gaussian beam is one. All real laser beams have values greater than one, although very high quality beams can have values very close to one. The
numerical aperture In optics, the numerical aperture (NA) of an optical system is a dimensionless number that characterizes the range of angles over which the system can accept or emit light. By incorporating index of refraction in its definition, NA has the proper ...
of a Gaussian beam is defined to be , where is the
index of refraction In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
of the medium through which the beam propagates. This means that the Rayleigh range is related to the numerical aperture by z_\mathrm = \frac .


Power and intensity


Power through an aperture

With a beam centered on an
aperture In optics, an aperture is a hole or an opening through which light travels. More specifically, the aperture and focal length of an optical system determine the cone angle of a bundle of rays that come to a focus in the image plane. An opt ...
, the
power Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may a ...
passing through a circle of radius in the transverse plane at position isMelles Griot. Gaussian Beam Optics
/ref> P(r,z) = P_0 \left 1 - e^ \right where P_0 = \frac \pi I_0 w_0^2 is the total power transmitted by the beam. For a circle of radius , the fraction of power transmitted through the circle is \frac = 1 - e^ \approx 0.865. Similarly, about 90% of the beam's power will flow through a circle of radius , 95% through a circle of radius , and 99% through a circle of radius .


Peak intensity

The peak intensity at an axial distance from the beam waist can be calculated as the limit of the enclosed power within a circle of radius , divided by the area of the circle as the circle shrinks: I(0,z) = \lim_ \frac . The limit can be evaluated using
L'Hôpital's rule In calculus, l'Hôpital's rule or l'Hospital's rule (, , ), also known as Bernoulli's rule, is a theorem which provides a technique to evaluate limits of indeterminate forms. Application (or repeated application) of the rule often converts an i ...
: I(0,z) = \frac \lim_ \frac = .


Complex beam parameter

The spot size and curvature of a Gaussian beam as a function of along the beam can also be encoded in the complex beam parameter Siegman, pp. 638–40.Garg, pp. 165–168. given by: q(z) = z + iz_\mathrm . Introducing this complication leads to a simplification of the Gaussian beam field equation as shown below. It can be seen that the reciprocal of contains the wavefront curvature and relative on-axis intensity in its real and imaginary parts, respectively: = = - i = - i . The complex beam parameter simplifies the mathematical analysis of Gaussian beam propagation, and especially in the analysis of optical resonator cavities using ray transfer matrices. Then using this form, the earlier equation for the electric (or magnetic) field is greatly simplified. If we call the relative field strength of an elliptical Gaussian beam (with the elliptical axes in the and directions) then it can be separated in and according to: u(x,y,z) = u_x(x,z)\, u_y(y,z) , where \begin u_x(x,z) &= \frac \exp\left(-i k \frac\right), \\ u_y(y,z) &= \frac \exp\left(-i k \frac\right), \end where and are the complex beam parameters in the and directions. For the common case of a circular beam profile, and , which yields u(r,z) = \frac\exp\left( -i k\frac\right) .


Beam optics

When a gaussian beam propagates through a
thin lens In optics, a thin lens is a lens with a thickness (distance along the optical axis between the two surfaces of the lens) that is negligible compared to the radii of curvature of the lens surfaces. Lenses whose thickness is not negligible are so ...
, the outgoing beam is also a (different) gaussian beam, provided that the beam travels along the cylindrical symmetry axis of the lens. The focal length of the lens f, the beam waist radius w_0, and beam waist position z_0 of the incoming beam can be used to determine the beam waist radius w_0' and position z_0' of the outgoing beam.


Lens equation

As derived by Saleh and Teich, the relationship between the ingoing and outgoing beams can be found by considering the
phase Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform * Phase space, a mathematic ...
that is added to each point (x,y) of the gaussian beam as it travels through the lens. Chapter 3, "Beam Optics" An alternative approach due to Self is to consider the effect of a thin lens on the gaussian beam wavefronts. The exact solution to the above problem is expressed simply in terms of the magnification M : \begin w_0' &= Mw_0\\ .2ex(z_0'-f) &= M^2(z_0-f). \end The magnification, which depends on w_0 and z_0, is given by : M = \frac where : r = \frac, \quad M_r = \left, \frac\. An equivalent expression for the beam position z_0' is : \frac+\frac = \frac. This last expression makes clear that the ray optics thin lens equation is recovered in the limit that \left, \left(\tfrac\right)\left(\tfrac\right)\\ll 1. It can also be noted that if \left, z_0+\frac\\gg f then the incoming beam is "well collimated" so that z_0'\approx f.


Beam focusing

In some applications it is desirable to use a converging lens to focus a laser beam to a very small spot. Mathematically, this implies minimization of the magnification M. If the beam size is constrained by the size of available optics, this is typically best achieved by sending the largest possible collimated beam through a small focal length lens, i.e. by maximizing z_R and minimizing f. In this situation, it is justifiable to make the approximation z_R^2/(z_0-f)^2\gg 1, implying that M\approx f/z_R and yielding the result w_0'\approx fw_0/z_R. This result is often presented in the form : \begin 2w_0' &\approx \frac\lambda F_\# \\ .2exz_0' &\approx f \end where : F_\# = \frac, which is found after assuming that the medium has index of refraction n\approx 1 and substituting z_R=\pi w_0^2/\lambda. The factors of 2 are introduced because of a common preference to represent beam size by the beam waist diameters 2w_0' and 2w_0, rather than the waist radii w_0' and w_0.


Wave equation

As a special case of
electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic field, electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, inf ...
, Gaussian beams (and the higher-order Gaussian modes detailed below) are solutions to the wave equation for an electromagnetic field in free space or in a homogeneous dielectric medium,Svelto, pp. 148–9. obtained by combining Maxwell's equations for the curl of and the curl of , resulting in: \nabla^2 U = \frac \frac, where is the speed of light ''in the medium'', and could either refer to the electric or magnetic field vector, as any specific solution for either determines the other. The Gaussian beam solution is valid only in the
paraxial In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system (such as a lens). A paraxial ray is a ray which makes a small angle (''θ'') to the optic ...
approximation, that is, where wave propagation is limited to directions within a small angle of an axis. Without loss of generality let us take that direction to be the direction in which case the solution can generally be written in terms of which has no time dependence and varies relatively smoothly in space, with the main variation spatially corresponding to the
wavenumber In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
in the direction: U(x, y, z, t) = u(x, y, z) e^ \, \hat \, . Using this form along with the paraxial approximation, can then be essentially neglected. Since solutions of the electromagnetic wave equation only hold for polarizations which are orthogonal to the direction of propagation (), we have without loss of generality considered the polarization to be in the direction so that we now solve a scalar equation for . Substituting this solution into the wave equation above yields the
paraxial approximation In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system (such as a lens). A paraxial ray is a ray which makes a small angle (''θ'') to the optical ...
to the scalar wave equation: \frac + \frac = 2ik \frac. Writing the wave equations in the
light-cone coordinates In physics, particularly special relativity, light-cone coordinates, introduced by Paul Dirac and also known as Dirac coordinates, are a special coordinate system where two coordinate axes combine both space and time, while all the others are spati ...
returns this equation without utilizing any approximation. Gaussian beams of any beam waist satisfy the paraxial approximation to the scalar wave equation; this is most easily verified by expressing the wave at in terms of the complex beam parameter as defined above. There are many other solutions. As solutions to a
linear system In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstraction o ...
, any combination of solutions (using addition or multiplication by a constant) is also a solution. The fundamental Gaussian happens to be the one that minimizes the product of minimum spot size and far-field divergence, as noted above. In seeking paraxial solutions, and in particular ones that would describe laser radiation that is ''not'' in the fundamental Gaussian mode, we will look for families of solutions with gradually increasing products of their divergences and minimum spot sizes. Two important orthogonal decompositions of this sort are the Hermite–Gaussian or Laguerre-Gaussian modes, corresponding to rectangular and circular symmetry respectively, as detailed in the next section. With both of these, the fundamental Gaussian beam we have been considering is the lowest order mode.


Higher-order modes


Hermite-Gaussian modes

It is possible to decompose a coherent paraxial beam using the orthogonal set of so-called ''Hermite-Gaussian modes'', any of which are given by the product of a factor in and a factor in . Such a solution is possible due to the separability in and in the
paraxial Helmholtz equation In mathematics, the eigenvalue problem for the Laplace operator is known as the Hermann von Helmholtz, Helmholtz equation. It corresponds to the linear partial differential equation \nabla^2 f = -k^2 f, where is the Laplace operator (or "Laplaci ...
as written in Cartesian coordinates. Thus given a mode of order referring to the and directions, the electric field amplitude at may be given by: E(x,y,z) = u_l(x,z) \, u_m(y,z) \, \exp(-ikz), where the factors for the and dependence are each given by: u_J(x,z) = \left(\frac\right)^ \!\! \left( \frac\right)^ \!\! \left(- \frac\right)^ \!\! H_J\!\left(\frac\right) \, \exp \left(\! -i \frac\right) , where we have employed the complex beam parameter (as defined above) for a beam of waist at from the focus. In this form, the first factor is just a normalizing constant to make the set of
orthonormal In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of un ...
. The second factor is an additional normalization dependent on which compensates for the expansion of the spatial extent of the mode according to (due to the last two factors). It also contains part of the Gouy phase. The third factor is a pure phase which enhances the Gouy phase shift for higher orders . The final two factors account for the spatial variation over (or ). The fourth factor is the
Hermite polynomial In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well as ...
of order ("physicists' form", i.e. ), while the fifth accounts for the Gaussian amplitude fall-off , although this isn't obvious using the complex in the exponent. Expansion of that exponential also produces a phase factor in which accounts for the wavefront curvature () at along the beam. Hermite-Gaussian modes are typically designated "TEM''lm''"; the fundamental Gaussian beam may thus be referred to as TEM00 (where ''TEM'' is '' transverse electro-magnetic''). Multiplying and to get the 2-D mode profile, and removing the normalization so that the leading factor is just called , we can write the mode in the more accessible form: \begin E_(x, y, z) = & E_0 \frac\, H_l \!\Bigg(\frac\Bigg)\, H_m \!\Bigg(\frac\Bigg) \times \\ & \exp \left( \right) \exp \left( \right) \times \\ & \exp \big(i \psi(z)\big) \exp(-ikz). \end In this form, the parameter , as before, determines the family of modes, in particular scaling the spatial extent of the fundamental mode's waist and all other mode patterns at . Given that , and have the same definitions as for the fundamental Gaussian beam described above. It can be seen that with we obtain the fundamental Gaussian beam described earlier (since ). The only specific difference in the and profiles at any are due to the Hermite polynomial factors for the order numbers and . However, there is a change in the evolution of the modes' Gouy phase over : \psi(z) = (N+1) \, \arctan \left( \frac \right), where the combined order of the mode is defined as . While the Gouy phase shift for the fundamental (0,0) Gaussian mode only changes by radians over all of (and only by radians between ), this is increased by the factor for the higher order modes. Hermite Gaussian modes, with their rectangular symmetry, are especially suited for the modal analysis of radiation from lasers whose cavity design is asymmetric in a rectangular fashion. On the other hand, lasers and systems with circular symmetry can better be handled using the set of Laguerre-Gaussian modes introduced in the next section.


Laguerre-Gaussian modes

Beam profiles which are circularly symmetric (or lasers with cavities that are cylindrically symmetric) are often best solved using the Laguerre-Gaussian modal decomposition. These functions are written in
cylindrical coordinates A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference di ...
using generalized Laguerre polynomials. Each transverse mode is again labelled using two integers, in this case the radial index and the azimuthal index which can be positive or negative (or zero): \begin u(r, \phi, z) = &C^_\frac\left(\frac\right)^ \exp\! \left(\! -\frac\right)L_p^ \! \left(\frac\right) \times \\ &\exp \! \left(\! - i k \frac\right) \exp(-i l \phi) \, \exp(i \psi(z)) , \end where are the generalized Laguerre polynomials. is a required normalization constant: C^_ = \sqrt \Rightarrow \int_0^d\phi\int_0^\infty rdr, u(r,\phi,z), ^2=1. and have the same definitions as above. As with the higher-order Hermite-Gaussian modes the magnitude of the Laguerre-Gaussian modes' Gouy phase shift is exaggerated by the factor : \psi(z) = (N+1) \, \arctan \left( \frac \right) , where in this case the combined mode number . As before, the transverse amplitude variations are contained in the last two factors on the upper line of the equation, which again includes the basic Gaussian drop off in but now multiplied by a Laguerre polynomial. The effect of the rotational mode number , in addition to affecting the Laguerre polynomial, is mainly contained in the ''phase'' factor , in which the beam profile is advanced (or retarded) by complete phases in one rotation around the beam (in ). This is an example of an
optical vortex An optical vortex (also known as a photonic quantum vortex, screw dislocation or phase singularity) is a zero of an optical field; a point of zero Intensity (physics), intensity. The term is also used to describe a beam of light that has such a ze ...
of topological charge , and can be associated with the
orbital angular momentum of light The orbital angular momentum of light (OAM) is the component of angular momentum of a light beam that is dependent on the field spatial distribution, and not on the polarization. It can be further split into an internal and an external OAM. The in ...
in that mode.


Ince-Gaussian modes

In
elliptic coordinates In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci F_ and F_ are generally taken to be fixed at -a and +a, respectively ...
, one can write the higher-order modes using Ince polynomials. The even and odd Ince-Gaussian modes are given byBandres and Gutierrez-Vega (2004) u_\varepsilon \left( \xi ,\eta ,z\right) = \frac\mathrm_^\left( i\xi ,\varepsilon \right) \mathrm _^\left( \eta ,\varepsilon \right) \exp \left -ik\frac-\left( p+1\right) \zeta\left( z\right) \right, where and are the radial and angular elliptic coordinates defined by \begin x &= \sqrt\;w(z) \cosh \xi \cos \eta ,\\ y &= \sqrt\;w(z) \sinh \xi \sin \eta . \end are the even Ince polynomials of order and degree where is the ellipticity parameter. The Hermite-Gaussian and Laguerre-Gaussian modes are a special case of the Ince-Gaussian modes for and respectively.


Hypergeometric-Gaussian modes

There is another important class of paraxial wave modes in
cylindrical coordinates A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference di ...
in which the
complex amplitude Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
is proportional to a
confluent hypergeometric function In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular ...
. These modes have a
singular Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular homology * SINGULAR, an open source Computer Algebra System (CAS) * Singular or sounder, a group of boar, ...
phase profile and are
eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
s of the
photon orbital angular momentum The orbital angular momentum of light (OAM) is the component of Angular momentum of light, angular momentum of a light beam that is dependent on the field spatial distribution, and not on the Polarization (waves), polarization. It can be further spl ...
. Their intensity profiles are characterized by a single brilliant ring; like Laguerre–Gaussian modes, their intensities fall to zero at the center (on the optical axis) except for the fundamental (0,0) mode. A mode's complex amplitude can be written in terms of the normalized (dimensionless) radial coordinate and the normalized longitudinal coordinate as follows: \begin u_(\rho, \phi, \Zeta) = &\sqrt\; \frac\, i^ \times \\ &\Zeta^\, (\Zeta + i)^\, \rho^ \times \\ &\exp\left(-\frac\right)\, e^\, _1F_1 \left(-\frac, , m, + 1; \frac\right) \end where the rotational index is an integer, and \ge-, m, is real-valued, is the gamma function and is a confluent hypergeometric function. Some subfamilies of hypergeometric-Gaussian (HyGG) modes can be listed as the modified Bessel-Gaussian modes, the modified exponential Gaussian modes,Karimi et al. (2007) and the modified Laguerre–Gaussian modes. The set of hypergeometric-Gaussian modes is overcomplete and is not an orthogonal set of modes. In spite of its complicated field profile, HyGG modes have a very simple profile at the beam waist (): u(\rho, \phi, 0) \propto \rho^e^.


See also

* Bessel beam *
Tophat beam In optics, a tophat (or top-hat) beam such as a laser beam or electron beam has a near-uniform fluence (energy density) within a circular disk. It is typically formed by diffractive optical elements from a Gaussian beam. Tophat beams are often used ...
*
Laser beam profiler A laser beam profiler captures, displays, and records the spatial intensity profile of a laser beam at a particular plane transverse to the beam propagation path. Since there are many types of lasers — ultraviolet, visible, infrared, contin ...
*
Quasioptics Quasioptics concerns the propagation of electromagnetic radiation where the wavelength is comparable to the size of the optical components (e.g. lenses, mirrors, and apertures) and hence diffraction effects may become significant. It commonly descr ...


Notes


References

* * * * * Chapter 5, "Optical Beams," pp. 267. * * * Chapter 3, "Beam Optics," pp. 80–107. * Chapter 16. * *


External links


Gaussian Beam Optics Tutorial, Newport
{{Lasers Physical optics Laser science Electromagnetic radiation