[Coherent light](Coherent%20light.md) in its [principle mode](TEM%20mode.md#Principle%20mode%20TEM%20_%2000) - i.e. light typically produced by a [laser](Lasers.md) - is modeled as a [light-field](Light-field.md) expressed as $\mathbf{E}(r,z)=E_0\frac{w_0}{w(z)}e^{-r^2(1/w^2(z)-ik/2R(z))-i(kz-\arctan{(2z/b)})}\hat{\mathbf{x}}$ where we often treat the field as a [scalar,](Light-field.md#Scalar%20light%20field) thus writing, $E(r,z)=E_0\frac{w_0}{w(z)}e^{-r^2(1/w^2(z)-ik/2R(z))-i(kz-\arctan{(2z/b)})}.$ Here the beam propagates along the $z$ axis while its radius extends into the $r$ axis. The remaining parameters are described [here.](Gaussian%20beam.md#Gaussian%20beam%20parameters) The overall geometry of the [light-field](Light-field.md) given in terms of some of the key [Gaussian beam parameters](Gaussian%20beam.md#Gaussian%20beam%20parameters) is shown below where the brighter color corresponds with the presence of a stronger light field. The field strength for a given cross section is a [Gaussian function](Gaussian%20function.md) with its domain in $r$ as shown below. In particular the cross section also models the appearance of the beam as it would look like while illuminating a spot on a clean surface.  <font size="2"> Here the light field distribution is plotted on the $z$ axis and its cross section on the $r$ axis where the geometry of this distribution depends on the [Gaussian beam parameters](Gaussian%20beam.md#Gaussian%20beam%20parameters) as shown here.</font> This is sometimes referred to as a _spherical Gaussian beam_ in order to distinguish from beams that manifest as [higher order modes.](Gaussian%20beam.md#Higher%20order%20modes) # Gaussian beam parameters Here we break down the constituent parameters that determine the form of the Gaussian [light-field](Light-field.md), where most of the relevant parameters appear in the [above exponential](Gaussian%20beam.md) while the [complex beam parameter](Gaussian%20beam.md#Complex%20beam%20parameter) and [complex beam parameter](Gaussian%20beam.md#Complex%20beam%20parameter) appear in the [derivation of the Gaussian beam expression](Gaussian%20beam.md#Derivation%20of%20the%20Gaussian%20beam%20expression) and are formed from combining other parameters. We consider the parameters in terms of a coordinate system in which $z=0$ at the _beam waist_- i.e. at the point where [beam radius](Gaussian%20beam.md#beam%20radius) is at its minimum. Here we summarize the main relations governing these parameters.  <font size="2"> Here the [light-field](Light-field.md) distribution for a Gaussian beam is plotted on the $z$ axis where many of the key [Gaussian beam parameters](Gaussian%20beam.md#Gaussian%20beam%20parameters) are shown and labeled.</font> ^bc83fa ## beam radius The [Gaussian beam radius](Gaussian%20beam%20radius.md) is given in terms of $z$ as  where the minimum radius at the waist is denoted by $w_0$ and $z_R$ is the _[[Rayleigh range]],_ which is one-half the [confocal parameter](Gaussian%20beam.md#confocal%20parameter). Thus we may also write  ## wave front curvature The _[wavefront curvature](Wavefront.md#wavefront%20curvature)_ is given by $R(z)$ and is given as . ## Gouy phase The [Gouy phase](Gouy%20phase.md), as we read it off of the [Gaussian beam](Gaussian%20beam.md) exponential is given as  ### Confocal parameter Given the [Gouy phase](Gaussian%20beam.md#Gouy%20phase) the denominator of its argument contains the so called _[confocal parameter](Confocal%20parameter.md)_ given as  where $z_R$ is the [[Rayleigh range]], $n$ is the [[Index of refraction]], $\lambda$ is the wavelength, and $w_0$ is the [[Beam radius]] at the beam waist. ## Complex beam parameter The _[complex beam parameter](Complex%20beam%20parameter.md)_, $q(z)$, isn't directly included in the expression governing the [[Gaussian beam]], since it is instead written by combining the [wave front curvature](Gaussian%20beam.md#wave%20front%20curvature) and the [beam radius](Gaussian%20beam.md#beam%20radius). This parameter is best described in a single sentence as containing the information that describes the overall shape of a [[Gaussian beam]]. It appears in the ansatz for solutions to the [cylindrically symmetric paraxial wave equation](Paraxial%20Wave%20Equation.md#In%20cylindrical%20coordinates) given as . It is convenient to express the complex beam parameter in terms of the [beam radius](Gaussian%20beam.md#beam%20radius) and [wave front curvature](Gaussian%20beam.md#wave%20front%20curvature) by writing it in its reciprocal form as  %% This section is confusing written. exactly how does the complex beam parameter relate here?%% ## Phase shift factor The [[phase shift factor]], $P(z)$ is a complex parameter that appears along with $q(z)$ in the ansatz for the [cylindrically symmetric paraxial wave equation](Paraxial%20Wave%20Equation.md#In%20cylindrical%20coordinates). # Ray optics # Derivation of the Gaussian beam expression The [Gaussian beam](Gaussian%20beam.md) is a [cylindrically symmetric](Paraxial%20Wave%20Equation.md#In%20cylindrical%20coordinates) solution to the [paraxial wave equation](Paraxial%20Wave%20Equation.md) which we derive from the following ansatz:  where [paraxial Wave Equation](Paraxial%20Wave%20Equation.md) is expressed in cylindrical coordinates as  # Higher order modes ## Hermite-Gaussian modes ## Laguerre-Gaussian modes ## Hypergeometric-Gaussian modes #Electromagnetism/Optics/waveOptics/GaussianBeamOptics #QuantumMechanics/quantumOptics