Gaussian beam
In optics, a Gaussian beam is an idealized beam of electromagnetic radiation whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This fundamental (or TEM00) transverse Gaussian mode describes the intended output of many lasers, as such a beam diverges less and can be focused better than any other. When a Gaussian beam is refocused by an ideal lens, a new Gaussian beam is produced. The electric and magnetic field amplitude profiles along a circular Gaussian beam of a given wavelength and polarization are determined by two parameters: the waist Template:Math, which is a measure of the width of the beam at its narrowest point, and the position Template:Mvar relative to the waist.<ref name="svelto153">Svelto, pp. 153–5.</ref>
Since the Gaussian function is infinite in extent, perfect Gaussian beams do not exist in nature, and the edges of any such beam would be cut off by any finite lens or mirror. However, the Gaussian is a useful approximation to a real-world beam for cases where lenses or mirrors in the beam are significantly larger than the spot size w(z) of the beam.
Fundamentally, the Gaussian is a solution of the paraxial Helmholtz equation, the wave equation for an electromagnetic field. Although there exist other solutions, the Gaussian families of solutions are useful for problems involving compact beams.
Mathematical formEdit
The equations below assume a beam with a circular cross-section at all values of Template:Mvar; this can be seen by noting that a single transverse dimension, Template:Mvar, appears. Beams with elliptical cross-sections, or with waists at different positions in Template:Mvar for the two transverse dimensions (astigmatic beams) can also be described as Gaussian beams, but with distinct values of Template:Math and of the Template:Math location for the two transverse dimensions Template:Mvar and Template:Mvar.
The Gaussian beam is a transverse electromagnetic (TEM) mode.<ref name="svelto158">Svelto, p. 158.</ref> The mathematical expression for the electric field amplitude is a solution to the paraxial Helmholtz equation.<ref name="svelto153" /> Assuming polarization in the Template:Mvar direction and propagation in the Template:Math direction, the electric field in phasor (complex) notation is given by:
<math display="block">{\mathbf E(r,z)} = E_0 \, \hat{\mathbf x} \, \frac{w_0}{w(z)} \exp \left( \frac{-r^2}{w(z)^2}\right ) \exp \left(\! -i \left(kz +k \frac{r^2}{2R(z)} - \psi(z) \right) \!\right)</math>
where<ref name="svelto153" /><ref>Template:Cite book</ref>
- Template:Mvar is the radial distance from the center axis of the beam,
- Template:Mvar is the axial distance from the beam's focus (or "waist"),
- Template:Mvar is the imaginary unit,
- Template:Math is the wave number (in radians per meter) for a free-space wavelength Template:Mvar, and Template:Mvar is the index of refraction of the medium in which the beam propagates,
- Template:Math, the electric field amplitude at the origin (Template:Math, Template:Math),
- Template:Math is the radius at which the field amplitudes fall to Template:Math of their axial values (i.e., where the intensity values fall to Template:Math of their axial values), at the plane Template:Mvar along the beam,
- Template:Math is the waist radius,
- Template:Math is the radius of curvature of the beam's wavefronts at Template:Mvar, and
- Template:Math is the Gouy phase at Template:Mvar, an extra phase term beyond that attributable to the phase velocity of light.
The physical electric field is obtained from the phasor field amplitude given above by taking the real part of the amplitude times a time factor: <math display=block>\mathbf E_\text{phys}(r,z,t) = \operatorname{Re}(\mathbf E(r,z) \cdot e^{i\omega t}),</math> where <math display=inline>\omega</math> is the angular frequency of the light and Template:Mvar is time. The time factor involves an arbitrary sign convention, as discussed at Template:Section link.
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 Template:Math.
The corresponding intensity (or irradiance) distribution is given by
<math display="block"> I(r,z) = { |E(r,z)|^2 \over 2 \eta } = I_0 \left( \frac{w_0}{w(z)} \right)^2 \exp \left( \frac{-2r^2}{w(z)^2}\right),</math>
where the constant Template:Mvar is the wave impedance of the medium in which the beam is propagating. For free space, Template:Math ≈ 377 Ω. Template:Math is the intensity at the center of the beam at its waist.
If Template:Math is the total power of the beam, <math display="block">I_0 = {2P_0 \over \pi w_0^2}.</math>
Evolving beam widthEdit
At a position Template:Mvar along the beam (measured from the focus), the spot size parameter Template:Mvar is given by a hyperbolic relation:<ref name="svelto153" /> <math display="block">w(z) = w_0 \, \sqrt{ 1+ {\left( \frac{z}{z_\mathrm{R}} \right)}^2 },</math> where<ref name="svelto153" /> <math display="block">z_\mathrm{R} = \frac{\pi w_0^2 n}{\lambda}</math> is called the Rayleigh range as further discussed below, and <math>n</math> is the refractive index of the medium.
The radius of the beam Template:Math, at any position Template:Mvar along the beam, is related to the full width at half maximum (FWHM) of the intensity distribution at that position according to:<ref name=zemax>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> <math display="block">w(z)={\frac {\text{FWHM}(z)}{\sqrt {2\ln2}}}.</math>
Wavefront curvatureEdit
The wavefronts have zero curvature (radius = ∞) at the waist. Wavefront curvature increases away from the waist, with the maximum rate of change occurring at the Rayleigh distance, Template:Math. Beyond the Rayleigh distance, Template:Math, the curvature again decreases in magnitude, approaching zero as Template:Math. The curvature is often expressed in terms of its reciprocal, Template:Mvar, the radius of curvature; for a fundamental Gaussian beam the curvature at position Template:Mvar is given by:
<math display="block">\frac{1}{R(z)} = \frac{z} {z^2 + z_\mathrm{R}^2} ,</math>
so the radius of curvature Template:Math is <ref name="svelto153" /> <math display="block">R(z) = z \left[{ 1+ {\left( \frac{z_\mathrm{R}}{z} \right)}^2 } \right].</math> 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.
Elliptical and astigmatic beamsEdit
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 Template:Mvar and Template:Mvar and distinct definitions of the Template:Math point. The Gouy phase is a single value calculated correctly by summing the contribution from each dimension, with a Gouy phase within the range Template:Math 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.
Gaussian as a decomposition into modesEdit
Arbitrary solutions of the paraxial Helmholtz equation can be decomposed as the sum of Hermite–Gaussian modes (whose amplitude profiles are separable in Template:Mvar and Template:Mvar using Cartesian coordinates), Laguerre–Gaussian modes (whose amplitude profiles are separable in Template:Mvar and Template:Mvar using cylindrical coordinates) or similarly as combinations of Ince–Gaussian modes (whose amplitude profiles are separable in Template:Mvar and Template:Mvar using elliptical coordinates).<ref name="siegman642">Siegman, p. 642.</ref><ref name="goubau">probably first considered by Goubau and Schwering (1961).</ref><ref name="ince-beams">Bandres and Gutierrez-Vega (2004)</ref> At any point along the beam Template:Mvar 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 which is why the net transverse profile due to a superposition of modes evolves in Template:Mvar, whereas the propagation of any single Hermite–Gaussian (or Laguerre–Gaussian) mode retains the same form along a beam.
Although there are other modal decompositions, Gaussians are 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 (cavity). "Gaussian beam" normally implies radiation confined to the fundamental (TEM00) Gaussian mode.
Beam parametersEdit
The geometric dependence of the fields of a Gaussian beam are governed by the light's wavelength Template:Mvar (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 waistEdit
The shape of a Gaussian beam of a given wavelength Template:Mvar is governed solely by one parameter, the beam waist Template:Math. This is a measure of the beam size at the point of its focus (Template:Math in the above equations) where the beam width Template:Math (as defined above) is the smallest (and likewise where the intensity on-axis (Template:Math) is the largest). From this parameter the other parameters describing the beam geometry are determined. This includes the Rayleigh range Template:Math and asymptotic beam divergence Template:Mvar, as detailed below.
Rayleigh range and confocal parameterEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The Rayleigh distance or Rayleigh range Template:Math is determined given a Gaussian beam's waist size:
<math display="block">z_\mathrm{R} = \frac{\pi w_0^2 n}{\lambda}.</math>
Here Template:Mvar is the wavelength of the light, Template:Mvar is the index of refraction. At a distance from the waist equal to the Rayleigh range Template:Math, the width Template:Mvar of the beam is Template:Math larger than it is at the focus where Template:Math, the beam waist. That also implies that the on-axis (Template:Math) intensity there is one half of the peak intensity (at Template:Math). That point along the beam also happens to be where the wavefront curvature (Template:Math) is greatest.<ref name="svelto153" />
The distance between the two points Template:Math is called the confocal parameter or depth of focus of the beam.<ref>Template:Cite journal</ref>
Beam divergenceEdit
Template:Further 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 Template:Math. That is where the intensity has dropped to Template:Math of its on-axis value. Now, for Template:Math the parameter Template:Math increases linearly with Template:Mvar. This means that far from the waist, the beam "edge" (in the above sense) is cone-shaped. The angle between that cone (whose Template:Math) and the beam axis (Template:Math) defines the divergence of the beam: <math display="block">\theta = \lim_{z\to\infty} \arctan\left(\frac{w(z)}{z}\right).</math>
In the paraxial case, as we have been considering, Template:Mvar (in radians) is then approximately<ref name="svelto153" /> <math display="block">\theta = \frac{\lambda}{\pi n w_0}</math>
where Template:Mvar is the refractive index of the medium the beam propagates through, and Template:Mvar 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 <math display="block">\Theta = 2 \theta\, .</math>
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 Template:Mvar, 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 (Template:Math) at the waist (and thus a large diameter where it is launched, since Template:Math is never less than Template:Math). This relationship between beam width and divergence is a fundamental characteristic of diffraction, and of the Fourier transform which describes Fraunhofer diffraction. 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.<ref>Siegman (1986) p. 630.</ref> From the above expression for divergence, this means the Gaussian beam model is only accurate for beams with waists larger than about Template:Math.
Laser beam quality is quantified by the beam parameter product (BPP). For a Gaussian beam, the BPP is the product of the beam's divergence and waist size Template:Math. 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 Template:Math ("M squared"). The Template:Math for a Gaussian beam is one. All real laser beams have Template:Math values greater than one, although very high quality beams can have values very close to one.
The numerical aperture of a Gaussian beam is defined to be Template:Math, where Template:Mvar is the index of refraction of the medium through which the beam propagates. This means that the Rayleigh range is related to the numerical aperture by <math display="block">z_\mathrm{R} = \frac{n w_0}{\mathrm{NA}} .</math>
Gouy phaseEdit
The Gouy phase is a phase shift gradually acquired by a beam around the focal region. At position Template:Mvar the Gouy phase of a fundamental Gaussian beam is given by<ref name="svelto153" /> <math display="block">\psi(z) = \arctan \left( \frac{z}{z_\mathrm{R}} \right).</math>
The Gouy phase results in an increase in the apparent wavelength near the waist (Template:Math). 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 which case the wavefronts' curvature (see previous section) changes substantially over the distance of a single wavelength. In all cases the wave equation is satisfied at every position.
The sign of the Gouy phase depends on the sign convention chosen for the electric field phasor.<ref name="gouy_phase_shift" /> With Template:Math dependence, the Gouy phase changes from Template:Math to Template:Math, while with Template:Math dependence it changes from Template:Math to Template:Math 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 Template:Mvar 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.<ref name="gouy_phase_shift"/>
Power and intensityEdit
Power through an apertureEdit
With a beam centered on an aperture, the power Template:Mvar passing through a circle of radius Template:Mvar in the transverse plane at position Template:Mvar is<ref name="melles griot">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> <math display="block">P(r,z) = P_0 \left[ 1 - e^{-2r^2 / w^2(z)} \right],</math> where <math display="block">P_0 = \frac{ 1 }{ 2 } \pi I_0 w_0^2</math> is the total power transmitted by the beam.
For a circle of radius Template:Math, the fraction of power transmitted through the circle is <math display="block">\frac{P(z)}{P_0} = 1 - e^{-2} \approx 0.865.</math>
Similarly, about 90% of the beam's power will flow through a circle of radius Template:Math, 95% through a circle of radius Template:Math, and 99% through a circle of radius Template:Math.<ref name="melles griot"/>
Peak intensityEdit
The peak intensity at an axial distance Template:Mvar from the beam waist can be calculated as the limit of the enclosed power within a circle of radius Template:Mvar, divided by the area of the circle Template:Math as the circle shrinks: <math display="block">I(0,z) = \lim_{r\to 0} \frac {P_0 \left[ 1 - e^{-2r^2 / w^2(z)} \right]} {\pi r^2} .</math>
The limit can be evaluated using L'Hôpital's rule: <math display="block">I(0,z)
= \frac{P_0}{\pi} \lim_{r\to 0} \frac { \left[ -(-2)(2r) e^{-2r^2 / w^2(z)} \right]} {w^2(z)(2r)}
= {2P_0 \over \pi w^2(z)} .</math>
Complex beam parameterEdit
Template:Main article The spot size and curvature of a Gaussian beam as a function of Template:Mvar along the beam can also be encoded in the complex beam parameter Template:Math<ref name="siegman638">Siegman, pp. 638–40.</ref><ref name="garg168">Garg, pp. 165–168.</ref> given by: <math display="block"> q(z) = z + iz_\mathrm{R} .</math>
The reciprocal of Template:Math contains the wavefront curvature and relative on-axis intensity in its real and imaginary parts, respectively:<ref name="siegman638" />
<math display="block">{1 \over q(z)} = {1 \over R(z)} - i {\lambda \over n \pi w^2(z)} .</math>
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 Template:Mvar the relative field strength of an elliptical Gaussian beam (with the elliptical axes in the Template:Mvar and Template:Mvar directions) then it can be separated in Template:Mvar and Template:Mvar according to: <math display="block">u(x,y,z) = u_x(x,z)\, u_y(y,z) ,</math>
where <math display="block">\begin{align} u_x(x,z) &= \frac{1}{\sqrt{{q}_x(z)}} \exp\left(-i k \frac{x^2}{2 {q}_x(z)}\right), \\ u_y(y,z) &= \frac{1}{\sqrt{{q}_y(z)}} \exp\left(-i k \frac{y^2}{2 {q}_y(z)}\right), \end{align}</math>
where Template:Math and Template:Math are the complex beam parameters in the Template:Mvar and Template:Mvar directions.
For the common case of a circular beam profile, Template:Math and Template:Math, which yields<ref>See Siegman (1986) p. 639. Eq. 29</ref> <math display="block">u(r,z) = \frac{1}{q(z)}\exp\left( -i k\frac{r^2}{2 q(z)}\right) .</math>
Beam opticsEdit
When a gaussian beam propagates through a thin lens, the outgoing beam is also a (different) gaussian beam, provided that the beam travels along the cylindrical symmetry axis of the lens, and that the lens is larger than the width of the beam. The focal length of the lens <math>f</math>, the beam waist radius <math>w_0</math>, and beam waist position <math>z_0</math> of the incoming beam can be used to determine the beam waist radius <math>w_0'</math> and position <math>z_0'</math> of the outgoing beam.
Lens equationEdit
As derived by Saleh and Teich, the relationship between the ingoing and outgoing beams can be found by considering the phase that is added to each point <math>(x,y)</math> of the gaussian beam as it travels through the lens.<ref name="fourier derivation of gaussian lens">Template:Cite book Chapter 3, "Beam Optics"</ref> An alternative approach due to Self is to consider the effect of a thin lens on the gaussian beam wavefronts.<ref name="wavefront derivation of gaussian lens">Template:Cite journal</ref>
The exact solution to the above problem is expressed simply in terms of the magnification <math>M</math>
- <math>
\begin{align} w_0' &= Mw_0\\[1.2ex] (z_0'-f) &= M^2(z_0-f). \end{align} </math>
The magnification, which depends on <math>w_0</math> and <math>z_0</math>, is given by
- <math>
M = \frac{M_r}{\sqrt{1+r^2}} </math>
where
- <math>
r = \frac{z_R}{z_0-f}, \quad M_r = \left|\frac{f}{z_0-f}\right|. </math>
An equivalent expression for the beam position <math>z_0'</math> is
- <math>
\frac{1}{z_0+\frac{z_R^2}{(z_0-f)}}+\frac{1}{z_0'} = \frac{1}{f}. </math>
This last expression makes clear that the ray optics thin lens equation is recovered in the limit that <math>\left|\left(\tfrac{z_R}{z_0}\right)\left(\tfrac{z_R}{z_0-f}\right)\right|\ll 1</math>. It can also be noted that if <math>\left|z_0+\frac{z_R^2}{z_0-f}\right|\gg f</math> then the incoming beam is "well collimated" so that <math>z_0'\approx f</math>.
Beam focusingEdit
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 <math>M</math>. 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 <math>z_R</math> and minimizing <math>f</math>. In this situation, it is justifiable to make the approximation <math>z_R^2/(z_0-f)^2\gg 1</math>, implying that <math>M\approx f/z_R</math> and yielding the result <math>w_0'\approx fw_0/z_R</math>. This result is often presented in the form
- <math>
\begin{align} 2w_0' &\approx \frac{4}{\pi}\lambda F_\# \\[1.2ex] z_0' &\approx f \end{align} </math>
where
- <math>
F_\# = \frac{f}{2w_0}, </math>
which is found after assuming that the medium has index of refraction <math>n\approx 1</math> and substituting <math>z_R=\pi w_0^2/\lambda</math>. The factors of 2 are introduced because of a common preference to represent beam size by the beam waist diameters <math>2w_0'</math> and <math>2w_0</math>, rather than the waist radii <math>w_0'</math> and <math>w_0</math>.
Wave equationEdit
As a special case of electromagnetic radiation, 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,<ref name="svelto148">Svelto, pp. 148–9.</ref> obtained by combining Maxwell's equations for the curl of Template:Mvar and the curl of Template:Mvar, resulting in: <math display="block"> \nabla^2 U = \frac{1}{c^2} \frac{\partial^2 U}{\partial t^2},</math> where Template:Mvar is the speed of light in the medium, and Template:Mvar 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 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 Template:Math direction in which case the solution Template:Mvar can generally be written in terms of Template:Mvar which has no time dependence and varies relatively smoothly in space, with the main variation spatially corresponding to the wavenumber Template:Mvar in the Template:Mvar direction:<ref name="svelto148" /> <math display="block"> U(x, y, z, t) = u(x, y, z) e^{-i(kz-\omega t)} \, \hat{\mathbf x} \, .</math>
Using this form along with the paraxial approximation, Template:Math can then be essentially neglected. Since solutions of the electromagnetic wave equation only hold for polarizations which are orthogonal to the direction of propagation (Template:Mvar), we have without loss of generality considered the polarization to be in the Template:Mvar direction so that we now solve a scalar equation for Template:Math.
Substituting this solution into the wave equation above yields the paraxial approximation to the scalar wave equation:<ref name="svelto148" /> <math display="block">\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 2ik \frac{\partial u}{\partial z}.</math> Writing the wave equations in the light-cone coordinates returns this equation without utilizing any approximation.<ref>Template:Cite journal</ref> Gaussian beams of any beam waist Template:Math satisfy the paraxial approximation to the scalar wave equation; this is most easily verified by expressing the wave at Template:Mvar in terms of the complex beam parameter Template:Math as defined above. There are many other solutions. As solutions to a linear system, 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 modesEdit
Hermite-Gaussian modesEdit
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 Template:Mvar and a factor in Template:Mvar. Such a solution is possible due to the separability in Template:Mvar and Template:Mvar in the paraxial Helmholtz equation as written in Cartesian coordinates.<ref>Siegman (1986), p645, eq. 54</ref> Thus given a mode of order Template:Math referring to the Template:Mvar and Template:Mvar directions, the electric field amplitude at Template:Math may be given by: <math display="block"> E(x,y,z) = u_l(x,z) \, u_m(y,z) \, \exp(-ikz), </math> where the factors for the Template:Mvar and Template:Mvar dependence are each given by: <math display="block"> u_J(x,z) = \left(\frac{\sqrt{2/\pi}}{ 2^J \, J! \; w_0}\right)^{\!\!1/2} \!\! \left( \frac{{q}_0}{{q}(z)}\right)^{\!\!1/2} \!\! \left(- \frac{{q}^\ast(z)}{{q}(z)}\right)^{\!\! J/2} \!\! H_J\!\left(\frac{\sqrt{2}x}{w(z)}\right) \, \exp \left(\! -i \frac{k x^2}{2 {q}(z)}\right) , </math> where we have employed the complex beam parameter Template:Math (as defined above) for a beam of waist Template:Math at Template:Mvar from the focus. In this form, the first factor is just a normalizing constant to make the set of Template:Math orthonormal. The second factor is an additional normalization dependent on Template:Mvar which compensates for the expansion of the spatial extent of the mode according to Template:Math (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 Template:Mvar.
The final two factors account for the spatial variation over Template:Mvar (or Template:Mvar). The fourth factor is the Hermite polynomial of order Template:Mvar ("physicists' form", i.e. Template:Math), while the fifth accounts for the Gaussian amplitude fall-off Template:Math, although this isn't obvious using the complex Template:Mvar in the exponent. Expansion of that exponential also produces a phase factor in Template:Mvar which accounts for the wavefront curvature (Template:Math) at Template:Mvar along the beam.
Hermite-Gaussian modes are typically designated "TEMlm"; the fundamental Gaussian beam may thus be referred to as TEM00 (where TEM is transverse electro-magnetic). Multiplying Template:Math and Template:Math to get the 2-D mode profile, and removing the normalization so that the leading factor is just called Template:Math, we can write the Template:Math mode in the more accessible form:
<math display="block">\begin{align}
E_{l, m}(x, y, z) ={}
& E_0 \frac{w_0}{w(z)}\, H_l \!\Bigg(\frac{\sqrt{2} \,x}{w(z)}\Bigg)\, H_m \!\Bigg(\frac{\sqrt{2} \,y}{w(z)}\Bigg) \times {} \exp \left( {-\frac{x^2+y^2}{w^2(z)}} \right) \exp \left( {-i\frac{k(x^2 + y^2)}{2R(z)}} \right) \times {} \exp \big(i \psi(z)\big) \exp(-ikz).
\end{align}</math>
In this form, the parameter Template:Math, 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 Template:Math. Given that Template:Math, Template:Math and Template:Math have the same definitions as for the fundamental Gaussian beam described above. It can be seen that with Template:Math we obtain the fundamental Gaussian beam described earlier (since Template:Math). The only specific difference in the Template:Mvar and Template:Mvar profiles at any Template:Mvar are due to the Hermite polynomial factors for the order numbers Template:Mvar and Template:Mvar. However, there is a change in the evolution of the modes' Gouy phase over Template:Mvar: <math display="block"> \psi(z) = (N+1) \, \arctan \left( \frac{z}{z_\mathrm{R}} \right), </math>
where the combined order of the mode Template:Mvar is defined as Template:Math. While the Gouy phase shift for the fundamental (0,0) Gaussian mode only changes by Template:Math radians over all of Template:Mvar (and only by Template:Math radians between Template:Math), this is increased by the factor Template:Math for the higher order modes.<ref name="gouy_phase_shift">Template:Cite encyclopedia</ref>
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.<ref name="goubau"/> These functions are written in cylindrical coordinates using generalized Laguerre polynomials. Each transverse mode is again labelled using two integers, in this case the radial index Template:Math and the azimuthal index Template:Mvar which can be positive or negative (or zero):<ref name="orbital momentum of light">Template:Cite journal</ref><ref>Template:Cite journal</ref>
<math display="block">\begin{align}
u(r, \phi, z) ={}
&C^{LG}_{lp}\frac{1}{w(z)}\left(\frac{r \sqrt{2}}{w(z)}\right)^{\! |l|} \exp\! \left(\! -\frac{r^2}{w^2(z)}\right)L_p^{|l|} \! \left(\frac{2r^2}{w^2(z)}\right) \times {} \\
&\exp \! \left(\! - i k \frac{r^2}{2 R(z)}\right) \exp(-i l \phi) \, \exp(i \psi(z)) ,
\end{align}</math>
where Template:Math are the generalized Laguerre polynomials. Template:Math is a required normalization constant:<ref name="LG_normalization">Note that the normalization used here (total intensity for a fixed Template:Math equal to unity) differs from that used in section #Mathematical form for the Gaussian mode. For Template:Math the Laguerre-Gaussian mode reduces to the standard Gaussian mode, but due to different normalization conditions the two formulas do not coincide.</ref> <math display="block">C^{LG}_{lp} = \sqrt{\frac{2 p!}{\pi(p+|l|)!}} \Rightarrow \int_0^{2\pi}d\phi\int_0^\infty dr\; r \,|u(r,\phi,z)|^2=1,</math>.
Template:Math and Template:Math 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 Template:Math: <math display="block">\psi(z) = (N+1) \, \arctan \left( \frac{z}{z_\mathrm{R}} \right) ,</math> where in this case the combined mode number Template:Math. 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 Template:Mvar but now multiplied by a Laguerre polynomial. The effect of the rotational mode number Template:Mvar, in addition to affecting the Laguerre polynomial, is mainly contained in the phase factor Template:Math, in which the beam profile is advanced (or retarded) by Template:Mvar complete Template:Math phases in one rotation around the beam (in Template:Mvar). This is an example of an optical vortex of topological charge Template:Mvar, and can be associated with the orbital angular momentum of light in that mode.
Ince-Gaussian modesEdit
In elliptic coordinates, one can write the higher-order modes using Ince polynomials. The even and odd Ince-Gaussian modes are given by<ref name="ince-beams"/>
<math display="block"> u_\varepsilon \left( \xi ,\eta ,z\right) = \frac{w_{0}}{w\left( z\right) }\mathrm{C}_{p}^{m}\left( i\xi ,\varepsilon \right) \mathrm{C} _{p}^{m}\left( \eta ,\varepsilon \right) \exp \left[ -ik\frac{r^{2}}{ 2q\left( z\right) }-\left( p+1\right) \zeta\left( z\right) \right] , </math> where Template:Mvar and Template:Mvar are the radial and angular elliptic coordinates defined by <math display="block">\begin{align} x &= \sqrt{\varepsilon /2}\;w(z) \cosh \xi \cos \eta ,\\ y &= \sqrt{\varepsilon /2}\;w(z) \sinh \xi \sin \eta . \end{align}</math> Template:Math are the even Ince polynomials of order Template:Mvar and degree Template:Mvar where Template:Mvar is the ellipticity parameter. The Hermite-Gaussian and Laguerre-Gaussian modes are a special case of the Ince-Gaussian modes for Template:Math and Template:Math respectively.<ref name=ince-beams/>
Hypergeometric-Gaussian modesEdit
There is another important class of paraxial wave modes in cylindrical coordinates in which the complex amplitude is proportional to a confluent hypergeometric function.
These modes have a singular phase profile and are eigenfunctions of the photon orbital angular momentum. 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 Template:Math and the normalized longitudinal coordinate Template:Math as follows:<ref name="Karimi et al. 2007">Karimi et al. (2007)</ref>
<math display="block">\begin{align}
u_{\mathsf{p}m}(\rho, \phi, \Zeta) {}={}
&\sqrt{\frac{2^{\mathsf{p} + |m| + 1}}{\pi\Gamma(\mathsf{p} + |m| + 1)}}\; \frac{\Gamma\left(\frac{\mathsf{p}}{2} + |m| + 1\right)}{\Gamma(|m| + 1)}\, i^{|m|+1} \times{} \\
&\Zeta^{\frac{\mathsf{p}}{2}}\, (\Zeta + i)^{-\left(\frac{\mathsf{p}}{2} + |m| + 1\right)}\, \rho^{|m|} \times{} \\
&\exp\left(-\frac{i\rho^2}{\Zeta + i}\right)\, e^{im\phi}\, {}_1F_1 \left(-\frac{\mathsf{p}}{2}, |m| + 1; \frac{\rho^2}{\Zeta(\Zeta + i)}\right)
\end{align}</math>
where the rotational index Template:Mvar is an integer, and <math> {\mathsf p}\ge-|m| </math> is real-valued, Template:Math is the gamma function and Template:Math 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,<ref name="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 (Template:Math): <math display="block">u(\rho, \phi, 0) \propto \rho^{\mathsf{p} + |m|}e^{-\rho^2 + im\phi}.</math>
See alsoEdit
NotesEdit
<references/>
ReferencesEdit
- Template:Cite journal
- Template:Cite book
- Template:Cite journal
- Template:Cite journal
- Template:Cite book Chapter 5, "Optical Beams," pp. 267.
- Template:Cite arXiv
- Template:Cite journal
- Template:Cite book Chapter 3, "Beam Optics," pp. 80–107.
- Template:Cite book Chapter 16.
- Template:Cite book
- Template:Cite book