Английская Википедия:Gaussian beam
In optics, a Gaussian beam is an ideal 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 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 Шаблон:Math. At any position Шаблон:Mvar relative to the waist (focus) along a beam having a specified Шаблон:Math, the field amplitudes and phases are thereby determined[1] as detailed below.
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 axial 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 form
The equations below assume a beam with a circular cross-section at all values of Шаблон:Mvar; this can be seen by noting that a single transverse dimension, Шаблон:Mvar, appears. Beams with elliptical cross-sections, or with waists at different positions in Шаблон:Mvar for the two transverse dimensions (astigmatic beams) can also be described as Gaussian beams, but with distinct values of Шаблон:Math and of the Шаблон:Math location for the two transverse dimensions Шаблон:Mvar and Шаблон:Mvar.
The Gaussian beam is a transverse electromagnetic (TEM) mode.[2] The mathematical expression for the electric field amplitude is a solution to the paraxial Helmholtz equation.[1] Assuming polarization in the Шаблон:Mvar direction and propagation in the Шаблон: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>
- Шаблон:Mvar is the radial distance from the center axis of the beam,
- Шаблон:Mvar is the axial distance from the beam's focus (or "waist"),
- Шаблон:Mvar is the imaginary unit,
- Шаблон:Math is the wave number (in radians per meter) for a free-space wavelength Шаблон:Mvar, and Шаблон:Mvar is the index of refraction of the medium in which the beam propagates,
- Шаблон:Math, the electric field amplitude (and phase) at the origin (Шаблон:Math, Шаблон:Math),
- Шаблон:Math is the radius at which the field amplitudes fall to Шаблон:Math of their axial values (i.e., where the intensity values fall to Шаблон:Math of their axial values), at the plane Шаблон:Mvar along the beam,
- Шаблон:Math is the waist radius,
- Шаблон:Math is the radius of curvature of the beam's wavefronts at Шаблон:Mvar, and
- Шаблон:Math is the Gouy phase at Шаблон: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 Шаблон:Mvar is time. The time factor involves an arbitrary sign convention, as discussed at Шаблон: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 Шаблон: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 Шаблон:Mvar is the wave impedance of the medium in which the beam is propagating. For free space, Шаблон:Math ≈ 377 Ω. Шаблон:Math is the intensity at the center of the beam at its waist.
If Шаблон:Math is the total power of the beam, <math display="block">I_0 = {2P_0 \over \pi w_0^2}.</math>
Evolving beam width
At a position Шаблон:Mvar along the beam (measured from the focus), the spot size parameter Шаблон:Mvar is given by a hyperbolic relation:[1] <math display="block">w(z) = w_0 \, \sqrt{ 1+ {\left( \frac{z}{z_\mathrm{R}} \right)}^2 },</math> where[1] <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 Шаблон:Math, at any position Шаблон:Mvar along the beam, is related to the full width at half maximum (FWHM) of the intensity distribution at that position according to:[4] <math display="block">w(z)={\frac {\text{FWHM}(z)}{\sqrt {2\ln2}}}.</math>
Wavefront curvature
The curvature of the wavefronts is largest at the Rayleigh distance, Шаблон:Math, on either side of the waist, crossing zero at the waist itself. Beyond the Rayleigh distance, Шаблон:Math, it again decreases in magnitude, approaching zero as Шаблон:Math. The curvature is often expressed in terms of its reciprocal, Шаблон:Mvar, the radius of curvature; for a fundamental Gaussian beam the curvature at position Шаблон: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 Шаблон:Math is [1] <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 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 Шаблон:Mvar and Шаблон:Mvar and distinct definitions of the Шаблон: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 Шаблон: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 modes
Arbitrary solutions of the paraxial Helmholtz equation can be decomposed as the sum of Hermite–Gaussian modes (whose amplitude profiles are separable in Шаблон:Mvar and Шаблон:Mvar using Cartesian coordinates), Laguerre–Gaussian modes (whose amplitude profiles are separable in Шаблон:Mvar and Шаблон:Mvar using cylindrical coordinates) or similarly as combinations of Ince–Gaussian modes (whose amplitude profiles are separable in Шаблон:Mvar and Шаблон:Mvar using elliptical coordinates).[5][6][7] At any point along the beam Шаблон: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 Шаблон: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 parameters
The geometric dependence of the fields of a Gaussian beam are governed by the light's wavelength Шаблон: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 waist
The shape of a Gaussian beam of a given wavelength Шаблон:Mvar is governed solely by one parameter, the beam waist Шаблон:Math. This is a measure of the beam size at the point of its focus (Шаблон:Math in the above equations) where the beam width Шаблон:Math (as defined above) is the smallest (and likewise where the intensity on-axis (Шаблон:Math) is the largest). From this parameter the other parameters describing the beam geometry are determined. This includes the Rayleigh range Шаблон:Math and asymptotic beam divergence Шаблон:Mvar, as detailed below.
Rayleigh range and confocal parameter
Шаблон:Main The Rayleigh distance or Rayleigh range Шаблон: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 Шаблон:Mvar is the wavelength of the light, Шаблон:Mvar is the index of refraction. At a distance from the waist equal to the Rayleigh range Шаблон:Math, the width Шаблон:Mvar of the beam is Шаблон:Math larger than it is at the focus where Шаблон:Math, the beam waist. That also implies that the on-axis (Шаблон:Math) intensity there is one half of the peak intensity (at Шаблон:Math). That point along the beam also happens to be where the wavefront curvature (Шаблон:Math) is greatest.[1]
The distance between the two points Шаблон:Math is called the confocal parameter or depth of focus of the beam.[8]
Beam divergence
Шаблон: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 Шаблон:Math. That is where the intensity has dropped to Шаблон:Math of its on-axis value. Now, for Шаблон:Math the parameter Шаблон:Math increases linearly with Шаблон:Mvar. This means that far from the waist, the beam "edge" (in the above sense) is cone-shaped. The angle between that cone (whose Шаблон:Math) and the beam axis (Шаблон: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, Шаблон:Mvar (in radians) is then approximately[1] <math display="block">\theta = \frac{\lambda}{\pi n w_0}</math>
where Шаблон:Mvar is the refractive index of the medium the beam propagates through, and Шаблон: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 Шаблон: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 (Шаблон:Math) at the waist (and thus a large diameter where it is launched, since Шаблон:Math is never less than Шаблон: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.[9] From the above expression for divergence, this means the Gaussian beam model is only accurate for beams with waists larger than about Шаблон: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 Шаблон: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 Шаблон:Math ("M squared"). The Шаблон:Math for a Gaussian beam is one. All real laser beams have Шаблон: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 Шаблон:Math, where Шаблон: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 phase
The Gouy phase is a phase shift gradually acquired by a beam around the focal region. At position Шаблон:Mvar the Gouy phase of a fundamental Gaussian beam is given by[1] <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 (Шаблон: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.[10] With Шаблон:Math dependence, the Gouy phase changes from Шаблон:Math to Шаблон:Math, while with Шаблон:Math dependence it changes from Шаблон:Math to Шаблон: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 Шаблон: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.[10]
Power and intensity
Power through an aperture
With a beam centered on an aperture, the power Шаблон:Mvar passing through a circle of radius Шаблон:Mvar in the transverse plane at position Шаблон:Mvar is[11] <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 Шаблон: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 Шаблон:Math, 95% through a circle of radius Шаблон:Math, and 99% through a circle of radius Шаблон:Math.[11]
Peak intensity
The peak intensity at an axial distance Шаблон:Mvar from the beam waist can be calculated as the limit of the enclosed power within a circle of radius Шаблон:Mvar, divided by the area of the circle Шаблон: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 parameter
Шаблон:Main article The spot size and curvature of a Gaussian beam as a function of Шаблон:Mvar along the beam can also be encoded in the complex beam parameter Шаблон:Math[12][13] given by: <math display="block"> q(z) = z + iz_\mathrm{R} .</math>
Introducing this complication leads to a simplification of the Gaussian beam field equation as shown below. It can be seen that the reciprocal of Шаблон:Math contains the wavefront curvature and relative on-axis intensity in its real and imaginary parts, respectively:[12]
<math display="block">{1 \over q(z)} = {1 \over z + iz_\mathrm{R}} = {z \over z^2 + z_\mathrm{R}^2} - i {z_\mathrm{R} \over z^2 + z_\mathrm{R}^2} = {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 Шаблон:Mvar the relative field strength of an elliptical Gaussian beam (with the elliptical axes in the Шаблон:Mvar and Шаблон:Mvar directions) then it can be separated in Шаблон:Mvar and Шаблон: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 Шаблон:Math and Шаблон:Math are the complex beam parameters in the Шаблон:Mvar and Шаблон:Mvar directions.
For the common case of a circular beam profile, Шаблон:Math and Шаблон:Math, which yields[14] <math display="block">u(r,z) = \frac{1}{q(z)}\exp\left( -i k\frac{r^2}{2 q(z)}\right) .</math>
Beam optics
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 equation
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.[15] An alternative approach due to Self is to consider the effect of a thin lens on the gaussian beam wavefronts.[16]
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 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 <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 equation
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,[17] obtained by combining Maxwell's equations for the curl of Шаблон:Mvar and the curl of Шаблон:Mvar, resulting in: <math display="block"> \nabla^2 U = \frac{1}{c^2} \frac{\partial^2 U}{\partial t^2},</math> where Шаблон:Mvar is the speed of light in the medium, and Шаблон: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 Шаблон:Math direction in which case the solution Шаблон:Mvar can generally be written in terms of Шаблон:Mvar which has no time dependence and varies relatively smoothly in space, with the main variation spatially corresponding to the wavenumber Шаблон:Mvar in the Шаблон:Mvar direction:[17] <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, Шаблон: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 (Шаблон:Mvar), we have without loss of generality considered the polarization to be in the Шаблон:Mvar direction so that we now solve a scalar equation for Шаблон:Math.
Substituting this solution into the wave equation above yields the paraxial approximation to the scalar wave equation:[17] <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.[18] Gaussian beams of any beam waist Шаблон:Math satisfy the paraxial approximation to the scalar wave equation; this is most easily verified by expressing the wave at Шаблон:Mvar in terms of the complex beam parameter Шаблон: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 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 Шаблон:Mvar and a factor in Шаблон:Mvar. Such a solution is possible due to the separability in Шаблон:Mvar and Шаблон:Mvar in the paraxial Helmholtz equation as written in Cartesian coordinates.[19] Thus given a mode of order Шаблон:Math referring to the Шаблон:Mvar and Шаблон:Mvar directions, the electric field amplitude at Шаблон: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 Шаблон:Mvar and Шаблон: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 Шаблон:Math (as defined above) for a beam of waist Шаблон:Math at Шаблон:Mvar from the focus. In this form, the first factor is just a normalizing constant to make the set of Шаблон:Math orthonormal. The second factor is an additional normalization dependent on Шаблон:Mvar which compensates for the expansion of the spatial extent of the mode according to Шаблон: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 Шаблон:Mvar.
The final two factors account for the spatial variation over Шаблон:Mvar (or Шаблон:Mvar). The fourth factor is the Hermite polynomial of order Шаблон:Mvar ("physicists' form", i.e. Шаблон:Math), while the fifth accounts for the Gaussian amplitude fall-off Шаблон:Math, although this isn't obvious using the complex Шаблон:Mvar in the exponent. Expansion of that exponential also produces a phase factor in Шаблон:Mvar which accounts for the wavefront curvature (Шаблон:Math) at Шаблон: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 Шаблон:Math and Шаблон:Math to get the 2-D mode profile, and removing the normalization so that the leading factor is just called Шаблон:Math, we can write the Шаблон: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 Шаблон: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 Шаблон:Math. Given that Шаблон:Math, Шаблон:Math and Шаблон:Math have the same definitions as for the fundamental Gaussian beam described above. It can be seen that with Шаблон:Math we obtain the fundamental Gaussian beam described earlier (since Шаблон:Math). The only specific difference in the Шаблон:Mvar and Шаблон:Mvar profiles at any Шаблон:Mvar are due to the Hermite polynomial factors for the order numbers Шаблон:Mvar and Шаблон:Mvar. However, there is a change in the evolution of the modes' Gouy phase over Шаблон:Mvar: <math display="block"> \psi(z) = (N+1) \, \arctan \left( \frac{z}{z_\mathrm{R}} \right), </math>
where the combined order of the mode Шаблон:Mvar is defined as Шаблон:Math. While the Gouy phase shift for the fundamental (0,0) Gaussian mode only changes by Шаблон:Math radians over all of Шаблон:Mvar (and only by Шаблон:Math radians between Шаблон:Math), this is increased by the factor Шаблон:Math for the higher order modes.[10]
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.[6] 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 Шаблон:Math and the azimuthal index Шаблон:Mvar which can be positive or negative (or zero):[20]
<math display="block">\begin{align}
u(r, \phi, z) ={}
&C^{LG}_{lp}\frac{w_0}{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 Шаблон:Math are the generalized Laguerre polynomials. Шаблон:Math is a required normalization constant: <math display="block">C^{LG}_{lp} = \sqrt{\frac{2 p!}{\pi(p+|l|)!}} \Rightarrow \int_0^{2\pi}d\phi\int_0^\infty rdr|u(r,\phi,z)|^2=1</math>.
Шаблон:Math and Шаблон: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 Шаблон: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 Шаблон: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 Шаблон:Mvar but now multiplied by a Laguerre polynomial. The effect of the rotational mode number Шаблон:Mvar, in addition to affecting the Laguerre polynomial, is mainly contained in the phase factor Шаблон:Math, in which the beam profile is advanced (or retarded) by Шаблон:Mvar complete Шаблон:Math phases in one rotation around the beam (in Шаблон:Mvar). This is an example of an optical vortex of topological charge Шаблон:Mvar, and can be associated with the orbital angular momentum of light in that mode.
Ince-Gaussian modes
In elliptic coordinates, one can write the higher-order modes using Ince polynomials. The even and odd Ince-Gaussian modes are given by[7]
<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 Шаблон:Mvar and Шаблон: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> Шаблон:Math are the even Ince polynomials of order Шаблон:Mvar and degree Шаблон:Mvar where Шаблон:Mvar is the ellipticity parameter. The Hermite-Gaussian and Laguerre-Gaussian modes are a special case of the Ince-Gaussian modes for Шаблон:Math and Шаблон:Math respectively.[7]
Hypergeometric-Gaussian modes
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 Шаблон:Math and the normalized longitudinal coordinate Шаблон:Math as follows:[21]
<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 Шаблон:Mvar is an integer, and <math> {\mathsf p}\ge-|m| </math> is real-valued, Шаблон:Math is the gamma function and Шаблон: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,[21] 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 (Шаблон:Math): <math display="block">u(\rho, \phi, 0) \propto \rho^{\mathsf{p} + |m|}e^{-\rho^2 + im\phi}.</math>
See also
Notes
- ↑ 1,0 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 Svelto, pp. 153–5.
- ↑ Svelto, p. 158.
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite web
- ↑ Siegman, p. 642.
- ↑ 6,0 6,1 probably first considered by Goubau and Schwering (1961).
- ↑ 7,0 7,1 7,2 Bandres and Gutierrez-Vega (2004)
- ↑ Шаблон:Cite journal
- ↑ Siegman (1986) p. 630.
- ↑ 10,0 10,1 10,2 Шаблон:Cite encyclopedia
- ↑ 11,0 11,1 Melles Griot. Gaussian Beam Optics
- ↑ 12,0 12,1 Siegman, pp. 638–40.
- ↑ Garg, pp. 165–168.
- ↑ See Siegman (1986) p. 639. Eq. 29
- ↑ Шаблон:Cite book Chapter 3, "Beam Optics"
- ↑ Шаблон:Cite journal
- ↑ 17,0 17,1 17,2 Svelto, pp. 148–9.
- ↑ Шаблон:Cite journal
- ↑ Siegman (1986), p645, eq. 54
- ↑ Шаблон:Cite journal
- ↑ 21,0 21,1 Karimi et al. (2007)
References
- Шаблон:Cite journal
- Шаблон:Cite book
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite book Chapter 5, "Optical Beams," pp. 267.
- Шаблон:Cite arXiv
- Шаблон:Cite journal
- Шаблон:Cite book Chapter 3, "Beam Optics," pp. 80–107.
- Шаблон:Cite book Chapter 16.
- Шаблон:Cite book
- Шаблон:Cite book
External links
