Gaussian beam

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 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 . The Gaussian beam is a transverse electromagnetic (TEM) mode. The mathematical expression for the electric field amplitude is a solution to the paraxial Helmholtz equation. • 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, • is the wave number (in radians 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 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 of the beam's wavefronts at , and • is the Gouy phase at , 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: \mathbf E_\text{phys}(r,z,t) = \operatorname{Re}(\mathbf E(r,z) \cdot e^{i\omega t}), where \omega is the angular frequency of the light and is time. The time factor involves an arbitrary sign convention, 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) distribution is given by 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), where the constant is the wave impedance 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 of the beam, I_0 = {2P_0 \over \pi w_0^2}. Evolving beam width has a diameter ( as used in the text) about 1.7 times the FWHM. At a position along the beam (measured from the focus), the spot size parameter is given by a hyperbolic relation: w(z)={\frac {\text{FWHM}(z)}{\sqrt {2\ln2}}}. Wavefront curvature The wavefronts have zero curvature (radius = ∞) at the waist. Wavefront curvature increases in magnitude away from the waist, reaching an extremum at the Rayleigh distance, (maximum for , minimum for ). Beyond the Rayleigh distance, , the curvature again decreases in magnitude, approaching zero as . The curvature is often expressed in terms of its reciprocal, , the radius of curvature; for a fundamental Gaussian beam the curvature at position is given by: \frac{1}{R(z)} = \frac{z} {z^2 + z_\mathrm{R}^2} , so the radius of curvature is 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 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 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 (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 . : beam waist; : depth of focus; : Rayleigh range; : total angular spread 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 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{R} = \frac{\pi w_0^2 n}{\lambda}. 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. 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_{z\to\infty} \arctan\left(\frac{w(z)}{z}\right). In the paraxial case, as we have been considering, (in radians) is then approximately 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 (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"). 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 of a Gaussian beam is defined to be , where 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 z_\mathrm{R} = \frac{n w_0}{\mathrm{NA}} . Gouy phase The Gouy phase is a phase shift 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{z}{z_\mathrm{R}} \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 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. 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. ==Power through an aperture==

If a Gaussian beam is centered on a circular aperture of radius at distance from the beam waist, the power that passes through the aperture is P(r,z) = P_0 \left[ 1 - e^{-2r^2 / w^2(z)} \right], For a circle of radius , the fraction of power transmitted through the circle is \frac{P(w(z),z)}{P_0} = 1 - e^{-2} \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 . ==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 given by: q(z) = z + iz_\mathrm{R} . The reciprocal of contains the wavefront curvature and relative on-axis intensity in its real and imaginary parts, respectively: u(r,z) = \frac{1}{q(z)}\exp\left( -i k\frac{r^2}{2 q(z)}\right) . ==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 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 that is added to each point (x,y) of the gaussian beam as it travels through the lens. 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{align} w_0' &= Mw_0\\[1.2ex] (z_0'-f) &= M^2(z_0-f). \end{align} The magnification, which depends on w_0 and z_0, is given by : M = \frac{M_r}{\sqrt{1+r^2}} where : r = \frac{z_R}{z_0-f}, \quad M_r = \left|\frac{f}{z_0-f}\right|. An equivalent expression for the beam position z_0' is : \frac{1}{z_0+\frac{z_R^2}{(z_0-f)}}+\frac{1}{z_0'} = \frac{1}{f}. This last expression makes clear that the ray optics thin lens equation is recovered in the limit that \left|\left(\tfrac{z_R}{z_0}\right)\left(\tfrac{z_R}{z_0-f}\right)\right|\ll 1. It can also be noted that if \left|z_0+\frac{z_R^2}{z_0-f}\right|\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{align} 2w_0' &\approx \frac{4}{\pi}\lambda F_\# \\[1.2ex] z_0' &\approx f \end{align} where : F_\# = \frac{f}{2w_0}, 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, 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, obtained by combining Maxwell's equations for the curl of and the curl of , resulting in: \nabla^2 U = \frac{1}{c^2} \frac{\partial^2 U}{\partial t^2}, 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 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 direction: 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, 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 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{\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) , 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. 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 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 "TEMlm"; 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{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} 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{z}{z_\mathrm{R}} \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. \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^ \! \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} where are the generalized Laguerre polynomials. is a required normalization constant: 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,. 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{z}{z_\mathrm{R}} \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 of topological charge , and can be associated with the orbital angular momentum of light in that mode. Ince–Gaussian modes The third complete family of solution for the paraxial wave equation is the Ince–Gaussian modes. They describe beams with elliptic transverse geometry characterized by the ellipticity \varepsilon. The Hermite–Gaussian and Laguerre–Gaussian modes are a special case of the Ince–Gaussian modes for \varepsilon = \infty and \varepsilon = 0 respectively. The Ince–Gaussian modes can be written using elliptic coordinates and Ince polynomials. The even and odd Ince–Gaussian modes \mathrm{IG}^\mathrm{e}_{p,m} and \mathrm{IG}^\mathrm{o}_{p,m} are given by: \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^ \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} where the rotational index is an integer, and {\mathsf p}\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, 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^{\mathsf{p} + |m|}e^{-\rho^2 + im\phi}. ==See also==