Ultrashort laser pulse phenomena, diels and rudolph. Deriving the equations behind a gaussian wave is straightforward but tedious. Exact solution of helmholtz equation for the case of non. Gaussian beam summary a gaussian beam is the paraxial limit solution to a wave equation equivalent to a point source shifted by an imaginary amount iz0 with z0 0 paraxial solution is just portion of spherical wave emanating from the origin q z i z0 measures complex distance from reference plane waist of guassian beam. Whilst a paraxial approach has some success close to the focal.
The gaussian beam solutions are the modes of the spherical mirror optical resonator see iii. When describing light as a wave phenomenon, the local propagation direction of the energy can be identified with a direction normal to the wavefronts except in situations with spatial walkoff. This implies that the electric and magnetic field components are mutually. It is also essential to establish the conditions under which the exact solution of maxwells equations can legitimately be approximated by the scalar gaussian wave. If a collimated gaussian beam with zr incident f is incident on a lens of focal length f along the lens axis its wavefront is nearly plane in front of the lens and hence the beam gets focused with its beam waist positioned to a good approximation.
A spatial restriction or squeeze of a wave solution. The exact monochromatic wave equation is the helmholtz equation. Paraxial optical beams can diverge at cone angles up to. The gaussian beam is an important solution of the helmholtz maxwell paraxial wave equation s the gaussian beam solutions are the modes of the spherical mirror optical resonator see iii. Understanding the paraxial gaussian beam formula comsol blog. One simple solution to the paraxial helmholtz equation. This approximate but analytic solution of the paraxial wave equation is known as gaussian beam 1,2. Department of physical optics, school of physics, university of sydney, sydney, new south. A r is a function of position which varies very slowly on a distance scale of a wavelength. Ludlow 2000 scalar field of nonparaxial gaussian beams. If you recall, in chapter 2 we solved the paraxial wave equation by introducing a. Nonparaxial theories of wave propagation are essential to model the interaction of highly focussed light with matter.
So far we have only dealth with two examples, but the conclusion is a general one. Recall from maxwells equations we obtained for a timeharmonic wave of freq. Gaussianbeam wave in both the plane of the transmitter and the plane of the. The scalar field of expression 1 is a solution of the wave equation and has many. They must satisfy the paraxial helmholtz equation derived earlier. Here we investigate the energy, momentum and propagation of the laguerre, hermite and incegaussian solutions lg, hg, and ig of the paraxial wave equation in an apertured nonparaxial regime. After reproducing beam and pulse expressions for the wellknown paraxial gaussian and axicon cases, we apply the method to analyse a laser beam with lorentzian transverse momentum distribution. Paraxial and exact wave equations exact wave equation paraxial wave equation a spherical wave emerging from the point r r 0. Recently, theoretical investigators have focused their attention on paraxial wave family of laser beams.
Paraxial waves, gaussian beams confocal parameter, beam divergence. From a physical point of view, the paraxial spherical wave is an acceptable approximate solution to describe the wave propagation. We shall see that special solutions to the electromagnetic wave equation exist that take the form of narrow beams called gaussian beams. Wave beam propagation in a weakly inhomogeneous isotropic. The gaussian beam is an important solution of the helmholtz maxwell paraxial wave equation s. The propagation of a gaussian beam in a homogeneous, isotropic, local, linear, and nonmagnetic dielectric medium is studied using the angular spectrum representation for the electric field. Gaussian beam wavefield computation uses high frequency asymptotic approximation to transform the helmholtz equation into a parabolic wave equation in raycentered coordinates s,n cerveny et al. The accuracy of the paraxial solution thus considered is assessed numerically in comparison with the corresponding exact solution of. Beam optics fundamentals of photonics wiley online library. This is still an exact solution to the paraxial wave equation, but.
This is a measure of the beam size at the point of its focus z0 in the above equations where the beam width wz as defined above is the smallest and likewise where the intensity onaxis r0 is the largest. Using the plane wave representation of the fundamental gaus. First, the phase is the same as that of the underlying gb, except for an excess phase zz. Paraxial optics with gaussian beams a gaussian beam is a function of z0 gaussian beam at any position is a function of q z i z0 a gaussian beam is locally a spherical wave with radius of curvature r consider a gaussian beam on a thin lens the size of the beam doesnt change on either side of the lens the curvature of the beam is altered. Iii22, the full distance between the 2 w 0 beam radii, within which the beam can be considered. Besides, let us also note that at the left part of the expression above we express the. Electromagnetic gaussian beams beyond the paraxial approximation. Energy, momentum and propagation of nonparaxial high. Laguerregaussian wave propagation in parabolic media. Note that rz does not obey ray tracing sign convention. Also, the amplitude of the wave should vary with z, asymptotically falling as 1z. Gaussian beams and the paraxial wave equation hermite gaussian beam modes taken from j. The paraxial gaussian beam formula is an approximation to the helmholtz equation derived from maxwells equations.
A free powerpoint ppt presentation displayed as a flash slide show on id. Full length article the gaussian wave solution of maxwell. The electric field associated with the gaussian beam inside the dielectric medium consists of the paraxial result and higherorder nongaussian correction terms. However, in our case that is, for lasers, this wave is not a convenient solution. Assume the modulation is a slowly varying function of z slowly here mean slow compared to the wavelength a variation of a can be written as so. The general solution to this is, of course, much more general than a gaussian. The transverse shape of these modes is described by the ince polynomials and is structurally stable under propagation. A new type of exact solutions of the full 3 dimensional spatial helmholtz equation for the case of non paraxial gaussian beams is presented here. We thus develop the paraxial wave equation, which forms the basis for gaussian beam propagation. A gaussian beam is basically a spherical wave field whose amplitude in the transverse direction has a gaussian pattern. A gaussian wave is usually considered in practice to be linearly polarised. Paraxial and exact wave equations 2 2 2 2 u u u 2jk 0 x y z 2 2 2 2 2 2 2 2 2 e e e 1 e x y z c t ex,y,z,t ux,y,z ejwt jkz exact wave equation paraxial wave equation 0 0 e jk r r e r r r a wellknown solution to the exact equation. Iii28 are exactly matched to the wavefront radii of the gaussian beam at those points and if the transverse size of the mirrors is substantially larger than the spot size of the beam at the mirrors, each of. Gaussian beams a paraxial wave is a plane wave ejkz modulated by a complex envelope ar that is a slowly varying function of position.
A new type of exact solutions of the full 3 dimensional spatial helmholtz equation for the case of nonparaxial gaussian beams is presented here. We derive focused laser pulse solutions to the electromagnetic wave equation in vacuum. Properties of the complex beam parameter by replacing the real valued radius of curvature rz for a spherical wave emerging from a real source point z 0 with a complex radius of curvature qz, we convert the paraxial spherical wave into a gaussian beam. Namely, we expect the transverse behavior of the wave function. Helmholtz equation, riccati equation, gaussian beam. An important solution of this equation that exhibits the characteristics of an optical beam is a wave called the gaussian beam. If the paraxial condition e22cr z paraxial gaussian beams. The beam power is principally concentrated within a small cylinder surrounding the beam axis. Helmholtz equation, paraxial approximation, gaussian beam. Derivation of ray matrix for concave mirror with radius r.
The shape of a gaussian beam of a given wavelength. Gaussian, hermitegaussian, and laguerregaussian beams. We see why the helmholtz equation may be regarded as a singular perturbation of the paraxial wave equation and how some of the difficulties arising in the solution of the former partial differential equation are related to this fact. Let us solve the paraxial wave equation assuming the solution has cylindrical symmetry. Ince gaussian modes of the paraxial wave equation and stable. Ifz is a slowly varying function of z as required by the paraxial approximation, the waves ig and c g have paraboloidal wave fronts with the same radius of curvature rz. For all wave solutions there is an interplay between the spatial extent of the solution with the range of wavenumbers required to model it. The paraxial helmholtz equation start with helmholtz equation consider the wave which is a plane wave propagating along z transversely modulated by the complex amplitude a. The gaussian beam is an important solution of the helmholtz maxwell paraxial wave equations the gaussian beam solutions are the modes of the spherical mirror optical resonator see iii. Optical resonator the optics of a laser beam is essentially that of the gaussian beam. Spectral solution of the helmholtz and paraxial wave. These solutions, as well as much of gaussian beam formalism, are wellknown to those familiar with laser design and engineering. Although we will not do so here see example 4 in worked examples, it can be.
The resultant beam is a rigorous solution of maxwells equations for all space that reduces to the conventional gaussian beam in the paraxial limit and that is physically realizable. Full length article the gaussian wave solution of maxwells. We present the incegaussian modes that constitute the third complete family of exact and orthogonal solutions of the paraxial wave equation in elliptic coordinates and that are transverse eigenmodes of stable resonators. This is the first important element to note, while the other portions of our discussion will focus on how the formula is derived and what types of assumptions are made from it. Energy, momentum and propagation of nonparaxial highorder. Strohaber s dissertation solutions to the paraxial wave equation in. Ince gaussian modes of the paraxial wave equation and. Electromagnetic gaussian beams beyond the paraxial approximation c.
It, like other solutions of the paraxial wave equation, has the form of eq. Optical resonators and gaussian beams paraxial wave. Unfortunately theres no particularly good way to fix this. As far as i understand it, the derivation of the gaussian beam as a solution to the wave equation relies crucially on the assumption that the rays comprising the beam meaning the wavefront normals are paraxial rays, which means by definition that. Complex argument hermitegaussian and laguerregaussian beams. Huygens integral gaussian spherical waves higherorder gaussian modes lowest order mode using differential approach. The complex envelope ar must satisfy the paraxial helmholtz equation. Paraxial wave equation an overview sciencedirect topics. They form two complete families of exact and orthogonal solutions of the paraxial wave equation pwe, they are transverse eigenmodes of stable resonators, and they do not change shape on propagation, i. The complex source point derivation used is only one of 4 different ways.
Osa gaussian beam propagation beyond the paraxial approximation. Lecture 24 gaussian beam optics university of toronto. Electromagnetic gaussian beams beyond the paraxial. Pdf gaussian symmetry of the paraxial wave equation. We consider appropriate representation of the solution for gaussian beams in a. Gaussian beams solution of scalar paraxial wave equation helmholtz equation is a gaussian beam, given by. Make a rigorous derivation of the fringe spacing on the screen as a function of. Paraxial waves, gaussian beams if the radii of curvature of the mirrors in fig. If a gaussian beam is focused down to a waist and than expands again fig. Hermitegaussian modes hgms and laguerre gaussian modes lgms exhibit three important properties. Osa incegaussian modes of the paraxial wave equation and. The solutions to the paraxial wave equation are the gaussian beam modes.
The usual textbook approach for deriving these modes is to solve the helmoltz electromagnetic wave equation within the paraxial approximation. Another way to see this is to notice that the paraxial wave equation is exactly the same as the timedependent schrodinger equation for a free particle in two dimensions, which youve already noticed. This is followed by a careful derivation of the paraxial wave equation. The electric field associated with the gaussian beam inside the dielectric medium consists of the paraxial result and higherorder non gaussian correction terms. Gaussian beams physical optics 31052017 maria dienerowitz. The gaussian beam approach to the problem of wave propagation is to obtain a local paraxial solution to the exact wave equation. The classical form of gaussian beams above could be also represented as below here pz is the complex phaseshift of the waves during their propagation along the z axis. Spectral solution of the helmholtz and paraxial wave equations. Sep 21, 2016 where k2 \pi\lambda for wavelength \lambda in vacuum the original idea of the paraxial gaussian beam starts with approximating the scalar helmholtz equation by factoring out the propagating factor and leaving the slowly varying function, i. Pdf starting from the expression of the optical paraxial wave equation we investigate a particular gaussian symmetry applying a lie group formalism. Can we use this as a guide to find a solution to the. In the present case the beam width is an oscillatory solution of the ermakov equation that depends.
1304 645 377 53 1172 1347 30 57 703 354 1194 815 183 415 1035 1419 995 1507 353 1311 1320 1018 1190 1045 1265 1212 804 904 127 570 1102 577 280 1417 1239 1235 1277 1082