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Gaussian Laser Beams via Oblate Spheroidal Waves
"... Gaussian beams provide the simplest mathematical description of the essential features of a focused optical beam, by ignoring higherorder effects induced by apertures elsewhere in the system. Wavefunctions ψ(x,t)=ψ(x)e −iωt forGaussianlaserbeams[1,2,3,4,5,6,7,10,11,12] of angular frequency ω are ty ..."
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Gaussian beams provide the simplest mathematical description of the essential features of a focused optical beam, by ignoring higherorder effects induced by apertures elsewhere in the system. Wavefunctions ψ(x,t)=ψ(x)e −iωt forGaussianlaserbeams[1,2,3,4,5,6,7,10,11,12] of angular frequency ω are typically deduced in the paraxial approximation, meaning that in the far zone the functions are accurate only for angles θ with respect to the beam axis that are at most a few times the characteristic diffraction angle θ0 = λ πw0
SecondOrder Paraxial Gaussian Beam
"... Many discussions of Gaussian beams emphasize a single electric field component, such as Ey = f(r, z)ei(kz−ωt) , of a cylindrically symmetric beam of angular frequency ω and wave number k = nω/c propagating along the z axis in a medium with index of refraction n. Here, we generalize to the case of a ..."
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Many discussions of Gaussian beams emphasize a single electric field component, such as Ey = f(r, z)ei(kz−ωt) , of a cylindrically symmetric beam of angular frequency ω and wave number k = nω/c propagating along the z axis in a medium with index of refraction n. Here, we generalize to the case of a beam with an elliptical cross section. Of course, the electric field must satisfy the freespace Maxwell equation ∇ · E =0. Iff(r, z) is not constant and Ex = 0, then we must have nonzero Ez. That is, the desired electric field has more than one vector component. To deduce all components of the electric and magnetic fields of a Gaussian beam from a single scalar wave function, we follow the suggestion of Davis [2] and seek solutions for a vector potential A that has only a single Cartesian component (such that ( ∇ 2 A)j = ∇ 2 Aj [4]). We work in the Lorenz gauge (and SI units), so that the electric scalar potential Φ is related to the vector potential A by ∇ · A = − n2 c2 ∂Φ ∂t = in2 ω c2 Φ=ik2 Φ. (1) ω The vector potential can therefore have a nonzero divergence, which permits solutions having only a single component. Of course, the electric and magnetic fields can be deduced from the potentials via E = −∇Φ − ∂A ∂t using the Lorenz condition (1), and
Ex ≈ 0,
"... When an electromagnetic wave in a medium with index of refraction n1 encounters an interface with a region of index n2 <n1 the wave can be totally reflected, with only an evanescent (surface) wave excited in the region of lower index. In this case, energy is (largely) transported parallel to the ..."
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When an electromagnetic wave in a medium with index of refraction n1 encounters an interface with a region of index n2 <n1 the wave can be totally reflected, with only an evanescent (surface) wave excited in the region of lower index. In this case, energy is (largely) transported parallel to the interface in the region of lower index. 1 Discuss the flow of energy when the incident wave has limited transverse extent. In particular, consider a weakly focused, linearly polarized Gaussian beam that is incident from the medium with lower index. Also discuss the flow of energy when a Gaussian beam is incident from a medium of index n1 onto a region of index n2 <n1 of thickness d beyond which the medium has index n1. 2