Superposition of Bessel beams: geometrical wavefronts, light rays, caustic, intensity patterns and experimental generation

Recently, it has been shown that if S(r, θ, ϕ) is a two-parameter solution of the eikonal and Laplace equations, then $\Psi (r,t)=\left(1/4\pi \right){\int }_{\Sigma }O(\theta ,\varphi )e^{i\left[k_{0}S\right(r,\theta ,\varphi )-\omega t]}d\theta d\varphi$ is an exact solution of the scalar wave equation in an isotropic optical medium. In particular, if we take $S(r,\theta ,\varphi )=(x\cos \varphi +y\sin \varphi )\sin \theta +z\cos \theta$ $+g(\theta ,\varphi )/k_0$, the resulting solution is a superposition of non-diffracting beams in free space with different cone angle θ for the k vector. In this paper, we analyze the geometrical properties of a particular example, when O(θ, ϕ) = 1 and g(θ,ϕ) = m(ϕ − π/2), which corresponds to the superposition of non-diffracting Bessel beams. That is, we compute its wavefronts, light rays and caustic. The Bessel beam is a special case of a non-diffracting beam, first proposed by Durnin in 1986, the superposition of two Bessel beams have been studied before in order to minimize the size of the central core. In this work, we present a beam resulting from the continuous superposition of Bessel beams with the same order m and different cone angle. To show the behavior of this beam, we simulate the intensity pattern at some different planes perpendicular to the Z and Y axes. Finally, we generate this beam experimentally, and we show that the caustic determines qualitatively the maximum of the corresponding intensity pattern. (paper)