Diffraction-free mode generation and propagation in optical waveguides

Propagation within optical waveguides is re-examined in terms of diffraction-free propagation. The concept of the general diffraction-free mode is introduced. It is suggested that the optimised photonic bandgap waveguide must generate such a mode for loss-free propagation to be achieved. The invention of the ‘‘Fresnel’’ waveguide is described. 2002 Elsevier Science B.V. All rights reserved. When the general wave equation is solved the most immediate solution that leads to zero diffraction in a propagating optical field in free-space is a plane wave [1]. Within a step-index optical fibre a propagating mode is often considered to be approximately planar since the effect of internal reflection prevents diffraction spreading the beam along its length. Consequently, whilst inside the waveguide the propagating field can be considered diffraction-free and hence the analogy with a plane wave. This interpretation and analogy is far from appropriate because the diffraction-free properties arise from the Bessel distribution of the optical field, which is the natural solution for a waveguide of cylindrical geometry [2]. It is well known that, in free-space, other solutions to the general wave equation exist for non-planar optical fields, which also propagate in free-space with zero diffraction [3,4]. These solutions tend to have Bessel distributions in the optical field since the propagating beam is usually treated as radial. In practice, unlimited diffraction-free propagation is not feasible because in theory Bessel light beams are infinite and possess infinite energy. However, when a Bessel beam is multiplied by a Gaussian profile (a so-called Bessel–Gauss beam), the beam now carries a finite power and is easier to realise experimentally [4]. It is proposed that these nondiffracting Bessel solutions are more accurate analogies to the diffraction-free like properties of a propagating fibre waveguide mode and offer some physical insight into the nature of waveguide propagation that can be used to further construct new waveguide designs. Generating so-called Bessel beams in free-space experimentally is extremely difficult and a diffraction-free, endlessly propagating, non-diverging laser beam has not yet been achieved. 15 June 2002 Optics Communications 207 (2002) 35–39 www.elsevier.com/locate/optcom Tel.: +612-9351-1934; fax: +612-9351-1911. E-mail address: [email protected] (J. Canning). 0030-4018/02/$ see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S0030-4018 (02 )01418-9 Approximations to a Bessel beam over a finite propagation length have been achieved using various optics, such as axicons, where the effective focal length is greater than the normal Rayleigh range [1,5]. Similar to free-space, the use of a core/ cladding and cladding external interface as a waveguide generates a modified Bessel distribution as internal reflection is used to overcome source diffraction at the input of a waveguide. However, once the restraint of confinement is removed, such as at the end of the waveguide, the Gaussian-like mode immediately diffracts with a diffraction angle, hd 1⁄4 k=ðpncosÞ, where s is the spot size, nco is the core refractive index and k is the wavelength [2]. The greater the confinement (i.e., smaller the spot size) the worse the diffraction. In this situation the output of an optical fibre is analogous to excitation of a point source at its end – whilst travelling within the infinitely extended aperture, zero diffraction can be generated. Since higher order Bessel solutions to the Helmholtz equation also have diffraction-free properties, the fibre refractive index profile can be used to generate higher order ‘‘Bessel’’ modes – in a multi-mode fibre the higher order modes do have higher order Bessel distributions and consequently they too can propagate almost indefinitely along an optical fibre. Preferential excitation of such modes is possible by numerous means including modifying the refractive index profile to allow higher order Bessel modes to propagate whilst filtering out the zero order mode – a single ring core structure, for example, where the core index is lowered will achieve this. Internal reflection is not the only means used to modify both the shape and direction of a freespace beam. For example, in so-called Bragg fibres the principle of Bragg diffraction allegedly plays a crucial role in allowing waveguide propagation within structures where internal reflection is distributed over many layers [6–9]. However, the solutions found in the literature for periodically uniform concentric ring structures do not conform readily to appropriate Bessel distributions. Following the logic thus far described, these structures should not guide strongly despite evidence of propagation within a so-called Bragg fibre being presented recently [9]. It was subsequently shown that the particular refractive index profile used in this work supported a cladding mode (principally by internal reflection from the cladding/air interface) and not a true core guided mode [10]. Further, if one analyses the reasons why these fibres are not ideal, then the necessary phase conditions required to have constructive interference from all interfaces within a concentric ring waveguide structure show that the rings must be spaced close to the Bessel distribution of a mode confined to a cylindrical box. That is their width is determined by the need to achieve a p (step-chirped) or 2p (graded chirp) phase change at each ring. The ring period in this case is naturally chirped and not uniform, completely analogous to the solutions already found in a class of optics falling under Fresnel lenses [11]. These converge to identical solutions using ray tracing methods and the ideal phase relationship can be approximated by annuli with equal area if the launched light has a spherical phase front (as most Gaussian laser beams do) [12]. It is notable that at the focus of a conventional Fresnel lens, the field distribution is described by an Airy function [11], itself made up of Bessel function. Thus a more appropriate description of a chirped Bragg fibre is the ‘‘Fresnel waveguide’’. The properties of a Fresnel waveguide can be analysed approximately in terms of the treatment available for Fresnel lenses. Since the refractive index difference between successive layers, Dn, plays the most crucial role, it determines the effective critical angle of the waveguide. Assuming the same Dn between alternating layers, the phase change, D/, between layers for a given length, L, is D/ 1⁄4 2pLDn=k. Since D/ 1⁄4 2p for graded Fresnel zones, L 1⁄4 k=Dn. Consequently, the critical angle of propagation can be defined in terms of the boundary radius, rb and minimum L: hc 1⁄4 tan 1 ðrb=LÞ (Fig. 2). If we assume the effective modal size is the equivalent spot size of a Fresnel lens of thickness L, from [11] the diffractionlimited spot diameter is given by 2x1=e2 1⁄4 1:64kðf =2rbÞ: Since the critical angle is defined in terms of a minimum length, the minimum focus will also be constrained to this length such that f 1⁄4 rb= tan hc. This is because light coupled in at an angle greater than the critical angle will not be 36 J. Canning / Optics Communications 207 (2002) 35–39