Diffraction-free mode generation and propagation in optical waveguides

Transcription

1 15 June 2002 Optics Communications 207 (2002) Diffraction-free mode generation and propagation in optical waveguides John Canning * Optical Fibre Technology Centre, University of Sydney, Australian Photonics Cooperative Research Centre, 206National Innovation Centre, ATP, Eveleigh, Sydney, NSW 1430, Australia Received 8 November 2001; received in revised form 6 March 2002; accepted 19 March 2002 Abstract 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. Ó 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 * Tel.: ; fax: address: j.canning@oftc.usyd.edu.au (J. Canning). 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 /02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S (02)

2 36 J. Canning / Optics Communications 207 (2002) 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, h d ¼ k=ðpn co sþ, where s is the spot size, n co 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/ ¼ 2pLDn=k. Since D/ ¼ 2p for graded Fresnel zones, L ¼ k=dn. Consequently, the critical angle of propagation can be defined in terms of the boundary radius, r b and minimum L: h c ¼ tan 1 ðr b =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 2x 1=e2 ¼ 1:64kðf =2r b Þ: 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 ¼ r b = tan h c. This is because light coupled in at an angle greater than the critical angle will not be

3 J. Canning / Optics Communications 207 (2002) Fig. 1. Idealised refractive index profile of a Fresnel waveguide. captured. Substituting and rearranging the critical angle can be redefined in terms of the modal spot size as h c ¼ tan 1 1:64k 4x 1=e 2 : ð1þ Hence at 1:5 lm, for x 1=e2 5 lm, the critical angle is 7. Compare this with a given step index fibre where the critical angle is h c ¼ cos 1 ðn cl =n co Þ¼6:7 for n co ¼ 1:46 and n cl ¼ 1:45. The required value of Dn can be worked out to be Dn 1:64k 2 =ð4r b x 1=e2 Þ: If it is assumed that the boundary radius is as small as the modal radius in the most efficient design then this means Dn 0:037. This value can be reduced using a larger spot size. Both the Bragg and Fresnel waveguide constructs treat the initial propagating solution as a plane wave this is true in the case for light launched from the focal plane of a microscope objective for example. Light launched from another waveguide will depend on the type of propagating solution and whether coupling occurs at the near-field or far-field. Fig. 2. Schematic illustration of Fresnel waveguide critical angle. It is known in micro-optics literature that constructing a diffractive lens with the phase condition satisfied only at the ring edges by using a chirped step index approach (i.e., binary Fresnel lens), only allows a maximum of 40.5% of light to be focussed [11]. This is because only the full phase change is accounted for and the phase change inbetween is not considered. To increase this amount requires tailoring of the step profile itself, usually generating a saw-tooth diffractive lens. When the profile is optimal for a given wavelength, then 100% efficiency can be obtained [11]. A Fresnel waveguide would ideally have a saw-tooth refractive index profile across its centre region as indicated in Fig. 1. In this situation, a saw-tooth chirped Fresnel fibre is maximising material usage over a graded index fibre in the same manner a Fresnel lens does over a bulk lens. Because of the two-dimensional and chirped nature of the problem the higher order coupling potential described previously for simple grating confined structures [6] needs further analysis. Further, a graded profile will clearly change the properties of a cylindrical box defined by sharp refractive index boundaries that are best described by a Bessel function and hence the functional form of the modal solution may be better described by other functions such as Laguerre Gaussians. Typical core/cladding index differences in a step-index fibre are 0.01so in principle it is technically feasible to make Fresnel waveguides with existing means. To achieve the profile in practice is extremely difficult although recently we demonstrated that the process of boil-off, i.e., the amount of medium which vaporises during preform deposition with each pass of a hot flame, during modified chemical vapor deposition

4 38 J. Canning / Optics Communications 207 (2002) (MCVD) preform fabrication leaves behind a gradient concentration of a particular dopant [13]. This was sufficient to enable etching of a fibre end for the fabrication of Fresnel lenses on fibre tips with a curved step profile. However, for waveguide propagation the index gradient arising from such a process using the same dopant was only It may conceivably be possible to deposit different dopants in steps with high resolution using for example, plasma enhanced MCVD deposition of optical fibre preforms. Alternatively, combinations of dopant variants and boil-off maybe employed. Fresnel waveguides can also be described as having soliton-like modal solutions where a sustained modal solution exists only when there is an efficient balance achieved between beam diffraction and waveguide diffraction. In this sense, the waveguide modes are truly diffraction-free modes. An intuitive understanding of zero order Bessel beams, for example, has been suggested on the basis that they can be treated as a superposition of plane waves whose wave vectors lie on a cone around the z-axis of propagation [4]. They therefore all suffer the same phase change for any propagation distance and therefore the overall interference pattern has one and the same shape at any distance. The confinement properties within a waveguide ensure just such behaviour for a propagating mode. Since diffraction is involved, there is some wavelength dependence although the properties can be tailored to be similar over a large bandwidth. Consequently, these are a class of photonic bandgap waveguides analogous to the ineffective Bragg fibres and, further, to the so-call photonic crystal fibres where a 2-D array of holes generates the more complex diffractive confinement necessary for waveguide propagation. Following the discussion thus far, a preliminary conclusion can be made generally the modal solutions in all bandgap waveguide types must be a diffraction-free solution. In addition to the possibility of significantly reduced beam divergence at the waveguide output, the soliton nature of the diffraction-free mode implies that Fresnel waveguides may suffer from reduced dispersion compared to conventional stepindex fibres. This has important applications in designing waveguides for both second and third harmonic generation and parametric frequency conversion processes where phase matching is critical. In the free-space case, Bessel beams, or quasi-bessel/gaussian beams [4] that may closer still approximate the Fresnel waveguide Bessel modes, are being investigated for similar applications. In conclusion, the general properties of waveguide propagation have been qualitatively re-examined in terms of achieving a soliton-like solution: using the waveguide structure to overcome the normal diffraction experienced when light is emitted by a source into free-space. The concept of diffraction-free bound modes is introduced and used to explain the criteria required for all waveguides, regardless of the confinement principle. Further, the Fresnel waveguide is developed to overcome the inherent flaws associated with conventional wisdom on Bragg fibres. Finally it is proposed that photonic crystal fibres made up of 2-D arrays also have complex modal solutions that fall into a class of diffraction-free Fresnel modes in 2-D. In certain confined structures, including in 3-D, these can be considered stationary soliton-like solutions when there is no waveguide path into and out of the structure. It is expected from the insight provided that optimised designs can be developed based on a physically intuitive approach rather than brute numerical solution of the full vectorial wave equation, offering the possibility that much simpler mathematical design routes may be exist. The study thus far has confined itself to waveguides of circular geometry where the diffraction-free solutions are in terms of Bessel functions, analogous to the solutions for the recently developed class of diffraction-free beams. Since waveguides of non-circular dimensions also exist, other non-bessel solutions to the waveequation may also generate diffraction-free properties within these particular structures for which there is no obvious analogy in free-space. References [1] W. Lauterborn, T. Kurz, M. Wiesenfeldt, Coherent Optics Fundamentals and Applications, Springer, Berlin, 1999.

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