Diffraction of Weierstrass scalar fractalwaves by circular apertures : symmetryand patterns, complexity and dimension

Christian, JM, Woodroofe, EP and McDonald, GS

Title Diffraction of Weierstrass scalar fractal waves by circular apertures : symmetry and patterns, complexity and dimension

Authors Christian, JM, Woodroofe, EP and McDonald, GS

Type Conference or Workshop Item

URL This version is available at: http://usir.salford.ac.uk/id/eprint/35368/

Published Date 2015

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Session: . . . 1

Diffraction of Weierstrass scalar fractal waves by circular apertures:symmetry and patterns, complexity and dimension

J. M. Christian1,∗, E. P. Woodroofe1, G. S. McDonald1

1Materials & Physics Research Centre, University of Salford, Greater Manchester M5 4WT, UK∗Email: j.christian@salford.ac.uk

Abstract

The diffraction of plane waves from simple hard-edged apertures constitutes a class of boundary-value problem that is well understood in optics,at least within the scalar approximation. Simi-larly, the diffraction of such waves from fractalapertures (amplitude or phase masks possess-ing structure across decades of spatial scale) hasalso received much attention in the literature.But the diffraction of fractal waves by simpleapertures constitutes an entirely new paradigm(in optics particularly, and wave physics moregenerally) that remains largely unexplored.

Here, we consider the diffraction of fractalwaves by a hard-edged circular aperture usinga range of analytical and semi-analytical meth-ods. Fast computational techniques are used toobtain Fresnel (near field) diffraction patterns,and specialist software assists with the investi-gation of their properties. Key issues to be ad-dressed include the fractal dimension of diffrac-tion patterns, and the asymptotic emergence ofFraunhofer (far field) predictions in an appro-priate limit.

Keywords: Fresnel diffraction, fractal waves,Weierstrass function.

1 Fractal Diffraction

We have recently proposed fractal diffraction asa context of fundamental physical importancewith enormous scope for potential applications[1]. Preliminary analyses investigated a Weier-strass function [2] for modelling fractal illumi-nation of the most elementary aperture imag-inable: the infinite single slit. The Weierstrassfunction has an intuitive interpretation, com-prising a set of periodic patterns whose ampli-tudes and spatial frequencies are connected in avery particular way. Moreover, each constituentpattern scale can be constructed from a super-position of two interfering plane waves.

The diffraction of a uniform wavefront by acircular aperture is another classic wave-basedproblem that has well-known solutions, both in

Fresnel and Fraunhofer regimes [3]. While weconsider the optical analogue with fractal illu-mination, our results are expected to be readilyapplicable to other fields, such as acoustics [4].

2 Diffraction Integral

The diffraction of a scalar optical field U(r) bya circular aperture of radius a is routinely de-scribed by the paraxial wave equation. For ahard-edged circular aperture, and where the in-cident wave Uin is azimuthally invariant, thediffracted wave at a distance L beyond the aper-ture is given by the formal solution,

U(r) =2πNF

iexp

(iπNFr

2)

×∫ +1

0dρ ρJ0(2πNFrρ) exp

(iπNFρ

2)Uin(ρ),

(1)

where the radial coordinate r is measured inunits of a. In this representation, the diffrac-tion pattern is uniquely parametrized by theaperture Fresnel number NF ≡ a2/λL.

3 Weierstrass Illumination

Here, we consider an illuminating field that hasthe form of an azimuthally invariant Weierstrasswave such that

Uin(r)

U0= 1 + ε

N∑n=0

1

γ(2−D0)ncos(κnr+φn), (2)

where U0 is a uniform plane-wave amplitudeand ε controls the strength of the fractal modu-lation. The spatial frequencies in Eq. (2) form aWeierstrass spectrum given by κn ≡ 2π(a/Λ)γn,where γ > 0 is a free parameter, and the set ofphases φn may be either deterministic or ran-dom. When N → ∞, the number 1 < D0 ≤ 2corresponds to the Hausdorff-Besicovich dimen-sion of Uin with values approaching 2 giving anincreasingly complex fractal curve.

The illuminating field is bandwidth-limitedwith a cut-off at n = N ; the spatial scalelengthsin Eq. (2) then range from the largest, Λ, to the

2 Session: . . .

smallest, Λγ−N . Placing a restriction on thenumber of spatial scales in Uin is important fortwo principal reasons. Firstly, no physical ob-ject can possess structure down to arbitrarily-small scales (since finite-size effects will even-tually come into play). Secondly, there tendsto exist a high-frequency cut-off beyond whichspatial scales cannot contribute to the diffractedintensity pattern [5] (so the basis on which oneintroduces finite-bandwidth considerations, andselects a value for N , are rooted in diffractiontheory).

4 Diffraction of Fractal Waves

Earlier analyses focusing on infinite-slit geome-tries have tended to use Young’s edge wavesas convenient spatial structures for understand-ing and quantifying fractal diffraction phenom-ena [1]. While edge waves can be used for cir-cular apertures and uniform illumination [3],such a formalism is not quite so readily de-ployed for fractal illumination (despite the one-dimensional nature of the system) and one mustinstead consider the diffraction integral more di-rectly. Substitution of Eq. (2) into Eq. (1) yieldsa formal expression for U(r) as a linear super-position of patterns with different scalelengths,

U(r)

U0=

2πNF

iexp

(iπNFr

2) [P (r; 0, 0)

+ ε

N∑n=0

1

γ(2−D0)nP (r;κn, φn)

], (3a)

where P is given by the integral

P (r;A,B) ≡∫ +1

0dρ ρJ0(2πNFrρ)

× exp(iπNFρ

2)

cos(Aρ+B)(3b)

and P (r; 0, 0) fully describes the diffraction pat-tern in the classic plane-wave problem [3].

A selection of new results will be discussed,with a combination of analytical methods andspecialist fractal analysis software [6] identify-ing trends in the Fresnel patterns (see Fig. 1).Attention is paid to the role played by NF incharacterizing these patterns, and we considerdifferent self-affine measures of dimension (suchas roughness-length, variogram, and rescaled-range). Asymptotic emergence of classic Fraun-hofer results will also be demonstrated, and awide range of potential applications highlighted.

Figure 1: (a) Illumination across a circular aper-ture corresponding to a plane wave (blue) and aWeierstrass function with fractal dimension D0 =1.6 and γ = 3 (green). (b) Diffracted intensitypatterns when NF = 1000.

References

[1] J. M. Christian, M. Mylova, and G. S. Mc-Donald, Diffraction of fractal optical fieldsby simple apertures, in Proceedings of theEuropean Optical Society Annual Meeting,Berlin, Germany, 15-19 September 2014(online).

[2] G. H. Hardy, Weierstrass’s nondifferen-tiable function, Transactions of the Amer-ican Mathematical Society 17 (2006), pp.301–325.

[3] D. S. Burch, Fresnel diffraction by a circu-lar aperture, American Journal of Physics53 (1985), pp. 255–260.

[4] R. D. Spence, The diffraction of sound bycircular disks and apertures, Journal of theAcoustical Society of America 20 (1948),pp. 380–386.

[5] G. H. C. New and T. Albaho, Spatial spec-trum of intensity profiles diffracted by aslit, Journal of Modern Optics 50 (2003),pp. 155–160.

[6] BENOIT 1.3, TruSoft International Inc.,www.trusoft-international.com.