Angular Talbot Effect José Azaña, Hugues Guillet De Chatellus

Angular Talbot effect José Azaña, Hugues Guillet de Chatellus To cite this version: José Azaña, Hugues Guillet de Chatellus. Angular Talbot effect. Physical Review Letters, American Physical Society, 2014, pp.112-213902-1 à 112-213902-5. 10.1103/PhysRevLett.112.213902. hal- 00998967 HAL Id: hal-00998967 https://hal.archives-ouvertes.fr/hal-00998967 Submitted on 3 Jun 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. week ending PRL 112, 213902 (2014) PHYSICAL REVIEW LETTERS 30 MAY 2014 Angular Talbot effect José Azaña1,* and Hugues Guillet de Chatellus2 1INRS—Énergie, Matériaux et Télécommunications, Montréal, Québec H5A1K6, Canada 2Laboratoire Interdisciplinaire de Physique, CNRS—Université Joseph Fourier, UMR 5588, BP 87—38402 Saint Martin d’Héres, France (Received 5 March 2014; published 29 May 2014) We predict the possibility of observing integer and fractional self-imaging (Talbot) phenomena on the discrete angular spectrum of periodic diffraction gratings illuminated by a suitable spherical wave front. Our predictions are experimentally validated, reporting what we believe to be the first observation of self-imaging effects in the far-field diffraction regime. DOI: 10.1103/PhysRevLett.112.213902 PACS numbers: 42.25.Fx, 42.30.Kq, 42.79.Dj The Talbot effect, also referred to as self-imaging, is a e.g., a spherical wave front (point source) under the Fresnel concept with a distinguished pedigree [1–4]. The Talbot approximation. We refer to this new class of self-imaging effect was first discovered and explained in the context of effects as “angular Talbot phenomena.” We anticipate that classical diffraction optics [1–3]. Moreover, the concept both integer and fractional self-images of the original can be easily transferred to a wide variety of problems [4], angular spectrum can be induced by properly fixing the which can be described using equivalent laws to those of curvature of the illumination beam, e.g., the distance the physics of diffraction of periodic wave fields [5], [6]. between the source focal point and grating location. Our Manifestations of the effect have been observed and applied theoretical predictions are experimentally confirmed. across many fundamental and applied areas, including Figure 1 shows a scheme of the problem under analysis, waveguide devices [7], x-ray diffraction and imaging [8], illustrating the predicted angular Talbot effect. We assume a electron microscopy [9], plasmonics [10], nonlinear dynam- 1D grating illuminated by a coherent monochromatic beam ics [11], matter-wave interactions [12], laser physics [13], of wavelength λ and associated wave number k 2π=λ. quantum mechanics [14,15], etc. Self-imaging of periodic ¼ temporal waveforms has been also reported in dispersive media [16]. Referring to its original optics description [1–3], Talbot phenomena can be observed when a coherent beam of monochromatic light is transmitted through a one- dimensional (1D) or two-dimensional (2D) periodic object, namely, a grating. Plane wave-front illumination is generally assumed. Following free-space diffraction, exact images of the original object are formed at specific distances from the object (integer Talbot). At other positions along the diffraction direction, the periodic pattern reappears but with a periodicity that is reduced by an integer number with respect to the input one (fractional Talbot). Similar self- imaging phenomena can be also observed using illumination with a Gaussian-like wave front [3,4]; in this case, the diffraction-induced “self-images” are transversally magni- fied or compressed versions of the original ones. Regardless of their specific manifestation, Talbot phenomena are intrinsically near-field diffraction effects; i.e., they are the result of wave interference in the near-field FIG. 1 (color). Schematic representation of angular Talbot diffraction region or equivalent regime [1–16]. Thus, it is phenomena observed by spherical wave-front illumination of a generally accepted that these phenomena cannot be pro- periodic diffraction grating. The detailed condition for d0 depend- duced or observed under far-field or Fraunhofer diffraction ing on the integer parameters s and m is given in Eq. (6). The figure represents the cases of a direct, integer angular Talbot conditions. In this Letter, we show, otherwise, that self- effect (s 2; m 1), whose diffraction is identical to the one imaging can be induced on the angular spectrum of periodic found under¼ plane¼ wave-front illumination, and a direct, frac- gratings, i.e., on their far-field diffraction patterns. The key tional angular Talbot effect (s 2; m 3), for which the same is to use illumination with a suitable parabolic wave front, beams are diffracted with one-third¼ of¼ the separation angle Δϕ. 0031-9007=14=112(21)=213902(5) 213902-1 © 2014 American Physical Society week ending PRL 112, 213902 (2014) PHYSICAL REVIEW LETTERS 30 MAY 2014 The grating extends along the direction x, perpendicular proportional to exp jαx2 , where j p−1 is the imagi- ð Þ ¼ to the light propagation direction z, with complex trans- nary unit, and α defines the curvature of theffiffiffiffiffiffi quadratic phase mission amplitude defined by profile. Under the Fresnel approximation (see Ref. [17], Chap. 4), this wave variation can be practically induced ∞ ∞ by illumination with a spherical wave front, ideally þ − Λ ∘ þ − Λ t x tpg x p g x tenv x δ x p ; generated from a single radiation point on the propagation ð Þ¼pX−∞ ð Þ¼ ð Þ ½ ð Þ pX−∞ ð Þ ¼ ¼ axis (x 0, z −d0), see Fig. 1. In this specific case, (1) ¼ ¼ α π=λd0. ¼Under parabolic wave-front illumination, the wave p 0; 1; 2; … ∘ where , denotes a convolution inte- complex amplitude e x at the grating location (z 0) gral, Λ ¼is theÆ gratingÆ period, g x is the complex- 01 is now given by ð Þ ¼ amplitude shape of the individual gratingð Þ aperture, δ x Λ ð Þ is the Dirac delta, and tp tenv p is a complex ∞ ¼ ð Þ 2 þ parameter that accounts for amplitude and phase varia- e01 x ∝ exp jαx tpg x − pΛ tions among consecutive apertures and the truncation ð Þ ð Þ p −∞ ð Þ X¼ associated to the finite grating length, i.e., the grating ∞ þ complex envelope tenv x . ≈ jp2αΛ2 t g x − pΛ ð Þ exp p (3) We recall first the well-known case of illumination with pX−∞ ð Þ ð Þ a plane wave front [17], which is assumed to propagate ¼ along z. In this case, the field complex amplitude e1 x at The latest approximation in Eq. (3) is valid if the the grating location (z 0) is proportional to the gratingð Þ transversal extension of the individual aperture g x is amplitude transmission¼ in Eq. (1), and the corresponding sufficiently short [18]. Equation (3) resembles the equationð Þ angular spectrum can then be written as that is obtained for the angular spectrum E2 kx of a grating illuminated by a plane wave front after free-spaceð Þ near-field E k FT e x diffraction, Eq. (4) below. In the classical grating diffrac- 1ð xÞ¼ f 1ð Þg ∞ tion case, the angularly equispaced diffracted beams (each þ − ∝ G kx Tenv kx qK0 approaching a plane wave and characterized by a different ð Þ q −∞ ð Þ order q) are phase shifted with respect to each other X¼ ∞ following a parabolic phase-shift distribution (∝ q2), as þ − GqTenv kx qK0 ; (2) directly induced by the near-field (NF) diffraction process ¼ q −∞ ð Þ [3–6], [17]. Mathematically, the wave angular spectrum X¼ after diffraction through a distance dNF is where the symbol ∝ expresses proportionality, FT holds ∞ for Fourier transform, q 0; 1; 2; …, K0 2π=Λ, ¼ Æ Æ ¼ ∝ þ 2 2 − Gq G qK0 , with G kx FT g x and Tenv kx E2 kx exp jq K0dNF=2k GqTenv kx qK0 : ¼ ð Þ ð Þ¼ f ð Þg ð Þ¼ ð Þ q −∞ ð Þ ð Þ FT tenv x .Equation(2) has been derived by applying X¼ well-knownf ð Þg FT rules on Eq. (1), including the convolution- (4) multiplication theorem and the periodic comb pair ∞ − Λ ∝ ∞ − Equivalently, in the problem under consideration, the FT pþ −∞ δ x p qþ −∞ δ kx qK0 (see f ¼ ð Þg ¼ ð Þ Ref.P[17], Chap. 2). Additionally,P the transversal width of periodically spaced apertures of the diffraction grating g x is assumed to be sufficiently short so that G kx is behave as consecutive spherical-like wave-front beam approximatelyð Þ constant along a frequency resolutionð givenÞ sources (each characterized by a different p). Equation (3) by the angular extension (bandwidth) of T k . Notice implies that these beams are also phase shifted with respect envð xÞ that the bandwidth of Tenv scales as the inverse of the total to each other following a similar parabolic phase-shift grating length. distribution (∝ p2). The mathematical equivalence between Equation (2) can be interpreted as a collection of Eqs. (3) and (4) establishes a duality between the angular monochromatic beams, each beam associated with a differ- spectrum domain (kx) in the conventional problem of ent diffraction order q and propagating with an angle ϕq diffraction of a periodic wave field, Eq. (4), and the trans- with respect to the axis z, sin ϕq qK0=k q λ=Λ .

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