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Phases of Talbot Patterns in Angular Self-Imaging Hugues Guillet De Chatellus, Eric Lacot, Olivier Hugon, Olivier Jacquin, Naïma Khebbache, José Azaña

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Phases of Talbot Patterns in Angular Self-Imaging Hugues Guillet De Chatellus, Eric Lacot, Olivier Hugon, Olivier Jacquin, Naïma Khebbache, José Azaña Phases of Talbot patterns in angular self-imaging Hugues Guillet de Chatellus, Eric Lacot, Olivier Hugon, Olivier Jacquin, Naïma Khebbache, José Azaña To cite this version: Hugues Guillet de Chatellus, Eric Lacot, Olivier Hugon, Olivier Jacquin, Naïma Khebbache, et al.. Phases of Talbot patterns in angular self-imaging. Journal of the Optical Society of America. A Optics, Image Science, and Vision, Optical Society of America, 2015, pp.10.1364. 10.1364/AO.99.099999. hal-01165168 HAL Id: hal-01165168 https://hal.archives-ouvertes.fr/hal-01165168 Submitted on 29 Jun 2015 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. Phase s of Talbot patterns in angular self - imaging Hugues Guillet de Chatellus ,1,2,* Eric Lacot, 1,2 Olivier Hugon ,1,2 Olivier Jacquin ,1,2 Naïma Khebbache ,3 and José Azaña 4 1Univ. Grenoble-Alpes, LIPhy, F-38000 Grenoble, France 2CNRS, LIPhy, F-38000 Grenoble, France 3Laboratory of Photonics Systems and Nonlinear Optics, Institute of Optics and Fine Mechanics, University of Sétif, 19000 Algeria 4INRS-Energie, Matériaux et Télécom., Montréal, Québec H5A1K6, Canada *Corresponding author: [email protected] Received Month X, XXXX; revised Month X, XXXX; accepted Month X, XXXX; posted Month X, XXXX (Doc. ID XXXXX); published Month X, XXXX The original Talbot (self-imaging) effect is observed in the vicinity of a grating of slits shined with a plane wave, and results in periodic images of the initial diffraction pattern (integer Talbot effect) and the appearance of images with a periodicity reduced by an integer factor (fractional Talbot effect). Most of the studies on Talbot effect so far have focused on the distribution of the intensity of the diffracted light. However the phases of the Talbot images, obtained in both the integer and fractional self-imaging cases, can be calculated in a closed form and display interesting auto-correlation properties. This paper reports what we believe to be the first experimental investigation of the phases of Talbot images beyond the integer self-imaging case. We address the problem of experimental measurement of the phases of the Talbot images in the equivalent frame of the angular Talbot effect, a recently reported manifestation of the Talbot effect in the far field. The phases of the Talbot images are measured by far-field holography and the obtained results are in excellent agreement with theoretical calculations. They also suggest the possibility of using the scheme for a precise “fractional ruler” aimed at distances’ measurements. © 2015 Optical Society of America OCIS codes: (050.1950) Diffraction gratings; (070.6760) Talbot and self-imaging effects. http://dx.doi/org/10.1364/AO.99.099999 Introduction TAIs take advantage of the possibility of generating a pure phase image from an amplitude grating and vice-versa, the possibility of creating an Since its discovery in optical diffraction [1, 2], Talbot effect has amplitude pattern from a pure phase grating [11-13]. In this context, been observed and interpreted in various domains of wave physics, the phases of the Talbot images have been studied theoretically, but the including quantum mechanics and electromagnetism in general [3-5]. theoretical investigations have been restricted to a specific subset of Whereas the original Talbot effect was first observed and explained in fractional Talbot conditions [14]. Interestingly, the sequence of phases the spatial domain, it was later observed in the temporal domain [6,7] arising in fractional Talbot images correspond to the arguments of so- and in the dual Fourier domains of spatial and temporal frequencies called Gauss sums, which in turn possess interesting autocorrelation (angular and spectral Talbot effects, respectively) [8,9]. The most properties [15,16]. For instance, considering that the phases of Talbot appealing features of the Talbot effect involve the possibility of creating patterns constitute a set of orthogonal phase codes, they have been an exact replica of a given periodic pattern (self-imaging, or integer proposed as an alternative to Hadamard matrices for phase-encoded Talbot effect) or the division of the periodicity of the original pattern by holographic multiplexing storage in volume-holographic media [17]. an integer factor (fractional Talbot effect). This property has led to applications in both the spatial domain (optical lithography and micro- Despite their intrinsic fundamental interest and strong potential patterning [10]) and the temporal domain (generation of trains of light for information coding and processing applications, to the best of our pulses with high repetition rate [6,7]). Essentially, the Talbot effect knowledge, no experimental measurements of the phases of the Talbot arises from the interference of waves with quadratic relative phases: in images have been reported yet, except for certain integer Talbot self- the original Talbot effect, the contributions of the individual slits of the images and the observation of phase anomalies and Gouy phase shifts grating to the intensity in the near field have a quadratic phase in the Talbot carpet [18, 19]. Moreover since the original Talbot effect is dependence in the paraxial approximation, while in the time domain, essentially a near-field effect, the experimental investigation of the the Talbot effect arises typically in dispersive lines that imprint a phases of the individual Talbot images can become quite challenging, quadratic spectral phase on the propagating pulses: the curvature of the requiring for instance the use of a high resolution interference parabola of the phases sets directly the specific integer or fractional microscope [19]. Talbot pattern that is observed at the system output. Most of the practical applications of the Talbot effect so far make use of the In this paper, we extend the state of the art on the Talbot effect, by intensity pattern, and the phase properties have triggered relatively reporting a simple technique that enables a measurement of the phases limited interest, except for the case of Talbot array illuminators (TAIs). of any fractional Talbot self-image. Practical limitations related to the where 605Z1 and T?@A 05Z1 are the /ourier transforms of J0E1 intrinsic near-field nature of the conventional Talbot effect are and t?@A 0E1 respective y. 8efining K\ C 2]/^ the Poisson overcome by carrying out the desired experimental characterization in summation formu a writes: an equivalent configuration, namely using the recently introduced UGVWY angular Talbot effect. This effect can be interpreted as the counterpart IG DFGH 0LM 1 T C I_ QFGH 0RS ` ab\1 041 of the spatial Talbot effect in the dual domain of angular frequencies – i.e. observed in the far field [8]. The concept we use for the target phase where the mu tip icative constant has been dropped and fina y: measurements is derived from far-field holography, where the initial wavefront interferes with the Talbot image under analysis: the Q0RS1 C X0RS1 I_ QFGH 0RS ` ab\1. 0,1 resulting intensity is then directly linked to the phase pattern of the Talbot image. As an additional consequence, our observations suggest a new concept for distance measurement using the employed 2. E ectric fie d in the /resne approximation interferometric scheme, referred here to as a “fractional ruler”, where the distance between a point source and the diffraction grating is Assuming that the incident p ane wave propagates a ong :, the directly measured on the interference pattern, precisely expressed in e ectric fie d diffracted by the grating at ocation 0x, :1 is proportiona to the form of a fraction of two integers . the convo ution product 0represented by c1 of the transmission function of the grating, by the impu se-response function of free space. The manuscript is structured as follows: in the first section, we In the /resne approximation, the atter can be written as: g recall theoretical results on the classical near-field Talbot effect and fUV W mn d 0E1 C d T gh where d C il jk TfUVe and R C "21%. In focus especially on the phase of the Talbot images. Subsequently, we e \ \ o the spatia frequency domain, the transfer function of free space is: outline the direct equivalence between the original (near-field) and the g UoeVW/qn fUVe angular (far-field) Talbot effects. In a third section, we present our pe0RS1 C p\T where p\ CT . Therefore, measurement scheme of the phases of the angular Talbot images, and fu the experimental results, which are found to show an excellent r0E, k1 s D0E1 c de0E1 C tQ vQ0RS1pe0RS1w 061 agreement with our theoretical predictions. which eads to: g xyhzW {| fUVWS r0E, k1 s p\ N X0RS1 I_ QFGH 0RS ` ab\1 T T PRS 1 Theoretical aspects of the near -field Talbot effect 0.1 In this part, we briefly recall the theory of the original (spatial) Consistent y with the above mentioned condition, Eq. 0.1 can be Talbot effect. Such effect can be observed in the problem of Fresnel great y simp ified under the assumption that the function 60RS1 5eeps λ approximate y constant over the angu ar frequency bandwidth of diffraction of a monochromatic plane wave of wavelength by a grating g UoeVW/qn of a large number of slits (N), ideally infinite along the direction y, and QFGH 0RS1. 4oreover the quadratic phase term e can a so been g Λ ~•€• g periodically separated by a distance equal to along the direction of x.
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