Measuring vector field correlations using diffraction Miguel Alonso, Katelynn Sharma, Gregory Costello, Esteban Vélez-Juárez, Thomas Brown To cite this version: Miguel Alonso, Katelynn Sharma, Gregory Costello, Esteban Vélez-Juárez, Thomas Brown. Mea- suring vector field correlations using diffraction. Optics Express, Optical Society of America -OSA Publishing, 2018, 26 (7), 10.1364/OE.26.008301. hal-01790608 HAL Id: hal-01790608 https://hal-amu.archives-ouvertes.fr/hal-01790608 Submitted on 27 Sep 2018 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. Vol. 26, No. 7 | 2 Apr 2018 | OPTICS EXPRESS 8301 Measuring vector field correlations using diffraction KATELYNN A.SHARMA,1,2 GREGORY COSTELLO,3 ESTEBAN VÉLEZ-JUÁREZ,4 THOMAS G.BROWN,1 AND MIGUEL A. ALONSO1,5,6,* 1The Institute of Optics, University of Rochester, 275 Hutchison Road, Rochester, NY 14627, USA 2Laboratory for Laser Energetics, University of Rochester, 250 East River Road, Rochester, NY 14623, USA 3Physics and Astronomy Department, Stonehill College, 320 Washington Street, Easton, MA 02357, USA 4Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Puebla 72000, Mexico 5Center for Coherence and Quantum Optics, University of Rochester, Rochester, NY 14627, USA 6Aix-Marseille Univ., CNRS, Centrale Marseille, Institut Fresnel, UMR 7249, 13397 Marseille Cedex 20, France *[email protected] Abstract: We present a method for efficiently measuring the 2 × 2 correlation matrix for paraxial partially coherent beams by using diffraction from small apertures and obstacles. Several representations for this matrix function of four spatial variables are discussed and illustrated with experimental results, including various alternative definitions of the spatial degree of coherence. © 2018 Optical Society of America OCIS codes: (030.1640) Coherence; (050.1940) Diffraction; (070.2580) Paraxial wave optics. References and links 1. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007), pp. 174–197. 2. H. P. Jauch and K. M. Baltes, “Reversing-wave-front interferometry of radiation from a diffusely illuminated phase grating,” Opt. Lett. 7, 127–129 (1982). 3. C. Iaconis and I. A. Walmsley, “Direct measurement of the two-point field correlation function,” Opt. Lett. 21, 1783–1785 (1996). 4. D. Mendlovic, G. Shabtay, A. W. Lohmann, and N. Konforti, “Display of spatial coherence,” Opt. Lett. 23, 1084–1086 (1998). 5. D. L. Marks, R. A. Stack and D. J. 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Express 18, 10777–10785 (2010). #319759 https://doi.org/10.1364/OE.26.008301 Journal © 2018 Received 26 Jan 2018; revised 5 Mar 2018; accepted 6 Mar 2018; published 22 Mar 2018 Corrected: 24 September 2018 Vol. 26, No. 7 | 2 Apr 2018 | OPTICS EXPRESS 8302 18. T. G. Brown and A. M. Beckley, “Stress engineering and the applications of inhomogeneously polarized optical fields," Frontiers of Optoelectronics 6, 89–96 (2013). 19. D. P. Brown and T. G. Brown, “Partially correlated azimuthal vortex illumination: Coherence and correlation measurements and effects in imaging," Opt. Express 281, 20418–20426 (2008). 20. D. P. Brown and T. G. Brown, “Coherence measurements applied to critical and Köhler vortex illumination,” Proc. SPIE 7184, 71840B (2009). 21. M. Santarsiero and R. Borghi, “Measuring spatial coherence by using a reversed-wavefront Young interferometer,” Opt. Lett. 31, 861–863 (2006). 1. Introduction The theory of spatial coherence provides a framework for modeling the propagation of statistically stationary partially coherent light [1]. In the case of quasimonochromatic light, the basic function in this theory is the mutual intensity, which provides the statistical correlation of the field at any pair of points. When the field is paraxial and its polarization is uniform, the vector character of light can be ignored and a scalar treatment suffices. However, in some cases the vector character of the electromagnetic field cannot be ignored and must be incorporated into the mutual intensity. For example, for paraxial beams, the mutual intensity is a 2 × 2 matrix composed of correlations between all transverse components of the electric field at two points x1 and x2 [1]: ∗ ∗ Jxx¹x1; x2º Jxy¹x1; x2º hEx¹x1ºEx¹x2ºi hEx¹x1ºEy¹x2ºi J¹x1; x2º = = ∗ ∗ ; (1) Jyx¹x1; x2º Jyy¹x1; x2º hEy ¹x1ºEx¹x2ºi hEy ¹x1ºEy¹x2ºi where Ex; Ey are the Cartesian components of the electric field, and the angular brackets denote a statistical correlation, corresponding to a temporal averaging performed by the detector. A full characterization of the beam requires the knowledge of all four complex elements of this matrix for all values of each of x1 = ¹x1; y1º and x2 = ¹x2; y2º over a transverse test plane. From this information, the propagation of the beam away from this plane can be modeled. Of course, in practice it is impossible to retrieve the mutual intensity matrix over all four spatial coordinates, and the best one can do is measure accurate estimates at a sufficiently dense sample of points within the region of the transverse plane where the beam is most significant. This task is simplified if the measurement technique provides simultaneously the mutual intensity for a large number of pairs of points. Several methods have been proposed to do this, based on the interferometric superposition of two copies of the wavefront [2–7]. Recently, a simple method for efficiently measuring the mutual intensity by using light diffraction around a small obstacle was proposed and demonstrated [8]. This approach has two attractive features. The first is that it provides simultaneous estimates for the field correlations at a large number of pairs of points, namely all pairs of points whose centroid is the centroid of the obstacle. The second is its simplicity, as it does not involve a complicated, vibration-sensitive interferometric setup in which the wavefront is separated and recombined. The only elements employed for the measurement are the obstacle (implemented with a spatial light modulator, or SLM), a lens to perform an optical Fourier transform, and a CCD detector. An extension of this approach was then proposed in which the lens is not required, even though this places some limitations in the range of validity of the method [9]. In this work we combine the advantages of the extended validity that results from using a lens [8] with the improvements that result from using not only diffraction from an obstacle but also from the complementary aperture [9]. In addition, we also incorporate polarization selection elements in order to allow measuring the complete matrix of correlations. This system is tested for fields with complex field correlations induced by an optical element with spatially varying birefringence. The resulting measurements are used to illustrate and compare with real data several proposed definitions of degree of coherence that incorporate the vector nature of light [10–15]. Vol. 26, No. 7 | 2 Apr 2018 | OPTICS EXPRESS 8303 2. Descriptions for the spatial coherence of a paraxial electromagnetic beam over a transverse plane As mentioned earlier, the mutual intensity matrix is a complex function over a large parameter space that includes four continuous spatial variables as well as two discrete