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Polarimetry by Classical Ghost Diffraction

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Polarimetry by Classical Ghost Diffraction Polarimetry by classical ghost diffraction Henri Kellock1,∗ Tero Setälä1, Ari T. Friberg1, and Tomohiro Shirai2 1Institute of Photonics, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland 2Research Institute of Instrumentation Frontier, National Institute of Advanced Industrial Science and Technology, AIST Tsukuba Central 2, 1-1-1 Umezono, Tsukuba 305-8568, Japan We present a technique for studying the polarimetric properties of a birefringent object by means of classical ghost diffraction. The standard ghost diffraction setup is modified to include polarizers for controlling the state of polarization of the beam in various places. The object is characterized by a Jones matrix and the absolute values of the Fourier transforms of its individual elements are measured. From these measurements the original complex-valued functions can be retrieved through iterative methods resulting in the full Jones matrix of the object. We present two different placements of the polarizers and show that one of them leads to better polarimetric quality, while the other place- ment offers the possibility to perform polarimetry without controlling the source’s state of polariza- tion. The concept of an effective source is introduced to simplify the calculations. Ghost polarimetry enables the assessment of polarization properties as a function of position within the object through simple intensity correlation measurements. I. INTRODUCTION applications (such as thin-film optics and nanophoton- ics) a simpler polarimetric characterization of the object Ghost imaging, or correlation imaging, is an imaging by means of its Jones matrix is adequate. The conven- technique where quantum entanglement and photon co- tional way of measuring the Jones matrix of a spatially incidences or classical intensity correlation statistics are uniform, planar object makes use of a coherent laser, used to provide information on an object through in- two polarizers and a detector [21]. In correlation imag- direct measurements [1, 2]. Setups can be arranged to ing light from a spatially incoherent source is split into form either the image or the far-field diffraction pattern two arms, a test arm containing the object followed by a of the object. The classical counterpart of ghost imag- non-resolving detector and a reference arm with a CCD ing has its merits in readily-available and cost-effective, camera. The intensities recorded at the end of the two bright light sources. It is able to emulate most aspects arms are correlated to obtain either the image or the far- of quantum ghost imaging [3–8], although classical cor- field diffraction pattern of the object. For polarimetry by relation imaging has limitations with regard to the vis- ghost diffraction we present two possible arrangements. ibility (contrast) of the resulting image [9–11]. Tech- In one, polarizers are placed after the source and be- niques have been developed to overcome these limita- fore the detectors. In the other arrangement the source’s tions, including higher-order correlations [9, 11], back- state of polarization is not controlled and the polarizers ground subtraction [10], differential ghost imaging [12] are located in front of the detectors in each arm. In both and computational ghost imaging [13, 14]. Using elec- cases measurements are done with different orientations tromagnetic theory also the effects that the degree of po- of the polarizers to individually select all the elements larization of the light has on correlation imaging have of the object’s Jones matrix. The advantage of ghost po- been studied recently [11, 15–17]. larimetry is that spatially varying, polarization-state al- In this paper we introduce a method for obtaining tering objects can be analyzed. the polarimetric properties of a (partially) transparent, spatially dependent, arbitrarily birefringent or dichroic object by means of correlation imaging. In contrast to quantum ellipsometry with two-photon polarization- entangled light in reflection geometry reported previ- ously [18–20], we consider light from a classical, spa- Section II recalls briefly the Jones calculus and the conventional measurement of the Jones matrix compo- arXiv:1404.3106v1 [physics.optics] 11 Apr 2014 tially incoherent, partially polarized source interacting with the object in transmission arrangement. We make nents. In section III we introduce the mathematical no- use of electromagnetic theory of optical coherence and tion of the effective source and incorporate it in our dis- employ a modified ghost diffraction setup, with the con- cussion on electromagnetic ghost diffraction. In section cept of an effective source used for convenience of anal- IV the standard ghost diffraction setup is modified in ysis. Whereas elaborate techniques for measuring an ob- two different ways so as to enable the measurement of ject’s polarization properties based on Mueller imaging the polarimetric properties of an unknown object. The and Stokes imaging have been developed [21], in many polarimetric qualities of these two setups are assessed in terms of image visibility. The main conclusions are summarized in section V. Appendix A gives further de- tails on the possible placements of the polarizers and on ∗ henri.kellock@uef.fi the constraints the placements put upon the source. 2 LPs where E0 is the coherent electric field of the laser source. i j objectlaser detector Orientating the LPs so that T0 = T and T1 = T , with i, j 2 fx, yg, we obtain, from equations (3) and (4), the intensity † 2 2 I1 = E1E1 = Tji jE0,ij , (5) Figure 1. Measuring the elements of the matrix T that de- scribes the response of the object. The setup consists of a spa- where the dagger denotes the Hermitian conjugate and 2 tially coherent laser source (I0), the object which is placed be- jE0,ij = I0,i are the intensities of the source components. tween two linear polarizers (LPs), T0 and T1, and a detector. These can be measured by removing T and T1 from the The measured intensity (I ) is compared to the case where the 1 setup and varying T0. Hence we can calculate jTjij with object and the second LP are removed. all i, j 2 fx, yg. Equation (5) demonstrates that only the absolute val- ues jT j of the object’s Jones matrix are obtained with II. JONES FORMALISM FOR POLARIMETRY ij this method. Using a considerably more complicated system one could measure all the complex transmission We employ spectral electromagnetic coherence theory matrix parameters Tij of a spatially independent object. to analyze polarimetry by ghost diffraction. A realiza- Such a setup involves linear polarizers oriented at ±45 tion of a random, beam-like electric field at position r degrees with respect to the x axis as well as circular po- and frequency w, propagating in the z direction is de- larizers in addition to the LPs introduced earlier [21]. If T noted by E(r) = [Ex(r) Ey(r)] , where the superscript the object has spatial dependence, the detector in figure T denotes the transpose and the frequency dependence 1 may in principle be replaced with a CCD array. is suppressed for brevity. On illuminating with a linear, polarimetric (birefringent, dichroic), planar optical ele- ment T(r) the output electric field Eout(r) is related to III. GHOST DIFFRACTION the input electric field Ein(r) through [22] After more than a decade of research, ghost imaging ( ) = ( ) ( ) Eout r T r Ein r , (1) and diffraction are now well-established optical tech- where niques [1, 2]. In particular, lensless ghost diffraction is employed to obtain the far-field pattern, or the Fourier T (r) T (r) transform, of the object [23–25]. Classical ghost imaging T(r) = xx xy (2) Tyx(r) Tyy(r) and diffraction are conveniently analyzed using optical coherence theory [26, 27]. In what follows we introduce is the optical element’s transmission (or Jones) matrix. also the novel idea of an ‘effective source’ which greatly We take Tij 2 C with i, j 2 fx, yg to be deterministic simplifies the calculations. and, in general, spatially dependent when T represents the object. For polarizers (and wave plates) the elements of T are constant. For example, A. Intensity and field correlations 1 0 0 0 Tx = , Ty = (3) We need to compute the correlations between the in- 0 0 0 1 tensities in the two arms of the ghost diffraction setup, i.e., the intensities I = E†E with a 2 f1, 2g. Be- are uniform linear polarizers (LPs) that only let the x or a a a sides the frequency w, in this section we further sup- y component of the light go through, unaltered. press the spatial dependencies of the functions for no- The conventional way to determine the polarimet- tational brevity. We begin by dividing I into the inten- ric response of an unknown optical element is to mea- a sities of the x and y components of the electric field as sure the components of the corresponding Jones matrix I = I + I . The intensity correlation can then be T individually. Normally, for a uniform object, this is a a,x a,y expressed as done with a setup schematically depicted in figure 1. A spatially coherent, uniformly polarized, monochromatic hI1 I2i = ∑ I1,i I2,j , (6) laser beam illuminates the object which is sandwiched i,j2fx,yg between two LPs, the ‘polarizer’ T0 and the ‘analyzer’ T1 and the transmitted intensity is recorded with a de- where h...i denotes the ensemble average. Assuming tector. To avoid effects of diffraction, the elements T0 that the field fluctuations of the source obey Gaussian and T1 and the detector are deep in the Fresnel zone of statistics, we may use the Gaussian moment theorem to the object T. According to equation (1) the field E1 at the write hI1,i I2,ji in terms of second-order field correlations detector is described by as [11, 15, 16, 28] 2 E1 = T1TT0E0, (4) I1,i I2,j = W11,iiW22,jj + W12,ij , (7) 3 source 0 0 where W(r0, r0) = W0(r0, r0) in the absence of the polar- GD BS 0 ∗ 0 T laser izer T0 and W(r0, r ) = T W0(r0, r )T in the presence ref.
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