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Left- and right-circularly polarized light in cascade conical diffraction Stephen D. Grant and Amin Abdolvand* School of Engineering, Physics and Mathematics, University of Dundee, Dundee DD1 4HN, UK *Corresponding author:
[email protected] Received November 8, 2012; accepted November 26, 2012; posted November 29, 2012 (Doc. ID 179568); published December 13, 2012 A cascade conical diffraction system consisting of three optically biaxial KGdWO4 2 crystals is considered. The effect of left- and right-handed circularly polarized incident light on the ring patterns produced away from the focal image plane of the system, the plane in which the incident beam waist would be focused if the crystals were isotropic, is investigated. Images and intensity distributions for scaled distances (ζ values) of 2.75, 3.00, and 3.25 from the focal image plane are presented. A discrepancy between the patterns produced depending on the handedness of the incident beam is observed in agreement with the recent theoretical predictions. © 2012 Optical Society of America OCIS codes: 260.1180, 260.1440, 260.5430.
Conical diffraction was first predicted in 1832 by Hamilton [1] and observed experimentally shortly thereafter by Lloyd [2]. A beam of light propagating along one of the optic axes of a biaxial crystal will travel as a cone of light and produce a hollow cylinder on its exit from the crystal. The image produced in the focal image plane of this cylinder gives rise to the Hamilton–Lloyd rings typical of conical diffraction, separated by the Poggendorff dark ring [1–8]. Renewed interest in the field [9–15] has led to the establishment of a more robust paraxial theory of conical diffraction [8] and with it a number of predictions on the behavior and parameters of a conically diffracted beam [4,7,8,13]. Recently, a paraxial theory for a general N-crystal cascade has been advised in which the relative orientation of crystals of different lengths was considered [13]. For instance, in a cascade system consisting of two crystals, conical diffraction leads to a pair of double rings, whose relative intensity depends on the angle between the crystals (this point will be elaborated on later in the text). It was also predicted, and shortly afterward observed by the authors [14], that the focus pattern of conical diffraction from an N-crystal cascade is the superposition of 2N−1 single-crystal concentric conical diffraction patterns (rings), whose ring radii are a combination of those from the individual crystals in cascade. The paraxial theory was also successful in explaining complex beams generated by crystal cascades [16]. In this Letter, we examine the effects of circularly polarized light of different handedness on the observed pattern away from the focal plane for a three-crystal cascade in accordance with the prediction in [13], which states that the patterns produced by left- and rightcircularly polarized incident light would be different away from the focal plane for a cascade system consisting of N > 1 crystals and remain circularly symmetrical in the focal plane for either left-circularly polarized light (LCPL) or right-circularly polarized light (RCPL) [13]. To do this the system shown in Fig. 1 was used. The linearly polarized incident beam from a 633 nm helium– neon laser was collimated using a two-lens telescopic system with linear magnification of 4 and then focused to a spot of radius ω0 ∼ 25 μm (1∕e value) at the beam 0146-9592/12/245226-03$15.00/0
waist using a lens of focal length 200 mm. A quarter-wave plate was used to produce circularly polarized light. RCPL was produced by placing the fast axis of the wave plate at 45° to the plane of polarization of the incident polarized beam, and rotating the wave plate through 90° to produce LCPL. The crystal cascade system consisted of three crystals of KGdWO4 2 manufactured by Conerefringent Optics S.L. The principle refractive indices of KGdWO4 2 at 632.8 nm are n1 2.01348, n2 2.04580, and n3 2.08608 [17]. Diffraction and refraction effects may be considered paraxial due to the small differences in these refractive indices [8]. The conical diffraction crystals (CDCs) were rectangular in shape and had a cross section of 4 × 3 mm and had lengths: L1 16.94 mm, L2 19.40 mm, and L3 24.50 mm. Each crystal was cut perpendicular to one of its optic axes so as to have the incident beam propagate along this optic axis. The angular misalignment between the axis and the normal surface for each crystal was: L1 1.25 mrad, L2 1.5 mrad, and L3 1.75 mrad. The strengths of the crystals, given by the dimensionless variable ρ, were 11.99, 13.74, and 17.35, respectively. In physical terms the variable ρ measures radial position of the conical diffraction rings in units of the beam width ω0 [8,13] and is given by the ratio of the ring radius produced by each crystal to the spot size in the focus of the lens (ω0 ).
Fig. 1. (Color online) Experimental setup. λ∕4 is a quarterwave plate, the lens had a focal length of 200 mm, CDC (1–3) are the conical diffraction crystals of increasing length and the beam profiler was used to capture the images. The arrows indicate the relative orientations of the crystals at 0, λ∕4, and λ∕2. © 2012 Optical Society of America
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Fig. 2. (Color online) Ring patterns and intensity profiles at (a) the ring plane and ζ values of 2.75, 3.00, and 3.25 for (b), (c), and (d), respectively. The image at the ring plane remains unchanged for both LCP and RCP while variations in intensity and the appearance of a central spot can be seen for LCP in each of the images away from the ring plane.
The crystals were individually aligned until the Hamilton–Lloyd rings were observed to force the beam along the optic axis and then carefully placed into a cascade configuration in which the crystals were arranged in order of ascending length and had their pseudovectors oriented at γ 1 0, γ 2 π∕4, and γ 3 π∕2 relative to one another, as illustrated in Fig. 1. These are different rotation angles (γ) of the crystals about their common optic axis [13,14]: for the n-th crystal, γ n is defined as the angle between the space-fixed and the principal direction perpendicular to the axis. The patterns produced were imaged in free space—directly and with no imaging optics—using a Spiricon SP620U beam profiler mounted on a mechanical travel translation stage. The results shown in Fig. 2 depict the various patterns produced for left- and right-circularly polarized incident light at various ζ values and their corresponding intensity profiles. The ζ value is the longitudinal coordinate of the electric field vector and in physical terms it is a measure of the propagation distance of the beam in units of the Rayleigh range fk0 ω0 2 g and is given by [13] ζ
N X 1 1 z l − 1 ; n μn kω2 n1
(1)
in which k is the wavenumber and equal to 2π∕λ, ω is the radius of the incident beam at the beam waist, z is the unscaled distance from the beam waist, ln is the length of crystal n, and μn is the mean refractive index were the crystal isotropic. Rearranging this equation and using the experimental parameters described, the values of z corresponding to ζ values of 2.75, 3.00, and 3.25 were found and images collected at these positions.
The intensity distributions displayed in Fig. 2 show the differences in the observed patterns for light of different handedness. The patterns observed at the ring plane remained the same for both LCPL and RCPL [Fig. 2(a)]. While the position of the concentric rings remains constant for ζ values 2.75, 3.00, and 3.25 [Figs. 2(b)–2(d), respectively] for both RCPL and LCPL in all cases, slight variations are observed at various points, for example at approximately 500 μm for ζ values of 3.00 and 3.25 [Figs. 2(c) and 2(d)] where a single peak for RCPL shows two close peaks for LCPL. The intensity of the patterns varies between the two cases with RCPL having higher intensity than LCPL in each instance while the ring positions are maintained. Of particular note is the appearance of a central spot when using LCPL for the ζ values used, which is absent in the right-handed case. This feature is observed for all of the ζ values used. It is theorized that the differences between the observed patterns result from the complex nature of the Belskii–Khapalyuk functions away from the ring plane and the noncommutativity of the matrices, which describe the transformation of the plane wave by each of the crystals in the cascade, the product of which gives the total transformation matrix for the cascade as a whole [13]. The appearance of the central spot for LCPL bears a peculiar similarity to the case where cancellations in the produced paths result in one of the radii being zero, and the “ring” produced is a reproduction of the incident Gaussian beam [11,13,14]. We have presented our results on the pattern produced by LCPL and RCPL in a cascade conical diffraction configuration and found that the results were in line with the predictions of recently advised paraxial theory [13]. For three positions away from the focal image plane the patterns produced were observed to be different for
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LCPL and RCPL for a three-crystal cascade system, the most pronounced feature of the differences being the appearance of a central spot for LCPL and its absence in the corresponding pattern using RCPL. By comparing experimental observations to the current theory, this work provides valuable qualitative verification of otherwise unreported effects. Quantitative comparison of the results is currently being considered. Professor A. Abdolvand is an EPSRC Career Acceleration Fellow at the University of Dundee (EP/I004173/1). The authors are very grateful to Prof. Sir M. V. Berry for fruitful discussions on the subject, and to Prof. W. A. Gillespie for providing access to the He–Ne laser used in these experiments. References 1. W. R. Hamilton, Trans. R. Irish Acad. 17, 1 (1833). 2. H. Lloyd, Trans. R. Irish Acad. 17, 145 (1833). 3. C. V. Raman, Nature 149, 552 (1942). 4. A. M. Belskii and A. P. Khapaluyk, Opt. Spectrosc. 44, 436 (1978).
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