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We present results of our observations on the free space evolution of conically diffracted beams from both single and cascade systems using various combinations of four biaxial crystals of the monoclinic double tungstate family [KGd(WO4)2]. Longitudinal shifts and radii of the Hamilton-Lloyd pair of rings were measured. In each case, the symmetric - forward and backward - evolution of the beam in free space from its focal image plane was monitored and quantified. Theoretical ring plane patterns based on a recently presented theoretical model are also compared to experimental patterns and found to be in good agreement.
© 2014 Optical Society of America
1. Introduction
A beam of light will propagate as a cone of light if it refracts into one of the optic axes of a biaxial crystal. On its emergence from the crystal, it then refracts into a hollow cylinder. When observed on a screen placed in the focal image plane of the hollow cylinder, a pair of rings, known as the Hamilton-Lloyd pair of rings, separated by the Poggendorff dark ring, will be seen. The conical diffraction phenomenon was first predicted in 1832 by W. R. Hamilton [1] and observed experimentally shortly thereafter by H. Lloyd [2], with further observations following later [3,4].
Recent numerous theoretical efforts, e.g [5–10], within the optics community were mainly towards extending the established theory from ray to wave propagation in biaxial crystals. These led to a range of predictions on intensity distributions of conically diffracted paraxial light beams that also work well for experimental observations on conically diffracted Gaussian beams. The majority of work has been focused on single crystal configurations [5–13]. The natural progression from single crystal conical diffraction was to consider the effect of placing the crystals in a cascade configuration in order to examine the effect that the number of crystals and their relative orientations had on the previously investigated parameters for single crystal conical diffraction [14–19].
The use of different polarisation states of incident light produces variations in the expected ring patterns in the focal plane of the system. For the pattern produced from a single crystal, the light at any two diagonally opposite points on the conically diffracted ring is orthogonally polarized with all intermediate points being a mixture of those two orthogonal polarisations. Therefore the use of linearly polarized light leads to a section of the ring with perpendicular polarisation being absent from the ring pattern. This effect is also present in cascade systems in a more complex manner.
In this paper, our aim is to introduce the full free space evolution of a conically diffracted Gaussian beam in both single and cascade configurations, and using the recently advised paraxial theory, provide simple means for assessing its practical applications and implications for optical design of novel devices based on the conical diffraction phenomenon and compare the theoretically predicted patterns, derived from the recently advised theoretical approach, with our experimentally registered patterns.
2. Experimental methods
The system to be investigated is shown in Fig. 1. A collimated beam from a diode (2 mW at λ0 = 635 nm) coupled to a monomode fiber was focused to a spot of radius ω0 ≈13.6 µm (1/e value) using a lens with a focal length of f = 100 mm. Four biaxial crystals of the monoclinic tungstate family, KGd(WO4)2, manufactured by Conerefringent Optics S.L., were used for the experiments. The three distinct principal refractive indices for KGd(WO4)2 at 632.8 nm were previously reported to be n1 = 2.01348, n2 = 2.04580, n3 = 2.08608 [20]. Since the differences of (n2 – n1) and (n3 – n2) are small compared with n2, all refraction and diffraction effects can be considered to be paraxial. The conical diffraction crystals (CDCs) had the shape of a matchstick with cross-section of 4 mm × 3 mm and the following lengths: L1 = 7.40 mm, L2 = 16.94 mm, L3 = 19.40 mm, and L4 = 24.50 mm.
Fig. 1 The collimated beam from a 635 nm laser diode was focused using a lens before passing through the conical diffraction crystals and arriving at the beam profiler. The two crystals depicted comprised the cascade system.
In order to observe the conical diffraction phenomenon, each crystal was cut so that the plane of the cut was perpendicular to one of its optic axes. Here the angular misalignment between the optic axis and the surface normal was 1.75 mrad, 1.25 mrad, 1.5 mrad, and 1.75 mrad for L1, L2, L3, and L4, respectively. For the single crystal conical diffraction experiments, each CDC was inserted between the lens and its focus (approximately 3 mm from the lens). The crystal’s position in the converging beam has no effect on the produced patterns; with only the radius of the beam at the focus influencing the pattern. Each CDC was then adjusted and aligned to force the beam to propagate along the optic axis until the familiar Hamilton-Lloyd pair of rings was directly observed. The rings were imaged using a Spiricon SP620U beam profiler (mounted on a mechanical travel translation stage) in free space with no imaging optics.
For the cascade system each crystal was individually aligned as for the single crystal experiments and then placed into a cascade configuration with the second crystal oriented at φ = π/2 relative to the first. The rotation angle φ is defined as the angle between the space-fixed axis and the principle direction perpendicular to the axis.
The cross-sectional images presented were obtained by extracting the central column from the image matrix (obtained by the beam profiler) using MATLAB. This series of single column arrays was then padded with zeroes to ensure uniform dimensions across all arrays and combined into a single matrix. This matrix was then smoothed using MATLAB image processing filters to produce the final image.
3. Theory
The key aspect to theoretical predictions for conical diffraction lies in the solutions to two functions which comprise the exact solutions to the paraxial theory [21]. However in this case Eq. (1) (Eq. 3.9 in [5]) will be sufficient as it offers an extremely accurate approximation for the ring pattern intensity, provided only the focus of the system is to be considered.
where I is the intensity of the rings, ρ0 and ρ is the ring radius and the radius vector in terms of the incident beam width respectively and ζ is the scaled propagation distance. The radial intensity obtained was then used to produce the ring pattern images shown in Fig. 2(a) top row. The bottom row represents the experimentally observed pattern.Fig. 2 The Ring plane patterns for (a) single crystal with unpolarized incident light, (b) single crystal with vertically polarized light, (c) cascade with unpolarized light and (d) cascade with vertically polarized light. The top row represents the theoretical patterns while the bottom row are experimental patterns.
To introduce linearly polarized incident light the pattern needs to be modulated about the ring with the area of the ring corresponding to the incident polarisation being the maximum and the orthogonally opposite area being a minimum. A straightforward methodology to do this is presented utilizing the following equation [17]:
where I is the incident intensity from (1), χ is the polar position about the ring, φ is the orientation of the crystal and Ф is the polarisation azimuth of the incident light. This results in the crescent shaped pattern shown in Fig. 2(b) - top row. The bottom row represents the experimentally observed pattern.The expansion from single crystal to cascade configurations is similarly described in [15] and [17]. Using both of the aforementioned methods in tandem produces an accurate depiction of the ring pattern detailing both of the pair of rings for both unpolarized and linearly polarized incident light as shown in Fig. 2(c) and 2(d) - top rows. The bottom row represents the experimentally observed pattern.
4. Results and discussion
The experimental parameters of interest in this work are the longitudinal shift from the focus of the system introduced by the crystal(s), Δ, the radius of the ring in the focal plane, R, and the variable Zf - denoting the distance from the ring plane to the point where the Raman spot has the highest axial intensity. In addition to these quantitative parameters, the cross sectional images outlining the overall shape of the beam following conical diffraction is of particular interest.
For the single crystal configuration the longitudinal shift is given by the well-known longitudinal shift equation in ray-optics, Δ = L (1-1/n2). The measured and theoretical values are shown in Table 1, along with those of the ring radius. The radius of the rings is proportional to the length of the crystal and is given by the product of the length and the semi angle of the cone of conical diffraction, here equal to A = 0.0176. The longitudinal values, as measured experimentally, are in each case within ± 0.4 mm of the theoretical values, corresponding to a maximum percentage error of less than 10%. The experimental ring radii values are within ± 6.5 mm of the theoretical values with a maximum percentage error of less than 2%. The single crystal ring pattern is shown in Fig. 3, as well as the cross section of the beam in free space showing two terminal Raman spots a distance of Zf away. The variable Zf is given by:
where λ is the wavelength of the incident beam, A is the semi angle of the cone of conical diffraction (here A = 0.0176), ω0 is the radius of the incident beam at the beam waist and L is the length of the CDC. The measured values for Zf, moving away from the crystal – right hand side of the ring plane, and moving towards the crystal – left hand side of the ring, are shown in Fig. 4(a).The theoretical values were calculated using (3) with the following constant values: λ = 635 nm, ω0 ≈13.60 µm, A = 0.0176 – the length of the crystals were used as the variables. These values are in good agreement with the experimentally measured values for Zf, with maximum % error of ≈15% - Fig. 4(a).
Table 1. Theoretical and Measured Values for Longitudinal Shifts and Ring Radii
Fig. 3 The conical diffraction pattern - the ring plane and free space evolution - for a single crystal (here L3) configuration.
Fig. 4 The 2Zf values – full free space evolution - for (a) single crystal and (b) cascade experiments.
The cross section itself shows the positions of the notable features within the full free space evolution of the beam. The ring plane sits at its center with Poggendorff splitting, seen a short distance from the ring plane, resulting in the increased separation between the bright and dark ring as the beam propagates, giving the beam its conical shape. The ring plane is also a symmetry plane and the conical pattern is observed in each direction resulting in the so called ‘bottle beam’ produced by the two intersecting, oppositely propagating cones between the Raman spots. For the cascade system, consisting of two CDCs, the longitudinal shift was merely the sum of the shift introduced by each individual crystal. The ring radii were found to be equal to the sum of the radii of the individual crystals for the outer ring and the difference between them for the inner ring. The values for the cascade configurations of (L1 + L2), (L1 + L3), and (L1 + L4) for both the inner and outer rings are shown in Table 2, with all values having a max % error within 2.5%.
Table 2. Theoretical and Measured Values for Inner and Outer Ring Radii
The images of the ring plane pattern is shown in Fig. 5 for the cascade system, along with the cross section showing an additional pair of rings when compared with the single crystal case. For the cascade configuration the Zf values were again found by using Eq. (3) where L is now the sum of the lengths of the crystals in the cascade and all other values remain the same as the single crystal case. These values are also in good agreement with the experimentally measured values for Zf with max % error of 9% as presented in Fig. 4(b). In comparing the theoretical images presented in Fig. 2 (top row) with the experimental images shown in Fig. 2 (bottom row) it can be seen that there is good agreement between the relative intensities of the pair of rings in the single crystal case and the relative sizes of the inner and outer pair of rings in the cascade case. Similarly the modulation method for the case of linearly polarized light presented in [17], as applied here in conjunction with the paraxial approximation in [5] is in good agreement with the experimental images shown for both single and cascade conical diffraction. Overall the theoretical and experimental values were found to be in close agreement showing the theory is accurate and straightforward enough to be of practical use in experimental investigation.
Fig. 5 The ring plane pattern and free space evolution for a two-crystal cascade - here consisting of L1 and L3 - and the cross section of the beam evolution in free space.
5. Conclusion
We presented and quantified results of our observations for the full free space evolution of a conically diffracted Gaussian beam for four crystals of various lengths individually and in a two-crystal cascade configuration. This work provides simple and valuable practical insight into the free space evolution of a conically diffracted Gaussian beam by comparing the recently advised paraxial theory with experimental observations, hopefully leading to a tangible expansion of the application space. The recently advised theory greatly simplifies the theoretical modelling and prediction of cascade ring patterns in the ring plane of the system from an arbitrary number of crystals when compared to the previously used diffracting wave theory, allowing for a practical model which accounts for both circularly and linearly polarized light accurately. Ideally this approach will allow for a more usable and robust theoretical foundation on which future applications and developments of novel devices can be built.
Acknowledgments
Amin Abdolvand is an EPSRC Career Acceleration Fellow (EP/I004173/1).
References and links
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