We propose a paraxial dual-cone model of conical refraction involving the interference of two cones of light behind the exit face of the crystal. The supporting experiment is based on beam selecting elements breaking down the conically refracted beam into two separate hollow cones which are symmetrical with one another. The shape of these cones of light is a product of a ‘competition’ between the divergence caused by the conical refraction and the convergence due to the focusing by the lens. The developed mathematical description of the conical refraction demonstrates an excellent agreement with experiment.
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Conical refraction (CR) is a well-known, fundamental optical phenomenon. It was first predicted in 1832 by W.R. Hamilton  who used a Fresnel wave-surface to calculate that if a narrow beam of light enters a biaxial crystal along one of the optic axes, then it will evolve as a hollow slanted cone and on the exit facet of the crystal it will refract as a hollow cylinder. This prediction was experimentally verified by H. Lloyd in the same year . CR was briefly investigated by C.V. Raman in the 1920’s and he published a short letter to Nature clarifying the difference between the ‘internal’ and ‘external’ forms of conical refraction . Raman was to revisit this phenomenon 20 years later when he published observations of new features in conically refracted beams  nowadays known as Raman spots, but still CR remained a mystery in the heart of classical optics . This area of optics has probably been neglected because there was no immediately obvious application for the effect and also due to the technical challenges involved in obtaining a biaxial crystal cut with the necessary precision.
In the last few years, however, new research has been published on conically refracting crystals (CRC’s) and their potential applications. Recent work has discovered that an ultra-efficient laser can be produced using CR [6,7] simultaneous ‘free’ and ‘forced’ second-harmonic generation with non-linear CRCs was demonstrated  and other researchers have noted the potential for CR to be used in quantum computing and cryptography . Also, generation of ‘bottle’ beams for optical manipulation with incoherent white light , flexible particle manipulation techniques  and the first-ever material ablation using the longitudinal electric field of femtosecond laser pulses  was demonstrated using CRCs.
By gaining a full understanding of the unique properties of CR then it follows that more CR based applications may be realized and commercially exploited. In this paper we propose a model suggesting a dual-cone structure of the CR beam and present its observational evidence.
The setup we used to perform conical refraction experiments is relatively simple for a modern optical laboratory; requiring only a source of laser light, a CRC, a few lenses and a screen/CCD. Such a setup is shown in Fig. 1 and uses a focusing lens to produce a converging beam which passes through the CRC and creates the hollow double-ring in the focal image plane (FIP). This ring is named after Lloyd, who first observed it. The distance from Lens 1 to the FIP is slightly longer than the focal length of Lens 1 due to the presence of the CRC causing a longitudinal shift of focus . Along the propagation axis, z, the Lloyd ring is preceded, and followed by, the Poggendorff rings which have clearly defined dark rings separating the bright ones. Preceding and following the Poggendorff rings are the two Raman spots finishing the evolution of the conically refracted beam, which is symmetrical about the FIP as shown schematically in Fig. 1.
2. Background theory
We model the evolution of the CR beam using the general paraxial solution derived by Belsky and Khapalyuk  and Berry  for a beam propagating through a slab of a conically-refracting transparent crystal of length l and ‘conicity’ χ defined as with refractive indexes n1<n2<n3 (differences between refractive index values are small). According to these, one can write the electric displacement vector D in the form:
From Eqs. (1)-(3), in the case of unpolarized light, utilizing the well-known relation to express Bessel functions as a simple combination of Hankel functions: Jm = ½(Hm(1) + Hm(2))  (which are the analogues of the exponent in cylindrical space), merging and re-separating components of B0 and B1 one can write the expressions for the cones that converge and diverge behind the exit plane of the conically-refracting crystal with:Eq. (5) and shown in Fig. 2(a) . The very important difference from the earlier formulae is that the mathematical separation of the components as I1 = |C1|2 and I2 = |C2|2 now has a clear physical meaning as the two cones of light converging and diverging behind the output facet of the CR crystal (shown in Fig. 2(b) and 2(c)). It is also important to note that the interference of the ‘cones’ C1 and C2 in the total intensity distribution cannot be omitted:
In contrast to the case of the functions B0 and B1 in Eq. (5), absence of the interference of the cones C1 and C2 in Eq. (8) immediately leads to the disappearance of the dark ring in the Lloyd plane. This demonstrates mathematically the interference nature of the Lloyd ring.
Our observational evidence is based on experiments we designed to separate the CR beam pattern in two cones. Our experimental setup consisted of a 2mW unpolarized He-Ne laser being focused through a 75mm lens into a KGW CRC 15mm in length and onto the CCD (the beam diameter at the focusing lens was ~0.7 mm). The CCD was mounted on a translatable stage which was used to control its position through the beam pattern emerging from the crystal facet. Shown in Fig. 2(d) is an experimentally obtained image of the beam demonstrating the classic CR pattern that agrees very well with the general paraxial solution (5) and the dual-cone model (8) [cf Fig. 2(a)].
The individual conical components of the CR pattern Eqs. (6) and (7) can be observed only by blocking off the converging and diverging beam components (cones C1 and C2) separately. This was accomplished by placement of a pinhole or opaque spot (i.e. the inverse of the pinhole) immediately after the crystal. Pinhole blocked the outer beam and allowed the inner beam to pass. Opaque spot blocked the inner beam whilst allowing the outer beam to pass. The results from these experiments are presented in Figs. 2(e) and 2(f) and demonstrate an excellent agreement with the calculation of the cones |C1|2 and |C2|2 in Figs. 2(b) and 2(c).
Discussion of the proposed dual-cone model can begin from elementary remarks on conical refraction that are well known from textbooks (see Figs. 3(a) and 3(b) ). In a biaxial crystal cut normally to one of the optic axes, all incident rays with wavefronts inclined at a small angle φ to this optic axis will give rise to pairs of rays inclined at angles of ½χ ± aφ to the central axis of the cone of internal conical refraction (which is itself inclined at the angle ½χ to the optic axis, as shown schematically in Fig. 3(c)), where χ is the apex angle of the cone of refraction and a is some constant. From simple consideration (e.g. Born & Wolf ), it follows that at angles of ½χ ± aφ, the intensity in the cone of rays is proportional to φ, and in particular that it is zero at φ = 0.
Therefore, the well-defined dark cone with apex angle χ will appear instead of Hamilton’s light cone. This dark cone will separate two light cones giving rise to the two bright circles appearing on the output plane of the crystal, with a dark circle between them (Fig. 3(c)).
Increasing the beam convergence angle φ to the value of the conicity of the crystal χ (which is indeed the case with shorter-focus lenses, still well within the paraxial condition), we immediately should expect to observe two things: i) a two-fold increase of the apex angle of the outer cone of the internal conical refraction (this cone is denoted as C2 in Fig. 3(d)) and ii) degradation of the inner cone to the ray of the external conical refraction (denoted as C1 in Fig. 3(d)). Important difference to the ‘classical’ external conical refraction (Fig. 3(b)) is the inclination of this ray (i.e. the degraded cone) at the angle ½χ to the optic axis. The dotted line in Figs. 3(c) and 3(d) indicates Hamilton’s ray trace.
In the dual-cone model presented here, the 1st and the 2nd Raman spots can be represented as the product of self-interference of conically converging beams, similar to Bessel beams, but with a more pronounced spherical shape of the wave-front. A recent piece of work  modeled the beam pattern of a conically refracted beam of circularly polarised light as a combination of two Bessel beams. This was also mentioned earlier in the work by Kazak et al. , as well as possibility for transformation of the order of the Bessel function . This was also a subject of research for other groups in recent years [20, 21].
Moreover, it is implicit from these results that the distinctive dark ring in the Lloyd plane is created by the interference of the two cones. Indeed, the appearance of the dark ring in the Lloyd plane could at first seem to be at odds with a dual-cone model - one would expect the cones to overlap and pass through each, thus creating only a single ring in the Lloyd plane. However, the ‘anticrossing’ of the cones can be easily explained as the interference of the converging and diverging beams in the plane of intersection. This can be proved experimentally by changing the optical path difference for the interfering cones [i.e. by introduction of an extra phase shift between C1 and C2 in (8)]. This can be demonstrated experimentally using the simple setup shown in Fig. 4 . A glass slide is accurately positioned in the plane of the 1st Raman spot so that only (part of) the outer cone passes through the slide while the inner cone travels unaffected. This enables us to change the optical path difference of part of one cone and thus alter the intensity profile of the Lloyd ring. As shown in Fig. 4(a), fine tilting of the glass slide introduces an extra phase shift Δφ for the outer light cone so that only one ring can be observed in the Lloyd plane (Δφ≈π/2) or so that the Lloyd distribution is inversed (Δφ≈π) or so that the intensity of both rings is equalized (Δφ≈3π/2). Numerical computation of the intensity distribution in the Lloyd plane with the dual-cone model with an extra phase shift Δφ introduced for the (part of) outer light cone C2 produces the same result as seen in Fig. 4(b).
This shows conclusively that the dark ring in the Lloyd plane is an interference pattern produced by two cones of light and provides an excellent agreement between the proposed dual-cone model and experimental results. The demonstrated technique also opens up the possibility of transforming the Lloyd ring profile which can be very useful for existing and future applications of CR beams in optical trapping and manipulation .
In conclusion, we have developed a dual-cone model of conical refraction involving the interference of two cones of light. The supporting experiment is based on beam selecting elements breaking down the CR beam pattern into two separate hollow cones of light that are symmetrical with one another. The shape of these cones is a product of a ‘competition’ between the divergence caused by the conical refraction (which is responsible for the cone’s opening angle) and the convergence due to the focusing by the lens (manifesting itself by the simultaneous narrowing of the cone ‘walls’ in the lens’s FIP). The developed mathematical description of conical refraction demonstrates an excellent agreement with experiment. Future work in this endeavor can focus on improving the mathematical model presented here which was created without consideration for polarization and further analysis of the phase relation between the cones C1 and C2 which is especially important in understanding the interference pattern of the Lloyd ring as seen in Fig. 4. Therefore, we anticipate that this work will inspire further research in this long neglected field of photonics which may result in a deeper insight into the old ‘optical curiosity’ of conical refraction and may lead to even more impressive applications.
This research was supported by FP7 Project HiCORE. We would like to thank Knowledge Transfer Partnerships (UK), Dr G. Malcolm (M-Squared Lasers, Glasgow) and Neil Stewart for helping to make this work possible.
References and links
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