Conical diffraction occurs when light is incident along the optic axis of a biaxial crystal. The light spreads out into a hollow cone inside the crystal, emerging as a hollow cylinder. The intensity distribution beyond the crystal is described using an adapted paraxial wave dispersion model. We show, experimentally and theoretically, how this results in a transition from conical diffraction for wavelengths at which the crystal is aligned to double refraction for misaligned wavelengths when using a white light source. The radius of the ring and location of the focal image plane (FIP) are also observed to have a wavelength dependency. The evolution of the conically diffracted beam beyond the FIP into the far field is studied and successfully described using a theoretical model.
©2013 Optical Society of America
Conical refraction was first predicted by William Rowan Hamilton  and later observed by Humphrey Lloyd at Trinity College Dublin . It occurs when light travels along an optic axis of a biaxial material. Such materials have three principal refractive indices The optic axes are special directions for which the dispersion surface  has no defined normal due to the intersection of the sheets of the dispersion surface at the ‘diabolical’ point . This singularity gives rise to a directional degeneracy for the Poynting vector, which now traces out a skewed cone. The approach formulated by Belsky and Khapalyuk  and later reformulated by Berry  leads to a model which demonstrates a double-ring profile in the focal image plane for the conical beam. This inclusion of diffraction theory gives an accurate description of the process and results in the switch of nomenclature from ‘conical refraction’ to ‘conical diffraction’. The phenomenon has found numerous applications such as the generation of radially polarised light beams  and a novel type of laser based on conical diffraction . Berry and Jeffrey have also formulated a paraxial theory to describe light propagating through a biaxial material in a direction not along, but very close to, the optic axis . This model will be used here as it will be seen that the direction of the optic axis (among other parameters) depends on wavelength. Thus, the paraxial theory can be used to obtain a theory with explicit wavelength dependence. We show that this theory accurately reproduces the complex images generated experimentally by the conical diffraction of white light.
The aim of this paper is to present experimental images of a conically diffracted white light beam and compare them to the theoretical intensity profiles. Also examined are the wavelength dependences of both the radius of the conical diffraction ring and the location of the focal image plane. This provides the foundation for future work on both the fundamental physics and applications of polychromatic conical diffraction, such as communication systems incorporating conically diffracted light of multiple wavelengths [10, 11].
2. Theoretical method
In conical diffraction, the Poynting vector at a diabolical point traces out a skewed cone with semi-angle 12] and , while Fig. 1(a) demonstrates what occurs graphically. Using the paraxial approximation, the radius of the emerging cylinder of light is thus where is the length of the biaxial material . In keeping with the convention used by Berry and Jeffrey , we define the following dimensionless parameters  for a Gaussian beamEq. (2) giving
The three principal refractive indices have wavelength dependencies which can be represented  using a form of the Sellmeier equation:16], which are the biaxial materials used in the following experiments. The wavelength-dependent refractive indices now give a wavelength-dependent semi-angle of the diffraction cone which in turn translates to a wavelength dependence in the radius of the ring given by
To achieve conical diffraction light must propagate down the optic axis of the crystal, otherwise double refraction occurs . The optic axis may be shown to make an angle4] as seen in Fig. 1(a). With the introduction of wavelength-dependent refractive indices it is clear that will vary with wavelength. This variance is shown explicitly in Fig. 1(b). When the crystal is aligned such that a wavelength passes down the optic axis, all other wavelengths are misaligned. This produces a gradual transition from conical diffraction to double refraction as the wavelength is shifted further from To describe this effect, it is useful to introduce a misalignment parameter which represents the extent of misalignment between the wavelength-dependent optic axis and the beam direction:9]. Such misalignment results in a breaking of symmetry in the plane orthogonal to the beam propagation direction, the plane. This leads to the introduction of a new complex coordinate system where the orientation of has been chosen to be orthogonal to both the optic axis and the direction of the incident beam as it enters the crystal. is orthogonal to both and the optic axis. Thus, the length and direction of may be given by12–14] in the formEq. (7) produces the familiar symmetric double ring structure associated with conical diffraction .
3. Experimental method
The white light source used was a polychromatic light emitting diode (LED). The source spectrum was measured using a spectrophotometer and is shown in Fig. 2(a). The experimental setup is shown in Fig. 2(b). A 100 µm pinhole was placed in front of the LED and an achromatic biconvex doublet lens of focal length 10cm was used to image the pinhole through a 3cm long slab of KGd(WO4)2. An iris was used to control the numerical aperture of the lens such that the image of the pinhole was below the diffraction limited resolution of the system. This effectively created a Gaussian spatial distribution for the input beam, which is required for the theory discussed in this paper. The intensity distribution in the focal image plane was recorded using a Sony ICX204AK colour charge-coupled device (CCD) with pixel size of 4.65 µm mounted on a rail allowing movement in the direction. In order to produce sharply defined rings a high value of is required. This was achieved by placing the lens a distance of 50 cm from the pinhole to produce a small beam waist. The diameter was measured to be μm giving a beam waist of μm.
Using Eq. (2) this value of gives for light at 500 nm. The paraxial theory is valid for which is the case here, and we expect to see well defined ring structures when In Fig. 3 the crystal has been aligned using a HeNe laser with peak emission at 632.8 nm, and then the white light source was introduced to produce the image of white light conical diffraction. It can be seen that only the wavelengths close to 632.8 nm demonstrate the full ring structure, with gradual transition to double refraction as the wavelength separation increases. This double refraction is most clear for blue light.
4. Variation of ring radius with wavelength
The radius of the ring produced by conical diffraction is dependent on the wavelength of the input light. To observe this dependence the same setup was used as in Fig. 2(b) but bandpass filters were placed in the beam to select a narrow spectrum of the white light. Each bandpass filter had a full-width at half-maximum (FWHM) of 40 nm and each was inserted in front of the CCD in turn. The crystal was then aligned to produce the sharpest and most symmetric ring structure in the FIP for that wavelength. To ensure the highest accuracy possible, the CCD was first moved into the far field as the beam is more sensitive to crystal misalignment there. The CCD was returned to the FIP and mounted on a 25 μm resolution translation stage. The CCD could then be moved in small increments through the FIP and the resulting images were examined in order to determine the one displaying the sharpest rings. This image was used for the measurement of the radius. The process was repeated for each of the bandpass filters, as well as for the HeNe laser. A number of profiles of each image were taken and the average radii of the Poggendorff dark rings were determined.
The results are shown in Fig. 4(a), along with the theoretical predictions calculated from the expression for given in Eq. (1) with wavelength dependence arising from the Sellmeier equation given by Eq. (4). It is clear that there is a discrepancy of approximately 70 μm between the predicted values of (red curve) and the measured values. This discrepancy arises since the value of the semi-angle of the cone of conical diffraction depends very sensitively on the values of and as given by Eq. (1),Fig. 4(a) uses the values given by Eq. (4) but with the value of increased by just throughout the visible spectrum. This small change of about 0.34% is sufficient to reconcile the theory with the observations. All of the refractive indices may have slight variations from those calculated using Eq. (4), however the aim of this example is to convince the reader that the discrepancy between the predicted and the measured radius values may be explained within the error of the measurements of the refractive index values.
5. Variation of FIP location with wavelength
The dependence of the location of the FIP on wavelength was also investigated. This was achieved by mounting the CCD on a translation stage with a resolution of 25 μm and placing the 650 nm bandpass filter in the beam. The achromatic focusing lens ensured that any wavelength dependece in the FIP location arose from the crystal alone. The crystal was aligned to achieve the most symmetric ring structure and the CCD was gradually translated until the sharpest ring structure was observed. Defining this location to be the FIP for 650 nm, the bandpass filter could then be replaced and the process was repeated for each of the bandpass filters. The location of the FIP was determined by examining the width of the rings in each image. The one containing the narrowest ring structure was determined to be at the FIP. Figure 5 shows these points sit on the predicted curve within error. Note the structure of the conical beam (images inset) when the 500 nm bandpass filter is used. This is misalignment due to the wavelength dependence of the optic axis direction. The crystal was not reoriented for different bandpass filters as moving it may have incurred errors in the location of the FIP.
6. Far-field white light observations
Since the value of ζ is proportional to as shown in Eq. (2), a beam waist of μm results in a large variation in ζ over a relatively small range of In order to observe the far-field evolution of the beam, the first step was to use a HeNe laser at 632.8 nm as the light source. This allowed very precise adjustment of the biaxial crystal to ensure the optic axis was aligned perfectly for conical diffraction at 632.8 nm. The LED was then returned as the source of light and a bandpass filter centred at 632.8 nm was inserted before the CCD. The CCD was moved in intervals of 1 cm starting at the FIP until reaching a value of ζ of approximately 45. The resultant images were stitched together using numerical tri-linear interpolation which allowed the full evolution of beam structure over ζ to be viewed from the side. We observe the spreading out of the rings for with the axial spike  becoming apparent between and Fig. 6 compares the observed beam evolution over ζ in the plane with a plot generated from Eq. (12) with and closely matches previous work [17, 18].
Of greater interest is what happens when the bandpass filter is removed and this process is repeated. We observe a gradually increasing misalignment of the optic axis with wavelength which translates to a transition from perfectly symmetrical conical diffraction for red light to asymmetrical beam evolution for decreasing wavelength described by Eq. (12). Figure 7 displays the observed patterns on the top row with increasing and hence ζ values, beginning at the FIP. The rows underneath are numerically generated intensity distributions for a given wavelength. The corresponding optic axis angles calculated from Eq. (5) are also shown.
The crystal was then aligned using a 500 nm bandpass filter and the beam evolution was again recorded as seen in Fig. 8. The beam exhibits a similar evolution as before but this time it is more apparent that light with 500 nm evolves as a mirror image to light with 500 nm. This is a manifestation of the sign reversal of at as given by Eq. (6).
While the theory does appear to fit quite well in Fig. 8 it is worth noting that some of the disagreement is due to the theoretical plot being for a single wavelength while the images at 650 nm are taken using a bandpass filter centred at with a 40 nm FWHM, and the crystal is aligned to This will inevitably result in more accentuated asymmetry than the theory predicts due to the presence of wavelengths further from than To see the extent of this effect note that if nm with nm, then
Some of the properties of a conically diffracted beam generated using a white light source have been examined and successfully described using theoretical models. Both the radius of the conically diffracted ring and the position of the FIP were shown to depend on the wavelength of the input light. The radius was observed to deviate from theory by a measurable amount, but by a factor sufficiently small to be within the range of errors of the measured refractive index values. The beam evolution beyond the FIP into the far-field was observed to depend strongly on wavelength, with asymmetries arising as a result of the polychromatic light source. A theoretical model was developed and applied to describe this beam evolution and was found to successfully predict the beam structure for the cases examined in the experiments. A method for incorporating explicit wavelength dependency in conical diffraction has been explained and this should provide a useful foundation for future work in this field.
References and links
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