| Conical Refraction
EXPERIMENTS AND LARGE-SCALE
Yuriy P. Mikhaylichenko
Department of Physics, Tomsk State University, Lenin St. 36, Tomsk
634050, Russia, email@example.com
Results of experiments on internal and external conical refraction on a rhombic sulfur single crystal are presented.
A high-transmission optical scheme is suggested with light focusing on the crystal surface which allows two fringes
of internal conical refraction about 1 m in diameter to be observed in laser radiation. Two fringes of external conical
refraction with unique polarization state are first observed, and conditions of light energy concentration along the
biradial are discussed. The focusing power of a plane-parallel biaxial crystal plate is measured.
Keywords: conical refraction, birefringence, biaxial
Conical refraction was predicted by Hamilton in 1832 and then confirmed by the Lloyd observations with an
aragonite crystal [1–7]. Schemes of conical refraction observations are shown in Fig. 1.
In 1839, Poggendorf  established that the internal conical refraction fringe is double, but this phenomenon was
explained by Voigt only in 1905 .
Difficulties of conical refraction observation are caused by small crystal sizes. Nevertheless, in the early twentieth
century, E. Leybold’s Nachfolger and M. Kohl firms produced devices for subjective observation of Lloyd internal and
external conical refraction with aragonite crystals. In , conical refraction on an aragonite specimen was observed with a
microscope, and in  the focusing properties of this crystal were discussed. Conical refraction on naphthalene crystals
was studied in [11–14], but the problem remained complicated and there was no clear understanding [15, 16].
With the advent of lasers, interest in conical refraction has increased, because unique distribution of polarization
was observed in conical refraction fringes [17–28].
In 1979, a large-scale demonstration of internal conical refraction on rhombic sulfur single crystals was arranged in
the Physical Laboratory of Tomsk State University, and results of continuation of these experiments by the author of the
present paper are given below.
1. Internal conical refraction
Experiments were carried out with rhombic sulfur single crystals. P. Drude, considering conical refraction on aragonite crystals, pointed out that sulfur crystals would be even better; however, treatment of these crystals was very difficult . Because sulfur single crystals are very fragile and sensitive to temperature gradients, glass treatment technology was used for their mechanical treatment. For rough treatment, a pig-iron slowly rotating disk with an abrasive powder was used. Then the specimen was manually polished using a pitch with a thin glass powder. To determine directions of optical axes, test specimens were polished.
Independently of Schell and Bloembergen , a method of narrow laser beam focusing onto the single crystal surface was suggested to obtain maximum image brightness. A conventional photo-objective with a focal length of 5 cm was used, which allowed us to limit the laser beam width by a field stop. For a 1:22 relative aperture, the light beam had angular divergence less than 10 min of arc. The second advantage of the focusing method is that a small focal spot observed on the first crystal surface allowed us to choose a well-polished fragment and not to use the field stop in the form of a pin hole in a foil. To choose a fragment with good surface quality, the crystal was clamped on a little coordinate table.
When a narrow light beam passed through the crystal, fringes of conical refraction were observed on the second crystal surface given that the crystal was properly oriented. Then these fringes were projected onto a screen by the same photo-objective or microscope objective to obtain higher magnification.
Hamilton–Lloyd internal (à) and external conical refraction (b). The angle ? of external
conical refraction in air approximately doubles as a result of beam refraction on the interface.
In our experiments, we used a He–Ne laser with output power from 2 to 20 mW (Fig. 2)
and a copper vapor laser with higher output power (? = 510.6 nm, Pout= 50 mW) (Fig.3).
Double fringe of internal conical refraction observed in linearly polarized He-Ne–laser
radiation. The brightness distribution in the fringes is determined by conical polarization. If the polaroid was located behind the crystal, a dark spot could be seen on the fringes. When the polaroid was rotated through an angle of
180°, the dark spot on the fringes rotated through 360° .
Double fringe of internal conical refraction in unpolarized laser radiation (? = 510.6 nm). The fringe diameter
on the screen is about 1 m. A person staying at the screen can be seen. The crystal plate is slightly
tilted to demonstrate changes in fringe shapes.
The sizes of the polished crystal plate allowed us to carry out an experiment on gradual transformation of internal conical refraction fringes into two sports of ordinary birefringence with
a continuous increase in the plate tilt angle (Fig. 4).
Change of internal conical refraction fringes with increase in the tilt angle of the crystal
plate relative to the incident beam.
2. External conical refraction
In textbooks and handbooks, conditions of observations of external conical refraction on single crystals are
illustrated by a scheme with a converging beam focused through a small aperture in the screen onto the first lens surface
(see Fig. 1b). The second field stop is placed on the other crystal surface at the exit of the external conical refraction cone.
In  it was reported that in this case, two concentric light fringes were observed, which was explained analogously to the case of internal conical refraction examined here. We now discuss these statements.
The Lloyd experiments demonstrated that to observe external conical refraction, the crystal must be illuminated by
a converging beam of rays. Based on the Voigt reasoning for light beam OM propagating inside the crystal along the
biradial (Fig. 5), splitting into an infinite number of rays at the exit from the crystal, and forming an external cone, we
conclude that the fringe of external conical refraction bifurcates. But this beam of rays propagating in the crystal along
biradial OM cannot be considered natural light, because it is formed from the cone of rays 1–2 due to bifurcation. To form
a full fringe of external conical refraction, we must illuminate the crystal at least through an annual field stop, which was done. However, under such conditions, two fringes of external conical refraction should be observed on the screen;
moreover, the vertex of the internal cone will be located on the crystal surface.
Scheme of observation of two external conical refraction fringes. Polarization states at
the opposite points of fringes are mutually perpendicular.
In our setup, a hollow cone of beams with incidence angles from 12 to 16° was formed by the annular field stop
0.1–0.5 mm wide . The annular field stop consisted of two elements and was inserted into the converging beam between
the focusing objective and the crystal. The first element of the field stop was a thin metal disk 3 mm in diameter glued to the
glass cover. A washer with internal aperture having the same diameter was placed behind it at a distance of several
millimeters. Mutual displacements of these elements allowed the beam cone width to be controlled, and their displacements
in the whole in the converging beam of rays along the optical axis of the setup enabled the required angle of beam cone
convergence to be obtained. Two narrow fringes of external conical refraction could be observed on the screen behind the
crystal for optimal adjustment (Fig.6). The diameters of these two fringes increased with distance from the crystal to the
screen, but their spacing remained constant and equal to 2 mm for our crystal. When the screen was removed at about 10 cm
from the crystal, these two fringes observed in laser radiation merged because of diffraction broadening.
On the second crystal surface, a contrast light fringe of about 4 mm in diameter was observed with a bright spot in
the fringe center which was the vertex of the outgoing internal cone of rays of external conical refraction. If only the central
spot was selected by the field stop, one fringe would be observed on the screen. We tried to verify the assumption that this
fringe of external refraction had a more complicated structure. However, we observed only diffraction broadening of the
internal fringe and the subsequent complete disappearance of the pattern when the field stop diameter D1 decreased from
1 mm to zero.
Figure 6. External Conical Refraction
Two fringes of external conical
refraction are observed in linearly polarized He-Ne–laser radiation. Letters a and b indicate
dark regions of the fringes. The gap between the light rings
is determined by the crystal plate thickness. Photographic film was
overexposed for more colour contrast.
The photograph (Fig. 6) in linearly polarized He-Ne–laser radiation shows a nonuniform brightness distribution in
the fringes caused by the unique polarization distribution. Polarization planes for two fringes of external conical refraction
were mutually perpendicular at points lying on one radius. Dark fragments were observed in the vicinity of point A on the
internal fringe and point B on the external fringe. Lloyd first observed a specific polarization distribution in the internal
fringe and called it conical polarization. He pointed out that when a tourmaline plate used as an analyzer was rotated
through 360°, a dark region of the fringe made two revolutions.
When the crystal was illuminated through the lens and annular field stop, all light energy was concentrated in the
focal spot O on the crystal surface. It is obvious that half of this energy subsequently propagated inside the crystal in the
light channel along the biradial, which suggests the energy concentration along this direction. The second half of energy i
spent on the formation of a larger fringe (rays 1–2 and 2–1). We now estimate the degree of energy concentration along the
biradial for the crystal illuminated through the annular field stop with radius B and width b. The field stop area is 2?Bb
When light is focused at point F in the crystal along the biradial, it will propagate as a narrow diverging cone of rays that on
the second crystal surface will produce a light spot in the vicinity of point M. Provided that the diameter of this spot i
determined by the width b of the annular field stop, its area can be expressed as ?(b/2) . The energy ratio will be 4B/b. Thus
this ratio for the pattern shown in Fig. 3 was equal to 40. Here it is of interest to consider additionally the phenomena caused
by energy concentration; however, in our experiments we observed no special features. It is obvious that an increase in the
energy density is limited by the diffraction effects. Raman  also pointed out the energy concentration in the biradial
In white light of a camera lamp, the fringe of external conical refraction can be examined individually on a small
screen; this was done in 1979 . The influence of dispersion was noticeable; it produced a reddish color of the internal
fringe edge and a bluish color of the external fringe. The same colors were also observed in white light for fringes of
internal conical refraction.
Thus, unlike internal refraction fringes, the internal fringe of external conical refraction has no doubled structure.
The statement that this fringe is formed when the beam propagating inside the crystal along the biradial is incident at point
M lying in the dip cusp (where it splits into a set of beams), used since times of the Hamilton publications, is most likely
incorrect. In  it was stated that “… this one internal cusp-ray must correspond to an external cone of rays, according to a
new theoretical law of light which may be called External Conical Refraction.” Of course, the point of wave surface self-crossing is a singular point, but not a cusp or bend. Probably, the following comparison will be pertinent. In mathematics,
the plane curve – one Descartes foil defined by the equation x3 + y3 - 3axy = 0
– is considered. This curve is reminiscent of
letter ?. Two branches of the curve are mutually intersected at the singular point – the origin of coordinates – and the curve
has two tangents at this point .
Strictly speaking, both Hamilton definitions of internal and external conical refraction are incorrect.
3. Focusing properties of biaxial crystals
Lloyd observed light of a distant lamp through a crystal and noticed a bright point surrounded by something resembling star rays in the axial direction. He also observed converging beams in the vicinity of the optical axis. Raman  also reported the focusing properties in 1921 and returned to this problem in 1941 when he published in Nature photographs of simplest luminous objects whose images were focused with a naphthalene plate . In  Raman referred to the Stokes papers published in 1877 and Volker papers published in 1904, where it was pointed out that the image of a point source is astigmatic.
Our experiments  demonstrated that a light spot was always observed in the center of fringes of internal conical
refraction. Its formation is influenced by the diffraction effects, but in our experiments with large image sizes, the original focusing properties of the wave front surface of the biaxial crystal in the vicinity of point M (where the wave surface forms a cone-shaped dip) are more pronounced.
To examine the focusing properties of the biaxial crystal in the direction of optical axis, the following experiments
were carried out. A crystal plate was illuminated in the normal direction with light of a focused beam to obtain fringes of
internal conical refraction. To estimate the effect, we analyzed ray trajectories in the figure plane. The angle between rays 3
and 4 was 2?. From Fig. 7a it can be seen that rays 3–1 and 4–2 are intersected at point F. If we select the region F with the
field stop D2, one fringe will be observed on the screen Sc. If beams 3 and 4 are incident at smaller angles, after refraction
they will be intersected at a greater distance from the crystal. Thus, when the plate is illuminated by a wide cone of rays,
they will be focused along the entire straight line MÑ. An approximate dependence of the focal length MF = L on the ray
incidence angle ? can be derived for ray 4. Ray 4–2 intersects straight line MC at point F; MN is the radius of the internal
refraction fringe equal to 2 mm for our crystal. Beam 4–2 after refraction intersects the second crystal surface at point K and
leaves the crystal at the angle ?. We obtain the formula, (1)
where d is the plate thickness and n is its refractive index. For small angles, this formula is simplified: . (2)
We measured the diameters of fringes D on matte screen Sc. The distance MC = L0 = 210 mm remained constant.
Only field stop D2 was displaced, which resulted in changes of distance L. If we substitute tan? expressed in terms of the
diameter D into Eq. (2), it assumes the form . (3)
||Fig. 7. Scheme of observations of the focusing properties of a plane-parallel biaxial crystal plate
(à). The dark Poggendorf fringe is seen on a palm. Illumination by a wide unpolarized laser beam
with power of 20 mW. The pattern corresponds to the position of the screen in plane P (b).
Figure 8 shows two experimental curves for larger and smaller fringe diameters D and the analytical curve which
passes between them.
Plots of larger and smaller fringe diameters D on
screen Sc and analytical curve calculated from Eq. (3).
The successful solution of the technical problem of polishing rhombic sulfur crystals has allowed experiments to be carried out that, in our opinion, are of the same importance as Lloyd, Poggendorf, Voigt, and Raman works. The author first obtained 1) large-scale patterns of conical refraction up to 1 m in diameter and 2) two fringes of external conical refraction with the use of an annular field stop and measured 3) the focusing power of the crystal plate. The degree of energy concentration in the biradial direction was estimated for external conical refraction. To prove the absence of double fringes of external conical refraction, the author performed experementum cruces with the annular field stop.
In the majority of handbooks, the definition of conical refraction is based on the Hamilton mathematical consideration of the wave surface and is of historical interest only. Fringes of external and internal conical refraction bifurcate because they are formed due to birefringence.
In our experiments, it has been demonstrated that the light intensity distribution in the double fringe of internal conical refraction is in agreement with the Voigt definition, and spacing of two fringes of external conical refraction is
determined by the crystal plate thickness. Therefore, five special features of double refraction can be distinguished in the
directions close to the optical axes of biaxial crystals: 1) the dark Poggendorf fringe corresponds to rays directed along the
binormal, 2) energy is concentrated inside the crystal along the biradial under conditions of formation of light fringes of
external conical refraction, 3) the wave surface of biaxial crystals possesses focusing properties, 4) ordered conical polarization arises in fringes of conical refraction, 5) obligatory attribute of conical refraction in actual practice is a light spot in the center of internal conical refraction fringes.
Fresnel defined the wave surface of biaxial crystals when he suggested that the distribution of light oscillations in the medium is determined by the elastic properties of ether in two transverse directions. His work gave impetus to the development of a new branch of science – elasticity theory (see the Navier, Cauchy, and Green works). It is natural that nowadays, investigations of conical refraction are extended. Thus, the intensity distributions over the light beam cross sections were examined in [19, 24–28, 32], and conical refraction on optically active crystals was studied in [21, 33]. Because conical refraction in crystal optics is described by the unified theory of oscillations, concepts of conical refraction are used in other fields of physics, including magnetic hydrodynamics [34, 35], investigations of quasi-optical electromagnetic beams [36, 37], and acoustic crystallography .
The author would like to acknowledge A. Arzhanik for his help in photography, M. Dreger for kindly providing a number of papers on conical refraction, and V. G. Bagrov for careful reading of the manuscript and valuable remarks.
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Yu. P. Mikhailichenko. Conical refraction: experiments and large-scale demonstrations. Russian Physics Journal, Vol. 50, No. 8, 2007, 788-795. Download