Tomsk State University. Physical Department.
Conical Refraction Back<
William Rowan Hamilton - 2005 Bicentenary

b. Aug. 3/4, 1805, Dublin, Ire.
d. Sept. 2, 1865, Dublin
Irish mathematician and astronomer who developed the theory of quaternions, a landmark in the development of algebra, and discovered the phenomenon of conical refraction.His unification of dynamics and optics, moreover, has had a lasting influence on mathematical physics, even though the full significance of his work was not fully appreciated until after the rise of quantum mechanics.
Sir William Rowan Hamilton, 1862

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Large scale experiments on conical refraction

Yuriy P. Mikhaylichenko

Department of Physics, Tomsk State University, Lenin St. 36, Tomsk 634050, Russia,

2004, December (with some correction)


Experimental results on conical refraction in a monocrystal of rhombic sulfur are presented. Double ring 1 m diameter of internal conical refraction was shown on a screen of auditorium. Two rings of external conical refraction with specific distribution of polarization have been obtained and there was shown the inner ring of external refraction does not display any double structure. Focusing features of a parallel-plane slab of biaxial crystal have been experimentally evaluated.

Keywords: conical refraction, birefringence, biaxial

1. Introduction

During its a brilliant history the conical refraction [1] met experimental difficulties arising from small sizes of the monocrystals being studied under a microscope. That is why the goal of this study was to effect a large-scale demonstration of the conical refraction when ring sizes would be up to 1 m. For this purpose samples of rhombic sulfur were prepared. In spite of its brittleness, our sample of sulfur has well preserved from 1978. The experiments on the conical refraction in naphthalene with Raman’s microscope are of particular interest [2]. Raman has chosen naphthalene because it has the largest angle of external conical refraction of φ = 13°44'. For aragonite the same angle is φ = 1°52' and for rhombic sulfur φ = 7°11'.These data are given for the green mercury line λ =546.1 nm. Getting available lasers caused a rebirth of interest in new studies of conical refraction. Thus, in [3] the photo of internal conical refraction single ring of 1 mm diameter is presented that was obtained in the crystal aragonite. The experimental results by Schell and Bloembergen are presented that have been obtained in aragonite [4] and artificial crystals of α-iodic acid [5] and an analytical consideration of the light radial distribution in double rings of the conical refraction is given. In 1994, the paper [6] appeared on experimental and theoretical consideration of internal and external conical refraction in KTP crystals.

2. Internal conical refraction

Our experiments were conducted with monocrystals of rhombic sulfur [7]. Of all prepared samples one of 16 mm thickness had the best optical characteristics. To get a bright picture the narrow laser beam was focused on the first crystal surface S1. The photographic lens of 5 cm focal length appeared to be convenient because it gave a possibility to adjust laser beam width with its diaphragm. At the aperture ratio 1:22 the central part of laser beam was separated providing its weakly angular divergence of about 10 minutes. To get a projection of the second surface of the crystal, with the double ring of internal refraction on it, onto a screen another similar photographic lens was used. In Figure 1 the rings of internal conical refraction in the He-Ne laser light are shown. The laser of .002 Watts power generated the single-mode Gaussian beam. The light is linearly polarized that results in specific intensity distribution in the rings.

Figure 1. Internal Conical Refraction

Double ring of the internal conical refraction on exit surface of the sulfur crystal slab. Illumination with Gaussian beam of a helium-neon laser (λ = 632.8 nm) power .002 Watts. Incident light beam had a line polarization. Photographic film was overexposed for more colour contrast.
In Figure 2a the double ring of internal conical refraction is shown in the light of Cu-vapour .02 Watts laser. The light is not polarized. To get the picture of larger size, the light source should be of more power. Optical beam propagators are derived from the Maxwell wave eqiation for a refractively biaxial medium and than numerical illustration using a Gaussian beam show diffractive counterparts of the ray optic effects of double refraction, internal conical refraction, and the transition between two [8].

Figure 2a.

Double ring of the internal conical refraction in projection onto a screen. Illumination with narrow nonpolarized laser beam (λ = 510.6 nm), .02 Watts power. Diameter of the Poggendorff dark circle is 1m. A man stands near for comparison. The crystal slab is slightly tilted to show a transformation of the rings. The central spot of internal conical refraction is clearly seen.

Figure 2b.

The transformation of the rings into  two rays.

3. External conical refraction

If the internal conical refraction with dark Poggendorff circle may be observed easily, to get separately two rings of external conical refraction appeared to be a more complex task.
The cone of external conical refraction is produced by the beam OM (Fig.3) that travels along the biradial. But the beam OM with an appropriate polarization distribution can be formed only by a hollow cone of incident from outside onto a crystal light. In picture two incident rays 1 and 2 of the cone are isolated that undergo ordinary double refraction. Thus, the ray 1 breaks down into two ones: the ray 1-1 and the ray 1-2 having mutually perpendicular polarization because the condition (D1-1·D1-2) = 0 should be met. From this it follows that a half of energy of the incident cone should travel along the biradial OM with energy density determined by the lens focusing features.
In our experimental set-up the hollow cone of rays with incident angles from 12° to 16° was formed with a ring-shaped diaphragm of 0.1 – 0.5 mm width. The angle of external conical refraction for sulfur is about 7° and at crossing the crystal-air surface it is almost doubled. To form the hollow cone the ring-shaped diaphragm composed of two elements was used. The diaphragm was located in the convergent beam between the focusing objective and the crystal. The first element of the diaphragm R-D is a thin metal disk 3 mm in diameter that is pasted on the cover glass. Behind it the disk with central aperture approximately of the same diameter was located. Relative moving of these elements makes it possible to adjust thickness of the cone wall, and their joint displacement along the optic axis of the setup in the convergent beam permits the cone angle to be adjusted. Two adjusting screws were used for smooth tilt and rotation of the crystal. At precision adjustment, the distinct light ring about 4 mm in diameter with bright central spot M was formed on the second surface S2 of the crystal (Fig.3). The photo in Figure 4 is obtained from a translucent mat screen located 1 cm behind the second surface. The diameters of two external refraction rings increase when the screen is moving away from the crystal, nevertheless the gap between them remains constant 2 mm because it depends on the crystal thickness. On further moving away of the screen up to 10 cm from the crystal two rings run together because of diffractional broadening.

Figure 3.

The diagram of external conical refraction observation. The slab is illuminated with hollow cone of convergent rays 1 and 2. R-D – ring diaphragm of 0.1 – 0.5 mm width. Each ray of the cone splits into two ones after birefringence. Behind the slab two narrow separated rings of external conical refraction are formed with the specific polarization distribution. Thus, the rays 1-1 and 1-2 have mutually perpendicular polarization. A half of energy propagates along the biradial OM.

Figure 4. External Conical Refraction

Two rings of external conical refraction in the light of He-Ne laser (λ = 632.8 nm). Incident beam had a line polarization, because of this on the rings two dark gaps a and b are visible. The gap between the rings is determined by the crystal plate thickness. Photographic film was overexposed for more colour contrast.
The light spot at M is a cone apex of going out rays of the external conical refraction. Being separated with a diaphragm it gives on a distant screen the single ring of external conical refraction. There was a hypothesis this ring to comprise a more complex structure; however, the diaphragm D2 contraction from 1 mm to 0 leads to the diffractional broadening of the ring and then full disappearance of the picture. In white light the ring of external conical refraction is faint, however it can be easily observed on a small screen when a lamp of movie projector is used.
The effect of dispersion is noticeable: inner rim of the ring is reddish and outer is bluish. The rings of internal conical refraction display similar colouring in white light.

4. Focusing features of the crystal

Experiments show that the light spot is always observed at the center of the internal conical refraction ring. Its origin may be attributed to peculiar focusing feature of the biaxial crystal in the vicinity of M and also to light diffraction and scattering on discontinuities. The double ring of the internal refraction in case of a wide illuminating beam shows in Figure 5. Such a pattern is observed on a screen in the plane P Fig.6.

Figure 5.

The dark Poggendorff ring on a palm. The projection with wide beam (λ = 510.6 nm) of nonpolarized light. The central light spot is clearly visible as result of diffraction and scattering on discontinuities. The convergent internal beam is seen. The picture corresponds to the screen position in the plane P in Figure 6.

Figure 6.

The diagram for measuring of the focusing characteristics of the crystal slab. The rays 3 and 4 of the incident focused beam undergo birefringence. Behind the slab the rays 3-1 and 4-2 intersect at the point F. Single ring would be visible on the screen if the zone F were separated with the diaphragm D2. The rays of other incident angles would be focused at other points of the focal line MC.
Raman in the small note [10] represented the pictures of focused patterns obtained in the crystals aragonite. In our paper [11] we made an estimate of focusing ability of the plane – parallel slab as follows. In Fig.6 the rays 3-1 and 4-2 intersect at point F. Being separated with the diaphragm D2 the zone F produces on the screen Sc the single ring. When the rays 3 and 4 fall at smaller angle the intersection point moves away from the crystal after their refraction. Thus, when the crystal slab is exposed to a wide cone beam the rays are focused along all line MF. Let us consider how the focal length MF = L depends on the angle of incidence A for the ray 4. The wave surface of this ray intersects the plane of the picture along the circle W1. The ray 4-2 intersects the straight line MC at point F. MN is the radius of the Poggendorff ring that for our crystal was R=2 mm. The ray 4 after refraction intersects the second surface of the crystal at point K and goes out at the angle A. Thus we come to the formula:


On the graphs 7 a pair of experimental curves is shown for the values of outer and inner ring diameters and the analytical curve that goes between them is also presented.

Figure 7.

Verification of the focusing characteristics of the crystal slab. Plots of the ring diameter on a screen versus distance L=MC are shown. The analytical curve D = D(L) - is the broken line. Unbroken lines correspond to the measurements of external and internal rims of the ring. Diameter of the movable diaphragm D2 = 1 mm.

5. Crystal preparation

Rhombic sulfur crystals were used in our experiments. In spite of extraordinary brittleness of the material we have been able to prepare some samples of which the greatest was of 16 mm thickness. The polishing technology for very soft minerals was applied to these crystals. Crystal faces were carefully polished manually against the pitch with a fine glass powder suspension obtained after 10 min. sedimentation. Directions of crystal axes were defined by the trial method because solely fragments of monocrystals were available. It took a week to finish the first crystal. The experiments were begun in 1978 and even though some cracks developed in the crystal and it has split up, the left part acts till now. Sulfur sublimes on artificial surfaces therefore before new experiments it needs to be slightly polished. It is hard to suppose items made of crystal sulfur to be applied in any appliances but for research experiments this material is of very interest.

6. Summary and conclusions

Often reference books give determinations of the conical refraction which presently are of the historical interest only. In reality the light ring of internal conical refraction can not be realized as a result of one ray splitting into infinitely large number of the rays forming the hollow light cone. It is known that an incident ray being directed along the binormal defines geometric locus of Poggendorff dark circle. To form the ring of external conical refraction it is insufficient one ray to propagate in the crystal strictly along the biradial. Affirmation about existence of such a ray would mean the double ring of external conical refraction to be formed that, however, was not observed experimentally. The ray for getting the ring of external conical refraction should be formed inside the crystal and this occurs when the crystal is illuminated with a focused beam or hollow cone of rays in a more strict experiment organization. In this case incident rays of the hollow cone are subject to birefringence and after going out of the crystal form two cones. A distinguishing feature of external conical refraction is that the apex of the central cone is positioned on the output surface at point M (Fig.3). Therefore, discussing the matter of light propagation in biaxial crystals it would be more accurate to say about birefringence features in conditions of internal and external conical refraction. The results of our experiments make it possible to underline two distinctions of the birefringence in conditions of internal and external refraction. The first: spacing between the rings of external refraction (in our experiments 2 mm) is determined by the crystal thickness instead of Poggendorf`s dark ring. The second: light polarization in the points of two external refraction rings lying on one radius is mutually perpendicular and in the neighboring points of internal refraction ring separated by a dark gap the polarization is identical. It should be noted that in real experiments the light spot of conical refraction is always formed in the center of the internal conical refraction ring. Melmore [12] said that Hamilton himself wrote to Lloyd on January 1, 1833: “It is much for theory to have predicted the facts of conical refraction, but I suspect that the exact laws of it depend on things as yet unknown”. Hamilton displayed an interest to the phenomena giving food for intellect up to the present.


Author is grateful to M. Dreger for some articles on conical refraction and 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