Conical diffraction (or refraction): an outstanding optical effect deep-rooted in the history of science
Figure 1. (a) internal conical refraction: a collimated beam propagating through a biaxial crystal parallel to one of the optical axes refracts as a skewed cone within the crystal and emerges as a hollow light cylinder. (b) External conical diffraction: a light cone propagating along the biradial refracts as a light filament through the crystal and emerges forming a divergent cone of light.
Conical refraction is an optical phenomenon predicted by Hamilton in 1832, on the basis of Fresnel’s equations, and confirmed experimentally by Lloyd[2,3] two months later. Conical refraction describes the propagation of a light beam in the direction of the optical axis of a biaxial crystal. It was the first physical effect ever predicted by theory and one of the two experimental demonstrations of the wave nature of light. However, subsequent experiments showed that conical refraction somewhat differs from Hamilton’s theory.
Only in 2004, 172 years after Sir Hamilton, Sir Michael Berry, [4, 5], proposed a complete and comprehensive theory of Conical refraction. He also renamed the effect – conical diffraction.
The complexity of the physics and the strangeness of conical diffraction puzzled researchers for more than a century.
“…the effect seems to occur nowhere in the natural universe”, (Sir Michael Berry).
Additionally, the beauty of the effect impressed all researchers involved in conical diffraction:
“. . . among the most beautiful and striking effects arising in crystal optics”, (Sir C.V. Raman).
Since its discovery, in 1832, conical refraction has attracted considerable theoretical and experimental interest, but “it can also be regarded as a curiosity”. Sir Michael Berry complained in 2007, “no practical application seems to have been found” .
It took only a few years after Berry’s publications, for several researchers to understand the potency of conical diffraction, as the basis of practical devices and systems. An enlightening review of applications of conical diffraction has been published lately by Turpin et al .
Figure 2. Intensity and polarization distribution (depicted with yellow double arrows) of conical refraction with input beams of circular (a) and linear vertical (b) polarization. The dark ring between the two bright ones is known as the Poggendorff dark ring (PDR) .
“In short, we traded all the beauty and elegance of Poggendorff rings and conical diffraction for a dull but efficient controllable beam shaping unit”
Gabriel Y. Sirat
Figure 3. Family of spatial distributions which can be generated by a biaxial crystal and polarizing optics. The upper horizontal axis indicates the input polarization state, and the vertical axis indicates the output polarization state.
Conical diffraction beam shaper
In 2006, Gabriel Y. Sirat was the first to recognize the specific properties of conical diffraction in thin crystals. He demonstrated that conical diffraction can be used as a practical tool to shape optical beams, and suggested its use, in 2010, to create a versatile beam shaper and apply it to super-resolution fluorescence microscopy [10, 11].
A thin biaxial crystal transforms the Point Spread Function (PSF) of a regular incident beam into an extensive family of light distributions, the choice of the distribution being controlled by the input and output polarizations.
Practically, beam shaping is achieved by enclosing the biaxial crystal between controllable polarizers; this simple optical set-up, similar to a polarimeter, has the ability to switch from one pattern to another pattern with a different topology in microseconds – or even faster. In addition, these patterns are perfectly co-localized, as they are produced by the same primary optical beam.
Figure 4. Beam shaper unit schematics, the biaxial crystal is placed in between polarizing optics to enable manipulation of both the input and output polarizations. In the optical path, prior to the crystal, a horizontal linear polarizer is used, after which a double Pockels cell acts as polarization generator (PSG). This allows for the generation of any incident light polarization. Past the crystal, another double Pockels cell unit with a vertical linear polarizer acts as polarization analyzer (PSA).
Figure 5. Typical light shapes used in the CODIM system.
Super-resolution using CODIM
The CODIM beam shaper is used as an add-on to a confocal module and the distributions are scanned on the sample, yielding several micro-images for each scan point. Due to the point scanning geometry, BioAxial system in fact requires a reduced quantity of diffuse light, compared to widefield based technique.
Figure 6. Measurement flow of CODIM system.
Figure 7. MTF of Bioaxial distributions compared to Airy pattern; (a) full MTF; (b) zoom on frequencies between 0.8 and 1.
The CODIM beam shaper generates compact, localized light distributions using the conical diffraction principle. Each micro-image contains a large amount of high frequencies, close to Abbe’s limit (up to a factor close to 3 compared to an Airy pattern) (Figure 7). These light distributions, projected on the sample, are analyzed using complex proprietary algorithms. This allows reconstruction of a super-resolved image, for general objects, with an improvement of resolution up to a factor of 2. Additionally these algorithms, leveraging positivity constraint and sparsity, allow for the resolution to be improved even further, for adequate samples.
Ultimately, the conjunction of much lower distribution peak power, use of a high quantum yield camera and a longer camera exposure time drastically reduce the peak power – and the energy – of light sent to the sample. Such is the reason for this method’s very low photobleaching and phototoxicity. This also avoids fluorophore saturation issues, making the method linear and quantitative.
WR Hamilton, « Third Supplement to an Essay on the Theory of Systems of Rays, » Trans. Royal Irish., Acad., Pp 1 -144 (1833).
H. Llyold, « On the Phenomena presented by Light in icts Passage along the Axes of Biaxial Crystals, « The London and Edinburgh Philosophical Magazine and Journal of Science ii, 112-120 (1833).
H. Llyold, « Further Experiments on the Phenomena presented by Light in icts Passage along the axes of Biaxal Crystals., » The London and Edinburgh Philosophical Magazine and Journal of Science H, 207-210 (1833).
MV Berry, « Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike, » Journal Of Optics A-Pure And Applied Optics 6, 289-300 (2004).
MV. Berry and Mr. Jeffrey, « Conical diffraction: Hamilton’s diabolical points at the heart of crystal optics, » Progress in Optics 50, 13 (2007).
Raman, C., V. Rajagopalan, and T. Nedungadi, Conical refraction in naphthalene crystals. Proceedings Mathematical Sciences, 1941. 14(3): p. 221-227.
Alex Turpin et al.: Conical refraction: fundamentals and applications, Laser Photonics Rev., 1–22 (2016).
A. Turpin, et al. « Blue-detuned optical ring trap for Bose-Einstein condensates based on conical refraction, » Opt. Express 23, 1638-1650 (2015).
Gabriel Y Sirat: “Optical devices based on internal conical diffraction”, US patent 8,514,685, Priority date 20 October 2006.
Gabriel Y Sirat: “Method and device for superresolution optical measurement using singular optics”, US patent 9,250,185, Priority date 15 October 2010.
Caron, Julien et al. “Conical Diffraction Illumination Opens the Way for Low Phototoxicity Super-Resolution Imaging.” Cell Adhesion & Migration 8.5 (2014).