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Virtual Journal for Biomedical Optics

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  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 8, Iss. 5 — Jun. 6, 2013
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A sub wavelength localization scheme in optical imaging using conical diffraction

Shani Rosen, Gabriel Y. Sirat, Har’el Ilan, and Aharon J. Agranat  »View Author Affiliations


Optics Express, Vol. 21, Issue 8, pp. 10133-10138 (2013)
http://dx.doi.org/10.1364/OE.21.010133


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Abstract

In this paper we present a scheme for the acquisition of high temporal resolution images of single particles with enhanced lateral localization accuracy. The scheme, which is implementable as a part of the illumination system of a standard confocal microscope, is based on the generation of a vector beam that is manipulated by polarimetry techniques to create a set of illumination PSFs with different spatial profiles. The combination of data collected in different illumination states enables the extraction of spatial information obscured by diffraction in the standard imaging system. An implementation of the scheme based on the utilization of the unique phenomenon of conical diffraction is presented, and the basic strategy it provides for enhanced localization in the diffraction limited region is demonstrated.

© 2013 OSA

1. Introduction

Single particle localization techniques are being widely used in optical imaging systems for the investigation of sub-cellular structures and dynamics. Applications of these techniques range from single particle tracking (SPT) [1

1. M. J. Saxton and K. Jacobson, “Single-particle tracking: applications to membrane dynamics,” Annu. Rev. Biophys. Biomol. Struct. 26(1), 373–399 (1997). [CrossRef] [PubMed]

] and trajectory mapping to acquisition of super-resolution images based on the use of photoactivated localization microscopy (PALM) [2

2. M. Fernández-Suárez and A. Y. Ting, “Fluorescent probes for super-resolution imaging in living cells,” Nat. Rev. Mol. Cell Biol. 9(12), 929–943 (2008). [CrossRef] [PubMed]

, 3

3. E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006). [CrossRef] [PubMed]

]. Imaging systems that are used for localization usually operate in the far-field, hence their resolution is fundamentally limited by the size of their diffracted spot, commonly referred to as the Airy disk [4

4. G. B. Airy, “On the diffraction of an object-glass with circular aperture,” Trans. Cambridge Philos. Soc. 5, 283–291 (1835).

].

It is widely recognized, that although the ability to resolve two or more particles is limited by diffraction to the Airy disk size, localization accuracy of a single particle is limited only by the signal-to-noise ratio (SNR) of the system, namely by the brightness and contrast of the imaged particle [5

5. N. Bobroff, “Position measurement with a resolution and noise-limited instrument,” Rev. Sci. Instrum. 57(6), 1152 –1157(1986). [CrossRef]

, 6

6. R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82(5), 2775–2783 (2002). [CrossRef] [PubMed]

]. This principle underlies the procedure of sub-wavelength localization, usually performed by the acquisition of a diffraction limited image of the particle on a pixilated camera, from which the image centroid is retrieved.

In this paper we present an alternative localization scheme that is based on the operation of a point scanning microscope, such as the confocal microscope, in which the out-of-focus light is inherently reduced, and better optical sectioning and resolution along the z-axis is provided. The standard pre-designed scan operation of these microscopes often does not support the visualization of fast cellular dynamics [7

7. J. B. Pawley, Handbook of Biological Confocal Microscopy, 3rd Ed. (Springer Science + Business Media, LLC, 2006).

], especially in SPT assignments, in which sub-wavelength scaled movements of the particle may readily be missed. We henceforth present a scheme that is designed to create an access to a fast localization process in point scanning microscopes, in which the particle’s location in two dimensions within the area illuminated by the beam (i.e. the Airy disk) is acquired simultaneously by a single point detector, independent of the scanning process. The scheme, reducing to practice the concept developed in [8

8. G. Y. Sirat, Patent application PCT/FR2011/000555.

], is implemented by the integration of a simple optical module to a standard point scanning microscope, through the use of the conical diffraction phenomenon.

2. Localization scheme outline

The basic tool used in our scheme for the differentiation between emission occurrences that originate in distinct [R,θ] locations inside the Airy disk area, is the formation of a unique temporal modulation of the light incident on each location during the excitation period. The spatially dependent temporal modulation within the Airy disk area is accomplished by the generation of a spatial polarization distribution in the illumination beam, in which a unique polarization state is applied to each [R,θ] location inside it. Before entering the microscope, the beam is manipulated by simple polarimetry techniques to create a set of continuously varying point spread function (PSF) profiles on the sample. The particle's emission will thus be temporally modulated according to the specific coordinates where it is located. The emission signal is collected by a photodetector which serves as a spatial integrator, and its read-out signal is subsequently Fourier transformed in time. This procedure, illustrated schematically in Fig. 1
Fig. 1 A schematic illustration of the signal generation procedure according to our scheme. An electric field polarization distribution is created inside the illumination beam (a), which is then manipulated by polarimetry techniques to obtain a continuous spatio-temporal modulation of the light incident on the sample (b). The emission signal, uniquely modulated according to the location [R1, θ1] of the particle, is integrated by the detector (c) and Fourier transformed (d), to create a frequency domain label by which I [t, R1, θ1] is identified and the particle’s location is reconstructed.
, results in the formation of a distinct frequency domain label by which the location of the particle inside the Airy disk may be reconstructed.

The proposed localization scheme is implemented using conical diffraction (CD), a phenomenon which occurs when light propagates along the optic axis of a biaxial crystal. CD was originally predicted by Hamilton in 1832 [9

9. W. R. Hamilton, “Third supplement to an essay on the theory of systems of rays,” Trans. Royal Irish. Acad. 1–144 (1833).

]. Later experiments [10

10. H. Llyold, “On the phenomena presented by light in its passage along the axes of biaxal crystals,” Phil. Mag. 1,112–120 and 207–210 (1833).

13

13. A. M. Belsky and A. P. Khapalyuk, “Internal conical refraction of limited light-beams in 2-axes crystals,” Opt. Spectrosc. 44, 746–751 (1978).

] showed this phenomenon to be more complex than the original prediction, extending beyond geometrical optics. An all-inclusive theoretical analysis of CD was given in 2004 [14

14. M. V. Berry, “Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A, Pure Appl. Opt. 6(4), 289–300 (2004). [CrossRef]

] and validated by a series of observations [15

15. M. V. Berry, M. R. Jeffrey, and J. G. Lunney, “Conical diffraction: observations and theory,” Proc. R. Soc. A. 462(2070), 1629–1642 (2006). [CrossRef]

].

Consider a laser beam with a wave number k that is propagating along one of the optic axes of a biaxial crystal with a conical radius parameter R0 defined by:

R0=ln2(n2n1)(n3n2)
(1)

wheren1,n2,n3 denote the three principal refractive indices of the crystal and l is the slab’s thickness. Under the paraxial approximation the field distribution inside the beam may be described in terms of two interacting waves, referred to here as the fundamental wave (EF)and the vortex wave(EV). By placing a lens behind the crystal, an image of the beam waist may be obtained at a plane which shall be referred to as the focal image plane [14

14. M. V. Berry, “Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A, Pure Appl. Opt. 6(4), 289–300 (2004). [CrossRef]

]. The field distribution in polar coordinates R and θ at this plane is given by:

E(R,θ)=EF(R)+EV(R,θ)=[B0(R)×I+B1(R)×(cosθsinθsinθcosθ)]|P
(2)

where|Pis the polarization vector of the incident beam and B0,B1are integrals given by:

B0(R)=kdUUa0(U)cos(kR0U)J0(kUR)B1(R)=kdUUa0(U)sin(kR0U)J1(kUR)
(3)

where the function a0(U) represents the incident light distribution in Fourier space.

The specific evolvement of the two waves in space is strongly dependent on the ratio between the conical radius R0 and the beam width parameterw0, which shall be designated byρ0. In a thick biaxial crystal configuration, for which ρ0is much greater than one, the fundamental and vortex waves build up to create the well-known cone-shaped beam inside the crystal, schematically shown in Fig. 2
Fig. 2 Geometry of conical diffraction for a circularly polarized incident beam. In a thick crystal configuration (Aa), the diffracted beam emerges out of the crystal as a hollow cylinder, forming the two Poggendorff rings at the focal image plane (Ab). In a thin crystal configuration (Ba), the diffracted beam is mostly confined to its original width, forming a unique spatial polarization distribution inside the Airy disk area at the focal image plane (Bb).
. This results in the emergence of a hollow cylinder outside the crystal, forming the two Poggendorff rings at the focal image plane [11

11. J. C. Poggendorff, “Ueber die konische refraction,” Pogg. Ann. 124(11), 461–462 (1839).

, 14

14. M. V. Berry, “Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A, Pure Appl. Opt. 6(4), 289–300 (2004). [CrossRef]

]. Striking as this phenomenon appears to be, to date no major application of CD has emerged [16

16. M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamilton's diabolical point at the heart of crystal optics,” Prog. Optics 50, 13–50 (2007). [CrossRef]

]. The development of high quality crystals that are biaxial in the linear regime, such as KTP [17

17. B. Boulanger and J. Zyss, Physical Properties of Crystals. Vol. D of International Tables for Crystallography, A. Authier, Ed. (Kluwer, Dordrecht, 1997).

], brings forward the possibility of harnessing CD for practical purposes. In particular, the use of CD in a thin biaxial crystal configuration has been suggested for the first time in [8

8. G. Y. Sirat, Patent application PCT/FR2011/000555.

] as a simple and available means for the creation of a wide spectrum polarization distribution inside a diffraction limited spot of a focused light beam.

In a thin biaxial crystal configuration [18

18. G. Y. Sirat, Patent application US 2009/0168613 A1.

] for which ρ0≤1, the fundamental wave builds up around the optical axis, resembling the incoming wave in its shape, while the vortex wave maintains a singularity at the optical axis and builds up in its periphery. For an appropriate selection of ρ0, the interference between the fundamental and vortex waves yields a vector beam that is mostly confined to the Airy disk area at the focal image plane. The spatial distribution of the polarization states inside the beam is illustrated in Fig. 2 for a circularly polarized incident beam. It generally manifests itself in a rotation of the linear polarization vector in the tangential direction, which occurs at half rate with respect to θ.

Polarization diversity obtained by CD is utilized according to our scheme for the creation of an illumination system in which the PSF spatial profile is temporally modulated by the use of a simple Stokes polarimeter [19

19. P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96(1-3), 108–140 (1980). [CrossRef]

] that is positioned after the crystal (see Fig. 3(a)
Fig. 3 (a) Schematics of the optical module integrated into the microscope’s illumination light path. (b) An illustration of the PSF intensity profiles incident on the sample at 16 equally spaced instances throughout the first half of the QWP rotation cycle.
). The polarimeter is comprised of a quarter wave plate (QWP) that is rotating at a frequency ω, and a linear polarizer (LP, the analyzer). The rotation of the QWP results in a continuous variation of the PSF intensity profile that is incident on the sample. Figure 3(b) illustrates this dynamics by freezing the QWP state at different instances throughout the first half of the rotation cycle. In terms of the Stokes parameters S0,S1,S2and S3, the modulated intensity incident on a specific [R,θ] location inside the illuminated Airy disk area, throughout the QWP rotation cycle is given by:

I(R,θ,t)=12[(S0(R,θ)+S1(R,θ)2)S3(R,θ)sin2ωt+S2(R,θ)2sin4ωt+S1(R,θ)2cos4ωt]
(4)

As revealed by Eq. (4), the location dependent temporally modulated signal is comprised of a linear combination of a dc component and components of the second and fourth harmonies of the QWP rotation frequency. The latter are shown in Fig. 4
Fig. 4 Location dependency of the frequency domain signals obtained in our system. (a) 2ω amplitude signal, (b) 4ω amplitude signal and (c) 4ω phase signal ranging between [-π,π]. Row A: calculated frequency domain signals based on the system's ρ0 parameter. Coordinates are normalized with respect to the Airy radius. Row B: Scanning measurement preformed on a 250nm diameter silver dot. The dashed black lines denote the Airy disk borderlines. Coordinates are given in microns.
for the considered conical beam as a function of the [R,θ] position inside the Airy disk. The 2ω amplitude parameter constitutes an equivalent opposite amplitude map with respect to the 4ω parameter, which together may be used to uniquely determine the radial position of the particle inside the Airy disk. The 4ω phase parameter carries the azimuthal position information. It changes linearly with respect to the azimuthal coordinate, and completes a 2π turn with it. Together these parameters form a distinct, measurable frequency domain label for each [R,θ] location inside the Airy disk, which may be utilized to reconstruct the lateral location of a single particle with high accuracy.

3. Experimental results and discussion

This localization scheme has been implemented in our lab and integrated into the illumination path of a reflected light microscope (Fig. 3). A 532nm Nd:YAG laser beam is used as the illumination source of a reflected light LEITZ ERGOLUX microscope, equipped with a 0.9 numerical aperture (NA) objective. The optical module integrated into the microscope's illumination path is comprised of a circular polarizer, a focusing lens (f≈50mm), a 200μm thick KTiOPO4 crystal for which ρ0≈0.6 and a Stokes polarimeter. The polarimeter is realized by the utilization of a rotating QWP, its rotation driven mechanically at 150-200Hz. Note that the rotating retarder effect may be achieved by electro-optic modulators at frequencies up to the GHz regime. Hence the rotation period does not impose a limit on the system’s temporal resolution. It should be also emphasized that the final polarization state of the illumination beam is not at all constrained by our system. By placing additional polarizing elements after the Stokes polarimeter, the polarization state of the beam entering the microscope may be freely manipulated according to experimental needs. In high NA imaging systems for example, a radial polarization state may be chosen for the beam in order to minimize the PSF size [20

20. S. Quabis, R. Dorn, M. Eberlerm, O. Glockl, and G. Leuchs, “The focus of light: theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72(1), 109–113 (2001). [CrossRef]

] such that resolution is not compromised.

The target used in our experiments is comprised of highly reflective nanometric silver dots fabricated on a dark background. The operation of the scheme is demonstrated by raster scanning the image plane in which a single silver dot is positioned through the illuminated area. The scanning is preformed with an incremental step of ~70nm using a piezoelectric flat scanner (Nanonics Flat. Scanner). As the dot is translated inside the illuminated Airy disk area, the temporally modulated signal obtained from it is collected by a photomultiplier detector (HAMAMATSU H9307). A software phase-locked loop continuously extracts both the amplitude and phase of the 2ω and 4ω frequency components, by which the location of the dot inside the Airy disk region is reconstructed for each translation step. Signals obtained from a 250nm diameter dot are shown in Fig. 4, where the 2ω and 4ω amplitude signals were summed over two measurements performed in orthogonal circular polarizations in order to compensate for vortex asymmetries in the original vector beam. Similar results were obtained for dots of 150 and 200 nm in diameter.

These results clearly show the local variations in amplitude and phase of the 2ω and 4ω frequency components, as predicted by Eq. (4). The 4ω phase signal is a robust manifestation of the system’s sub-diffraction limit localization capabilities; its high sensitivity to small shifts of the illumination center of gravity in the vicinity of the Airy disk’s center is indicated by the distinct, sharp phase variations in this area. The phase completes a 2π turn, approximately linearly dependent in the azimuthal coordinate inside the Airy disk.

In the radial direction, an approximately linear transition from a pure 2ω signal to a pure 4ω signal occurs inside the Airy disk region. (Deviations from this relation are attributed to alignment imperfections in the optical system). The reconstruction of the dot's location from the amplitude and phase signals leads to a 35nm mean localization accuracy. This experiment demonstrates the simplicity of our scheme, by which a good assessment of the [R,θ] location of a single particle inside the Airy disk region may be obtained in a straightforward way.

4. Conclusions

Paradigms and tools for the improvement of spatio-temporal precision of optical imaging system are constantly being introduced. These should preferably be applicable to a large variety of systems, imposing minimum restrictions on the sample’s preparation and the system’s modes of operation. In the scheme proposed here, sub-wavelength localization capabilities are generated by the aid of simple polarimetry techniques applied to a focused vector beam, enabling its implementation as an add-on module to the standard point scanning microscope, thus retaining its simplicity. Importantly, the system’s localization accuracy must not come at the expense of its temporal resolution as long as the transition between different polarization states is achieved at fast enough rates that don’t limit the microscope’s operation. Its integration may thus be designed to immensely improve fast rate imaging and tracking capabilities.

The implementation of the system described in this paper is designed to optimize its sub-diffraction limit localization accuracy. Yet the ability to rapidly control and adjust the illumination PSF of a point scanning microscope using CD provides a much more powerful tool that may be utilized for the extraction of otherwise inaccessible information about the sample. Future work will describe in details the different methods by which the combination of data collected from a series of illumination states of the system may be used for the fast rate acquisition of super-resolution images.

References and links

1.

M. J. Saxton and K. Jacobson, “Single-particle tracking: applications to membrane dynamics,” Annu. Rev. Biophys. Biomol. Struct. 26(1), 373–399 (1997). [CrossRef] [PubMed]

2.

M. Fernández-Suárez and A. Y. Ting, “Fluorescent probes for super-resolution imaging in living cells,” Nat. Rev. Mol. Cell Biol. 9(12), 929–943 (2008). [CrossRef] [PubMed]

3.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006). [CrossRef] [PubMed]

4.

G. B. Airy, “On the diffraction of an object-glass with circular aperture,” Trans. Cambridge Philos. Soc. 5, 283–291 (1835).

5.

N. Bobroff, “Position measurement with a resolution and noise-limited instrument,” Rev. Sci. Instrum. 57(6), 1152 –1157(1986). [CrossRef]

6.

R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82(5), 2775–2783 (2002). [CrossRef] [PubMed]

7.

J. B. Pawley, Handbook of Biological Confocal Microscopy, 3rd Ed. (Springer Science + Business Media, LLC, 2006).

8.

G. Y. Sirat, Patent application PCT/FR2011/000555.

9.

W. R. Hamilton, “Third supplement to an essay on the theory of systems of rays,” Trans. Royal Irish. Acad. 1–144 (1833).

10.

H. Llyold, “On the phenomena presented by light in its passage along the axes of biaxal crystals,” Phil. Mag. 1,112–120 and 207–210 (1833).

11.

J. C. Poggendorff, “Ueber die konische refraction,” Pogg. Ann. 124(11), 461–462 (1839).

12.

C. V. Raman, “'Conical refraction in biaxial crystals,” Nature 107(2702), 747–747 (1921). [CrossRef]

13.

A. M. Belsky and A. P. Khapalyuk, “Internal conical refraction of limited light-beams in 2-axes crystals,” Opt. Spectrosc. 44, 746–751 (1978).

14.

M. V. Berry, “Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A, Pure Appl. Opt. 6(4), 289–300 (2004). [CrossRef]

15.

M. V. Berry, M. R. Jeffrey, and J. G. Lunney, “Conical diffraction: observations and theory,” Proc. R. Soc. A. 462(2070), 1629–1642 (2006). [CrossRef]

16.

M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamilton's diabolical point at the heart of crystal optics,” Prog. Optics 50, 13–50 (2007). [CrossRef]

17.

B. Boulanger and J. Zyss, Physical Properties of Crystals. Vol. D of International Tables for Crystallography, A. Authier, Ed. (Kluwer, Dordrecht, 1997).

18.

G. Y. Sirat, Patent application US 2009/0168613 A1.

19.

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96(1-3), 108–140 (1980). [CrossRef]

20.

S. Quabis, R. Dorn, M. Eberlerm, O. Glockl, and G. Leuchs, “The focus of light: theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72(1), 109–113 (2001). [CrossRef]

OCIS Codes
(110.0110) Imaging systems : Imaging systems
(260.1180) Physical optics : Crystal optics
(260.1960) Physical optics : Diffraction theory
(220.2945) Optical design and fabrication : Illumination design

ToC Category:
Imaging Systems

History
Original Manuscript: January 2, 2013
Revised Manuscript: March 1, 2013
Manuscript Accepted: March 1, 2013
Published: April 16, 2013

Virtual Issues
Vol. 8, Iss. 5 Virtual Journal for Biomedical Optics

Citation
Shani Rosen, Gabriel Y. Sirat, Har’el Ilan, and Aharon J. Agranat, "A sub wavelength localization scheme in optical imaging using conical diffraction," Opt. Express 21, 10133-10138 (2013)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-8-10133


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References

  1. M. J. Saxton and K. Jacobson, “Single-particle tracking: applications to membrane dynamics,” Annu. Rev. Biophys. Biomol. Struct.26(1), 373–399 (1997). [CrossRef] [PubMed]
  2. M. Fernández-Suárez and A. Y. Ting, “Fluorescent probes for super-resolution imaging in living cells,” Nat. Rev. Mol. Cell Biol.9(12), 929–943 (2008). [CrossRef] [PubMed]
  3. E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science313(5793), 1642–1645 (2006). [CrossRef] [PubMed]
  4. G. B. Airy, “On the diffraction of an object-glass with circular aperture,” Trans. Cambridge Philos. Soc.5, 283–291 (1835).
  5. N. Bobroff, “Position measurement with a resolution and noise-limited instrument,” Rev. Sci. Instrum.57(6), 1152 –1157(1986). [CrossRef]
  6. R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J.82(5), 2775–2783 (2002). [CrossRef] [PubMed]
  7. J. B. Pawley, Handbook of Biological Confocal Microscopy, 3rd Ed. (Springer Science + Business Media, LLC, 2006).
  8. G. Y. Sirat, Patent application PCT/FR2011/000555.
  9. W. R. Hamilton, “Third supplement to an essay on the theory of systems of rays,” Trans. Royal Irish. Acad.1–144 (1833).
  10. H. Llyold, “On the phenomena presented by light in its passage along the axes of biaxal crystals,” Phil. Mag. 1,112–120 and 207–210 (1833).
  11. J. C. Poggendorff, “Ueber die konische refraction,” Pogg. Ann.124(11), 461–462 (1839).
  12. C. V. Raman, “'Conical refraction in biaxial crystals,” Nature107(2702), 747–747 (1921). [CrossRef]
  13. A. M. Belsky and A. P. Khapalyuk, “Internal conical refraction of limited light-beams in 2-axes crystals,” Opt. Spectrosc.44, 746–751 (1978).
  14. M. V. Berry, “Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A, Pure Appl. Opt.6(4), 289–300 (2004). [CrossRef]
  15. M. V. Berry, M. R. Jeffrey, and J. G. Lunney, “Conical diffraction: observations and theory,” Proc. R. Soc. A. 462(2070), 1629–1642 (2006). [CrossRef]
  16. M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamilton's diabolical point at the heart of crystal optics,” Prog. Optics50, 13–50 (2007). [CrossRef]
  17. B. Boulanger and J. Zyss, Physical Properties of Crystals. Vol. D of International Tables for Crystallography, A. Authier, Ed. (Kluwer, Dordrecht, 1997).
  18. G. Y. Sirat, Patent application US 2009/0168613 A1.
  19. P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci.96(1-3), 108–140 (1980). [CrossRef]
  20. S. Quabis, R. Dorn, M. Eberlerm, O. Glockl, and G. Leuchs, “The focus of light: theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B72(1), 109–113 (2001). [CrossRef]

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