Abstract

We experimentally address the wave-vector and polarization dependence of the internal conical refraction phenomenon by demonstrating that an input light beam of elliptical transverse profile refracts into two beams after passing along one of the optic axes of a biaxial crystal, i.e. it exhibits double refraction instead of refracting conically. Such double refraction is investigated by the independent rotation of a linear polarizer and a cylindrical lens. Expressions to describe the position and the intensity pattern of the refracted beams are presented and applied to predict the intensity pattern for an axicon beam propagating along the optic axis of a biaxial crystal.

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  1. J. G. O’Hara, “The prediction and discovery of conical refraction by William Rowan Hamilton and Humphrey Lloyd,” Proc. Roy. Ir. Acad.82 (2), 231–257 (1982).
  2. M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamiltons diabolical point at the heart of crystal optics,” Prog. Opt.50, 13–50 (2007).
    [CrossRef]
  3. A. Abdolvand, K.G. Wilcox, T. K. Kalkandjiev, and Edik U. Rafailov, “Conical refraction Nd : KGd(WO4)2 laser,” Opt. Express18, 2753–2759 (2010).
    [CrossRef] [PubMed]
  4. D. P. O’Dwyer, K. E. Ballantine, C. F. Phelan, J. G. Lunney, and J. F. Donegan, “Optical trapping using cascade conical refraction of light,” Opt. Express20, 21119–21125 (2012).
    [CrossRef]
  5. A. Turpin, Yu. V. Loiko, T. K. Kalkandjiev, and J. Mompart, “Free-space optical polarization demultiplexing and multiplexing by means of conical refraction,” Opt. Lett.37, 4197–4199 (2012).
    [CrossRef] [PubMed]
  6. D. L. Portigal and E. Burstein, “Internal Conical Refraction,” J. Opt. Soc. Am.59, 1567–1573 (1969).
    [CrossRef]
  7. E. Lalor, “An Analytical Approach to the Theory of Internal Conical Refraction,” J. Math. Phys.13, 449–454 (1972).
    [CrossRef]
  8. A. J. Schell and N. Bloembergen, “Laser studies of internal conical diffraction. I. Quantitative comparison of experimental and theoretical conical intensity distribution in aragonite,” J. Opt. Soc. Am.68, 1093–1098 (1978).
    [CrossRef]
  9. A. M. Belskii and A. P. Khapalyuk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc.44, 436–439 (1978).
  10. A. M. Belskii and A. P. Khapalyuk, “Propagation of confined light beams along the beam axes (axes of single ray velocity) of biaxial crystals,” Opt. Spectrosc.44, 312–315 (1978).
  11. A. M. Belsky and M. A. Stepanov, “Internal conical refraction of light beams in biaxial gyrotropic crystals,” Opt. Commun.204, 1–6 (2002).
    [CrossRef]
  12. J. P. Fève, B. Boulanger, and G. Marnier, “Experimental study of internal and external conical refractions in KTP,” Optics Commun.105, 243–252 (1994).
    [CrossRef]
  13. M.V. Berry, “Conical diffraction from an N-crystal cascade,” J. Opt.12, 075704 (2010).
    [CrossRef]
  14. A. Abdolvand, “Conical diffraction from a multi-crystal cascade: experimental observations,” Appl. Phys B103, 281–283 (2011).
    [CrossRef]
  15. Y. Loiko, M. A. Bursukova, T. K. Kalkanjiev, E. U. Rafailov, and J. Mompart, “Fermionic transformation rules for spatially filtered light beams in conical refraction,” in Complex Light and Optical Forces V, D. L. Andrews, E. J. Galvez, and J. Glückstad, eds., Proc. SPIE7950, 79500D (2011).
    [CrossRef]
  16. T. K. Kalkandjiev and M. A. Bursukova, “Conical refraction: an experimental introduction,” in Photon Management III, J. T. Sheridan and F. Wyrowski, eds., Proc. SPIE6994, 69940B (2008).
    [CrossRef]

2012 (2)

2011 (1)

A. Abdolvand, “Conical diffraction from a multi-crystal cascade: experimental observations,” Appl. Phys B103, 281–283 (2011).
[CrossRef]

2010 (2)

2007 (1)

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

2002 (1)

A. M. Belsky and M. A. Stepanov, “Internal conical refraction of light beams in biaxial gyrotropic crystals,” Opt. Commun.204, 1–6 (2002).
[CrossRef]

1994 (1)

J. P. Fève, B. Boulanger, and G. Marnier, “Experimental study of internal and external conical refractions in KTP,” Optics Commun.105, 243–252 (1994).
[CrossRef]

1982 (1)

J. G. O’Hara, “The prediction and discovery of conical refraction by William Rowan Hamilton and Humphrey Lloyd,” Proc. Roy. Ir. Acad.82 (2), 231–257 (1982).

1978 (3)

A. J. Schell and N. Bloembergen, “Laser studies of internal conical diffraction. I. Quantitative comparison of experimental and theoretical conical intensity distribution in aragonite,” J. Opt. Soc. Am.68, 1093–1098 (1978).
[CrossRef]

A. M. Belskii and A. P. Khapalyuk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc.44, 436–439 (1978).

A. M. Belskii and A. P. Khapalyuk, “Propagation of confined light beams along the beam axes (axes of single ray velocity) of biaxial crystals,” Opt. Spectrosc.44, 312–315 (1978).

1972 (1)

E. Lalor, “An Analytical Approach to the Theory of Internal Conical Refraction,” J. Math. Phys.13, 449–454 (1972).
[CrossRef]

1969 (1)

Abdolvand, A.

A. Abdolvand, “Conical diffraction from a multi-crystal cascade: experimental observations,” Appl. Phys B103, 281–283 (2011).
[CrossRef]

A. Abdolvand, K.G. Wilcox, T. K. Kalkandjiev, and Edik U. Rafailov, “Conical refraction Nd : KGd(WO4)2 laser,” Opt. Express18, 2753–2759 (2010).
[CrossRef] [PubMed]

Ballantine, K. E.

Belskii, A. M.

A. M. Belskii and A. P. Khapalyuk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc.44, 436–439 (1978).

A. M. Belskii and A. P. Khapalyuk, “Propagation of confined light beams along the beam axes (axes of single ray velocity) of biaxial crystals,” Opt. Spectrosc.44, 312–315 (1978).

Belsky, A. M.

A. M. Belsky and M. A. Stepanov, “Internal conical refraction of light beams in biaxial gyrotropic crystals,” Opt. Commun.204, 1–6 (2002).
[CrossRef]

Berry, M. V.

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

Berry, M.V.

M.V. Berry, “Conical diffraction from an N-crystal cascade,” J. Opt.12, 075704 (2010).
[CrossRef]

Bloembergen, N.

Boulanger, B.

J. P. Fève, B. Boulanger, and G. Marnier, “Experimental study of internal and external conical refractions in KTP,” Optics Commun.105, 243–252 (1994).
[CrossRef]

Burstein, E.

Bursukova, M. A.

T. K. Kalkandjiev and M. A. Bursukova, “Conical refraction: an experimental introduction,” in Photon Management III, J. T. Sheridan and F. Wyrowski, eds., Proc. SPIE6994, 69940B (2008).
[CrossRef]

Y. Loiko, M. A. Bursukova, T. K. Kalkanjiev, E. U. Rafailov, and J. Mompart, “Fermionic transformation rules for spatially filtered light beams in conical refraction,” in Complex Light and Optical Forces V, D. L. Andrews, E. J. Galvez, and J. Glückstad, eds., Proc. SPIE7950, 79500D (2011).
[CrossRef]

Donegan, J. F.

Fève, J. P.

J. P. Fève, B. Boulanger, and G. Marnier, “Experimental study of internal and external conical refractions in KTP,” Optics Commun.105, 243–252 (1994).
[CrossRef]

Jeffrey, M. R.

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

Kalkandjiev, T. K.

Kalkanjiev, T. K.

Y. Loiko, M. A. Bursukova, T. K. Kalkanjiev, E. U. Rafailov, and J. Mompart, “Fermionic transformation rules for spatially filtered light beams in conical refraction,” in Complex Light and Optical Forces V, D. L. Andrews, E. J. Galvez, and J. Glückstad, eds., Proc. SPIE7950, 79500D (2011).
[CrossRef]

Khapalyuk, A. P.

A. M. Belskii and A. P. Khapalyuk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc.44, 436–439 (1978).

A. M. Belskii and A. P. Khapalyuk, “Propagation of confined light beams along the beam axes (axes of single ray velocity) of biaxial crystals,” Opt. Spectrosc.44, 312–315 (1978).

Lalor, E.

E. Lalor, “An Analytical Approach to the Theory of Internal Conical Refraction,” J. Math. Phys.13, 449–454 (1972).
[CrossRef]

Loiko, Y.

Y. Loiko, M. A. Bursukova, T. K. Kalkanjiev, E. U. Rafailov, and J. Mompart, “Fermionic transformation rules for spatially filtered light beams in conical refraction,” in Complex Light and Optical Forces V, D. L. Andrews, E. J. Galvez, and J. Glückstad, eds., Proc. SPIE7950, 79500D (2011).
[CrossRef]

Loiko, Yu. V.

Lunney, J. G.

Marnier, G.

J. P. Fève, B. Boulanger, and G. Marnier, “Experimental study of internal and external conical refractions in KTP,” Optics Commun.105, 243–252 (1994).
[CrossRef]

Mompart, J.

A. Turpin, Yu. V. Loiko, T. K. Kalkandjiev, and J. Mompart, “Free-space optical polarization demultiplexing and multiplexing by means of conical refraction,” Opt. Lett.37, 4197–4199 (2012).
[CrossRef] [PubMed]

Y. Loiko, M. A. Bursukova, T. K. Kalkanjiev, E. U. Rafailov, and J. Mompart, “Fermionic transformation rules for spatially filtered light beams in conical refraction,” in Complex Light and Optical Forces V, D. L. Andrews, E. J. Galvez, and J. Glückstad, eds., Proc. SPIE7950, 79500D (2011).
[CrossRef]

O’Dwyer, D. P.

O’Hara, J. G.

J. G. O’Hara, “The prediction and discovery of conical refraction by William Rowan Hamilton and Humphrey Lloyd,” Proc. Roy. Ir. Acad.82 (2), 231–257 (1982).

Phelan, C. F.

Portigal, D. L.

Rafailov, E. U.

Y. Loiko, M. A. Bursukova, T. K. Kalkanjiev, E. U. Rafailov, and J. Mompart, “Fermionic transformation rules for spatially filtered light beams in conical refraction,” in Complex Light and Optical Forces V, D. L. Andrews, E. J. Galvez, and J. Glückstad, eds., Proc. SPIE7950, 79500D (2011).
[CrossRef]

Rafailov, Edik U.

Schell, A. J.

Stepanov, M. A.

A. M. Belsky and M. A. Stepanov, “Internal conical refraction of light beams in biaxial gyrotropic crystals,” Opt. Commun.204, 1–6 (2002).
[CrossRef]

Turpin, A.

Wilcox, K.G.

Appl. Phys B (1)

A. Abdolvand, “Conical diffraction from a multi-crystal cascade: experimental observations,” Appl. Phys B103, 281–283 (2011).
[CrossRef]

J. Math. Phys. (1)

E. Lalor, “An Analytical Approach to the Theory of Internal Conical Refraction,” J. Math. Phys.13, 449–454 (1972).
[CrossRef]

J. Opt. (1)

M.V. Berry, “Conical diffraction from an N-crystal cascade,” J. Opt.12, 075704 (2010).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Commun. (1)

A. M. Belsky and M. A. Stepanov, “Internal conical refraction of light beams in biaxial gyrotropic crystals,” Opt. Commun.204, 1–6 (2002).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Opt. Spectrosc. (2)

A. M. Belskii and A. P. Khapalyuk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc.44, 436–439 (1978).

A. M. Belskii and A. P. Khapalyuk, “Propagation of confined light beams along the beam axes (axes of single ray velocity) of biaxial crystals,” Opt. Spectrosc.44, 312–315 (1978).

Optics Commun. (1)

J. P. Fève, B. Boulanger, and G. Marnier, “Experimental study of internal and external conical refractions in KTP,” Optics Commun.105, 243–252 (1994).
[CrossRef]

Proc. Roy. Ir. Acad. (1)

J. G. O’Hara, “The prediction and discovery of conical refraction by William Rowan Hamilton and Humphrey Lloyd,” Proc. Roy. Ir. Acad.82 (2), 231–257 (1982).

Prog. Opt. (1)

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

Other (2)

Y. Loiko, M. A. Bursukova, T. K. Kalkanjiev, E. U. Rafailov, and J. Mompart, “Fermionic transformation rules for spatially filtered light beams in conical refraction,” in Complex Light and Optical Forces V, D. L. Andrews, E. J. Galvez, and J. Glückstad, eds., Proc. SPIE7950, 79500D (2011).
[CrossRef]

T. K. Kalkandjiev and M. A. Bursukova, “Conical refraction: an experimental introduction,” in Photon Management III, J. T. Sheridan and F. Wyrowski, eds., Proc. SPIE6994, 69940B (2008).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Experimental set-up of light refraction along the optic axis of a biaxial crystal (BC): the input beam is focused by a lens and passes through the BC. The refraction pattern is obtained at the Lloyd plane of the system, which coincides with the focal plane. (a) An input Gaussian beam focused with a spherical lens (focal length of 150mm) yields the well-known ring of CR (c) at the Lloyd (focal) plane of the system. (b) An elliptical beam (EB) beam, obtained with a cylindrical lens (focal length of 150mm) that focuses in the horizontal direction (see Fig. 2 for details), yields the the double refraction pattern shown in figure (d).

Fig. 2
Fig. 2

(a) An elliptical beam with linear polarization is generated when a circularly polarized collimated Gaussian beam passes through the linear polarizer (with polarization plane given by azimuthal angle ϕE) and is focused by a cylindrical lens (focal length of 150mm), which determines the wave-vectors plane (given by azimuthal angle ϕK). The resulting patterns are captured by a CCD camera at the Lloyd plane behind the BC. The orientation of the crystal is characterized by the orientation of the plane of its optic axes (given by azimuthal angle ϕG). (b) Elliptical beam at the focal plane of the lens when the BC is removed. The beam is parameterized by the azimuthal angles ϕE and ϕK related to the polarization and wave-vector planes, respectively. All angles are measured from the horizontal x-axis of the laboratory system of coordinates.

Fig. 3
Fig. 3

Transverse intensity patterns obtained after rotating (a) the polarizer, i.e. varying ϕE, or (b) cylindrical lens, i.e. varying ϕK. ϕ means ϕE in (a) and ϕK in (b) and it is varied in the range [0°, 157.5°] with steps of 22.5°, while ϕG = 0°. Yellow (a) and green (b) lines at the bottom right corner indicate the polarization E-plane (a) and the wave-vector K-plane (b), respectively. (c) Splitting of an input EB beam (dashed ellipse), where R± denote the position of the two output refracted beams at the Lloyd plane. Dashed ring in (c) denotes the otherwise expected CR in case of input beam of Gaussian profile.

Fig. 4
Fig. 4

Normalized intensities of the two refracted beams after the splitting of the elliptical beam as a function of the variation of the parameters (a) ϕG/2E (ϕK = 0°) or (b) ϕG +ϕK (ϕE = 0°). Black solid (I+) and red dashed (I) curves show the analytical fitting given by Eqs.(5), while symbols represent the corresponding experimental data. The error in angle measurements is ±0.5°.

Fig. 5
Fig. 5

Experimental set-up for axicon beam propagation along the optic axis of a biaxial crystal. The axicon lens (apex angle of 179.5°) generates a conical beam from an input linearly polarized Gaussian beam which is then focused by a spherical lens (focal length of 150mm) along the optic axis of the BC.

Fig. 6
Fig. 6

Refraction of linearly polarized axicon annular beam along optic axis of a biaxial crystal. (a) Schematic representation of linearly polarized axicon annular beam with each point characterized by the azimuthal angles ϕk = ϕ (short green lines) and all of them have the same polarization plane ϕE (short orange lines). (b) Experimentally observed transverse intensity pattern at the Lloyd plane consisting of two concentric rings when a linearly polarized axicon beam with ϕE = 0 propagates along the optic axis of the biaxial crystal with ϕG = 0. (c) Corresponding theoretical simulation from Eqs. (6) and (7).

Equations (7)

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ϕ + = ϕ K , ϕ = ϕ K + π .
R ± ( ϕ K ) = S ( ϕ K ) + G ± R 0 u ( ϕ K ) ,
I + { ϕ E , ϕ G , ϕ K = 0 } = I 0 cos 2 ( ϕ G 2 ϕ E ) ; I { ϕ E , ϕ G , ϕ K = 0 } = I 0 sin 2 ( ϕ G 2 ϕ E ) ,
I + { ϕ E = 0 , ϕ G , ϕ K } = I 0 cos 2 ( ϕ G 2 + ϕ K 2 ) ; I { ϕ E = 0 , ϕ G , ϕ K } = I 0 sin 2 ( ϕ G 2 + ϕ K 2 ) .
I + = I 0 cos 2 ( ω 2 ) , I = I 0 sin 2 ( ω 2 ) , ω = ϕ G ϕ χ , ϕ χ = 2 ϕ E ϕ K .
R ± ( ϕ ) = G + ( | R a x | ± | R 0 | ) u ( ϕ ) .
I + = I 0 R 0 R + cos 2 ( ϕ ϕ 0 2 ) , I = I 0 R 0 R sin 2 ( ϕ ϕ 0 2 ) ,

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