## Wave-vector and polarization dependence of conical refraction |

Optics Express, Vol. 21, Issue 4, pp. 4503-4511 (2013)

http://dx.doi.org/10.1364/OE.21.004503

Acrobat PDF (1687 KB)

### 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.

© 2013 OSA

## 1. Introduction

*i.e.*decomposition of an input beam into ordinary (o) and extraordinary (e) beams with orthogonal linear polarizations. The extraordinary beam is laterally shifted with respect to the ordinary one, being this shift equal to zero when the input beam propagates along the optic axis. The intensity distribution between the o- and e- beams, expressed by the Malus law, depends only on one parameter: the relative orientation of the crystal with respect to the polarization plane of the input beam.

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*(*WO*_{4})_{2} laser,” Opt. Express **18**, 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. Express **20**, 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]

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]

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]

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 B **103**, 281–283 (2011). [CrossRef]

*et al.*[15

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. SPIE*7950, 79500D (2011). [CrossRef]

## 2. Conical refraction of spatially anisotropic beams

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]

**50**, 13–50 (2007). [CrossRef]

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

*ϕ*refract to a plane with the same azimuthal angle at the Lloyd plane,

*i.e.*into two opposite points on the CR ring.

## 3. Transformation rules of conical refraction

*w*= 1mm is obtained from a 640nm diode laser coupled to a monomode fiber. Then a linear polarizer is introduced to fix the polarization plane in a well defined direction. The resulting linearly polarized beam is focused by a cylindrical lens of 150mm focal length with its flat face oriented strictly perpendicular to the beam propagation direction. The cylindrical lens only focuses the Gaussian beam in one direction, so that it transforms its transverse circular shape to an elliptical one with a ratio 3/100 of the semi-axes of the ellipse. As a consequence, the divergence of the generated EB is different along the focused and unfocused directions (

*w*= 30

_{f}*μ*m,

*θ*= 6.8mrad;

_{f}*w*= 1000

_{uf}*μ*m,

*θ*= 0.2mrad, where

_{uf}*f*and

*uf*subscripts refer to

*focused*and

*unfocused*directions, respectively). The EB is characterized by its polarization plane, represented by the azimuthal angle

*ϕ*, and by its plane of wave-vectors (or

_{E}*K*-plane), represented by azimuthal angle

*ϕ*; see Fig. 2(b). Different EBs are obtained by rotating either the cylindrical lens or the linear polarizer. The BC is 28mm long and it was cut from a monoclinic centrosymmetric KGd(WO

_{K}_{4})

_{2}crystal. Its polished entrance surfaces (cross-section 6×4mm

^{2}, parallelism 10 arc seconds) are perpendicular to one of the optic axes (misalignment angle < 1.5mrad) and positioned strictly perpendicular to the beam propagation direction, so that the incoming beam to the BC passes along its optic axis. Both the orientation of the optic axis of the BC and the cylindrical lens are well controlled in the

*θ*and

*φ*directions in 3D spherical coordinates by a micrometer positioning system. The resulting pattern is captured by a CCD camera at the Lloyd (focal) plane behind the BC [16

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. SPIE*6994, 69940B (2008). [CrossRef]

### 3.1. Position of the refracted beams

**G**=

*R*

_{0}(cos

*ϕ*, sin

_{G}*ϕ*) that belongs to the plane of the crystal optic axes and is perpendicular to them [16

_{G}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. SPIE*6994, 69940B (2008). [CrossRef]

**G**| ≡

*R*

_{0}(in our set-up

*R*

_{0}= 476

*μ*m), given by a product of the length and the conical refraction semiangle of the BC (17mrad for KGd(WO

_{4})

_{2}) [16

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. SPIE*6994, 69940B (2008). [CrossRef]

*ϕ*(

_{E}*ϕ*= 0°) (a) or

_{K}*ϕ*(

_{K}*ϕ*= 0°) (b) from 0 to 157.5° in 22.5° intervals, when the crystal orientation remains fixed at

_{E}*ϕ*= 0°. The geometric center of the refracted beams coincides with the center of the otherwise expected CR ring, since it is shifted from the position of the initial EB by the vector

_{G}**G**. This shift is shown schematically in Fig. 3(c), and it is subtracted in Fig. 3(a) and Fig. 3(b).

*ϕ*, but rotates linearly with

_{E}*ϕ*. Moreover, the azimuthal angles

_{K}*ϕ*of the refracted beams are defined by the wave-vector plane,

_{±}*ϕ*, of the input EB, namely, Since

_{K}*ϕ*and

_{K}*ϕ*+

_{K}*π*describe the same wave-vector plane, the latter expressions mean conservation of the wave-vectors

*K*-planes. Rotation of the crystal,

*i.e.*change of

*ϕ*, does not affect the angles

_{G}*ϕ*, but it affects the position of the center of the refracted beams and redistributes the intensity between the refracted beams as it will be shown in the next subsection. Summarizing, in the

_{±}*xy*laboratory coordinates, the positions of the refracted beams

**R**

*at the Lloyd plane can be written as follows: where*

_{±}**u**(

*ϕ*) ≡ (cos(

_{K}*ϕ*), sin (

_{K}*ϕ*)).

_{K}**S**denotes the position at the Lloyd plane where the initial EB would be focused in the absence of the BC, see Fig. 3(c). In other words, the two refracted beams are located at diagonal positions of the otherwise expected CR ring. EBs obtained from the same Gaussian beam have the same initial position

**S**,

*i.e.*

**S**does not depend on

*ϕ*in this case. As a final comment, Eqs. (2) generalize the geometrical approach [2

_{E,K,G}**50**, 13–50 (2007). [CrossRef]

### 3.2. Relative intensity distribution of the refracted beams

*ϕ*,

_{G}*ϕ*and

_{E}*ϕ*, associated to the crystal orientation represented by

_{K}**G**and to the polarization and wave-vectors planes of the incident EBs. Below we show that only one combination of these angles governs the relative intensity distribution between the two refracted beams. With this purpose, we have repeated the experiments shown in Fig. 3(a) and Fig. 3(b) for different orientations of the BC. Symbols (black crosses and red circles) in Fig. 4(a) and Fig. 4(b) show the corresponding experimental results for the intensities

*I*of the two refracted beams normalized with respect to the intensity of the incident beam. Black solid and red dashed curves represent their analytical fittings given by the following expressions:

_{±}*I*

_{+}and

*I*

_{−}are the intensities of the beams refracted at angles

*ϕ*

_{+}and

*ϕ*

_{−}and located diagonally at the both ends of the CR ring at positions

*R*

_{+}and

*R*

_{−}respectively, following Eq. (2).

*ω*: From Eq. (5) it follows that, with respect to the relative energy distribution the only significant parameters of the input EB is

*ϕ*. Therefore, the intensity splitting under CR can be expressed in terms of the difference between the parameter

_{χ}*ϕ*and the orientation of the BC, given by

_{χ}*ϕ*. We would like to highlight here that the expression

_{G}*ϕ*= 2

_{χ}*ϕ*−

_{E}*ϕ*that we have obtained experimentally, corrects the theoretical result derived in [13

_{K}13. M.V. Berry, “Conical diffraction from an N-crystal cascade,” J. Opt. **12**, 075704 (2010). [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. SPIE*7950, 79500D (2011). [CrossRef]

*ϕ*=

_{χ}*ϕ*

**, where**

_{G}*ϕ*

**denotes the orientation of an initial biaxial crystal that produces a CR ring with lateral shift given by vector**

_{G}**G**

_{0}. Then, a pinhole placed at angle

*ϕ*on the CR ring produces a CR filtered beam with polarization

*ϕ*=

_{K}*ϕ*that also splits into two beams after passing along the optic axis of a biaxial crystal.

## 4. Application of the transformation rules of CR to an axicon beam

*θ*and

*φ*directions in 3D spherical coordinates by a micrometer positioning system. Each infinitesimally thin azimuthal sector of the axicon lens characterized by azimuthal angle

*ϕ*forms a thin prism that produces a wave with particular wave-vector whose transverse projection comprise an angle

*ϕ*=

_{k}*ϕ*. The axicon lens generates, therefore, a continuous collection of beams with

*ϕ*∈ [0, 2

*π*). After the axicon, the refracted beams, following Snell law, have the same inclination angle

*θ*

_{0}with respect to the

*z*-axis. At the focal plane of the lens they form a ring such that each point can be characterized by wave-vector plane,

*ϕ*, and polarization plane,

_{k}*ϕ*. In other words, each point of the axicon ring is an EB. In this case all these EB have their polarization plane fixed at

_{E}*ϕ*and their wave-vector plane

_{E}*ϕ*is varying continuously along the ring as shown in Fig. 6(a). Behind the BC, the refraction pattern can be calculated by applying Eq. (2) to every point of the input axicon annular beam taking into account the initial positions as given by

**S**(

*ϕ*) =

*R*(cos

_{ax}*ϕ*, sin

*ϕ*) (where

*R*is the radius of the axicon light ring). Therefore, from Eq. (2) one can obtain the refracted pattern for an axicon beam: These expressions, with

_{ax}*ϕ*scanned from 0 to 2

*π*, parameterize two concentric rings with radii

*R*±

_{ax}*R*

_{0}laterally shifted by

**G**relatively to the axicon ring axis. The intensity distribution is calculated from Eq. (5). All points of the incident axicon beam have the same intensity, which is distributed between the two refracted rings as follows: where we have taken

*ω*=

*ϕ*−

*ϕ*

_{0}being

*ϕ*

_{0}≡ 2

*ϕ*−

_{E}*ϕ*a constant parameter. In addition, since both rings have different radii, normalization factors

_{G}*R*

_{0}/

*R*have been introduced to

_{±}*I*to assure energy conservation. Fig. 6(a) shows the intensity pattern of the axicon beam with polarization and

_{±}*K*-plane distribution. The experimental refraction pattern behind the BC is shown in Fig. 6(b). The pattern is formed by two concentric rings oppositely polarized, with polarization distribution analogous to that one obtained in a cascaded CR configuration [16

*Photon Management III*, J. T. Sheridan and F. Wyrowski, eds., *Proc. SPIE*6994, 69940B (2008). [CrossRef]

## 5. Conclusions

## Acknowledgments

## References and links

1. | J. G. O’Hara, “The prediction and discovery of conical refraction by William Rowan Hamilton and Humphrey Lloyd,” Proc. Roy. Ir. Acad. |

2. | M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamiltons diabolical point at the heart of crystal optics,” Prog. Opt. |

3. | A. Abdolvand, K.G. Wilcox, T. K. Kalkandjiev, and Edik U. Rafailov, “Conical refraction |

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. Express |

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. |

6. | D. L. Portigal and E. Burstein, “Internal Conical Refraction,” J. Opt. Soc. Am. |

7. | E. Lalor, “An Analytical Approach to the Theory of Internal Conical Refraction,” J. Math. Phys. |

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. |

9. | A. M. Belskii and A. P. Khapalyuk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc. |

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. |

11. | A. M. Belsky and M. A. Stepanov, “Internal conical refraction of light beams in biaxial gyrotropic crystals,” Opt. Commun. |

12. | J. P. Fève, B. Boulanger, and G. Marnier, “Experimental study of internal and external conical refractions in KTP,” Optics Commun. |

13. | M.V. Berry, “Conical diffraction from an N-crystal cascade,” J. Opt. |

14. | A. Abdolvand, “Conical diffraction from a multi-crystal cascade: experimental observations,” Appl. Phys B |

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 |

16. | T. K. Kalkandjiev and M. A. Bursukova, “Conical refraction: an experimental introduction,” in |

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(260.1180) Physical optics : Crystal optics

(260.1440) Physical optics : Birefringence

**ToC Category:**

Physical Optics

**History**

Original Manuscript: December 21, 2012

Revised Manuscript: January 16, 2013

Manuscript Accepted: January 16, 2013

Published: February 13, 2013

**Citation**

A. Turpin, Yu. V. Loiko, T. K. Kalkandjiev, H. Tomizawa, and J. Mompart, "Wave-vector and polarization dependence of conical refraction," Opt. Express **21**, 4503-4511 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-4-4503

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### References

- 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).
- 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]
- 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]
- 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]
- 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]
- D. L. Portigal and E. Burstein, “Internal Conical Refraction,” J. Opt. Soc. Am.59, 1567–1573 (1969). [CrossRef]
- E. Lalor, “An Analytical Approach to the Theory of Internal Conical Refraction,” J. Math. Phys.13, 449–454 (1972). [CrossRef]
- 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).
- A. M. Belsky and M. A. Stepanov, “Internal conical refraction of light beams in biaxial gyrotropic crystals,” Opt. Commun.204, 1–6 (2002). [CrossRef]
- 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]
- M.V. Berry, “Conical diffraction from an N-crystal cascade,” J. Opt.12, 075704 (2010). [CrossRef]
- A. Abdolvand, “Conical diffraction from a multi-crystal cascade: experimental observations,” Appl. Phys B103, 281–283 (2011). [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]
- 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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