Abstract

Conical diffraction is observed in biaxial materials when a beam of light is directed along one of the two optic axis directions. When the beam is directed close to but not along an optic axis, a rich interference pattern is observed beyond the material. We observe some of the previously predicted low intensity interference patterns, representing a qualitatively new optical phenomenon in biaxial materials.

© 2015 Optical Society of America

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References

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  1. W. R. Hamilton, “Third supplement to an essay on the theory of systems of rays,” Trans. R. Irish Acad. 17, 137–139 (1837).
  2. A. M. Belsky and A. P. Khapalyuk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc. 44, 746–751 (1978).
  3. M. V. Berry, “Conical diffraction asymptotics: fine structure of the Poggendorff rings and axial spike,” J. Opt. A, Pure Appl. Opt. 6(4), 289–300 (2004).
    [Crossref]
  4. M. V. Berry, M. R. Jeffrey, and J. G. Lunney, “Conical diffraction: observations and theory,” Proc. Royal Soc. A 462, 1629–1642 (2006).
    [Crossref]
  5. R. T. Darcy, D. McCloskey, K. E. Ballantine, J. G. Lunney, P. R. Eastham, and J. F. Donegan, “Conical diffraction intensity profiles generated using a top-hat input beam,” Opt. Express 22(9), 11290–11300 (2014).
    [Crossref] [PubMed]
  6. A. Turpin, Y. V. Loiko, T. K. Kalkandjiev, H. Tomizawa, and J. Mompart, “Wave-vector and polarization dependence of conical refraction”, Opt. Express 21(4), 4503–4511 (2013).
    [Crossref] [PubMed]
  7. V. Peet and D. Zolotukhin, “Free-space evolution of focused Gaussian beams transformed by conical diffraction in a biaxial crystal”, Opt. Commun. 283(15), 3011–3016 (2010).
    [Crossref]
  8. M. V. Berry and M. R. Jeffrey, “Hamilton’s diabolical point at the heart of crystal optics,” Prog. Opt. 50, 13–50 (2007).
    [Crossref]
  9. K. F. Warnick and D. V. Arnold, “Secondary dark rings of internal conical refraction,” Phys. Rev. E 55, 6092–6096 (1997).
  10. C. V. Raman, V. S. Rajagopalan, and T. M. K. Nedungadi, “Conical refraction in naphthalene crystals,” Nature 147, 268 (1941).
    [Crossref]
  11. T.A. King, W. Hogervorst, N.S. Kazak, N.A. Khilo, and A.A. Ryzhevich, “Formation of higher-order Bessel light beams in biaxial crystals,” Opt. Comm. 187, 407–414 (2001).
    [Crossref]
  12. D. P. O’Dwyer, C. F. Phelan, Y. P. Rakovich, P. R. Eastham, J. G. Lunney, and J. F. Donegan, “The creation and annihilation of optical vortices using cascade conical diffraction,” Opt. Express 19(3), 2580–2588 (2011).
    [Crossref]
  13. A. Abdolvand, K. G. Wilcox, T. K. Kalkandjiev, and E. U. Rafailov, “Conical refraction Nd:KGd(WO4)2 laser, Opt. Express 18(3), 2753–2759 (2010).
    [Crossref] [PubMed]
  14. M. V. Berry and M. R. Jeffrey, “Conical diffraction complexified: dichroism and the transition to double refraction,” J. Opt. A, Pure Appl. Opt. 8(12), 1043–1051 (2006).
    [Crossref]
  15. M. R. Jeffrey, Conical Diffraction: Complexifying Hamilton’s Diabolical Legacy (University of Bristol, 2007).
  16. M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguiló, and F. Díaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68, 187–197 (1999).
    [Crossref]
  17. R. T. Darcy, D. McCloskey, K. E. Ballantine, B. D. Jennings, J. G. Lunney, P. R. Eastham, and J. F. Donegan, “White light conical diffraction,” Opt. Express 21(17), 20394–20403 (2013).
    [Crossref] [PubMed]
  18. H. Lloyd, “On the phenomena presented by light in its passage along the axes of biaxial crystals,” Trans. R. Irish Acad. 17, 145–158 (1837).

2014 (1)

2013 (2)

2011 (1)

2010 (2)

A. Abdolvand, K. G. Wilcox, T. K. Kalkandjiev, and E. U. Rafailov, “Conical refraction Nd:KGd(WO4)2 laser, Opt. Express 18(3), 2753–2759 (2010).
[Crossref] [PubMed]

V. Peet and D. Zolotukhin, “Free-space evolution of focused Gaussian beams transformed by conical diffraction in a biaxial crystal”, Opt. Commun. 283(15), 3011–3016 (2010).
[Crossref]

2007 (1)

M. V. Berry and M. R. Jeffrey, “Hamilton’s diabolical point at the heart of crystal optics,” Prog. Opt. 50, 13–50 (2007).
[Crossref]

2006 (2)

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

M. V. Berry and M. R. Jeffrey, “Conical diffraction complexified: dichroism and the transition to double refraction,” J. Opt. A, Pure Appl. Opt. 8(12), 1043–1051 (2006).
[Crossref]

2004 (1)

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

2001 (1)

T.A. King, W. Hogervorst, N.S. Kazak, N.A. Khilo, and A.A. Ryzhevich, “Formation of higher-order Bessel light beams in biaxial crystals,” Opt. Comm. 187, 407–414 (2001).
[Crossref]

1999 (1)

M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguiló, and F. Díaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68, 187–197 (1999).
[Crossref]

1997 (1)

K. F. Warnick and D. V. Arnold, “Secondary dark rings of internal conical refraction,” Phys. Rev. E 55, 6092–6096 (1997).

1978 (1)

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

1941 (1)

C. V. Raman, V. S. Rajagopalan, and T. M. K. Nedungadi, “Conical refraction in naphthalene crystals,” Nature 147, 268 (1941).
[Crossref]

1837 (2)

W. R. Hamilton, “Third supplement to an essay on the theory of systems of rays,” Trans. R. Irish Acad. 17, 137–139 (1837).

H. Lloyd, “On the phenomena presented by light in its passage along the axes of biaxial crystals,” Trans. R. Irish Acad. 17, 145–158 (1837).

Abdolvand, A.

Aguiló, M.

M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguiló, and F. Díaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68, 187–197 (1999).
[Crossref]

Arnold, D. V.

K. F. Warnick and D. V. Arnold, “Secondary dark rings of internal conical refraction,” Phys. Rev. E 55, 6092–6096 (1997).

Ballantine, K. E.

Belsky, A. M.

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

Berry, M. V.

M. V. Berry and M. R. Jeffrey, “Hamilton’s diabolical point at the heart of crystal optics,” Prog. Opt. 50, 13–50 (2007).
[Crossref]

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

M. V. Berry and M. R. Jeffrey, “Conical diffraction complexified: dichroism and the transition to double refraction,” J. Opt. A, Pure Appl. Opt. 8(12), 1043–1051 (2006).
[Crossref]

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

Darcy, R. T.

Díaz, F.

M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguiló, and F. Díaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68, 187–197 (1999).
[Crossref]

Donegan, J. F.

Eastham, P. R.

Hamilton, W. R.

W. R. Hamilton, “Third supplement to an essay on the theory of systems of rays,” Trans. R. Irish Acad. 17, 137–139 (1837).

Hogervorst, W.

T.A. King, W. Hogervorst, N.S. Kazak, N.A. Khilo, and A.A. Ryzhevich, “Formation of higher-order Bessel light beams in biaxial crystals,” Opt. Comm. 187, 407–414 (2001).
[Crossref]

Jeffrey, M. R.

M. V. Berry and M. R. Jeffrey, “Hamilton’s diabolical point at the heart of crystal optics,” Prog. Opt. 50, 13–50 (2007).
[Crossref]

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

M. V. Berry and M. R. Jeffrey, “Conical diffraction complexified: dichroism and the transition to double refraction,” J. Opt. A, Pure Appl. Opt. 8(12), 1043–1051 (2006).
[Crossref]

M. R. Jeffrey, Conical Diffraction: Complexifying Hamilton’s Diabolical Legacy (University of Bristol, 2007).

Jennings, B. D.

Kalkandjiev, T. K.

Kazak, N.S.

T.A. King, W. Hogervorst, N.S. Kazak, N.A. Khilo, and A.A. Ryzhevich, “Formation of higher-order Bessel light beams in biaxial crystals,” Opt. Comm. 187, 407–414 (2001).
[Crossref]

Khapalyuk, A. P.

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

Khilo, N.A.

T.A. King, W. Hogervorst, N.S. Kazak, N.A. Khilo, and A.A. Ryzhevich, “Formation of higher-order Bessel light beams in biaxial crystals,” Opt. Comm. 187, 407–414 (2001).
[Crossref]

King, T.A.

T.A. King, W. Hogervorst, N.S. Kazak, N.A. Khilo, and A.A. Ryzhevich, “Formation of higher-order Bessel light beams in biaxial crystals,” Opt. Comm. 187, 407–414 (2001).
[Crossref]

Lloyd, H.

H. Lloyd, “On the phenomena presented by light in its passage along the axes of biaxial crystals,” Trans. R. Irish Acad. 17, 145–158 (1837).

Loiko, Y. V.

Lunney, J. G.

McCloskey, D.

Mompart, J.

Nedungadi, T. M. K.

C. V. Raman, V. S. Rajagopalan, and T. M. K. Nedungadi, “Conical refraction in naphthalene crystals,” Nature 147, 268 (1941).
[Crossref]

Nikolov, V.

M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguiló, and F. Díaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68, 187–197 (1999).
[Crossref]

O’Dwyer, D. P.

Peet, V.

V. Peet and D. Zolotukhin, “Free-space evolution of focused Gaussian beams transformed by conical diffraction in a biaxial crystal”, Opt. Commun. 283(15), 3011–3016 (2010).
[Crossref]

Phelan, C. F.

Pujol, M. C.

M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguiló, and F. Díaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68, 187–197 (1999).
[Crossref]

Rafailov, E. U.

Rajagopalan, V. S.

C. V. Raman, V. S. Rajagopalan, and T. M. K. Nedungadi, “Conical refraction in naphthalene crystals,” Nature 147, 268 (1941).
[Crossref]

Rakovich, Y. P.

Raman, C. V.

C. V. Raman, V. S. Rajagopalan, and T. M. K. Nedungadi, “Conical refraction in naphthalene crystals,” Nature 147, 268 (1941).
[Crossref]

Rico, M.

M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguiló, and F. Díaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68, 187–197 (1999).
[Crossref]

Ryzhevich, A.A.

T.A. King, W. Hogervorst, N.S. Kazak, N.A. Khilo, and A.A. Ryzhevich, “Formation of higher-order Bessel light beams in biaxial crystals,” Opt. Comm. 187, 407–414 (2001).
[Crossref]

Solans, X.

M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguiló, and F. Díaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68, 187–197 (1999).
[Crossref]

Solé, R.

M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguiló, and F. Díaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68, 187–197 (1999).
[Crossref]

Tomizawa, H.

Turpin, A.

Warnick, K. F.

K. F. Warnick and D. V. Arnold, “Secondary dark rings of internal conical refraction,” Phys. Rev. E 55, 6092–6096 (1997).

Wilcox, K. G.

Zaldo, C.

M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguiló, and F. Díaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68, 187–197 (1999).
[Crossref]

Zolotukhin, D.

V. Peet and D. Zolotukhin, “Free-space evolution of focused Gaussian beams transformed by conical diffraction in a biaxial crystal”, Opt. Commun. 283(15), 3011–3016 (2010).
[Crossref]

Appl. Phys. B (1)

M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguiló, and F. Díaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68, 187–197 (1999).
[Crossref]

J. Opt. A, Pure Appl. Opt. (2)

M. V. Berry and M. R. Jeffrey, “Conical diffraction complexified: dichroism and the transition to double refraction,” J. Opt. A, Pure Appl. Opt. 8(12), 1043–1051 (2006).
[Crossref]

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

Nature (1)

C. V. Raman, V. S. Rajagopalan, and T. M. K. Nedungadi, “Conical refraction in naphthalene crystals,” Nature 147, 268 (1941).
[Crossref]

Opt. Comm. (1)

T.A. King, W. Hogervorst, N.S. Kazak, N.A. Khilo, and A.A. Ryzhevich, “Formation of higher-order Bessel light beams in biaxial crystals,” Opt. Comm. 187, 407–414 (2001).
[Crossref]

Opt. Commun. (1)

V. Peet and D. Zolotukhin, “Free-space evolution of focused Gaussian beams transformed by conical diffraction in a biaxial crystal”, Opt. Commun. 283(15), 3011–3016 (2010).
[Crossref]

Opt. Express (5)

Opt. Spectrosc. (1)

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

Phys. Rev. E (1)

K. F. Warnick and D. V. Arnold, “Secondary dark rings of internal conical refraction,” Phys. Rev. E 55, 6092–6096 (1997).

Proc. Royal Soc. A (1)

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

Prog. Opt. (1)

M. V. Berry and M. R. Jeffrey, “Hamilton’s diabolical point at the heart of crystal optics,” Prog. Opt. 50, 13–50 (2007).
[Crossref]

Trans. R. Irish Acad. (2)

W. R. Hamilton, “Third supplement to an essay on the theory of systems of rays,” Trans. R. Irish Acad. 17, 137–139 (1837).

H. Lloyd, “On the phenomena presented by light in its passage along the axes of biaxial crystals,” Trans. R. Irish Acad. 17, 145–158 (1837).

Other (1)

M. R. Jeffrey, Conical Diffraction: Complexifying Hamilton’s Diabolical Legacy (University of Bristol, 2007).

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Figures (6)

Fig. 1
Fig. 1 A two-dimensional slice of a light beam undergoing conical diffraction inside a biaxial material of length l.
Fig. 2
Fig. 2 Averaged radial intensity profile (blue dots) taken from an experimental image (inset) compared to theoretical intensity plots for ρ0 = 36 (solid black line), ρ0 = 46 (dashed black line), and ρ0 = 56 (solid red line).
Fig. 3
Fig. 3 The experimental apparatus used to observe low intensity features in the conically diffracted pattern when u ≠ 0. The distances z1 and z2 could be varied to achieve different magnifications. In the diagram, the FIP (ζ = 0) is being imaged, but any ζ value could be selected by moving lens l1 using the translation stage.
Fig. 4
Fig. 4 Comparison of (a) theoretical and (b) experimental logarithmic intensity plots with ρ0 = 56.2, u = 3.7, and ζ = 14.2. The dashed circles are centred about ρ = {0,0} with radius ρ0. The solid circles are centred about ρc with radius Rc as given by Eq. (18).
Fig. 5
Fig. 5 Comparison of (a) theoretical, and (b) experimental logarithmic intensity plots with ρ0 = 56.2,u = 1.8, and ζ = 31. The white arcs in the upper part of the images show the remnants of the Warnick-Arnold rings, visible as faint intensity maxima at the locations shown. The white dots in the lower part of the images are the predicted locations of intensity minima from Eq. (19) for 0 ≤ n ≤ 15, which occur along an arc of the circle given by Eq. (17).
Fig. 6
Fig. 6 Comparison of (a) theoretical, and (b) experimental logarithmic intensity plots with ρ0 = 56.2,u = 1.8, and ζ = 42. The white dots in the lower part of the images are the predicted locations of intensity minima from Eq. (19) for 0 n ≤ 4, which occur along an arc of the circle in Eq. (17).

Equations (19)

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A = 1 2 arctan [ n 2 2 ( n 1 2 n 2 2 ) ( n 2 2 n 3 3 ) ] .
R 0 A l
ρ 0 = A l w = R 0 w , ρ = R w .
ζ l + ( z 1 ) n 2 n 2 k 0 w 2 .
D ( B 0 + B 1 cos ϕ ˜ B 1 sin ϕ ˜ B 1 sin ϕ ˜ B 0 B 1 cos ϕ ˜ ) ,
B 0 ( ρ ˜ , ζ ˜ , ρ 0 ) = 0 d κ ˜ κ ˜ exp { 1 2 i ζ ˜ κ ˜ 2 } J 0 ( κ ˜ ρ ˜ ) cos ( κ ˜ ρ 0 ) ,
B 1 ( ρ ˜ , ζ ˜ , ρ 0 ) = 0 d κ ˜ κ ˜ exp { 1 2 i ζ ˜ κ ˜ 2 } J 1 ( κ ˜ ρ ˜ ) sin ( κ ˜ ρ 0 ) ,
ρ ˜ = { ξ + i u , η } , ζ ˜ = ζ 1 ,
cos ϕ ˜ = ξ + i u ρ ˜ , sin ϕ ˜ = η ρ ˜ ,
ρ ˜ = ( ξ + i u ) 2 + η 2 .
A + B 0 + B 1 , A B 0 B 1 .
J v ( x ) 2 π x cos ( x π 4 v π 2 )
A ± 2 π ρ ˜ 0 d κ ˜ κ ˜ exp { 1 2 i ζ ˜ κ ˜ 2 } cos { κ ˜ ( ρ ˜ ρ 0 ) π 4 } .
A ± μ ± ζ ˜ ( ρ 0 ρ ˜ ρ ˜ ) 1 / 2 exp { i 2 ζ ˜ ( ρ ˜ ρ 0 ) 2 } , { μ + = sgn [ Im ( ρ 0 ρ ˜ 1 + i ζ ) ] μ = i
| cos ϕ ˜ | 2 + | sin ϕ ˜ | 2 2 u 2 / | ρ ˜ 2 |
exp { 2 i ρ 0 ρ ˜ ζ ˜ } = i , Im ρ ˜ ζ ˜ = 0.
( ρ ρ c ) ( ρ ρ c ) = R c 2 ,
ρ c = { u ( ζ 2 1 ) 2 ζ , 0 } , R c = u ( ζ 2 + 1 ) 2 ζ .
Re ρ ˜ ζ ˜ = ( n + 1 4 ) π ρ 0 , n = 1 , 0 , 1 ,

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