Abstract

Internal conical diffraction by biaxial crystals with aligned optic axes, known as cascade conical diffraction is investigated. Formulae giving the intensity distributions for a cascade conically diffracted Gaussian beam are shown to compare well with experiment for the cases of two biaxial crystals with the same and different lengths and with the second crystal rotated with respect to the first. The effects of placing half wave-plates between crystals are also investigated.

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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, 1–144 (1837).
  2. H. Lloyd, “On the phenomena presented by light in its passage along the axes of biaxial crystals,” Philos. Mag. 1, 112–120 (1833).
  3. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media 2nd ed. (Pergamon Press, 1985).
  4. M. Born and E. Wolf, Principles of Optics 7th ed. (Cambridge University Press, 1999).
  5. J. G. Lunney and D. W. Weaire, “The ins and outs of conical refraction,” Europhys. News 37(3), 26–29 (2006).
    [CrossRef]
  6. A. M. Belskii and A. P. Khapaluyk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc. 44, 436–439 (1978).
  7. M. V. Berry, “Conical diffraction asymptotics: Fine structure of Poggendorf rings and axial spike,” J. Opt. A, Pure Appl. Opt. 6(4), 289–300 (2004).
    [CrossRef]
  8. M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamilton’s diabolical point at the heart of crystal optics,” Prog. Opt. 50, 13–50 (2007).
    [CrossRef]
  9. M. V. Berry, M. R. Jeffrey, and J. G. Lunney, “Conical diffraction: Observation and theory,” Proc. R. Soc. A 462(2070), 1629–1642 (2006).
    [CrossRef]
  10. C. F. Phelan, D. P. O’Dwyer, Y. P. Rakovich, J. F. Donegan, and J. G. Lunney, “Conical diffraction and Bessel beam formation with a high optical quality biaxial crystal,” Opt. Express 17(15), 12891–12899 (2009).
    [CrossRef] [PubMed]
  11. 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] [PubMed]
  12. V. Peet, “Biaxial crystal as a versatile mode converter,” J. Opt. 12(9), 095706 (2010).
    [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, “Conical diffraction from an N-crystal cascade,” J. Opt. 12(7), 075704 (2010).
    [CrossRef]
  15. A. Abdolvand, “Conical diffraction from a multi-crystal cascade: experimental observations,” Appl. Phys. B 103(2), 281–283 (2011).
    [CrossRef]
  16. M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguilo, and F. Diaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68(2), 187–197 (1999).
    [CrossRef]

2011 (2)

2010 (3)

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, “Biaxial crystal as a versatile mode converter,” J. Opt. 12(9), 095706 (2010).
[CrossRef]

M. V. Berry, “Conical diffraction from an N-crystal cascade,” J. Opt. 12(7), 075704 (2010).
[CrossRef]

2009 (1)

2007 (1)

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

J. G. Lunney and D. W. Weaire, “The ins and outs of conical refraction,” Europhys. News 37(3), 26–29 (2006).
[CrossRef]

2004 (1)

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

1999 (1)

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

1978 (1)

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

1837 (1)

W. R. Hamilton, “Third supplement to an essay on the theory of systems of rays,” Trans. R. Irish Acad. 17, 1–144 (1837).

1833 (1)

H. Lloyd, “On the phenomena presented by light in its passage along the axes of biaxial crystals,” Philos. Mag. 1, 112–120 (1833).

Abdolvand, A.

A. Abdolvand, “Conical diffraction from a multi-crystal cascade: experimental observations,” Appl. Phys. B 103(2), 281–283 (2011).
[CrossRef]

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]

Aguilo, M.

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

Belskii, A. M.

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

Berry, M. V.

M. V. Berry, “Conical diffraction from an N-crystal cascade,” J. Opt. 12(7), 075704 (2010).
[CrossRef]

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

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

Diaz, F.

M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguilo, and F. Diaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68(2), 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, 1–144 (1837).

Jeffrey, M. R.

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

Kalkandjiev, T. K.

Khapaluyk, A. P.

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

Lloyd, H.

H. Lloyd, “On the phenomena presented by light in its passage along the axes of biaxial crystals,” Philos. Mag. 1, 112–120 (1833).

Lunney, J. G.

Nikolov, V.

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

O’Dwyer, D. P.

Peet, V.

V. Peet, “Biaxial crystal as a versatile mode converter,” J. Opt. 12(9), 095706 (2010).
[CrossRef]

Phelan, C. F.

Pujol, M. C.

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

Rafailov, E. U.

Rakovich, Y. P.

Rico, M.

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

Solans, X.

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

Solé, R.

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

Weaire, D. W.

J. G. Lunney and D. W. Weaire, “The ins and outs of conical refraction,” Europhys. News 37(3), 26–29 (2006).
[CrossRef]

Wilcox, K. G.

Zaldo, C.

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

Appl. Phys. B (2)

A. Abdolvand, “Conical diffraction from a multi-crystal cascade: experimental observations,” Appl. Phys. B 103(2), 281–283 (2011).
[CrossRef]

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

Europhys. News (1)

J. G. Lunney and D. W. Weaire, “The ins and outs of conical refraction,” Europhys. News 37(3), 26–29 (2006).
[CrossRef]

J. Opt. (2)

V. Peet, “Biaxial crystal as a versatile mode converter,” J. Opt. 12(9), 095706 (2010).
[CrossRef]

M. V. Berry, “Conical diffraction from an N-crystal cascade,” J. Opt. 12(7), 075704 (2010).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

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

Opt. Express (3)

Opt. Spectrosc. (1)

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

Philos. Mag. (1)

H. Lloyd, “On the phenomena presented by light in its passage along the axes of biaxial crystals,” Philos. Mag. 1, 112–120 (1833).

Proc. R. Soc. A (1)

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

Prog. Opt. (1)

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

Trans. R. Irish Acad. (1)

W. R. Hamilton, “Third supplement to an essay on the theory of systems of rays,” Trans. R. Irish Acad. 17, 1–144 (1837).

Other (2)

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media 2nd ed. (Pergamon Press, 1985).

M. Born and E. Wolf, Principles of Optics 7th ed. (Cambridge University Press, 1999).

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram showing conical diffraction by one biaxial crystal, and the coordinate system used in the text.

Fig. 2
Fig. 2

Cascade conical diffraction setup consisting of two biaxial crystals in series with their optic axes aligned and a relative rotation of one about the optic axis.

Fig. 3
Fig. 3

The experimental arrangement used to observe the intensity profiles in the focal image plane for cascade conical diffraction by a pair of crystals. We consider the effects of rotating one crystal about the optic axis as well as introducing a half-wave plate (HWP) between the crystals.

Fig. 4
Fig. 4

Experimental images in the focal image plane for conical diffraction by a pair of crystals with a relative rotation about the optic axis, as shown in Fig. 2. The second crystal is rotated relative to the first by (a) 0°, (b) 15°, (c) 45° and (d) 90°.

Fig. 5
Fig. 5

Intensity profiles from experiment (dashed line) compared with theory (solid line) for cascade conical diffraction with a relative rotation of α = 45° (a) and 15° (b) for the two crystals.

Fig. 6
Fig. 6

Experimental (b,d) and theoretical (a,c) images in the focal image plane following conical diffraction by two identical crystals separated by a half-wave plate (angles α=β=0 ). Experimental images are obtained from the setup in Fig. 3, and theoretical images calculated using Eq. (9). (c) and (d) are close-ups of the central region of (a) and (b).

Fig. 7
Fig. 7

The experimental arrangement used to observe conical diffraction by a pair of crystals with different geometrical ring radii R0.

Fig. 8
Fig. 8

(a) Intensity in the focal image plane generated by cascade conical diffraction from two aligned crystals of different lengths, obtained using the apparatus of Fig. 7. (b) Intensity profile from the experiment (dashed line) compared with paraxial theory (solid lines).

Equations (11)

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E(R,Z)= k 2π dP e ikP·R exp( 1 2 ik P 2 Z){ cos(kP R 0 )Iisin(kP R 0 )M( φ P ) }a(P) ,
E(R,ϕ,Z)={ B 0 I+ B 1 M(ϕ) } e in
B 0 =k dPPexp( 1 2 ik P 2 Z)cos(kP R 0 ) J 0 ( kPR)a(P),
B 1 =k dPPexp( 1 2 ik P 2 Z)sin(kP R 0 ) J 1 ( kPR)a(P).
a(P)exp( 1 2 ik P 2 Z)C(P, R 0 )a(P)
E(R,ϕ,Z)= 1 2 {( B 0 ( R 1 + R 2 )+ B 0 ( R 1 R 2 ))I +( B 0 ( R 1 + R 2 ) B 0 ( R 1 R 2 ))R(α) +( B 1 ( R 1 + R 2 )+ B 1 ( R 1 R 2 ))M(ϕ) +( B 1 ( R 1 + R 2 ) B 1 ( R 1 R 2 ))M(ϕ+α)} e in .
E(R,ϕ,Z)={ G 0 ×(R(α)+I)+ B 0 (2 R 0 )×(R(α)+I) + B 1 (2 R 0 )×(M(ϕ)+M(ϕ+α))} e in
exp( 1 2 ik P 2 Z 2 )C(P)( cos(2β) sin(2β) sin(2β) cos(2β) )C(P)a(P).
E(R,ϕ,Z)={ 1 2 ( B 0 (2 R 0 )+ G 0 ) M(2β) + B 1 (2 R 0 )cos(2βϕ)I 1 2 ( B 2 (2 R 0 ) G 2 )M(2ϕ2β) } e in ,
B 2 (2 R 0 )=k 0 Pcos(2kP R 0 )a(P)exp( 1 2 ik P 2 Z 2 ) J 2 (kPR) dP, G 2 =k 0 Pa(P)exp( 1 2 ik P 2 Z 2 ) J 2 (kPR) dP, G 0 =k 0 Pa(P)exp( 1 2 ik P 2 Z 2 ) J 0 (kPR) dP,
0 2π d φ p exp(ikPRcos( φ p ϕ))cos(n φ p ) =2π( i n ) J n (kPR)cos(nϕ), 0 2π d φ p exp(ikPRcos( φ p ϕ))sin(n φ p ) =2π( i n ) J n (kPR)sin(nϕ).

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