Abstract

Conical diffraction occurs when light is incident along the optic axis of a biaxial crystal. The light spreads out into a hollow cone inside the crystal, emerging as a hollow cylinder. The intensity distribution beyond the crystal is described using an adapted paraxial wave dispersion model. We show, experimentally and theoretically, how this results in a transition from conical diffraction for wavelengths at which the crystal is aligned to double refraction for misaligned wavelengths when using a white light source. The radius of the ring and location of the focal image plane (FIP) are also observed to have a wavelength dependency. The evolution of the conically diffracted beam beyond the FIP into the far field is studied and successfully described using a theoretical model.

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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. H. Lloyd, “On the phœnomena presented by light in its passage along the axes of biaxal crystals,” Philos. Mag.2, 112–120 (1833).
  3. N. A. Khilo, “Conical diffraction and transformation of Bessel beams in biaxial crystals,” Opt. Commun.286, 1–5 (2013).
    [CrossRef]
  4. M. V. Berry and M. R. Jeffrey, “Hamilton's diabolical point at the heart of crystal optics,” Prog. Opt.50, 13–50 (2007).
    [CrossRef]
  5. A. M. Belsky and A. P. Khapalyuk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc.44, 746–751 (1978).
  6. 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]
  7. C. F. Phelan, J. F. Donegan, and J. G. Lunney, “Generation of a radially polarized light beam using internal conical diffraction,” Opt. Express19(22), 21793–21802 (2011).
    [CrossRef] [PubMed]
  8. A. Abdolvand, K. G. Wilcox, T. K. Kalkandjiev, and E. U. Rafailov, “Conical refraction Nd:KGd(WO4)2 laser,” Opt. Express18(3), 2753–2759 (2010).
    [CrossRef] [PubMed]
  9. 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]
  10. A. Turpin, Y. Loiko, T. K. Kalkandjiev, and J. Mompart, “Free-space optical polarization demultiplexing and multiplexing by means of conical refraction,” Opt. Lett.37(20), 4197–4199 (2012).
    [CrossRef] [PubMed]
  11. A. Turpin, Y. V. Loiko, T. K. Kalkandjiev, H. Tomizawa, and J. Mompart, “Wave-vector and polarization dependence of conical refraction,” Opt. Express21(4), 4503–4511 (2013).
    [CrossRef] [PubMed]
  12. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).
  13. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon Press 1984).
  14. A. Belafhal, “Theoretical intensity distribution of internal conical refraction,” Opt. Commun.178(4-6), 257–265 (2000).
    [CrossRef]
  15. M. V. Berry, M. R. Jeffrey, and J. G. Lunney, “Conical diffraction: observations and theory,” Proc. R. Soc. A 462, 1629–1642 (2006).
    [CrossRef]
  16. M. C. Pujol, M. Rico, C. Zaldo, R. Sole, V. Nikolov, X. Solans, M. Aguilo, and F. Diaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2,” Appl. Phys. B68(2), 187–197 (1999).
    [CrossRef]
  17. 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]
  18. 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. Express17(15), 12891–12899 (2009).
    [CrossRef] [PubMed]

2013 (2)

2012 (1)

2011 (1)

2010 (2)

A. Abdolvand, K. G. Wilcox, T. K. Kalkandjiev, and E. U. Rafailov, “Conical refraction Nd:KGd(WO4)2 laser,” Opt. Express18(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]

2009 (1)

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 (1)

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 Poggendorff rings and axial spike,” J. Opt. A, Pure Appl. Opt.6(4), 289–300 (2004).
[CrossRef]

2000 (1)

A. Belafhal, “Theoretical intensity distribution of internal conical refraction,” Opt. Commun.178(4-6), 257–265 (2000).
[CrossRef]

1999 (1)

M. C. Pujol, M. Rico, C. Zaldo, R. Sole, V. Nikolov, X. Solans, M. Aguilo, and F. Diaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2,” Appl. Phys. B68(2), 187–197 (1999).
[CrossRef]

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).

1837 (1)

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

1833 (1)

H. Lloyd, “On the phœnomena presented by light in its passage along the axes of biaxal crystals,” Philos. Mag.2, 112–120 (1833).

Abdolvand, A.

Aguilo, M.

M. C. Pujol, M. Rico, C. Zaldo, R. Sole, V. Nikolov, X. Solans, M. Aguilo, and F. Diaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2,” Appl. Phys. B68(2), 187–197 (1999).
[CrossRef]

Belafhal, A.

A. Belafhal, “Theoretical intensity distribution of internal conical refraction,” Opt. Commun.178(4-6), 257–265 (2000).
[CrossRef]

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 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 Poggendorff 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. Sole, V. Nikolov, X. Solans, M. Aguilo, and F. Diaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2,” Appl. Phys. B68(2), 187–197 (1999).
[CrossRef]

Donegan, J. F.

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).

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 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]

Kalkandjiev, T. K.

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.

N. A. Khilo, “Conical diffraction and transformation of Bessel beams in biaxial crystals,” Opt. Commun.286, 1–5 (2013).
[CrossRef]

Lloyd, H.

H. Lloyd, “On the phœnomena presented by light in its passage along the axes of biaxal crystals,” Philos. Mag.2, 112–120 (1833).

Loiko, Y.

Loiko, Y. V.

Lunney, J. G.

Mompart, J.

Nikolov, V.

M. C. Pujol, M. Rico, C. Zaldo, R. Sole, V. Nikolov, X. Solans, M. Aguilo, and F. Diaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2,” Appl. Phys. B68(2), 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. Sole, V. Nikolov, X. Solans, M. Aguilo, and F. Diaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2,” Appl. Phys. B68(2), 187–197 (1999).
[CrossRef]

Rafailov, E. U.

Rakovich, Y. P.

Rico, M.

M. C. Pujol, M. Rico, C. Zaldo, R. Sole, V. Nikolov, X. Solans, M. Aguilo, and F. Diaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2,” Appl. Phys. B68(2), 187–197 (1999).
[CrossRef]

Solans, X.

M. C. Pujol, M. Rico, C. Zaldo, R. Sole, V. Nikolov, X. Solans, M. Aguilo, and F. Diaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2,” Appl. Phys. B68(2), 187–197 (1999).
[CrossRef]

Sole, R.

M. C. Pujol, M. Rico, C. Zaldo, R. Sole, V. Nikolov, X. Solans, M. Aguilo, and F. Diaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2,” Appl. Phys. B68(2), 187–197 (1999).
[CrossRef]

Tomizawa, H.

Turpin, A.

Wilcox, K. G.

Zaldo, C.

M. C. Pujol, M. Rico, C. Zaldo, R. Sole, V. Nikolov, X. Solans, M. Aguilo, and F. Diaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2,” Appl. Phys. B68(2), 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. Sole, V. Nikolov, X. Solans, M. Aguilo, and F. Diaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2,” Appl. Phys. B68(2), 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 Poggendorff rings and axial spike,” J. Opt. A, Pure Appl. Opt.6(4), 289–300 (2004).
[CrossRef]

Opt. Commun. (3)

A. Belafhal, “Theoretical intensity distribution of internal conical refraction,” Opt. Commun.178(4-6), 257–265 (2000).
[CrossRef]

N. A. Khilo, “Conical diffraction and transformation of Bessel beams in biaxial crystals,” Opt. Commun.286, 1–5 (2013).
[CrossRef]

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 (4)

Opt. Lett. (1)

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).

Philos. Mag. (1)

H. Lloyd, “On the phœnomena presented by light in its passage along the axes of biaxal crystals,” Philos. Mag.2, 112–120 (1833).

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. (1)

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

Other (3)

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

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

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon Press 1984).

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Figures (8)

Fig. 1
Fig. 1

(a) Dispersion surface close to a diabolical point showing the angle θ the optic axis makes with the k 3 direction. When light is incident along the optic axis, the singularity at the diabolical point results in a directional degeneracy of the Poynting vector (S), which is shown tracing a skewed cone of semi-angle A with respect to the wavevector k. Part (b) demonstrates the wavelength dependence of the optic axis angle θ as given by Eq. (5).

Fig. 2
Fig. 2

(a) Spectrum of the white LED used in the experiment recorded using a spectro-photometer. (b) Experimental setup used for white light conical diffraction experiments.

Fig. 3
Fig. 3

Experimental image of the conically diffracted beam in the FIP where the crystal is aligned to 632.8 nm and no filters are used. The transition to double refraction for blue light is clear. The image is 1.5 mm × 1.5 mm.

Fig. 4
Fig. 4

(a) Plot of R 0 (λ) (red) calculated using the formula for A given in Eq. (1) compared with the measured radii (points) for different bandpass filters. The point at 632.8 nm was recorded using a HeNe laser as a light source. The green curve shows the predicted radius with the value of n 1 increased by 0.007 throughout the visible spectrum. Part (b) of Fig. 4 shows the corresponding experimental images in the FIP for each of the data points in part (a) obtained using bandpass filters (wavelengths labelled). Each image is 1.16 mm in both directions.

Fig. 5
Fig. 5

Results for the relative location of the focal image plane with respect to wavelength. The data recorded (red dots) agree well with the theoretical plot (blue line) calculated from Eq. (3). The point at 650 nm sits exactly on the curve as all other measurements were taken relative to this point. Image (i) (inset) shows what is observed when the CCD is at the FIP for 650 nm and the 650 nm bandpass filter is replaced with the 500 nm bandpass filter. The CCD in (ii) has been moved 200 μm and is deemed to be at the FIP for 500 nm due to the sharp beam structure.

Fig. 6
Fig. 6

(a) is a composite image stitched together using a series of slices perpendicular to the beam propagation direction. Image (b) is a numerically generated plot using the same parameters as the experiment and agrees very well with what is observed. The FIP occurs at ζ=0 and this is where the double ring structure is at its sharpest. The intensity in the numerically generated plot has been artificially increased to allow the fine structure of the beam to be seen.

Fig. 7
Fig. 7

Comparison of measured white light images (top row) and the theoretical intensity distributions given by Eq. (12). The crystal is aligned such that the optic axis for light at 632.8 nm and the beam direction coincide. The theoretical plots show the expected pattern that would be observed if all but one wavelength of light were blocked. There is good agreement between the predicted distributions and the observations suggesting the theoretical model is accurate. Note that the white light images for ζ 632 4.6. have been artificially intensified to make the structure more visible. The theoretical plots at 450 nm and 500 nm were also intensified for ζ 632 9.1. Also note the mirroring of the structure evolution for 650 nm. This is a manifestation of the sign reversal of μ(λ, λ 0 ) which occurs on opposite sides of the aligned wavelength, which in this case is 632.8 nm. This effect becomes clearer in Fig. 8.

Fig. 8
Fig. 8

A bandpass filter at 650nm was placed in front of the CCD for the images on the top row while the optic axis of the crystal remained aligned to 500nm. For the observed images at 450nm, the white light images were deconstructed into RGB values and only the blue values above a certain threshold were chosen. This plot shows very good agreement between theory and experiment.

Equations (13)

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A= 1 2 arctan[ n 2 2 ( n 1 2 n 2 2 )( n 2 2 n 3 2 ) ].
ρ 0 = Al w , ρ= R w , ζ= l+(zl) n 2 n 2 k 0 w 2 ,
z FIP =l( 1 1 n 2 ).
n i (λ)= A i + B i 1 ( C i λ ) 2 D i λ 2 ,
θ=arctan ( n 1 2 n 2 2 )/( n 2 2 n 3 2 )
μ(λ, λ 0 ) ρ 0 (λ) ρ 0 ( λ 0 ),
| ρ ˜ |= ρ ˜ (ξ,η,λ)= (ξ+iμ(λ, λ 0 )) 2 + η 2 ,
cos ϕ ˜ (ξ,η,λ)= ξ+iμ(λ, λ 0 ) ρ ˜ (ξ,η,λ) , sin ϕ ˜ (ξ,η,λ)= η ρ ˜ (ξ,η,λ) .
D( B 0 + B 1 cos ϕ ˜ B 1 sin ϕ ˜ B 1 sin ϕ ˜ B 0 B 1 cos ϕ ˜ ) d 0 ,
B 0 ( ρ ˜ , ζ ˜ ; ρ 0 )= 0 d κ κexp[ 1 2 i ζ ˜ κ 2 ] J 0 (κ ρ ˜ )cos( κ ρ 0 ),
B 1 ( ρ ˜ , ζ ˜ ; ρ 0 )= 0 d κ κexp[ 1 2 i ζ ˜ κ 2 ] J 1 (κ ρ ˜ )sin( κ ρ 0 ),
I= D ¯ D=| B 0 ( ρ ˜ , ζ ˜ ,λ) | 2 +| B 1 ( ρ ˜ , ζ ˜ ,λ) | 2 (|cos ϕ ˜ (ξ,η,λ) | 2 +|sin ϕ ˜ (ξ,η,λ) | 2 ).
A( λ ) n 2 2 [ ( n 1 2 n 2 2 )( n 2 2 n 3 2 ) ] 1 2 ,

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