Conical diffraction intensity profiles generated using a top-hat input beam

R. T. Darcy; D. McCloskey; K. E. Ballantine; J. G. Lunney; P. R. Eastham; J. F. Donegan

doi:10.1364/OE.22.011290

Optics Express
Vol. 22,
Issue 9,
pp. 11290-11300
(2014)
•https://doi.org/10.1364/OE.22.011290

Conical diffraction intensity profiles generated using a top-hat input beam

R. T. Darcy, D. McCloskey, K. E. Ballantine, J. G. Lunney, P. R. Eastham, and J. F. Donegan

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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, 11290-11300 (2014)

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Table of Contents Category
- Diffraction and Gratings
Optics & Photonics Topics
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The topics in this list come from the Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Diffraction (050.1940)
- Crystal optics (260.1180)

About this Article
History
- Original Manuscript: February 25, 2014
- Revised Manuscript: April 27, 2014
- Manuscript Accepted: April 28, 2014
- Published: May 2, 2014

Abstract

The phenomenon of internal conical diffraction has been studied extensively for the case of laser beams with Gaussian intensity profiles incident along an optic axis of a biaxial material. This work presents experimental images for a top-hat input beam and offers a theoretical model which successfully describes the conically diffracted intensity profile, which is observed to differ qualitatively from the Gaussian case. The far-field evolution of the beam is predicted to be particularly interesting with a very intricate structure, and this is confirmed experimentally.

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References

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Year
Author
Publication

W. R. Hamilton, “Third supplement to an essay on the theory of systems of rays,” Trans. R. Irish Acad. 17, 137–139 (1837).
A. M. Belsky and A. P. Khapalyuk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc. 44, 746–751 (1978).
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]
M. V. Berry, M. R. Jeffrey, and J. G. Lunney, “Conical diffraction: observations and theory,” Proc. R. Soc. A 462, 1629–1642 (2006).
[Crossref]
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]
C. F. Phelan, K. E. Ballantine, P. R. Eastham, J. F. Donegan, and J. G. Lunney, “Conical diffraction of a Gaussian beam with a two crystal cascade,” Opt. Express 20(12), 13201–13207 (2012).
[Crossref] [PubMed]
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]
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]
D. P. O’Dwyer, C. F. Phelan, K. E. Ballantine, Y. P. Rakovich, J. G. Lunney, and J. F. Donegan, “Conical diffraction of linearly polarised light controls the angular position of a microscopic object,” Opt. Express 18(26), 27319–27326 (2010).
[Crossref]
A. Turpin, V. Shvedov, C. Hnatovsky, Y. V. Loiko, J. Mompart, and W. Krolikowski, “Optical vault: A reconfigurable bottle beam based on conical refraction of light,” Opt. Express 21(22), 26335–26340 (2013).
[Crossref] [PubMed]
M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
[Crossref]
L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984).
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. 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]
M. R. Jeffrey, Conical Diffraction: Complexifying Hamilton’s Diabolical Legacy (University of Bristol, 2007).
V. Peet, “The far-field structure of Gaussian light beams transformed by internal conical refraction in a biaxial crystal,” Opt. Commun. 311, 150–155 (2013).
[Crossref]
M. V. Berry, “Exact nonparaxial transmission of subwavelength detail using superoscillations,” J. Phys. A: Math. Theor. 46, 205203 (2013).
[Crossref]
A. Ashkin, Optical Trapping and Manipulation of Neutral Particles Using Lasers (World Scientific, 2007).
M. V. Berry, “Conical diffraction from an N-crystal cascade,” J. Opt. 12, 75704–75712 (2010).
[Crossref]
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. Commun. 187, 407–414 (2001).
[Crossref]

2013 (4)

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]

V. Peet, “The far-field structure of Gaussian light beams transformed by internal conical refraction in a biaxial crystal,” Opt. Commun. 311, 150–155 (2013).
[Crossref]

M. V. Berry, “Exact nonparaxial transmission of subwavelength detail using superoscillations,” J. Phys. A: Math. Theor. 46, 205203 (2013).
[Crossref]

A. Turpin, V. Shvedov, C. Hnatovsky, Y. V. Loiko, J. Mompart, and W. Krolikowski, “Optical vault: A reconfigurable bottle beam based on conical refraction of light,” Opt. Express 21(22), 26335–26340 (2013).
[Crossref] [PubMed]

M. V. Berry, “Conical diffraction from an N-crystal cascade,” J. Opt. 12, 75704–75712 (2010).
[Crossref]

2009 (1)

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]

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, M. R. Jeffrey, and J. G. Lunney, “Conical diffraction: observations and theory,” Proc. R. Soc. A 462, 1629–1642 (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. Commun. 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]

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).

Aguiló, M.

Ashkin, A.

A. Ashkin, Optical Trapping and Manipulation of Neutral Particles Using Lasers (World Scientific, 2007).

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, “Exact nonparaxial transmission of subwavelength detail using superoscillations,” J. Phys. A: Math. Theor. 46, 205203 (2013).
[Crossref]

M. V. Berry, “Conical diffraction from an N-crystal cascade,” J. Opt. 12, 75704–75712 (2010).
[Crossref]

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. R. Soc. A 462, 1629–1642 (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]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
[Crossref]

Darcy, R. T.

Díaz, F.

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).

Hnatovsky, C.

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. Commun. 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. R. Soc. A 462, 1629–1642 (2006).
[Crossref]

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

Jennings, B. D.

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. Commun. 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. Commun. 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. Commun. 187, 407–414 (2001).
[Crossref]

Krolikowski, W.

Landau, L. D.

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

Lifshitz, E. M.

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

Loiko, Y. V.

Lunney, J. G.

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

McCloskey, D.

Mompart, J.

Nikolov, V.

O’Dwyer, D. P.

Peet, V.

V. Peet, “The far-field structure of Gaussian light beams transformed by internal conical refraction in a biaxial crystal,” Opt. Commun. 311, 150–155 (2013).
[Crossref]

Phelan, C. F.

Pitaevskii, L. P.

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

Pujol, M. C.

Rakovich, Y. P.

Rico, M.

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. Commun. 187, 407–414 (2001).
[Crossref]

Shvedov, V.

Solans, X.

Solé, R.

Turpin, A.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
[Crossref]

Zaldo, C.

Appl. Phys. B (1)

J. Opt. (1)

M. V. Berry, “Conical diffraction from an N-crystal cascade,” J. Opt. 12, 75704–75712 (2010).
[Crossref]

J. Opt. A, Pure Appl. Opt. (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]

J. Phys. A: Math. Theor. (1)

M. V. Berry, “Exact nonparaxial transmission of subwavelength detail using superoscillations,” J. Phys. A: Math. Theor. 46, 205203 (2013).
[Crossref]

Opt. Commun. (2)

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. Commun. 187, 407–414 (2001).
[Crossref]

V. Peet, “The far-field structure of Gaussian light beams transformed by internal conical refraction in a biaxial crystal,” Opt. Commun. 311, 150–155 (2013).
[Crossref]

Opt. Express (6)

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).

Proc. R. Soc. A (1)

M. V. Berry, M. R. Jeffrey, and J. G. Lunney, “Conical diffraction: observations and theory,” Proc. R. 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. (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 (4)

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

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
[Crossref]

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

A. Ashkin, Optical Trapping and Manipulation of Neutral Particles Using Lasers (World Scientific, 2007).

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Fig. 1 A light beam undergoing conical diffraction within a biaxial material. This diagram is a two-dimensional slice taken along the plane where R = 0.

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Fig. 2 Plot (a) is a Gaussian intensity profile calculated using the electric displacement field given by Eq. (9). The value w is the 1/e radius of the beam. Plot (b) shows a top-hat profile calculated using the electric displacement field given by Eq. (11). The red dots in (b) show a profile, taken from an experimental image, of the beam used in the experiments which demonstrates a reasonably good approximation of a top-hat beam.

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Fig. 3 Intensity profiles at the FIP generated using Eq. (8) in the case of (a) a Gaussian input beam, as given by Eq. (9), and (b) a top-hat input beam, as given by Eq. (11).

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Fig. 4 The experimental setup used to study the fine structure of the beam profile in the FIP which is imaged onto the CCD. The values of u and v can be adjusted independently to give the image a desired magnification.

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Fig. 5 (a) An experimental image of the conically diffracted Gaussian beam in the FIP. The structure is distinct from that formed when using a top-hat input beam as seen in (b). The radius of the Poggendorff dark ring in image (b) was determined to be 360 ± 10 μm. Both images are 0.8 mm × 0.8 mm.

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Fig. 6 The maximum transmissible transverse wavevector component κ_max is determined by the radius of the iris R_lim and its position z_iris. This transverse wavevector is given by κ_max = n₂k₀w sinα where α = arctan(R_lim/z_iris).

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Fig. 7 (a) Comparison of the observed intensity profile (red dots) with the theoretical profile (solid blue) where κ_max = 69. (b) Comparison of the observed intensity profile (red dots) with the theoretical profile (solid blue) where κ_max = 19 corresponding to an iris of diameter 1.5 mm present before the imaging lens in Fig. 4.

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Fig. 8 Theoretical plot of the far-field evolution of a conically diffracted Gaussian beam generated using Eq. (8).

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Fig. 9 The experimental setup used to study how the conically diffracted top-hat beam evolves in space. The pinhole is imaged using a biconvex lens of focal length f to a point z = 0 beyond the crystal, thus using Eq. (4) the FIP occurs outside the crystal at a distance v + l (1 − 1/n₂) from the lens. z increases in the direction of beam propagation, and hence so do Z and ζ as given by Eq. (3).

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Fig. 10 Comparison of the theoretical evolution of the beam (a) with the observed evolution (b) in the case of a 50 μm radius top-hat beam. Image (b) was generated by stitching together a series of images taken at 1 mm increments in the direction of beam propagation.

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Fig. 11 A logarithmic plot of the far-field evolution of a conically diffracted top-hat beam, calculated using Eq. (8), in order to show the very complicated and intricate structure of the beam. It also demonstrates how the oscillating lobes along the ρ = 0 line are formed from many converging rings.

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Equations (21)

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A = \frac{1}{2} arctan [n_{2}^{2} \sqrt{(n_{1}^{- 2} - n_{2}^{- 2}) (n_{2}^{- 2} - n_{3}^{- 2})}] .

R_{0} = \frac{l}{2} \frac{sin (2 A)}{cos (2 A)} = \frac{l}{2} tan (2 A) \approx A l .

\begin{matrix} ρ_{0} \equiv \frac{A l}{w} = \frac{R_{0}}{w}, & ρ \equiv \frac{R}{w}, & ζ \equiv \frac{l + (z - l) n_{2}}{n_{2} k_{0} w^{2}} = \frac{Z}{n_{2} k_{0} w^{2}}, \end{matrix}

z_{FIP} = l (1 - 1 / n_{2}) .

B_{0} (ρ, ρ_{0}, ζ) = \int_{0}^{\infty} d κ κ a (κ) exp (- \frac{1}{2} i ζ κ^{2}) J_{0} (κ ρ) cos (κ ρ_{0}),

B_{1} (ρ, ρ_{0}, ζ) = \int_{0}^{\infty} d κ κ a (κ) exp (- \frac{1}{2} i ζ κ^{2}) J_{1} (κ ρ) sin (κ ρ_{0}),

a (κ) = \int_{0}^{\infty} d ρ ρ D_{0} (ρ) J_{0} (κ ρ) .

I (ρ, ρ_{0}, ζ) = {| B_{0} (ρ, ρ_{0}, ζ) |}^{2} + {| B_{1} (ρ, ρ_{0}, ζ) |}^{2} .

D_{0 G} (ρ) = exp (- ρ^{2} / 2) .

a_{G} (κ) = exp (- κ^{2} / 2) .

D_{0 T} (ρ) = Θ (1 - ρ),

Θ (x) \equiv {\begin{matrix} 0, & x < 0 \\ 1, & x \geq 0 \end{matrix} .

a_{T} (κ) = J_{1} (κ) / κ .

κ_{max} \equiv n_{2} k_{0} w sin [arctan (\frac{R_{lim}}{z_{iris}})] .

I (ρ = 0, ζ) \approx \frac{π ρ_{0}^{2}}{2 ζ^{3}} {| a (\frac{ρ_{0}}{ζ}) |}^{2} .

I (ρ = 0, ζ) \approx \frac{π ρ_{0}^{2}}{2 ζ^{3}} {| \frac{ζ}{ρ_{0}} J_{1} (\frac{ρ_{0}}{ζ}) |}^{2} = \frac{π}{2 ζ} {| J_{1} (\frac{ρ_{0}}{ζ}) |}^{2} .

J_{ν} (x) \approx \sqrt{\frac{2}{π x}} cos (x - \frac{π}{4} - \frac{ν π}{2}), x ≫ 1,

\to J_{1} (\frac{ρ_{0}}{ζ}) \approx \sqrt{\frac{2 ζ}{π ρ_{0}}} cos (\frac{ρ_{0}}{ζ} - \frac{3 π}{4}), ζ ≪ ρ_{0} .

\Rightarrow I (ρ = 0, ζ) \approx \frac{π}{2 ζ} \frac{2 ζ}{π ρ_{0}} {cos}^{2} (\frac{ρ_{0}}{ζ} - \frac{3 π}{4}) = \frac{1}{ρ_{0}} {cos}^{2} (\frac{ρ_{0}}{ζ} - \frac{3 π}{4}) .

\frac{\partial}{\partial ζ} I (ρ = 0, ζ) = \frac{2}{ζ^{2}} cos (\frac{ρ_{0}}{ζ} + \frac{π}{4}) sin (\frac{ρ_{0}}{ζ} + \frac{π}{4}) = 0,

\Rightarrow ζ_{\pm} = \frac{ρ_{0}}{π (n \pm \frac{1}{4})}, n \in ℤ^{+} .

Abstract

References

2013 (4)

2012 (1)

2011 (1)

2010 (2)

2009 (1)

2007 (1)

2006 (1)

2004 (1)

2001 (1)

1999 (1)

1978 (1)

1837 (1)

Aguiló, M.

Ashkin, A.

Ballantine, K. E.

Belsky, A. M.

Berry, M. V.

Born, M.

Darcy, R. T.

Díaz, F.

Donegan, J. F.

Eastham, P. R.

Hamilton, W. R.

Hnatovsky, C.

Hogervorst, W.

Jeffrey, M. R.

Jennings, B. D.

Kazak, N. S.

Khapalyuk, A. P.

Khilo, N. A.

King, T. A.

Krolikowski, W.

Landau, L. D.

Lifshitz, E. M.

Loiko, Y. V.

Lunney, J. G.

McCloskey, D.

Mompart, J.

Nikolov, V.

O’Dwyer, D. P.

Peet, V.

Phelan, C. F.

Pitaevskii, L. P.

Pujol, M. C.

Rakovich, Y. P.

Rico, M.

Ryzhevich, A. A.

Shvedov, V.

Solans, X.

Solé, R.

Turpin, A.

Wolf, E.

Zaldo, C.

Appl. Phys. B (1)

J. Opt. (1)

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

J. Phys. A: Math. Theor. (1)

Opt. Commun. (2)

Opt. Express (6)

Opt. Spectrosc. (1)

Proc. R. Soc. A (1)

Prog. Opt. (1)

Trans. R. Irish Acad. (1)

Other (4)

Cited By

Figures (11)

Equations (21)

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