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

Optics Express, Vol. 22, Issue 9, pp. 11290-11300 (2014)

http://dx.doi.org/10.1364/OE.22.011290

Acrobat PDF (2163 KB)

### 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.

© 2014 Optical Society of America

## 1. Introduction

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. R. Soc. A **462**, 1629–1642 (2006). [CrossRef]

7. 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]

8. 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]

9. 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]

10. 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]

## 2. Theory

*n*

_{1}<

*n*

_{2}<

*n*

_{3}. Light incident along one of the two optic axes in such a material undergoes internal conical diffraction and spreads out as a hollow cone with semi angle The geometrical optics description of this may be found in [11

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

*A*is small, on the order of 10

^{−2}rad, and hence a paraxial approximation may be used to determine the radius of the ring of light as it emerges from such a material of length

*l*: A rigorous examination of the validity of this paraxial approximation when modelling conical diffraction is shown in [13

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

_{4})

_{2}whose principal refractive indices have been determined by Pujol

*et al*. [14

14. M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguiló, and F. Díaz, “Crystalline structure and optical spectroscopy of Er^{3+}-doped KGd(WO_{4})_{2} single crystals,” Appl. Phys. B **68**, 187–197 (1999). [CrossRef]

*n*

_{1}= 2.0109,

*n*

_{2}= 2.0414, and

*n*

_{3}= 2.0950.

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

*k*

_{0}= 2

*π/λ*,

*R*is measured along the radius of the beam with

*R*= 0 at the centre of the beam, and

*z*is measured along the propagation direction of the beam with

*z*= 0 at the location of the focused image of the source in the absence of the crystal. The terms in these equations may be seen in Fig. 1. While these parameters are defined for Gaussian beams with a 1/

*e*intensity radius of

*w*, they are also useful for top-hat beams where

*w*has been defined to be the radius of the beam. When

*ζ*= 0 the conically diffracted rings are most sharply defined and this location is known as the focal image plane (FIP). Letting

*ζ*= 0 in Eq. (3) yields the location

*z*

_{FIP}of the FIP measured from

*z*= 0:

*D*

_{0}(

*ρ*) enters a biaxial material along the optic axis it is conically diffracted. If the beam is radially symmetric and circularly polarised (or unpolarised), the equations describing this transformation are [3

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]

*J*(

_{ν}*x*) is the

*ν*

^{th}order Bessel function of the first kind and

*a*(

*κ*) is the Fourier transform of the input profile

*D*

_{0}(

*ρ*) given by The intensity distribution after the crystal is then given by [3

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]

*x*) is the unit step function defined by The top-hat intensity profile is plotted in Fig. 2(b). The corresponding Fourier transform of a top-hat beam is then obtained using Eq. (7):

*ζ*= 0. These profiles are plotted in Fig. 3. The use of a top-hat input beam is predicted to produce a strikingly different profile from the case of using a Gaussian input beam. Most notable in Fig. 3(b) is the presence of a singularity and a wedge-shaped feature with an instantaneous diminution, as predicted by Berry [3

**6**(4), 289–300 (2004). [CrossRef]

## 3. Fine structure of the radial intensity profile at the FIP

_{4})

_{2}was placed as close as possible to the pinhole and the crystal was aligned so that the beam propagated along an optic axis. A biconvex lens of focal length

*f*= 3 cm was placed

*u*= 3.11 cm after the FIP where

*ζ*= 0. This location occurs inside the crystal as can be found using Eq. (4). The image of the FIP was formed 86 cm after the lens, where a colour charge-coupled device (CCD) of pixel size 4.65 μm was placed to record the profiles generated. The magnification produced by the lens was calculated to be

*m*= |

*v/u*| = 28 which was sufficient to almost fill the CCD chip with the entire singularity and wedge structure. The unmagnified FIP radius was determined to be

*R*

_{0}= 360 ± 10 μm which corresponds to

*ρ*

_{0}= 7.2 when using a pinhole with a 50 μm radius. Note that this value differs from the predicted value of

*R*

_{0}= 430 μm obtained using Eq. (2), a discrepancy which is discussed in more detail in [8

8. 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]

*et al.*[8

8. 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]

*κ*

_{max}being the maximum transverse wavevector component reaching the imaging device. Consider an iris of radius

*R*

_{lim}centred on the conically diffracted beam and placed a distance

*z*

_{iris}from the entrance face of the crystal, as seen in Fig. 6. The maximum transverse wavevector component

*κ*

_{max}passing through this iris is then

*R*

_{lim}= 1.5 mm. Since the beam is free to propagate transversely in the crystal until it emerges, the value of

*z*

_{iris}is simply the length of the crystal, 22 mm. Using these values in Eq. (14) gives

*κ*

_{max}= 69 which was subsequently used in Eq. (8) to generate the plot shown in Fig. 7(a). This is compared with the observed intensity profile obtained by averaging several radial profiles and shows very good agreement. The rapid oscillations of the profile for 0.9 ≤

*ρ/ρ*

_{0}≤ 1.1 are somewhat smoothed out in the experimental profile due to the effect of this averaging.

*z*

_{iris}= 40 mm from the entrance face of the crystal. When the iris was at its minimum setting, the limiting radius was

*R*

_{lim}= 0.75 mm which is smaller than the radius of the crystal, and so using Eq. (14) we find

*κ*

_{max}= 19. Equation (8) was subsequently used to calculate the theoretical intensity profile shown in Fig. 7(b), compared with the experimentally observed profile, again showing very good agreement. The inner ring is less intense than the case with no iris as seen in Fig. 7(a), and it is also broader. The oscillations of the wedge-shaped feature have become more pronounced with longer periods than in Fig. 7(a).

## 4. Far-field propagation

5. 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]

16. 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]

*ζ*increases and eventually the inner ring converges to produce a high intensity region in the centre of the beam

*ρ*= 0 known as the ‘axial spike’. When

*ρ*

_{0}≫ 1, the peak intensity of this axial spike occurs at

*ζ*≤ 10 is shown in Fig. 8 generated using Eq. (8) with

*ρ*

_{0}= 7.2 and

*a*(

*κ*) given by Eq. (10).

*f*= 10 cm was placed a distance

*u*= 20 cm after the pinhole. This produced an unmagnified image of the pinhole a distance

*v*= 20 cm after the lens. The profile of the laser beam at this point

*z*= 0 approximated a top-hat very well as shown by the red dots in Fig. 2(b). When a 22 mm long slab of KGd(WO

_{4})

_{2}was inserted into the beam between the lens and

*z*= 0, the FIP occurred outside the crystal as determined by Eq. (4). A CCD with a pixel size of 6 μm was mounted on a rail allowing movement in the

*ζ*direction.

*ζ*= 0 and moving away from the crystal in 1 mm increments to

*Z*= 8 cm corresponding to

*ζ*= 2.8. The images were then stitched together using numerical interpolation software which allowed the beam propagation to be viewed from the side. The resultant intensity profile is shown in Fig. 10(b). A theoretical profile was generated to show how the beam evolves in this region using Eq. (8) with

*a*(

*κ*) given by Eq. (13). The theoretical profile is shown in Fig. 10(a). There is good agreement between theory and experiment and the presence of the predicted oscillatory axial spike is obvious; a feature which does not occur for the Gaussian input beam as seen in Fig. 8.

*ρ*= 0 line for 0 <

*ζ*< 0.4 which appear and disappear over extremely short distances.

*ρ*= 0 taken from Jeffrey [15]: Using the Fourier transform of the top-hat beam

*a*(

_{T}*κ*) as given in Eq. (13) gives It will now be useful to use the following approximation for a

*ν*

^{th}order Bessel function of the first kind, which is valid when

*x*≫ 1: The form of the expression obtained in Eq. (19) brings us to an interesting conclusion—since

*ζ*does not appear in the factor before the oscillatory cos

^{2}term, the intensity of the maxima along

*ρ*= 0 is constant when

*ζ*≪

*ρ*

_{0}. Furthermore, we may now find the extrema of the function by taking the derivative and finding the values of

*ζ*for which we get zero: A simple calculation taking the derivative of Eq. (20) shows that

*ζ*

_{−}corresponds to local maxima, while

*ζ*

_{+}corresponds to local minima. Examining Eq. (21) reveals that as

*n*→ ∞ with

*ζ*→ 0, the separation between adjacent

*ζ*

_{±}values decreases at approximately the rate of 1/

*n*

^{2}. This means the intensity along

*ρ*= 0 is an oscillatory function whose frequency increases rapidly as

*ζ*→ 0. This feature suggests these beams may be used in super-resolution lensing [17

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

*ζ*= 0. Further work is anticipated in this potential application.

## 5. Conclusion

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

20. 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]

## Acknowledgments

## References and links

1. | W. R. Hamilton, “Third supplement to an essay on the theory of systems of rays,” Trans. R. Irish Acad. |

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

3. | M. V. Berry, “Conical diffraction asymptotics: fine structure of the Poggendorff rings and axial spike,” J. Opt. A, Pure Appl. Opt. |

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

5. | 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 |

6. | 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 |

7. | 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 |

8. | 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 |

9. | 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 |

10. | 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 |

11. | M. Born and E. Wolf, |

12. | L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, |

13. | M. V. Berry and M. R. Jeffrey, “Hamilton’s diabolical point at the heart of crystal optics,” Prog. Opt. |

14. | M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguiló, and F. Díaz, “Crystalline structure and optical spectroscopy of Er |

15. | M. R. Jeffrey, |

16. | V. Peet, “The far-field structure of Gaussian light beams transformed by internal conical refraction in a biaxial crystal,” Opt. Commun. |

17. | M. V. Berry, “Exact nonparaxial transmission of subwavelength detail using superoscillations,” J. Phys. A: Math. Theor. |

18. | A. Ashkin, |

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

20. | 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. |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(260.1180) Physical optics : Crystal optics

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: February 25, 2014

Revised Manuscript: April 27, 2014

Manuscript Accepted: April 28, 2014

Published: May 2, 2014

**Citation**

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)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-9-11290

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### References

- 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, 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, 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, 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, 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, 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, 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, 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, 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, E. Wolf, Principles of Optics (Cambridge University, 1999). [CrossRef]
- L. D. Landau, E. M. Lifshitz, L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984).
- M. V. Berry, 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, 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, A. A. Ryzhevich, “Formation of higher-order Bessel light beams in biaxial crystals,” Opt. Commun. 187, 407–414 (2001). [CrossRef]

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