Conical diffraction and Bessel beam formation with a high optical quality biaxial crystal

C. F. Phelan; D. P. O’Dwyer; Y. P. Rakovich; J. F. Donegan; J. G. Lunney

doi:10.1364/OE.17.012891

Abstract

The manipulation of a Gaussian laser beam using conical diffraction in a high optical quality biaxial crystal of KGd(WO₄)₂ has been examined in detail with emphasis on the experimental techniques involved and intuitive explanations of the notable features. Two different optical arrangements were used to form the Pogendorff double-ring light pattern in the focal image plane. The formation of both diverging and non-diverging zeroth and first order Bessel beams was investigated. The various intensity distributions and polarization properties were measured and compared with the predictions of well-established theory.

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References

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Publication

W. R. Hamilton, “Third supplement to an essay on the theory of systems of rays,” Transactions of the Royal Irish Academy 17, 1–144 (1837).
H. Lloyd, “On the phenomena presented by light in its passage along the axes of biaxial crystals” Phil. Mag. 1,112–120 and 207–210 (1833).
L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media 2nd Ed (Pergamon Press 1984) Chap. 11.
M. Born, and E. Wolf, Principles of Optics 7th Ed (Cambridge University Press 1999) Chap. 15.
J. G. Lunney and D. W. Weaire, “The ins and outs of conical refraction,” Europhys. News 37(3), 26–29 (2006).
[Crossref]
A. M. Belskii and A. P. Khapaluyk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc. 44, 312–315 (1978).
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]
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: Observations and theory,” Proc. R. Soc. A 462(2070), 1629–1642 (2006).
[Crossref]
M. V. Berry, M. R. Jeffrey, and M. J. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. 7(11), 685–690 (2005).
[Crossref]
L. Allen, S. M. Barnett, and M. J. Padgett, Optical angular momentum (Bristol: Institute of physics publishing 2003).
J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]
R. M. Hermann and T. A. Wiggins, “Production and uses of diffraction free beams,” J. Opt. Soc. Am. A 8(6), 932–942 (1991).
[Crossref]
S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” N. J. Phys. 8(5), 75 (2006).
[Crossref]
L. P. Deng, H. Wang, and K. Wang, “Quantum CNOT gates with orbital angular momentum and polarization of a single photon quantum logic,” J. Opt. Soc. Am. B 24(9), 2517–2520 (2007).
[Crossref]
J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref] [PubMed]
M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Santos, 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]

2007 (2)

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]

L. P. Deng, H. Wang, and K. Wang, “Quantum CNOT gates with orbital angular momentum and polarization of a single photon quantum logic,” J. Opt. Soc. Am. B 24(9), 2517–2520 (2007).
[Crossref]

2006 (3)

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” N. J. Phys. 8(5), 75 (2006).
[Crossref]

M. V. Berry, M. R. Jeffrey, and J. G. Lunney, “Conical diffraction: Observations 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]

2005 (1)

M. V. Berry, M. R. Jeffrey, and M. J. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. 7(11), 685–690 (2005).
[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]

2002 (1)

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref] [PubMed]

1999 (1)

M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Santos, 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]

1991 (1)

R. M. Hermann and T. A. Wiggins, “Production and uses of diffraction free beams,” J. Opt. Soc. Am. A 8(6), 932–942 (1991).
[Crossref]

1987 (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

1978 (1)

A. M. Belskii and A. P. Khapaluyk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc. 44, 312–315 (1978).

1837 (1)

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

1833 (1)

H. Lloyd, “On the phenomena presented by light in its passage along the axes of biaxial crystals” Phil. Mag. 1,112–120 and 207–210 (1833).

Aguiló, M.

M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Santos, 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]

Barnett, S. M.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref] [PubMed]

Belskii, A. M.

A. M. Belskii and A. P. Khapaluyk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc. 44, 312–315 (1978).

Berry, M. V.

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: Observations and theory,” Proc. R. Soc. A 462(2070), 1629–1642 (2006).
[Crossref]

M. V. Berry, M. R. Jeffrey, and M. J. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. 7(11), 685–690 (2005).
[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]

Courtial, J.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref] [PubMed]

Deng, L. P.

L. P. Deng, H. Wang, and K. Wang, “Quantum CNOT gates with orbital angular momentum and polarization of a single photon quantum logic,” J. Opt. Soc. Am. B 24(9), 2517–2520 (2007).
[Crossref]

Díaz, F.

M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Santos, 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]

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

Franke-Arnold, S.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref] [PubMed]

Gröblacher, S.

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” N. J. Phys. 8(5), 75 (2006).
[Crossref]

Hamilton, W. R.

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

Hermann, R. M.

R. M. Hermann and T. A. Wiggins, “Production and uses of diffraction free beams,” J. Opt. Soc. Am. A 8(6), 932–942 (1991).
[Crossref]

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: Observations and theory,” Proc. R. Soc. A 462(2070), 1629–1642 (2006).
[Crossref]

M. V. Berry, M. R. Jeffrey, and M. J. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. 7(11), 685–690 (2005).
[Crossref]

Jennewein, T.

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” N. J. Phys. 8(5), 75 (2006).
[Crossref]

Khapaluyk, A. P.

A. M. Belskii and A. P. Khapaluyk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc. 44, 312–315 (1978).

Leach, J.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref] [PubMed]

Lloyd, H.

H. Lloyd, “On the phenomena presented by light in its passage along the axes of biaxial crystals” Phil. Mag. 1,112–120 and 207–210 (1833).

Lunney, J. G.

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

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

Mansuripur, M. J.

M. V. Berry, M. R. Jeffrey, and M. J. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. 7(11), 685–690 (2005).
[Crossref]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

Nikolov, V.

M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Santos, 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]

Padgett, M. J.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref] [PubMed]

Pujol, M. C.

M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Santos, 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]

Rico, M.

M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Santos, 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]

Santos, X.

M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Santos, 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. Santos, 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]

Vaziri, A.

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” N. J. Phys. 8(5), 75 (2006).
[Crossref]

Wang, H.

L. P. Deng, H. Wang, and K. Wang, “Quantum CNOT gates with orbital angular momentum and polarization of a single photon quantum logic,” J. Opt. Soc. Am. B 24(9), 2517–2520 (2007).
[Crossref]

Wang, K.

L. P. Deng, H. Wang, and K. Wang, “Quantum CNOT gates with orbital angular momentum and polarization of a single photon quantum logic,” J. Opt. Soc. Am. B 24(9), 2517–2520 (2007).
[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]

Weihs, G.

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” N. J. Phys. 8(5), 75 (2006).
[Crossref]

Wiggins, T. A.

R. M. Hermann and T. A. Wiggins, “Production and uses of diffraction free beams,” J. Opt. Soc. Am. A 8(6), 932–942 (1991).
[Crossref]

Zaldo, C.

M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Santos, 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]

Zeilinger, A.

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” N. J. Phys. 8(5), 75 (2006).
[Crossref]

Appl. Phys. B (1)

M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Santos, 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]

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. A, Pure Appl. Opt. (2)

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]

M. V. Berry, M. R. Jeffrey, and M. J. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. 7(11), 685–690 (2005).
[Crossref]

J. Opt. Soc. Am. A (1)

R. M. Hermann and T. A. Wiggins, “Production and uses of diffraction free beams,” J. Opt. Soc. Am. A 8(6), 932–942 (1991).
[Crossref]

J. Opt. Soc. Am. B (1)

L. P. Deng, H. Wang, and K. Wang, “Quantum CNOT gates with orbital angular momentum and polarization of a single photon quantum logic,” J. Opt. Soc. Am. B 24(9), 2517–2520 (2007).
[Crossref]

N. J. Phys. (1)

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” N. J. Phys. 8(5), 75 (2006).
[Crossref]

Opt. Spectrosc. (1)

A. M. Belskii and A. P. Khapaluyk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc. 44, 312–315 (1978).

Phil. Mag. (1)

H. Lloyd, “On the phenomena presented by light in its passage along the axes of biaxial crystals” Phil. Mag. 1,112–120 and 207–210 (1833).

Phys. Rev. Lett. (2)

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref] [PubMed]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

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(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]

Transactions of the Royal Irish Academy (1)

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

Other (3)

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media 2nd Ed (Pergamon Press 1984) Chap. 11.

M. Born, and E. Wolf, Principles of Optics 7th Ed (Cambridge University Press 1999) Chap. 15.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical angular momentum (Bristol: Institute of physics publishing 2003).

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Fig. 1 The system of cylindrical co-ordinates used to describe the cylinder of light that propagates in space after having been refracted into a cone of semi-angle A within the crystal.

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Fig. 2 The intensity due to B₀ (a), B₁ (b) and the conically diffracted Gaussian field, B₀ + B₁ (c) as the beam propagates away from the focal image. The vertical axis measures distance from the focal image plane in mm and the horizontal axis measures radial distance in mm. (d) shows the normalized greyscale.

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Fig. 3 Circularly polarized light is focused into the crystal with a x10 objective to achieve a narrow waist (10μm). The focal image plane which for this setup occurs somewhere near the crystals entrance face (due to the short focal length of the objective) is imaged onto the CCD with a 10cm focal length lens.

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Fig. 4 Experimental image of the focal image plane, the excellent quality of the crystal is obvious from the image above

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Fig. 5 The different phase properties of the fields described by B₀ & B₁ . 5(a) shows, over one optical cycle, the how field vectors of the B₀ component combine with the field vectors of the B₁ field to give the half turn of polarization associated with conical refraction; 5(b)shows a snapshot of the B₀ and B₁ electric field profiles in the focal image plane (B₁ dashed)

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Fig. 6 This optical arrangement used to analyze conical diffraction requires no imaging lens

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Fig. 7 A comparison of experiment with theory (dashed line) in the focal image plane; Radial distance is measured from the centre of the cylinder associated with conical refraction (Fig. 1)

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Fig. 8 CCD pictures of diverging Bessel beams; B₀ in (a); B₁ in (b)

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Fig. 9 (a) Exact far field B₀ vs experimental data (dashed) at 10.75 cm from FIP; (b) same data for B₁

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Fig. 10 Optical setup for non-diverging Bessel beams

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Equations (5)

Equations on this page are rendered with MathJax. Learn more.

A = \sqrt{(n_{3} - n_{2}) (n_{2} - n_{1})}

\begin{array}{l} E_{x} \propto \exp (- \frac{r^{2}}{w^{2}}) \\ E_{y} \propto i \exp (- \frac{r^{2}}{w^{2}}) \end{array}

\begin{array}{l} E_{x} (r, φ, z) \propto B_{0} (r, z) + B_{1} (r, z) (\cos φ + i \sin φ) \\ E_{y} (r, φ, z) \propto i B_{0} (r, z) + B_{1} (r, z) (\sin φ - i \cos φ) \end{array}

B_{0} = k \int_{0}^{\infty} p \cos (k p R_{0}) J_{0} (k p r) \exp (- \frac{1}{4} k^{2} p^{2} w^{2}) \exp (- \frac{1}{2} i k p^{2} Z)

B_{1} = k \int_{0}^{\infty} p \sin (k p R_{0}) J_{1} (k p r) \exp (- \frac{1}{4} k^{2} p^{2} w^{2}) \exp (- \frac{1}{2} i k p^{2} Z)

Abstract

References

2007 (2)

2006 (3)

2005 (1)

2004 (1)

2002 (1)

1999 (1)

1991 (1)

1987 (1)

1978 (1)

1837 (1)

1833 (1)

Aguiló, M.

Barnett, S. M.

Belskii, A. M.

Berry, M. V.

Courtial, J.

Deng, L. P.

Díaz, F.

Durnin, J.

Eberly, J. H.

Franke-Arnold, S.

Gröblacher, S.

Hamilton, W. R.

Hermann, R. M.

Jeffrey, M. R.

Jennewein, T.

Khapaluyk, A. P.

Leach, J.

Lloyd, H.

Lunney, J. G.

Mansuripur, M. J.

Miceli, J. J.

Nikolov, V.

Padgett, M. J.

Pujol, M. C.

Rico, M.

Santos, X.

Solé, R.

Vaziri, A.

Wang, H.

Wang, K.

Weaire, D. W.

Weihs, G.

Wiggins, T. A.

Zaldo, C.

Zeilinger, A.

Appl. Phys. B (1)

Europhys. News (1)

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

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

N. J. Phys. (1)

Opt. Spectrosc. (1)

Phil. Mag. (1)

Phys. Rev. Lett. (2)

Proc. R. Soc. A (1)

Prog. Opt. (1)

Transactions of the Royal Irish Academy (1)

Other (3)

Cited By

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Equations (5)

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