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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 12 — Jun. 4, 2012
  • pp: 13201–13207
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Conical diffraction of a Gaussian beam with a two crystal cascade

C. F. Phelan, K. E. Ballantine, P. R. Eastham, J. F. Donegan, and J. G. Lunney  »View Author Affiliations


Optics Express, Vol. 20, Issue 12, pp. 13201-13207 (2012)
http://dx.doi.org/10.1364/OE.20.013201


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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.

© 2012 OSA

1. Introduction

Internal conical refraction is a phenomenon of crystal optics in which a beam of light that is refracted into the optic axis of a biaxial crystal propagates as a cone of light in the crystal, and then refracts into a hollow cylinder as it exits the crystal [1

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

5

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

]. Several attempts have been made to extend the theory of conical refraction to deal with situations involving realistic light beams rather than idealized rays propagating in biaxial crystals. The most fruitful approach is to make the assumption of paraxiality. This approach has led to detailed predictions of the intensity distributions of conically diffracted paraxial light beams [6

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

8

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]

]. These predictions have been shown to agree well with theory for the case of the conically diffracted Gaussian beam [9

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

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]

]. The propagation of paraxial light beams along the optic axes of successive biaxial crystals, known as cascade conical diffraction, has been receiving interest recently for the creation and annihilation of optical vortices [11

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]

], as a versatile beam shaping tool [12

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

] and in connection with a novel type of laser based on conical diffraction [13

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]

]. Berry has provided a paraxial theory for a general N-crystal cascade in which the relative orientation of crystals of differing lengths is considered [14

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

]. For a two-crystal cascade, conical diffraction leads to a pair of Poggendorf (double) rings, whose relative intensity depends on the angle between the crystals. In the degenerate case, where the crystals are identical the smaller ring becomes a central spot. Abdolvand [15

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

] has provided a first set of experimental results on cascade conical refraction. Here we compare the intensity profiles predicted by the paraxial theory for this case with experiments on crystals of both the same and different lengths. In addition we consider the generalization to include a half-wave plate between the crystals, which can also lead to beams comprising a central spot and an outer ring. Our results demonstrate that the paraxial theory provides an excellent account of the complex beams generated by crystal cascades.

2. Background theory

We are considering the case of conical diffraction of a Gaussian beam by propagation along the optic axis of a biaxial crystal of length l and with principal refractive indices n1<n2<n3. We define our coordinate system, as shown in Fig. 1
Fig. 1 Schematic diagram showing conical diffraction by one biaxial crystal, and the coordinate system used in the text.
, such that conical refraction displaces the center of the incident beam along the x axis, and measure angles from this axis. A paraxial light beam defined by the Fourier transform of the x- and y- components of its electric fielda(P), whereP={P,φP}is the normalized transverse wave-vector in polar coordinates, is transformed by conical diffraction into:
E(R,Z)=k2πdPeikP·Rexp(12ikP2Z){cos(kPR0)Iisin(kPR0)M(φP)}a(P),
(1)
where M(φP)=(cos(φP)sin(φP)sin(φP)cos(φP)) andIis the identity matrix. R={R,ϕ}is the position in the transverse plane and Z = l + (z - l)n2 is the propagation distance from the image in the crystal of the incident beam waist (or other reference plane at whicha(P)is specified). k=n2k0is the crystal wave-number with k0 being the free space wave number and R0 is the radius of the refracted cylinder predicted by geometrical optics. Making the assumption that the incident beam is uniformly polarized, i.e., replacinga(P)witha(P)ein whereeinis the incident polarization vector, and performing the integration over φP we have:
E(R,ϕ,Z)={B0I+B1M(ϕ)}ein
(2)
where

B0=kdPPexp(12ikP2Z)cos(kPR0)J0(kPR)a(P),
(3)
B1=kdPPexp(12ikP2Z)sin(kPR0)J1(kPR)a(P).
(4)

The generalization to a cascade of several crystals [14

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

] follows on noting that the transformation described by Eq. (1) can be written in Fourier space as
a(P)exp(12ikP2Z)C(P,R0)a(P)
(5)
whereC(P,R0)cos(kPR0)Iisin(kPR0)M(φp).

We now consider a configuration in which there are two biaxial crystals and the second crystal is rotated about the optic axis by an angle α relative to the first, as shown in 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.
. Note that such a rotation displaces the beam, because conical diffraction induces a translation of the center of the beam in the transverse (x, y) plane (Fig. 1), and if the crystals are rotated, so are the displacements induced by the two crystals (Fig. 2). In what follows we will ignore this effect and consider the position of the center of the beam being described to be at the origin of the coordinate system. The crystals are assumed to have equal principal refractive indices and hence equal cone refraction angles. The crystals have lengths l1 and l2 and geometrical ring radii R1 and R2. The transformation generated by the second crystal is Eq. (5) with C(P,R0)R(α)C({P,φpα},R2)R(α), with rotation matrix R(α)=(cosαsinαsinαcosα). This gives for the final beam

E(R,ϕ,Z)=12{(B0(R1+R2)+B0(R1R2))I+(B0(R1+R2)B0(R1R2))R(α)+(B1(R1+R2)+B1(R1R2))M(ϕ)+(B1(R1+R2)B1(R1R2))M(ϕ+α)}ein.
(6)

The coordinate Z is now defined by Z = l1 + l2 + (z –(l1 + l2))n2 to take into account propagation through the two crystals. If we assume that the two crystals are the same length Eq. (6) reduces to
E(R,ϕ,Z)={G0×(R(α)+I)+B0(2R0)×(R(α)+I)+B1(2R0)×(M(ϕ)+M(ϕ+α))}ein
(7)
whereG0represents the incident Gaussian beam (taking into account propagation) and we have used the fact thatB0(R0=0)=G0andB1(R0=0)=0.

A half wave plate (HWP) inserted between the crystals has the effect of interchanging the polarisations of theB0andB1components before propagation through the second crystal. This interchange of polarisations will occur regardless of the orientation of the HWP. A HWP with its fast axis making an angle β with the positive x axis is represented by the Jones matrix(cos(2β)sin(2β)sin(2β)cos(2β)). If the HWP is placed between two biaxial crystals with α set to 0° and with equal lengths l then a beam that propagates through this system is given (in Fourier space) by:

exp(12ikP2Z2)C(P)(cos(2β)sin(2β)sin(2β)cos(2β))C(P)a(P).
(8)

Multiplying out the matrices and integrating over φP leads to:
E(R,ϕ,Z)={12(B0(2R0)+G0)M(2β)+B1(2R0)cos(2βϕ)I12(B2(2R0)G2)M(2ϕ2β)}ein,
(9)
where
B2(2R0)=k0Pcos(2kPR0)a(P)exp(12ikP2Z2)J2(kPR)dP,G2=k0Pa(P)exp(12ikP2Z2)J2(kPR)dP,G0=k0Pa(P)exp(12ikP2Z2)J0(kPR)dP,
(10)
and we have used the following identities for the integrals over φP;

02πdφpexp(ikPRcos(φpϕ))cos(nφp)=2π(in)Jn(kPR)cos(nϕ),02πdφpexp(ikPRcos(φpϕ))sin(nφp)=2π(in)Jn(kPR)sin(nϕ).
(11)

3. Experiment compared with theory

Figure 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°.
shows CCD images of the FIP profiles recorded with the experimental setup shown in Fig. 3 without the half-wave plate between the two crystals for four different values of the angle between the crystals, α. The predicted increase of intensity of the central spot relative to the ring is evident. Theory is compared with experiment in 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.
for rotations of 15° and 45°. It is noteworthy that the agreement with theory is less accurate for the smaller rotation angle. This could be because the peak intensity of the central spot grows rapidly relative to the peak intensity of the outer ring as the crystals are rotated away from 0°.

The predictions regarding the effect of placing a half wave plate between the crystals were also confirmed by the experimental setup shown in Fig. 3. The results are shown in 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 (anglesα=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).
. The notable features are a central spot that is elongated in one direction and a ring profile with an angular modulation. The two maxima in the ring’s intensity distribution are at the same (diametrically opposite) angular positions as the maxima in the modulated central spot.

A similar experimental setup, shown in Fig. 7
Fig. 7 The experimental arrangement used to observe conical diffraction by a pair of crystals with different geometrical ring radii R0.
, was used to observe the beam produced by propagation through two biaxial crystals with different geometric ring radii. This time the crystal lengths were 3 cm and 2 cm respectively. These crystal lengths implied the geometrical ring radii associated with each crystal would be 0.53 mm and 0.37 mm respectively. Hence theory predicts that the FIP intensity distribution will consist of two concentric double ring patterns with dark ring radii of 0.9 and 0.16 mm. The same laser source was used in this experiment as in the case of equal length crystals but in this experiment a longer focal length lens was used to form the FIP on the CCD without an imaging lens (this point is discussed in [10

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]

]). The minimum waist of the Gaussian formed by the lens was 70 μm.

Figure 8(a)
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).
shows a CCD image of the beam recorded using the optical arrangement shown in Fig. 7 with the predicted pair of concentric double rings clearly visible. Theory is plotted against experiment in Fig. 8(b).

4. Conclusion

Paraxial conical diffraction theory can be extended to cases involving more than one biaxial crystal as well as biaxial crystals separated by wave plates in a relatively straightforward manner. The dependence of beam shape on both relative crystal orientation and both polarisation of the incident beam as well as manipulation of the polarisation between crystals facilitates the generation of a wide variety of beam geometries. The dependence of the conically diffracted beams on Gaussian waist and geometric ring radius explored in other publications can be combined with the effects explored in this paper further adding to the versatility of the biaxial crystal as a beam shaping device.

Acknowledgments

We acknowledge the support of Science Foundation Ireland under research grant numbers 08/IN.1/I1862 and SFI/SIRG/I1592.

References and links

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]

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(260.1180) Physical optics : Crystal optics
(260.1440) Physical optics : Birefringence

ToC Category:
Physical Optics

History
Original Manuscript: April 27, 2012
Revised Manuscript: May 20, 2012
Manuscript Accepted: May 21, 2012
Published: May 25, 2012

Citation
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, 13201-13207 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-13201


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References

  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. News37(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. A462(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. Express17(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. Express19(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. Express18(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. B103(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. B68(2), 187–197 (1999). [CrossRef]

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