Abstract

In 1832 Hamilton predicted conical refraction, concluding that if a beam propagates along an optic axis of a biaxial crystal, a hollow cone of light will emerge. Nearly two centuries on, cascade conical refraction involving multiple crystals has not been investigated. We empirically investigate a unique two-crystal configuration, and use this to demonstrate an ultra-efficient conical refraction Nd:KGd(WO4)2 laser providing multi-watt output with excellent beam quality independent of resonator design with a slope efficiency close to the theoretical maximum, offering a new route for power and brightness-scaling in solid-state bulk lasers.

©2010 Optical Society of America

1. Introduction

Shortly after prediction of the conical refraction (CR) phenomenon [1], Lloyd observed the hollow light cone using a natural biaxial crystal and sunlight [2]. Modern studies refer to conical refraction as conical diffraction since its theoretical description requires the inclusion of wave effects [3, 4]. The optical effects and devices based on conical refraction phenomenon are of fundamental and practical importance in the field of photonics since most of the known crystal structures are optically biaxial. Yet there are relatively few studies of the phenomenon available [228]. Recent interest is driven by the availability of modern crystal growth, cutting and polishing technologies having advanced to a stage where producing crystals with the correct orientation is now possible.

To observe the CR phenomenon one must cut an optically biaxial crystal perpendicular to one of its optic axes. The spatial evolution of an incident Gaussian beam and its transformation under the effect of CR is shown in Fig. 1(a) . The light ring is observed at the Lloyd plane, which is also called the focal image plane [4]. After the Lloyd plane the beam then progresses to a series of rings first observed by Poggendorff, before evolving to an axial spike first noted by Raman [59]. Finally, the beam returns to the original profile in the far field. The Lloyd plane is also a symmetry plane [26], Fig. 1(b). The centre of the ring in the Lloyd plane is laterally shifted by an amount, denoted here by C, which depends on the crystal length, d, and a factor representing the crystal’s ability for conical refraction [3, 26]. The direction of this lateral shift can be defined as a property of the crystal orientation. A pseudovector, Λ, can also be empirically defined as being perpendicular to both the beam propagation direction and the direction of the lateral shift obeying a right hand rule [26]. Another feature is related to the longitudinal shift of the Lloyd plane. The longitudinal shift, Δ in Fig. 1(a), is given by [1315, 26]:

Δ=d(11n),
where n is the refractive index of the crystal in the propagation direction of the photons.

 figure: Fig. 1

Fig. 1 The features of single-crystal conical refraction: (a) spatial evolution of a focused beam upon its passage along the optic axis of a biaxial crystal. The dashed red lines are the imagined continuation of the beam to its focus. (b) The Lloyd plane is also a symmetry plane. Existence of two focal planes can be seen. Here, an expanded beam from a Helium Neon laser was brought to a focus by a lens (f=100 mm). The focal spot size was measured to be 18 μm in radius. A Nd:KGd(WO4)2 crystal cut for CR and of length d = 17 mm, was positioned before the focus of the incident beam so that the beam radius on the entrance facet was measured to be ~ 375 μm. Further alignment by rotating the crystal about an axis perpendicular to the plane of the optic axes results in the direct observation on a charge-coupled device (CCD) with no imaging optics of a ring pattern located at the Lloyd plane. The spatial evolution of the beam in free space was measured by recording CCD images at 2 mm intervals. The entire evolution occurred over a distance of ~ 30 mm either side of the Lloyd ring plane.

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Cascade conical refraction (CCR) represents the passage of a beam through two or more CR crystals. In contrast to the single crystal conical refraction, the transformation rules of the CCR are yet to be theoretically investigated. Recently, we presented our observations on CCR consisting of two CR crystals of different length [26].

Following our recent observations based on this effect [28], we demonstrate an ultra-efficient conical refraction Nd:KGd(WO4)2 laser, disobeying Gaussian laser theory. We empirically investigate a unique two-crystal CCR configuration, which can help to understand the behavior of the laser. Based on the experimental observations, we conclude that such “conical refraction lasers” (CRLs) are unusually stable, with a cavity mode that self-adapts to the pump beam mode in a very simple resonator. CRLs produce excellent beam quality and operate with a slope efficiency limited only by quantum defect heating in the gain medium and cavity losses.

2. Experimental methods

In order to prepare the conical refraction crystal active medium, a Nd:KGd(WO4)2 crystal, commonly known as Nd:KGW, with dimensions of 3 mm × 4 mm × 17 mm was cut with measured angular misalignment between the optic axis and the surface normal of less than 2 mrad. The volume concentration of the Nd3+ ions in the crystal was 1.9 × 1020 cm−3 (3 at. % doping). The end facets of the crystal were antireflection coated at both the expected lasing wavelength of 1067 nm and optical pumping wavelength of 808 nm.

The laser cavity was formed around the active medium (mounted in a metallic water-cooled holder at 20 °C) using a concave mirror (input mirror) with high reflectivity at the lasing wavelength and a flat mirror (output coupler) with 3% transmission at the lasing wavelength. The cavity was end-pumped by an unpolarised, multimode (core diameter of 100 µm, M2 = 40) fiber coupled diode laser at a wavelength of 808 nm focused on the entrance facet of the crystal. The pump mode diameter was varied using a two-lens system. To ensure alignment for conical refraction the CR active medium was first aligned with a He-Ne laser until the Lloyd ring was observed on a CCD. The pump was then adjusted to be co-linear with the Helium Neon beam until the observation of a Lloyd ring on the CCD from the pump beam. When the laser was operating above threshold the cavity mirrors were further adjusted until the laser output was also co-linear with the He-Ne beam. The beam quality of the different CRLs were assessed using two techniques; by coupling the laser output into a single mode fiber with a core diameter of 6 μm using an aspheric lens with a focal length of 6.2 mm, and also by using a standard z-scan technique with a CCD based beam profiler to obtain the beam quality factor (M2).

For demonstration of the CCR, two CR crystals of identical lengths (22 mm ± 20 nm) were mounted with opposite orientations of their psudovectors Λ relative to each other. The collimated, unmodified output of a Helium Neon laser with a beam diameter of 1.5 mm was passed through both crystals. The first crystal was located 15 cm from the laser and the second a further 10 cm away. Each crystal was individually aligned for CR by observing the Lloyd ring on a CCD with no imaging optics located 40 cm from the laser. When both crystals were in the beam path the final output was recorded on the CCD and compared with the profile of the original beam from the Helium Neon laser.

3. Results and discussion

The laser was end-pumped through the concave high reflector using a multimode fibre coupled diode laser at 808 nm (Fig. 2 ). In the initial experiment the total mirror separation was set to 50 mm. The laser provided 3.3 W of output power at 1067 nm for 5 W of incident pump power at 808 nm, with a lasing threshold of 400 mW of input pump power. The single pass pump absorption in the Nd:KGW was measured to be 98% of the incident pump power. The linear fit to the measured values for output power versus incident pump power revealed an optical-to-optical slope efficiency of 74% (Fig. 3 ). The quantum defect of the crystal, defined as the proportion of the pumping photon energy that is not turned into the lasing photon energy, is approximately 24%. This limits the theoretical maximum efficiency achievable for this gain medium. Hence, we conclude that the efficiency of the laser was only limited by the very low (~ 0.1%) cavity loss that is not transmitted through the output coupler. We observed a single, linearly polarized output with a circularly symmetric distribution profile with measured M2 ≤ 1 (Fig. 3-insert) As further evidence of the excellent beam quality, we were able to couple over 85% of the laser beam at full power into a single mode fibre. The polarization of the output beam was measured to be along the principal dielectric axis Nm, i.e. along the largest stimulated emission cross-section of Nd3+ in KGW.

 figure: Fig. 2

Fig. 2 Conical refraction laser cavity. HR is the 75 mm radius of curvature concave high reflector through which the active medium was optically pumped along the 17 mm axis. OC is the plane output coupler with a transmission of 3% at the lasing wavelength. For all experiments the pump beam was focused on the entrance facet of the crystal. L denotes the distance between the input mirror and the entrance facet of the crystal. This distance was varied in the second set of the experiments while the pump beam focus was kept on the entrance facet of the crystal. In the third set of experiments the cavity was set to 50 mm and left unchanged whilst the pump mode diameter was varied between 235 and 1570 µm.

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 figure: Fig. 3

Fig. 3 Performance of the conical refraction laser. Measured output power versus incident pump power for two laser cavities of different lengths. The pump mode diameter was set to 400 µm. Characteristics of the 50 mm cavity are shown in black and the 80 mm cavity in red. Both configurations have a slope efficiency of 74%. Inset is the 3-D profile of the laser output from the 50 mm cavity at the maximum output.

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We investigated the dependence of the laser output characteristics on the laser cavity mirror separation. The total mirror separation was varied from 50 mm to 80 mm. The efficiency and output beam profile remained constant, independent of the cavity length (Fig. 3). Furthermore the position of the crystal, L in Fig. 2, was varied in the 80 mm mirror separation cavity, from the crystal entrance facet being ~ 5 mm from the curved input mirror, to the other end of the cavity being ~ 60 mm from the input mirror. Provided that the pump focus was maintained on the entrance facet of the crystal and the laser was correctly aligned, no variation in the performance of the laser in terms of output power and beam profile was observed.

This behaviour is in stark contrast to the established Gaussian laser theory, where the diameter of the fundamental Gaussian cavity mode varies strongly with the different cavity configurations demonstrated here. Simple calculations assuming a Gaussian cavity mode show that the mode diameter in the crystal varies from ~ 160 µm to over 450 µm upon the change of the position of the crystal inside the cavity [29]. This in turn should severely affect the efficiency of the laser output, with the theoretical maximum varying from 37% to 71% [30].

We have also investigated the laser performance for various pump mode diameters of 235, 400, 800, 1140 and 1570 µm shown in Fig. 4(a) . The best performance was obtained for both the 235 and 400 µm pump mode diameters with a slope efficiency of ~ 74%. As the pump diameter increases beyond 400 µm the efficiency decreases in a near-linear fashion, as can be seen in Fig. 4(b). It is clear that in all cases the efficiency of the laser exceeds that of the maximum theoretical Gaussian efficiency. The beam quality of the lasers was assessed and in all cases the measured M2 ≤ 1 was observed.

 figure: Fig. 4

Fig. 4 Performance of the conical refraction laser for various pump mode diameters on the crystal. (a) The length of the cavity was always 50 mm. Lasing action was observed with all pump mode diameters investigated. A maximum output power of 3.3 W with an incident pump power of 5 W was obtained for both the 235 and 400 µm pump mode diameters. This corresponds to a slope efficiency of ~ 74%. As the pump diameter increases beyond 400 µm the efficiency of the laser decreases in a near-linear fashion as shown in next figure. (b) Graph of measured and calculated laser slope efficiency versus pump mode diameters on the crystal. The black squares show the measured slope efficiency of the CR laser. The calculated theoretical maximum slope efficiency of the lasers if they were operating as Gaussian lasers are shown in red dots - dashed line. The blue triangles represent the calculated theoretical maximum slope efficiency where only the losses due to the quantum defect heating in the active medium and cavity losses are considered.

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Our comparative experimental studies undertaken using an ordinary Nd:KGW crystal with identical dimensions to the Nd:KGW cut for CR are in agreement with the observations. A maximum output power of 800 mW was achieved from this conventional laser whilst maintaining a good beam quality when the laser was fully optimised. Above this output power level the beam quality rapidly degraded. This is in contrast to our observation of the CRL providing constant slope efficiency and excellent beam quality irrespective of cavity configurations.

The maximum theoretical slope efficiency, ηs, of a laser is governed by [30]:

ηs=ηpηcηtηq,
where ηp is the pump efficiency, ηc is the cavity efficiency, ηt is the transverse efficiency and ηq is the quantum efficiency. The pump power was measured at the front facet of the crystal with a single pass absorption of 98%, hence ηp = 0.98. The maximum quantum efficiency achievable with Nd:KGW, ηq = 0.757. The cavity efficiency (ηc) is reduced by cavity losses that do not contribute to the output power, e.g. scattering losses and crystal defects. These losses increase with the increasing mode area (Fig. 4 (b)-blue triangles).

It is clear that the efficiency we experimentally observe is only limited by these losses and the maximum quantum efficiency. The transverse efficiency, which is a measure of mismatch between pump and laser modes is of paramount importance in solid-state bulk lasers. However, there is no dependency in the case of CRLs, as ηt must equal unity for the experimentally observed slope efficiencies to be physically possible. Therefore, the CR laser mode is self-adapting to the pump beam mode and the cavity mode, hence rules governing the operation of CRLs are different.

The behaviour of the laser can be partially understood should one considers a unique cascade scheme consisting of two CR crystals of identical length (to within 20 nm), with opposite orientations of the pseudovector Λ (Fig. 5 ). When both crystals are correctly aligned, after the first crystal the Lloyd ring can be observed. However, after the beam passes through both crystals, the observed beam has an identical beam profile to the initial Gaussian laser beam before the first crystal. We consider this observation as direct evidence of transformation of an annular beam between the two crystals into the original Guassian beam after the second crystal.

 figure: Fig. 5

Fig. 5 Two-crystal cascade conical refraction. Experimental setup of cascade conical refraction where the two CR crystals have identical length and are orientated with opposite orientations of the pseudovector Λ. When both crystals are in place in the cascade scheme, after the first crystal the Lloyd ring was observed but after the second crystal instead of observing a Lloyd ring, the observed beam had an identical beam profile to the initial Gaussian beam before the first crystal. This is direct evidence of transformation of an annular beam between the two CR crystals to the original Guassian beam after the second crystal (CR 2).

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In our laser, the plane output coupler introduces a π phase shift to the reflected 1067 nm laser beam. As the reflected beam passes back through the gain medium, this phase shift has the same effect as changing the crystal’s pseudovector orientation to its opposite value.

In a previous attempt to demonstrate a laser based on Yb:KGW [25] cut for CR, the authors did not observe the fundamental difference in the behaviour of their lasers in comparison to an ordinary laser. We believe this is due to the use of a curved output coupler as well as the quality of the crystal cut with an estimated angular misalignment between the optic axis and the surface normal of approximately 100 mrad as compared to our value of ~2 mrad.

4. Conclusion

In summary, we demonstrated a unique case of cascade conical refraction, which forms the basis of the presented CR lasers. Our results show that a laser based on CR active medium, i.e. a typical rare earth ion-doped biaxial crystal gain medium cut for beam propagation along the CR axis, provides a new generation of ultra-efficient solid-state lasers with the prospect for power and brightness scaling.

We provided a descriptive explanation of the phenomenon based on the experimental observations. However, the exact theory of laser operation based on conical refraction is not yet well understood. To determine the factors responsible for the unusual observations require a further detailed experimental and theoretical study of the transformation rules of cascade conical refraction, which cannot be deduced by analogy with the known optical effects.

We believe that our observations are of paramount importance in the field of photonics, especially since conical refraction has traditionally been considered as “little more than a curious optical phenomenon which had no conceivable application” [31].

Acknowledgments

This work was partially conducted under the aegis of the Advanced Components Cooperation for Optoelectronics Research and Development (ACCORD), funded by the European Union (WP-4, Project 310).

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 (1833).

2. H. Lloyd, “On the phenomenon presented by light in its passage along the axis of biaxial crystals,” Trans. R. Irish Acad. 17, 145–158 (1833).

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

4. M. V. Berry, “Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A 6, 289–300 (2004).

5. C. V. Raman, V. S. Rajagopalan, and T. M. K. Nedungadi, “Conical refraction in naphthalene crystals,” Proc. Indian Ins. Sci. A 14, 221–227 (1941).

6. C. V. Raman and T. M. K. Nedungadi, “Optical images formed by conical refraction,” Nature 149(3785), 552–553 (1942). [CrossRef]  

7. J. C. Poggendorff, “Ueber die konische Refraction,” Pogg. Ann. 124(11), 461–462 (1839). [CrossRef]  

8. R. Potter, “An examination of the phaenomena of conical refraction in biaxial crystals,” Philos. Mag. 18, 343–353 (1841).

9. W. Haidinger, “Die konische Refraction am Diopsid, nebst Bemerkungen ber einige Erscheinungen der konischen Refraction an Arragonit,” Ann. Phys. Chem. 172(11), 469–487 (1855). [CrossRef]  

10. S. Melmore, “Conical refraction,” Nature 150(3804), 382–383 (1942). [CrossRef]  

11. D. L. Portigal and E. Burstein, “Effect of optical activity or Faraday rotation on internal conical refraction,” J. Opt. Soc. Am. 62(7), 859–864 (1972). [CrossRef]  

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

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

14. A. M. Belsky and M. A. Stepanov, “Internal conical refraction of coherent light beams,” Opt. Commun. 167(1-6), 1–5 (1999). [CrossRef]  

15. A. M. Belsky and M. A. Stepanov, “Internal conical refraction of light beams in biaxial gyrotropic crystals,” Opt. Commun. 204(1-6), 1–6 (2002). [CrossRef]  

16. M. V. Berry and M. R. Jeffrey, “Chiral conical diffraction,” J. Opt. A 8, 363–372 (2006).

17. M. V. Berry and M. R. Jeffrey, “Conical diffraction complexified: dichroism and the transition to double refraction,” J. Opt. A 8, 1043–1051 (2006).

18. 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]  

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

20. N. Bloembergen and H. Shih, “Conical refraction in nonlinear optics,” Opt. Commun. 1(2), 70–72 (1969). [CrossRef]  

21. J. P. F`eve, B. Boulanger, and G. Marnier, “Experimental study of internal and external conical refraction in ktp,” Opt. Commun. 105, 243–252 (1994). [CrossRef]  

22. M. R. Jeffrey, “The spun cusp complexified: complex ray focusing in chiral conical diffraction,” J. Opt. A 9, 634–641 (2007).

23. O. N. Naida, “Tangential conical refraction in a three-dimensional inhomogeneous weakly anisotropic medium,” Sov. Phys. JETP 50, 239–245 (1979).

24. B. S. Perkal’skis and Y. P. Mikhailichenko, “Demonstration of conical refraction,” Izv Vyss Uch Zav Fiz 8, 103–105 (1979).

25. J. Hellström, H. Henricsson, V. Pasiskevicius, U. Bünting, and D. Haussmann, “Polarization-tunable Yb:KGW laser based on internal conical refraction,” Opt. Lett. 32(19), 2783–2785 (2007). [CrossRef]   [PubMed]  

26. T. K. Kalkandjiev, and M. A. Bursukova, “Conical refraction: an experimental introduction,” Proc. SPIE 6994, 69940B1–69940B10 (2008).

27. 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), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-12891. [CrossRef]   [PubMed]  

28. A. Abdolvand, K. G. Wilcox, T. K. Kalkandjiev, and E. U. Rfailov, “Solid-state conical refraction laser,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science and Photonic Application Systemes Technologies, Technical Digest (CD) (Optical Society of America, 2009), Post-deadline paper CPDB1. http://www.opticsinfobase.org/abstract.cfm?URI=CLEO-2009-CPDB1

29. W. Koechner, Solid-State Laser Engineering, 5th ed. (Springer-Verlag, 1999), pp. 198–205.

30. O. Svelto, Principles of lasers, 4th ed. (Plenum Press, 1998), pp. 249–272.

31. J. G. O’Hara, “The prediction and discovery of conical refraction by William Rowan Hamilton and Humphrey Lloyd (1832-1833),” Proc. R. Ir. Acad. [B] 82A, 231–257 (1982).

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, 1–144 (1833).
  2. H. Lloyd, “On the phenomenon presented by light in its passage along the axis of biaxial crystals,” Trans. R. Irish Acad. 17, 145–158 (1833).
  3. M. V. Berry, M. R. Jeffrey, and J. L. Lunney, “Conical diffraction: observations and theory,” Proc. R Soc. A 462(2070), 1629–1642 (2006).
    [Crossref]
  4. M. V. Berry, “Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A 6, 289–300 (2004).
  5. C. V. Raman, V. S. Rajagopalan, and T. M. K. Nedungadi, “Conical refraction in naphthalene crystals,” Proc. Indian Ins. Sci. A 14, 221–227 (1941).
  6. C. V. Raman and T. M. K. Nedungadi, “Optical images formed by conical refraction,” Nature 149(3785), 552–553 (1942).
    [Crossref]
  7. J. C. Poggendorff, “Ueber die konische Refraction,” Pogg. Ann. 124(11), 461–462 (1839).
    [Crossref]
  8. R. Potter, “An examination of the phaenomena of conical refraction in biaxial crystals,” Philos. Mag. 18, 343–353 (1841).
  9. W. Haidinger, “Die konische Refraction am Diopsid, nebst Bemerkungen ber einige Erscheinungen der konischen Refraction an Arragonit,” Ann. Phys. Chem. 172(11), 469–487 (1855).
    [Crossref]
  10. S. Melmore, “Conical refraction,” Nature 150(3804), 382–383 (1942).
    [Crossref]
  11. D. L. Portigal and E. Burstein, “Effect of optical activity or Faraday rotation on internal conical refraction,” J. Opt. Soc. Am. 62(7), 859–864 (1972).
    [Crossref]
  12. A. Belafhal, “Theoretical intensity distribution of internal conical refraction,” Opt. Commun. 178(4-6), 257–265 (2000).
    [Crossref]
  13. A. M. Belskii and A. P. Khapalyuk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt Spectrosc (USSR) 44, 436–439 (1978).
  14. A. M. Belsky and M. A. Stepanov, “Internal conical refraction of coherent light beams,” Opt. Commun. 167(1-6), 1–5 (1999).
    [Crossref]
  15. A. M. Belsky and M. A. Stepanov, “Internal conical refraction of light beams in biaxial gyrotropic crystals,” Opt. Commun. 204(1-6), 1–6 (2002).
    [Crossref]
  16. M. V. Berry and M. R. Jeffrey, “Chiral conical diffraction,” J. Opt. A 8, 363–372 (2006).
  17. M. V. Berry and M. R. Jeffrey, “Conical diffraction complexified: dichroism and the transition to double refraction,” J. Opt. A 8, 1043–1051 (2006).
  18. 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]
  19. M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A 7, 685–690 (2005).
  20. N. Bloembergen and H. Shih, “Conical refraction in nonlinear optics,” Opt. Commun. 1(2), 70–72 (1969).
    [Crossref]
  21. J. P. F`eve, B. Boulanger, and G. Marnier, “Experimental study of internal and external conical refraction in ktp,” Opt. Commun. 105, 243–252 (1994).
    [Crossref]
  22. M. R. Jeffrey, “The spun cusp complexified: complex ray focusing in chiral conical diffraction,” J. Opt. A 9, 634–641 (2007).
  23. O. N. Naida, “Tangential conical refraction in a three-dimensional inhomogeneous weakly anisotropic medium,” Sov. Phys. JETP 50, 239–245 (1979).
  24. B. S. Perkal’skis and Y. P. Mikhailichenko, “Demonstration of conical refraction,” Izv Vyss Uch Zav Fiz 8, 103–105 (1979).
  25. J. Hellström, H. Henricsson, V. Pasiskevicius, U. Bünting, and D. Haussmann, “Polarization-tunable Yb:KGW laser based on internal conical refraction,” Opt. Lett. 32(19), 2783–2785 (2007).
    [Crossref] [PubMed]
  26. T. K. Kalkandjiev, and M. A. Bursukova, “Conical refraction: an experimental introduction,” Proc. SPIE 6994, 69940B1–69940B10 (2008).
  27. 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), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-12891 .
    [Crossref] [PubMed]
  28. A. Abdolvand, K. G. Wilcox, T. K. Kalkandjiev, and E. U. Rfailov, “Solid-state conical refraction laser,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science and Photonic Application Systemes Technologies, Technical Digest (CD) (Optical Society of America, 2009), Post-deadline paper CPDB1. http://www.opticsinfobase.org/abstract.cfm?URI=CLEO-2009-CPDB1
  29. W. Koechner, Solid-State Laser Engineering, 5th ed. (Springer-Verlag, 1999), pp. 198–205.
  30. O. Svelto, Principles of lasers, 4th ed. (Plenum Press, 1998), pp. 249–272.
  31. J. G. O’Hara, “The prediction and discovery of conical refraction by William Rowan Hamilton and Humphrey Lloyd (1832-1833),” Proc. R. Ir. Acad. [B] 82A, 231–257 (1982).

2009 (1)

2007 (3)

J. Hellström, H. Henricsson, V. Pasiskevicius, U. Bünting, and D. Haussmann, “Polarization-tunable Yb:KGW laser based on internal conical refraction,” Opt. Lett. 32(19), 2783–2785 (2007).
[Crossref] [PubMed]

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. R. Jeffrey, “The spun cusp complexified: complex ray focusing in chiral conical diffraction,” J. Opt. A 9, 634–641 (2007).

2006 (3)

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

M. V. Berry and M. R. Jeffrey, “Chiral conical diffraction,” J. Opt. A 8, 363–372 (2006).

M. V. Berry and M. R. Jeffrey, “Conical diffraction complexified: dichroism and the transition to double refraction,” J. Opt. A 8, 1043–1051 (2006).

2005 (1)

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

2004 (1)

M. V. Berry, “Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A 6, 289–300 (2004).

2002 (1)

A. M. Belsky and M. A. Stepanov, “Internal conical refraction of light beams in biaxial gyrotropic crystals,” Opt. Commun. 204(1-6), 1–6 (2002).
[Crossref]

2000 (1)

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

1999 (1)

A. M. Belsky and M. A. Stepanov, “Internal conical refraction of coherent light beams,” Opt. Commun. 167(1-6), 1–5 (1999).
[Crossref]

1994 (1)

J. P. F`eve, B. Boulanger, and G. Marnier, “Experimental study of internal and external conical refraction in ktp,” Opt. Commun. 105, 243–252 (1994).
[Crossref]

1982 (1)

J. G. O’Hara, “The prediction and discovery of conical refraction by William Rowan Hamilton and Humphrey Lloyd (1832-1833),” Proc. R. Ir. Acad. [B] 82A, 231–257 (1982).

1979 (2)

O. N. Naida, “Tangential conical refraction in a three-dimensional inhomogeneous weakly anisotropic medium,” Sov. Phys. JETP 50, 239–245 (1979).

B. S. Perkal’skis and Y. P. Mikhailichenko, “Demonstration of conical refraction,” Izv Vyss Uch Zav Fiz 8, 103–105 (1979).

1978 (1)

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

1972 (1)

1969 (1)

N. Bloembergen and H. Shih, “Conical refraction in nonlinear optics,” Opt. Commun. 1(2), 70–72 (1969).
[Crossref]

1942 (2)

S. Melmore, “Conical refraction,” Nature 150(3804), 382–383 (1942).
[Crossref]

C. V. Raman and T. M. K. Nedungadi, “Optical images formed by conical refraction,” Nature 149(3785), 552–553 (1942).
[Crossref]

1941 (1)

C. V. Raman, V. S. Rajagopalan, and T. M. K. Nedungadi, “Conical refraction in naphthalene crystals,” Proc. Indian Ins. Sci. A 14, 221–227 (1941).

1855 (1)

W. Haidinger, “Die konische Refraction am Diopsid, nebst Bemerkungen ber einige Erscheinungen der konischen Refraction an Arragonit,” Ann. Phys. Chem. 172(11), 469–487 (1855).
[Crossref]

1841 (1)

R. Potter, “An examination of the phaenomena of conical refraction in biaxial crystals,” Philos. Mag. 18, 343–353 (1841).

1839 (1)

J. C. Poggendorff, “Ueber die konische Refraction,” Pogg. Ann. 124(11), 461–462 (1839).
[Crossref]

1833 (2)

W. R. Hamilton, “Third supplement to an essay on the theory of systems of rays,” Trans. R. Irish Acad. 17, 1–144 (1833).

H. Lloyd, “On the phenomenon presented by light in its passage along the axis of biaxial crystals,” Trans. R. Irish Acad. 17, 145–158 (1833).

Belafhal, A.

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

Belskii, A. M.

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

Belsky, A. M.

A. M. Belsky and M. A. Stepanov, “Internal conical refraction of light beams in biaxial gyrotropic crystals,” Opt. Commun. 204(1-6), 1–6 (2002).
[Crossref]

A. M. Belsky and M. A. Stepanov, “Internal conical refraction of coherent light beams,” Opt. Commun. 167(1-6), 1–5 (1999).
[Crossref]

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 and M. R. Jeffrey, “Chiral conical diffraction,” J. Opt. A 8, 363–372 (2006).

M. V. Berry and M. R. Jeffrey, “Conical diffraction complexified: dichroism and the transition to double refraction,” J. Opt. A 8, 1043–1051 (2006).

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

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

M. V. Berry, “Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A 6, 289–300 (2004).

Bloembergen, N.

N. Bloembergen and H. Shih, “Conical refraction in nonlinear optics,” Opt. Commun. 1(2), 70–72 (1969).
[Crossref]

Boulanger, B.

J. P. F`eve, B. Boulanger, and G. Marnier, “Experimental study of internal and external conical refraction in ktp,” Opt. Commun. 105, 243–252 (1994).
[Crossref]

Bünting, U.

Burstein, E.

Donegan, J. F.

F`eve, J. P.

J. P. F`eve, B. Boulanger, and G. Marnier, “Experimental study of internal and external conical refraction in ktp,” Opt. Commun. 105, 243–252 (1994).
[Crossref]

Haidinger, W.

W. Haidinger, “Die konische Refraction am Diopsid, nebst Bemerkungen ber einige Erscheinungen der konischen Refraction an Arragonit,” Ann. Phys. Chem. 172(11), 469–487 (1855).
[Crossref]

Hamilton, W. R.

W. R. Hamilton, “Third supplement to an essay on the theory of systems of rays,” Trans. R. Irish Acad. 17, 1–144 (1833).

Haussmann, D.

Hellström, J.

Henricsson, H.

Jeffrey, M. R.

M. R. Jeffrey, “The spun cusp complexified: complex ray focusing in chiral conical diffraction,” J. Opt. A 9, 634–641 (2007).

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 and M. R. Jeffrey, “Conical diffraction complexified: dichroism and the transition to double refraction,” J. Opt. A 8, 1043–1051 (2006).

M. V. Berry and M. R. Jeffrey, “Chiral conical diffraction,” J. Opt. A 8, 363–372 (2006).

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

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

Khapalyuk, A. P.

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

Lloyd, H.

H. Lloyd, “On the phenomenon presented by light in its passage along the axis of biaxial crystals,” Trans. R. Irish Acad. 17, 145–158 (1833).

Lunney, J. G.

Lunney, J. L.

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

Mansuripur, M.

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

Marnier, G.

J. P. F`eve, B. Boulanger, and G. Marnier, “Experimental study of internal and external conical refraction in ktp,” Opt. Commun. 105, 243–252 (1994).
[Crossref]

Melmore, S.

S. Melmore, “Conical refraction,” Nature 150(3804), 382–383 (1942).
[Crossref]

Mikhailichenko, Y. P.

B. S. Perkal’skis and Y. P. Mikhailichenko, “Demonstration of conical refraction,” Izv Vyss Uch Zav Fiz 8, 103–105 (1979).

Naida, O. N.

O. N. Naida, “Tangential conical refraction in a three-dimensional inhomogeneous weakly anisotropic medium,” Sov. Phys. JETP 50, 239–245 (1979).

Nedungadi, T. M. K.

C. V. Raman and T. M. K. Nedungadi, “Optical images formed by conical refraction,” Nature 149(3785), 552–553 (1942).
[Crossref]

C. V. Raman, V. S. Rajagopalan, and T. M. K. Nedungadi, “Conical refraction in naphthalene crystals,” Proc. Indian Ins. Sci. A 14, 221–227 (1941).

O’Dwyer, D. P.

O’Hara, J. G.

J. G. O’Hara, “The prediction and discovery of conical refraction by William Rowan Hamilton and Humphrey Lloyd (1832-1833),” Proc. R. Ir. Acad. [B] 82A, 231–257 (1982).

Pasiskevicius, V.

Perkal’skis, B. S.

B. S. Perkal’skis and Y. P. Mikhailichenko, “Demonstration of conical refraction,” Izv Vyss Uch Zav Fiz 8, 103–105 (1979).

Phelan, C. F.

Poggendorff, J. C.

J. C. Poggendorff, “Ueber die konische Refraction,” Pogg. Ann. 124(11), 461–462 (1839).
[Crossref]

Portigal, D. L.

Potter, R.

R. Potter, “An examination of the phaenomena of conical refraction in biaxial crystals,” Philos. Mag. 18, 343–353 (1841).

Rajagopalan, V. S.

C. V. Raman, V. S. Rajagopalan, and T. M. K. Nedungadi, “Conical refraction in naphthalene crystals,” Proc. Indian Ins. Sci. A 14, 221–227 (1941).

Rakovich, Y. P.

Raman, C. V.

C. V. Raman and T. M. K. Nedungadi, “Optical images formed by conical refraction,” Nature 149(3785), 552–553 (1942).
[Crossref]

C. V. Raman, V. S. Rajagopalan, and T. M. K. Nedungadi, “Conical refraction in naphthalene crystals,” Proc. Indian Ins. Sci. A 14, 221–227 (1941).

Shih, H.

N. Bloembergen and H. Shih, “Conical refraction in nonlinear optics,” Opt. Commun. 1(2), 70–72 (1969).
[Crossref]

Stepanov, M. A.

A. M. Belsky and M. A. Stepanov, “Internal conical refraction of light beams in biaxial gyrotropic crystals,” Opt. Commun. 204(1-6), 1–6 (2002).
[Crossref]

A. M. Belsky and M. A. Stepanov, “Internal conical refraction of coherent light beams,” Opt. Commun. 167(1-6), 1–5 (1999).
[Crossref]

Ann. Phys. Chem. (1)

W. Haidinger, “Die konische Refraction am Diopsid, nebst Bemerkungen ber einige Erscheinungen der konischen Refraction an Arragonit,” Ann. Phys. Chem. 172(11), 469–487 (1855).
[Crossref]

Izv Vyss Uch Zav Fiz (1)

B. S. Perkal’skis and Y. P. Mikhailichenko, “Demonstration of conical refraction,” Izv Vyss Uch Zav Fiz 8, 103–105 (1979).

J. Opt. A (5)

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

M. R. Jeffrey, “The spun cusp complexified: complex ray focusing in chiral conical diffraction,” J. Opt. A 9, 634–641 (2007).

M. V. Berry and M. R. Jeffrey, “Chiral conical diffraction,” J. Opt. A 8, 363–372 (2006).

M. V. Berry and M. R. Jeffrey, “Conical diffraction complexified: dichroism and the transition to double refraction,” J. Opt. A 8, 1043–1051 (2006).

M. V. Berry, “Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A 6, 289–300 (2004).

J. Opt. Soc. Am. (1)

Nature (2)

S. Melmore, “Conical refraction,” Nature 150(3804), 382–383 (1942).
[Crossref]

C. V. Raman and T. M. K. Nedungadi, “Optical images formed by conical refraction,” Nature 149(3785), 552–553 (1942).
[Crossref]

Opt Spectrosc (USSR) (1)

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

Opt. Commun. (5)

A. M. Belsky and M. A. Stepanov, “Internal conical refraction of coherent light beams,” Opt. Commun. 167(1-6), 1–5 (1999).
[Crossref]

A. M. Belsky and M. A. Stepanov, “Internal conical refraction of light beams in biaxial gyrotropic crystals,” Opt. Commun. 204(1-6), 1–6 (2002).
[Crossref]

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

N. Bloembergen and H. Shih, “Conical refraction in nonlinear optics,” Opt. Commun. 1(2), 70–72 (1969).
[Crossref]

J. P. F`eve, B. Boulanger, and G. Marnier, “Experimental study of internal and external conical refraction in ktp,” Opt. Commun. 105, 243–252 (1994).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Philos. Mag. (1)

R. Potter, “An examination of the phaenomena of conical refraction in biaxial crystals,” Philos. Mag. 18, 343–353 (1841).

Pogg. Ann. (1)

J. C. Poggendorff, “Ueber die konische Refraction,” Pogg. Ann. 124(11), 461–462 (1839).
[Crossref]

Proc. Indian Ins. Sci. A (1)

C. V. Raman, V. S. Rajagopalan, and T. M. K. Nedungadi, “Conical refraction in naphthalene crystals,” Proc. Indian Ins. Sci. A 14, 221–227 (1941).

Proc. R Soc. A (1)

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

Proc. R. Ir. Acad. [B] (0)

J. G. O’Hara, “The prediction and discovery of conical refraction by William Rowan Hamilton and Humphrey Lloyd (1832-1833),” Proc. R. Ir. Acad. [B] 82A, 231–257 (1982).

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]

Sov. Phys. JETP (1)

O. N. Naida, “Tangential conical refraction in a three-dimensional inhomogeneous weakly anisotropic medium,” Sov. Phys. JETP 50, 239–245 (1979).

Trans. R. Irish Acad. (2)

W. R. Hamilton, “Third supplement to an essay on the theory of systems of rays,” Trans. R. Irish Acad. 17, 1–144 (1833).

H. Lloyd, “On the phenomenon presented by light in its passage along the axis of biaxial crystals,” Trans. R. Irish Acad. 17, 145–158 (1833).

Other (4)

A. Abdolvand, K. G. Wilcox, T. K. Kalkandjiev, and E. U. Rfailov, “Solid-state conical refraction laser,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science and Photonic Application Systemes Technologies, Technical Digest (CD) (Optical Society of America, 2009), Post-deadline paper CPDB1. http://www.opticsinfobase.org/abstract.cfm?URI=CLEO-2009-CPDB1

W. Koechner, Solid-State Laser Engineering, 5th ed. (Springer-Verlag, 1999), pp. 198–205.

O. Svelto, Principles of lasers, 4th ed. (Plenum Press, 1998), pp. 249–272.

T. K. Kalkandjiev, and M. A. Bursukova, “Conical refraction: an experimental introduction,” Proc. SPIE 6994, 69940B1–69940B10 (2008).

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Figures (5)

Fig. 1
Fig. 1 The features of single-crystal conical refraction: (a) spatial evolution of a focused beam upon its passage along the optic axis of a biaxial crystal. The dashed red lines are the imagined continuation of the beam to its focus. (b) The Lloyd plane is also a symmetry plane. Existence of two focal planes can be seen. Here, an expanded beam from a Helium Neon laser was brought to a focus by a lens (f=100 mm). The focal spot size was measured to be 18 μm in radius. A Nd:KGd(WO4)2 crystal cut for CR and of length d = 17 mm, was positioned before the focus of the incident beam so that the beam radius on the entrance facet was measured to be ~ 375 μm. Further alignment by rotating the crystal about an axis perpendicular to the plane of the optic axes results in the direct observation on a charge-coupled device (CCD) with no imaging optics of a ring pattern located at the Lloyd plane. The spatial evolution of the beam in free space was measured by recording CCD images at 2 mm intervals. The entire evolution occurred over a distance of ~ 30 mm either side of the Lloyd ring plane.
Fig. 2
Fig. 2 Conical refraction laser cavity. HR is the 75 mm radius of curvature concave high reflector through which the active medium was optically pumped along the 17 mm axis. OC is the plane output coupler with a transmission of 3% at the lasing wavelength. For all experiments the pump beam was focused on the entrance facet of the crystal. L denotes the distance between the input mirror and the entrance facet of the crystal. This distance was varied in the second set of the experiments while the pump beam focus was kept on the entrance facet of the crystal. In the third set of experiments the cavity was set to 50 mm and left unchanged whilst the pump mode diameter was varied between 235 and 1570 µm.
Fig. 3
Fig. 3 Performance of the conical refraction laser. Measured output power versus incident pump power for two laser cavities of different lengths. The pump mode diameter was set to 400 µm. Characteristics of the 50 mm cavity are shown in black and the 80 mm cavity in red. Both configurations have a slope efficiency of 74%. Inset is the 3-D profile of the laser output from the 50 mm cavity at the maximum output.
Fig. 4
Fig. 4 Performance of the conical refraction laser for various pump mode diameters on the crystal. (a) The length of the cavity was always 50 mm. Lasing action was observed with all pump mode diameters investigated. A maximum output power of 3.3 W with an incident pump power of 5 W was obtained for both the 235 and 400 µm pump mode diameters. This corresponds to a slope efficiency of ~ 74%. As the pump diameter increases beyond 400 µm the efficiency of the laser decreases in a near-linear fashion as shown in next figure. (b) Graph of measured and calculated laser slope efficiency versus pump mode diameters on the crystal. The black squares show the measured slope efficiency of the CR laser. The calculated theoretical maximum slope efficiency of the lasers if they were operating as Gaussian lasers are shown in red dots - dashed line. The blue triangles represent the calculated theoretical maximum slope efficiency where only the losses due to the quantum defect heating in the active medium and cavity losses are considered.
Fig. 5
Fig. 5 Two-crystal cascade conical refraction. Experimental setup of cascade conical refraction where the two CR crystals have identical length and are orientated with opposite orientations of the pseudovector Λ. When both crystals are in place in the cascade scheme, after the first crystal the Lloyd ring was observed but after the second crystal instead of observing a Lloyd ring, the observed beam had an identical beam profile to the initial Gaussian beam before the first crystal. This is direct evidence of transformation of an annular beam between the two CR crystals to the original Guassian beam after the second crystal (CR 2).

Equations (2)

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Δ = d ( 1 1 n ) ,
η s = η p η c η t η q ,

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