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
Shortly after prediction of the conical refraction (CR) phenomenon , Lloyd observed the hollow light cone using a natural biaxial crystal and sunlight . 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 [2–28]. 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 . 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 [5–9]. Finally, the beam returns to the original profile in the far field. The Lloyd plane is also a symmetry plane , 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 . Another feature is related to the longitudinal shift of the Lloyd plane. The longitudinal shift, Δ in Fig. 1(a), is given by [13–15, 26]:
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 .
Following our recent observations based on this effect , 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.
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 . This in turn should severely affect the efficiency of the laser output, with the theoretical maximum varying from 37% to 71% .
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.
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 :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.
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  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.
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” .
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).
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