## Abstract

The far-field pattern of Gaussian beams transformed by conical refraction in biaxial crystal is analyzed. It is shown that one of the two outgoing beam components acquires, under certain conditions, a profile with a dominating central peak. The width of this peak can be made significantly smaller than the width of the parent diffraction-limited Gaussian beam at the same propagation distance. The formation of such structurally-stable sub-diffraction beam core improves the beam directivity. Another component is a charge-one optical vortex, that forms the annular shell of the beam and carries the rest of the beam power.

©2010 Optical Society of America

## 1. Introduction

The concept of “diffraction-free” propagation of laser beam has been extensively studied during the last decades. After the first demonstration of diffraction-free properties of zero-order Bessel beams [1, 2], several techniques have been developed to produce such beams possessing a considerably enhanced directivity as compared with the ordinary Gaussian beams. The ideal Bessel beam with an infinitely long line focus cannot physically exist because it would have an infinite transverse extent and an infinite energy. Therefore, in laboratory realizations, the length of the diffraction-free region of Bessel beams is always limited. Beyond this length, the Bessel beam profile decays abruptly and transforms into a diverging cone with a ring-shaped intensity distribution. This feature of physically realizable Bessel beams is in a sharp contrast with Gaussian beams, which preserve their profile under propagation. Another feature of Bessel and other conical beams is that they usually require a relatively large aperture of the input beam and, for a given cone angle, the longer is the diffraction-free region, the larger is input beam size needed to maintain the line focus under propagation.

Recently, it was demonstrated [3] that the problem of an abrupt change of intensity at the boundary of the non-diffracting region can be solved with the aid of a combination of lenses and axicons that produces a conical superposition of waves with a *z*-dependent cone angle. It allows for the formation Bessel-like beams with a much greater quasi non-diffracting propagation. However, such Bessel-like beams with sub-diffraction structures were again obtained at the expense of an increased diameter of the outgoing beam, when the overall transverse size of the Bessel-like beam as a whole exceeded significantly the size of the initial unperturbed Gaussian beam. In this context, it would be interesting to analyze the possibility of transforming a Gaussian beam into some other light configuration which, after some evolution upon propagation, would have a structurally-stable sub-diffraction profile, but without a transverse spread beyond the profile of the parent Gaussian beam at the same propagation distance. In this paper, the formation of such embedded sub-diffraction structures of a Gaussian beam is outlined theoretically and demonstrated experimentally.

## 2. Theory

Usually, the beams with non-diffracting or quasi non-diffracting propagation are formed from conical superposition of multiple waves, all inclined with respect to the propagation axis. For a Bessel beam, this is a superposition of plane waves, and for a Bessel-like beam with a *z*-dependent cone angle this is a superposition of diverging spherical waves. The latter case was considered previously in [4] and termed as generalized Bessel-Gauss beams. All these conical superpositions lead to an inevitable spread of the produced light configuration off the beam axis since the propagation directions of individual waves are inclined with respect to the common propagation direction. To avoid this walk-off effect which increases the overall beam size, the inclination angle of the light cone must be set to zero. It gives a cylindrical superposition of waves where all individual waves have their wave vectors directed parallel to the beam axis. For such geometry, the beam as a whole diverges upon propagation entirely due to diffraction without any additional walk-off effects.

With an input Gaussian beam, the required cylindrical superposition is obtained if an optical system transforms the input beam into an annular superposition of multiple Gaussian beams with wave vectors parallel to the propagation axis, and the centers of individual Gaussian beams are placed along a circumference with radius *a* (see Fig. 2 in [4]). This excitation geometry has been analyzed in [4]. In the simplest case of equal amplitudes and phases of individual Gaussian beams, the resulting field emerges as a modified Bessel-Gauss beam where, apart from a constant factor, the intensity *ℐ* is written as

where *I*
_{0}(*x*) is the modified Bessel function of the first kind of zero order, *w*
^{2}(*z*) = *w*
^{2}
_{0} (1 + (*z*/*z _{R}*)

^{2}),

*z*=

_{R}*πw*

^{2}

_{0}/

*λ*,

*F*(

*z*) = 1/

*w*

^{2}(

*z*) −

*ik*/2

*R*(

*z*), and

*R*(

*z*) =

*z*+

*z*

^{2}

_{R}/

*z*. In the far field, the argument of the modified Bessel function tends to a pure imaginary value −

*ikra*/

*z*, and the Bessel function of

*I*-type can be replaced by the Bessel function of

*J*-type through the relation

*I*

_{0}(

*ix*) =

*J*

_{0}(

*x*). For large

*z*the beam radius

*w*(

*z*) ≫

*a*and the equation (1) gives the familiar profile of a Bessel-Gauss (BG) beam

The on-axis intensity of this BG-beam approaches in the far-field the intensity of the parent Gaussian beam and the BG-beam is embedded entirely into an envelope, which the Gaussian beam would have without transforming into annular superposition of beams. It means that after some evolution under propagation, the outgoing beam approaches the profile of initial unperturbed Gaussian beam and there is no additional beam broadening.

The characteristic size of the central peak of the BG-beam is given by the radius *r*
_{0}(*z*) of the first zero of *J*
_{0} function *r*
_{0}(*z*) ≈ 2.4*z*/*ka*. Hence, in the far field, the ratio *r*
_{0}(*z*)/*w*(*z*) ≈ 1.2*w*
_{0}/*a* is independent of the propagation distance, and the formed BG-beam acquires the same structural stability as the Gaussian beam. For a suitable choice of *w*
_{0}/*a*, the ratio *r*
_{0}(*z*)/*w*(*z*) ≤ 1. The central peak of such BG-beam dominates the profile of the outgoing beam since the annular side-maxima are strongly suppressed by the wings of the Gaussian function. The width of this well-defined central peak can be made significantly smaller than the width of the parent Gaussian beam improving thus the directivity of the beam.

In the case here, the required annular superposition of beams was obtained by employing the effect of internal conical refraction in a biaxial crystal. When a collimated light beam propagates along one of the optical axes of a biaxial crystal, the light spreads inside the crystal into a slanted cone and exits the crystal as a hollow light cylinder (Fig. 1). A characteristic parameter of conical refraction for a given crystal sample is the radius *R*
_{0} that corresponds to the radius of the light cylinder when *w*
_{0} ≪ *R*
_{0}. For a collimated Gaussian input beam, the light pattern beyond the crystal can be considered as an annular beam that emerges from an imaginary ring-like origin with a radius *a* ~ *R*
_{0} located at the beam waist position (see Fig. 1). Direct implementation of the effect for producing the required superposition of Gaussian beams, however, faces some complications because the cross-section of the light cylinder differs, in general, from that expected for such superposition. It is especially pronounced for *w*
_{0} ≪ *R*
_{0} when the cross-section of the light cylinder reveals a fine structure of two bright rings (the so-called Poggendorff rings) separated by a dark ring. Both Poggendorff rings have asymmetric profiles and the most intense outer ring carries up to 80% of the total beam power [5, 6]. However, if *w*
_{0} ≃ *R*
_{0}, the fine-structure of the light cylinder smears out and the beam structure is simplified significantly. It is just the case when the produced light pattern could be approximated by an annular superposition of Gaussian beams.

It has been known from the paraxial theory of conical refraction phenomenon [5, 6, 7, 8, 9] that for a circularly polarized Gaussian beam, propagating along the optical axis of a biaxial crystal, the emerging light field builds up from two components given by diffraction integrals

where *J*
_{0} and *J*
_{1} are the Bessel functions, *k* is the wave number of the beam along the axis, and *k*
**p** = *k*{*p _{x}*,

*p*} are the off-axis wave vectors with

_{y}*p*≪ 1 for a paraxial beam. The term

*B*

_{0}retains the incident polarization, while the term

*B*

_{1}has the opposite circularity and possesses a helical phase surface with an axial phase dislocation.

Figure 2 shows a comparison of the beam profiles calculated from Eq. (1) for an annular superposition of Gaussian beams, and the corresponding profiles of *B*
_{0} component of the conical refraction pattern calculated from the diffraction integral (3). These profiles were obtained for three values of *w*
_{0} at an imaginary origin at *z* = 0, where the difference between the two approaches is most pronounced. The diffraction integral (3) was calculated for *R*
_{0}=0.24 mm that corresponds to the used biaxial crystal (see below). Since the radius of the light cylinder at *w*
_{0} ≃ *R*
_{0} differs significantly from *R*
_{0} [5, 6, 7, 8, 9, 10], the radius *a* in equation (1) was varied as a fitting parameter. The best results for a given range of *R*
_{0} and *w*
_{0} were obtained for *a* = (1.4 − 1.5)*R*
_{0}. As seen from the numerical results, the general structure of the conical refraction pattern at *w*
_{0} ≃ *R*
_{0} can be rather well reproduced by an annular superposition of Gaussian beams, and the agreement between the two approaches is improved progressively with an increased ratio of *w*
_{0}/*R*
_{0}. If *w*
_{0}/*R*
_{0} = 1 (top panel in Fig. 2), the conical refraction pattern has a weak on-axis peak as a residual sub-structure. For *w*
_{0}/*R*
_{0} = 1.5 (middle panel in Fig. 2), this sub-structure smears out and the agreement between the two calculated profiles is improved. Finally, for *w*
_{0}/*R*
_{0} = 2 (bottom panel in Fig. 2), both approaches give nearly identical profiles.

## 3. Results and discussion

Figure 3 shows a schematic view of the experimental setup. A monoclinic biaxial KGW crystal from Conerefringent Optics was placed near the output aperture of a He-Ne laser *λ* = 632.8 nm. Actually, the crystal can be placed anywhere along the beam axis since the far-field pattern of conical refraction is independent of crystal position. The laser beam had a waist of *w*
_{0} ≃ 0.3 mm, the divergence of about 1.6 mrad full angle 2*θ*, and the beam quality parameter *M*
^{2} ≃ 1.2. The crystal, cut along one of its optical axes, was 14 mm long with a cross-section of 4×3 mm^{2}, and had a conical refraction radius of *R*
_{0} = 0.24 mm. A set of linear polarizers and *λ* = 4 plates was used to control the polarization of the input beam and to select the components of the output beam with different circular polarizations. To change the Gaussian beam parameter *w*
_{0}, a lens *f* = 40 cm was used so that the laser output was transformed into a Gaussian beam with *w*
_{0} ≃ 0.2 mm and 2*θ* ≃ 2.4 mrad. The far-field profiles of the outgoing beam were captured by a CCD camera.

Without polarization selection of *B*
_{0} and *B*
_{1} components, the far-field profile of the outgoing beam was identical to that for the parent Gaussian beam. With polarization selection, however, the two components of conical refraction pattern with significantly different profiles emerged. Fig. 4 displays the far-field profiles of beams. For an initial Gaussian beam with *M*
^{2} > 1, the width of profile exceeds slightly the width of the corresponding diffraction-limited Gaussian beam. When the crystal was inserted into the beam and the *B*
_{0} component was selected, the far-field beam profile was squeezed toward the beam axis so that its full width at half maximum (FWHM) becomes smaller than the FWHM of diffraction-limited Gaussian beam. In terms of the beam divergence, the angle 2*θ* ≃ 1.6 mrad for the parent beam was reduced to about 1 mrad when the *B*
_{0} beam was formed. The latter value is smaller than the divergence 2*θ* ≃ 1.3 mrad for the corresponding diffraction-limited Gaussian beam. Being corrected for losses on optical elements, the on-axis intensity of the *B*
_{0} beam is close to that for the parent Gaussian beam.

The central peak of *B*
_{0} was squeezed further when the Gaussian beam waist *w*
_{0} was reduced (bottom panel in Fig. 4). The FWHM of this peak was by about 2 times smaller than the FWHM of the corresponding diffraction-limited Gaussian beam, and the divergence was about a half of the diffraction limit. In this case, a weak ring containing about 10% of the *B*
_{0} beam power appears around the central peak. A very weak ring could be seen also at a close inspection of the beam profile at *w*
_{0}=0.3 mm (top panel of Fig. 4), but that ring contained a negligible fraction of the *B*
_{0} power.

When turned to the selection of *B*
_{1} component, the far-field beam acquires a doughnut-shape profile. This component is a vortex beam possessing an on-axis singularity [5, 6, 7, 9, 11, 12]. The phase structure of the doughnut-shaped outgoing beam was checked by interference with the reference Gaussian beam, and verified as a single-charge optical vortex, as expected. The vortex beam shown in the top panel of Fig. 4 carries nearly a half, and that in the bottom panel of Fig. 4 nearly two thirds of the total beam power. Such operation mode of a laser together with a biaxial crystal may serve for a simple and effective conversion of the lowest-order Gaussian beams into optical vortex beams.

Once established, the beam profiles shown in Fig. 4 were very stable upon propagation and, even after passing dozens of meters, the same squeezing of Gaussian profile was observed when forming the *B*
_{0} beam. Thus, this effect can be considered as a significant improving of the beam directivity, when the transverse size of the beam profile is reduced below the diffraction limit while retaining the on-axis intensity. With a reduced *w*
_{0}, the central peak of *B*
_{0} and the corresponding dark central spot of *B*
_{1} become progressively narrower, accompanied by the appearance of more pronounced secondary rings.

A numerical simulation of the experimental observations was carried out using equation (2) for the far-field BG-beam arising from the annular superposition of Gaussian beams, and diffraction integral (3) for the corresponding *B*
_{0} component of conical refraction pattern. Additionally, the profile of *B*
_{1} component was calculated from the diffraction integral (4). The calculation results are presented in Fig. 4. In general, the numerical results have shown a very good agreement with the experimental observations. For *w*
_{0} =0.3 mm (top panel in Fig. 4), the far-field ratio *r*
_{0}(*z*)/*w*(*z*) ≃ 1 and both approaches demonstrate practically identical profiles of a single peak with the FWHM reduced by about 1.6 times below the FWHM of the corresponding diffraction-limited Gaussian beam. For *w*
_{0} = 0.2 mm (bottom panel in Fig. 4), the far-field ratio *r*
_{0}(*z*)/*w*(*z*) ≃ 0.67. In this case, a nearly identical central peak emerges again from both approaches. The FWHM of this peak is by 2.1 times smaller than the FWHM of the corresponding diffraction-limited Gaussian beam. The secondary ring maximum, however, for the calculated BG-beam profile is remarkably weaker than that observed in the experiment. Here, the diffraction integral (3) reproduces experimental observations much better. This difference between the two approaches is increased when *w*
_{0} is further reduced. It is easily understandable since the conical refraction pattern deviates progressively from the pattern of annular superposition of Gaussian beams when the ratio *w*
_{0}/*R*
_{0} is reduced (see Fig. 2).

The calculated profiles of vortex *B*
_{1} component in Fig. 4 are in a very good agreement with the experimental observation, and here the paraxial theory of conical refraction demonstrates again its accuracy in describing the phenomenon. As expected, the sum of *B*
_{0} and *B*
_{1} intensities matches exactly the profile of the diffraction-limited Gaussian beam at a given propagation distance. Formally, the far-field Gaussian beam transformed by conical refraction at *w*
_{0} ≃ *R*
_{0} can, thus, be considered as a superposition of two embedded components: a sub-diffraction core and an annular shell. The sub-diffraction core is the *B*
_{0} beam component with its dominating central peak with a width smaller than that of the diffraction-limited Gaussian envelope. The shell is the vortex *B*
_{1} component carrying the rest of the total beam power.

In recent studies [13, 14], it was reported that solid-state lasers operated under conditions of conical refraction in the active element are able to generate high-quality near-Gaussian beams with *M*
^{2} ≤ 1. The results of the present study may give a simple qualitative explanation of these observations. The pump mode diameter in [13, 14] was close to the conical refraction radius *R*
_{0} of the used doped biaxial crystal. Hence, the laser output formation proceeded in conditions when *w*
_{0} ~ *R*
_{0}, and it is just the case considered here, when the conical refraction pattern evolves to a light structure containing an on-axis sub-diffraction core. The formation and amplification of this sub-diffraction core may be the key issue in producing high-quality beams with *M*
^{2} ≤ 1 from the lasers operated under conditions of intracavity conical refraction.

An annular superposition of Gaussian beams forms in the far field a structurally stable BG-beam with a constant ratio of *r*
_{0}(*z*)/*w*(*z*). The structural stability of the *B*
_{0} and *B*
_{1} beam components is a more complex issue since the integrals (3) and (4) cannot be expressed in an explicit form, and their asymptotics at *z* → ∞ have not been analyzed. When resorting to numerical simulations and experiment, no deformation of *B*
_{0} and *B*
_{1} profiles shown in Fig. 4 was found for a propagation distance of up to 50 meters. The profiles demonstrate a natural divergence together with the parent Gaussian envelope, but they preserved their established structure and relative widths.

The experiments and numerical simulations have also revealed the general trend that exhibits the evolution of *B*
_{0} and *B*
_{1} beams under propagation. These profiles evolve in the far field into a Gaussian envelope modulated in the radial direction by squared cosine and sine functions for *B*
_{0} and *B*
_{1}, respectively. It is especially pronounced when *w*
_{0} ≪ *R*
_{0} and the far-field pattern contains many ring maxima, but even for *w*
_{0} ≃ *R*
_{0}, the corresponding *B*
_{0} and *B*
_{1} profiles are very well approximated by a product of a Gaussian function and squared cosine and sine, so that the sum ∣*B*
_{0}∣^{2} + ∣*B*
_{1}∣^{2} gives exactly the Gaussian envelope. These observations require a further detailed analysis.

## 4. Conclusion

When a Gaussian laser beam propagates along one of the optical axes of a biaxial crystal, the effect of conical refraction transforms the emerging light into a hollow light cylinder. Under certain conditions, this light configuration can be approximated by a superposition of multiple Gaussian beams placed along a circumference. According to the analysis of [4], such superposition produces a modified Bessel-Gauss beam of *I*-type that evolves into a Bessel-Gauss beam of *J*-type in the far field. Since all individual Gaussian beams have their wave vectors parallel to the common propagation axis, the beam as a whole diverges upon propagation entirely due to diffraction, and the far-field beam becomes embedded into the profile of the initial unperturbed Gaussian beam.

Experiments and numerical simulation have shown that the approximation of the conical refraction pattern by an equivalent annular superposition of Gaussian beams is quite accurate when *w*
_{0} ≃ *R*
_{0}. In this case, the far-field pattern of the outgoing beam contains a component with a dominating on-axis peak. The width of this peak can be made significantly smaller than the width of the parent diffraction-limited Gaussian beam at the same propagation distance, while preserving the on-axis intensity. Another beam component is a charge-one optical vortex, which forms the annular shell of the outgoing beam and carries the rest of the total beam power. Without polarization selection, the two superimposed components form a Gaussian profile identical to that of the parent Gaussian beam. With circularly-polarized input beam, the core and the shell have opposite circular polarizations and can be easily separated. The variable transverse size of the bright or dark central spot of these beam components can be used, for example, for a precise long-range alignment with Gaussian beams transformed by conical refraction. The formation of sub-diffraction beam structures may be the reason of high-quality beams with *M*
^{2} ≤ 1 reported recently for lasers operated under conditions of intracavity conical refraction.

## Acknowledgements

This work was supported by the Estonian Science Foundation under Grant No. 7971.

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