1. Introduction

Recently considerable effort has been invested in understanding the propagation of electromagnetic fields characterized by a definite transverse localization but strongly different from the classical families of linearly polarized Hermite-Gauss or Laguerre-Gauss modes. In order to understand the properties of these beams - such as divergence, focusing ability or polarization state rotation - a thorough analysis is necessary, based on basic wave equations or complete diffraction integrals. Most formulae derived for the propagation of Gaussian beams are not applicable. Neither are the most common beam quality indicators for Gaussian beams applicable. Such a situation is encountered, for instance, when examining beams generated by the annular wave-guide diffusion cooled CO₂ lasers [1–2] leading to the study present here.

A similar geometry, with a ring shaped beam cross-section, is actually encountered in several other systems as a result of specific laser designs. This is the case of axially pumped Nd:YAG [3] or dye lasers [4] and also of Concentric-Circle Grating Surface Emitting (CCGSE) semiconductor lasers [5]. This last case gave origin to a series of fruitful developments and observations. Jordan and Hall [6] derived a free-space azimuthal paraxial wave equation supporting azimuthal Bessel-Gauss (ABG) beams, i.e. beams with a Bessel-Gauss amplitude distribution and electric field everywhere directed in the azimuthal direction of the transverse plane. Later, Hall [7] generalized the solution of the Helmoltz wave-equation to the general vector case and Greene and Hall [8] studied the properties of vector Bessel- Gauss beams as solutions of the aforementioned vector wave-equation. This family of modes includes the azimuthal Bessel-Gauss beam as the lowest order mode. The focusing properties of this set of beams has also been studied by Greene and Hall in [9].

The field distributions of our study are generated as modes of an annular waveguide [10]; thus they do not represent pure eigenfunctions of the free-space propagation problem. Nevertheless our analysis confirms some results already presented in the series of papers previously cited [6–9]. Moreover, as is detailed in what follows, we further obtain evidence of phenomena that appear of particular significance for the case of high power laser beams, designed for material processing applications.

First of all, it is noteworthy that a fundamental annular mode with azimuthal polarization maintains its shape, with an on-axis hole, during propagation and thus also during focusing. This behavior differs from that of a linearly polarized constant phase “donut” mode, that shows an on-axis peak in its far field. Secondly, as a consequence of its rotation invariance, the fundamental azimuthal mode also maintains its polarization state during propagation, as does the linearly polarized donut mode. This result is extended in [11] to the focusing of fundamental radially polarized circularly symmetric annular modes, with the sole exception of the generation of on-axis longitudinal components when focusing with high numerical apertures.

Higher order modes generated by the annular waveguide have markedly different propagation characteristics in free-space. In this particular case, with the metallic coaxial guide generating high order azimuthal modes having azimuthal polarization, neither the azimuthal field distribution nor the polarization state are preserved during propagation. Instead, a characteristic contrast reversal is observed, as noted in ref. [9], along with a polarization transformation from azimuthal to mainly radial.

This last result appears particularly interesting. Indeed it has been shown [12] that, given the higher absorption of the S-polarized wave in metallic reflection, a beam with radial polarization can double the cutting performance of the analogous beam with circular polarization. This latter case is, at present, the typical polarization state produced in metal cutting systems from standard linearly polarized laser sources.

2. Propagation modeling

Our analysis is based on numerical algorithms performing Fresnel diffraction integral propagations [10,13]. For initial conditions, we use the annular field distributions defined in [10], which correspond to the eigen-functions of the coaxial-tube waveguide.

In our previous studies [13] we analyzed the propagation of high azimuthal order modes and the effects of phase corrections at particular propagation planes, making use of scalar propagation integrals. These studies were motivated by the specific strategy adopted for attaining a good quality beam from the quite peculiar transverse field distribution of our laser. Heat diffusion constraints define the design of our wave-guide laser, giving rise to a large and thin annular cross-section. The main consequence of this is very little loss discrimination in our resonator between modes with different azimuthal number and the consequent emission of a low quality multi-mode beam [1]. To solve this problem we made use of an intra-cavity spatial filter to select a single transverse mode or at least a reduced family of frequency degenerate modes and a phase plate outside the cavity to transfer a large fraction of the output power from this high order mode to the azimuthal fundamental one [2].

To understand the role of polarization, we now compare the behavior of fields that have the same transverse amplitude distribution but different polarization state. Uniform linearly polarized fields are treated with a scalar propagator, while azimuthally polarized beams are propagated with a vector-propagator, operating on a two orthogonal component complex electric field. Metallic reflection boundary conditions together with the typical geometrical dimensions of our guides (characterized by a large diameter-to-thickness ratio) are responsible for the generation of fields with an azimuthal polarization state. We will thus concentrate our analysis on the propagation of two cases: beams with an initial uniform linear polarization state (LPB); and beams with an initial linear polarization everywhere directed parallel to the azimuthal versor in the transverse plane (APB). This purely azimuthal polarization state, even on higher order modes, is a direct consequence of a wave-guided beam generation. It does not find correspondence in the vector field solutions of the free space propagation problem, discussed in ref. [8].

 

Fig. 1. Near- and Far-Field plots of LPB (first column) and APB (second column) corresponding to a mode with azimuthal number equal to four. Beam dimensions in the nearfield are 28 mm internal diameter, 32 mm outer diameter. The Far-field figure corresponds to a full angle of 13.25 mrad. (Near-Fields on the first row, Far-Fields on the second.)

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3. Numerical results

First of all we show in Fig.1 the effect of polarization in the propagation of a pure mode of our optical waveguide [10]. For sake of clarity, we considered modes with a relatively low transverse number, so that a reasonable resolution is maintained along the whole propagation distance. In Fig.1 the near- and far-field intensity plots of the LPB and APB are reported, with azimuthal mode-number equal to 4 and with a thickness-to-diameter ratio corresponding to that of our experimental beams [1–2].

To better evidence the structure of the principal lobes of these beams we enhance the thickness-to-diameter-ratio by transforming the beam in the same manner as a w-axicon does, and we repeated our propagation process. The results are shown in Fig.2 and in the related movies. A comparison of Fig.1 and 2 indicates that the radial beam compression process doesn’t affect the properties of the main far-field lobes, as we will discuss in what follows. Of course, the compression process does change the beam divergence causing the necessity of longer propagation distances to obtain a self-similar far-field pattern. Furthermore, it reduces the importance of secondary radial rings, which are quite intense and numerous in the case of narrow annular near-field cross sections (Fig.1). Again in Fig.2, the left-hand column (like the first movie) shows the behavior of the LPB while the right-hand one (together with the second movie) refers to the APB.

 

Fig. 2. Propagation movies of compressed LPB (left-hand column — Movie-1 (830 kb)) and APB (right-hand column— Movie-2 (860 kb)) of order 4. Beams are propagated for 6 m after collimation with a concave 12 m R.O.C. mirror. Beam dimensions in the near-field are 4 mm internal diameter, 32 mm outer diameter. Snapshots show the near-field (first row) and far-field (second row) intensity patterns of the two different beams.

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The movies show that the linearly polarized mode diffracts in the radial direction only, maintaining the same nodal lines on all the transverse planes. On the contrary the APB diffracts in the azimuthal direction too, filling in the nodal lines and producing a peaked distribution which exhibits contrast reversal in the far field. During this propagation next-neighboring lobes diverge radially and interfere to produce field distributions with elliptical polarization in the intermediate planes. As propagation proceeds towards the far-field, these distributions gradually re-organize into main lobes with a linear polarization oriented parallel to the radial direction. Apparently these lobes are not divided by nodal lines but rather by point singularities around which polarization rotates.

 

Fig.3. Vector Plots of compressed beams having LP and AP on the initial planes. Beam dimensions in the near-field are 4 mm internal diameter, 32 mm outer diameter. The far-field figures correspond to a full angle of 3.2 mrad.

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This last property can be observed in Fig.3 where the near- and far-field vector field plots are shown for both the LPB (left column) and the APB (right column). The different effect of propagation for this two fields is clearly seen. In the case of uniform linear polarization the lobes with alternating phase interfere in such a way as to maintain the same nodal-lines geometry at any propagation distance (see the first movie). On the contrary the differently oriented fields of the APB lobes produce non-zero contributions along the originally nodal transverse directions. Precisely these directions have the maximum intensity in the Far-field and a linear polarization oriented in the radial direction. The intermediate distributions (visible in the second movie), in which the regularly peaked distribution in the azimuthal direction is smeared out, exhibit a general elliptical polarization. Incidentally we note here a striking result, which is not completely clear in the figures due to space limitations. The secondary diffraction rings seen in the bottom row of Fig.1 not only maintain their azimuthal orientation also in the case of an APB, but also they maintain their polarization state that is consequently different from that of the principal far-field ring.

4. The phase-correction process

In this last section we comment on the application of the phase correction process, studied in Ref.[2] and [13], on the propagation of high order LPBs and APBs. Numerical results are shown in Fig.4 and in the related movies. In this phase correction process a profile modulated mirror is used to suppress the π-dephasing between adjacent lobes. This correction transforms field distributions originating from high order waveguide modes into distributions similar to the fundamental LPB and APB, with the sole exception of a residual azimuthal amplitude modulation.

In our numerical results, the corrected beams consistently produced far-field distributions with a main lobe resembling that of the fundamental mode. This lobe naturally exhibited the polarization of the fundamental mode. This means that when the lobes of a high order azimuthal mode are re-phased, the Far-field will have a substantially azimuthal polarization, whereas the polarization is mainly radial in the case of de-phased (uncorrected) lobes.

The regular patterning visible in the early part of the movies is a consequence of the sharp-edged phase correction producing cusps in the pure azimuthal mode distributions.

 

Fig. 4. Propagation movies of compressed and phase-corrected LPB (left-hand column — Movie-3 (817 kb)) and APB (right-hand column— Movie-4 (850 kb)) of order 4. Beam dimensions and propagation distance are as in Fig.2. Only far-fields are shown in the snapshots (near-fields being equal to those in the first row of fig.2).

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5. Conclusions

The numerical propagation studies presented here demonstrate the complex behavior of vector beams with a high modal number. In particular we have shown that changes in near-field polarization and phase distributions can profoundly alter far-field intensity patterns and polarization states. Hence all these properties must be account for when one tries to asses even simple quantities such as divergence or focusing ability for beams such as those produced by annular diffusion-cooled lasers. A quite different effectiveness from normal expectations is also to be taken into account in applications such as metal cutting, as a consequence of these polarization changes.

Since one can usually change the polarization state of a fundamental Gaussian-like beam without affecting the far-field intensity distribution, these properties are too often disregarded in many applications.