A recent article by Biss and Brown [1] adapts the theory of Török et al. [2, 3] to calculate the electric energy density due to radially and azimuthally polarized beams focused through a dielectric interface. The paper presents both numerical results and conclusions based on these results, some of which, we believe, are in error. The authors consider two cases of focusing through a dielectric interface: one when the numerical aperture is NA=0.85 and one where the numerical aperture is NA=1.4. In the latter case the published results, Figs. 2(a), 3(a), and 5, are in error. Therefore, the conclusions based on these results are incorrect as well.

Considering the field distribution in the focal region of a high numerical aperture lens embedded in a homogeneous medium of refractive index n ₁, it is well-known that the numerical aperture determines the extent of the region in which the field is effectively confined to a spot. When a plane dielectric interface is placed in the vicinity of focus, separating the medium the lens is embedded in from another medium of refractive index n ₂, the effective numerical aperture of this focusing arrangement can at most be equal to n ₂. Those plane wave components of the angular spectrum for which the angle of incidence is beyond the critical angle (NA > n ₂) suffer total internal reflection resulting in their conversion to evanescent waves. These decay exponentially as one moves away from the interface hence their contribution to the field distribution in the second medium is confined to a thin region (typically a fraction of the wavelength) in the vicinity of the interface [4]. At larger distances from the interface the field distribution can in essence be represented by that due to a lens of numerical aperture NA=n ₂ embedded in a homogeneous medium using a pupil plane mask (apodizer) to account for the Fresnel transmission of plane wave components through the interface. The results obtained by Biss and Brown [1], where the field remains confined over a region of several wavelengths and effects of the interface can be observed even far beyond the Gaussian focus, are highly non-physical.

 

Fig. 1. (694 KB) Movie showing the intensity distribution in the x-z meridional plane due to a strongly confined radially polarized Bessel-Gauss beam. The beam is focused (NA=1.4) through an interface separating media n ₁=1.518 and n ₂=1.0. The black broken line represents the position of the interface. The color scale is a logarithmic base 10 scale.

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Fig. 2. FWHM of the lateral intensity distribution at the location of axial maximum for different interface positions (n ₁=1.518,n ₂=1.0,NA=1.4). The red and magenta triangles correspond to FWHM values for x-polarized incident light along the x and y-axes, respectively. Blue stars and green squares correspond to the FWHM values of the total intensity and the longitudinal component, respectively, for radially polarized incident light.

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In Movie 1, we show the time averaged intensity distribution obtained by focusing a radially polarized Bessel-Gauss beam on a glass-air interface. This case corresponds to that of Fig. 2(a) of Biss and Brown [1]. The time-evolution in the movie corresponds to the axial position of the interface, changing from 2λ (behind the Gaussian focus) to -10λ (in front of the Gaussian focus). Careful examination reveals a number of interesting phenomena. However, contrary to the results presented in [1], the spot remains confined to a well-defined volume in space. In Fig. 2 we show for x-polarized incident illumination the full width at half maximum (FWHM) of the lateral intensity distribution taken at the point of maximum intensity along the optic axis in the x (red) and y (magenta) directions, together with the FWHM corresponding to the overall intensity (blue line) and |E_z|² (green line) of a radially polarized incident illumination. Note that the irregular behaviour of the FWHM values corresponding to the linearly polarized arrangement has been observed and explained before [5].

 

Fig. 3. Intensity distribution for a focused radially polarized Bessel-Gauss beam (n ₁=1.518,n ₂=1.0,NA=1.4), where only 20 points were used for the numerical integration in (a) |E_x|²+|E_y|² and (b) |E_z|², and 1000 points in (c) |E_x|²+|E_y|² and (d) |E_z|², respectively.

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The error in the results presented in the paper [1] is, most likely, caused by the lack of convergence of the numerical integration. The authors state that they used a Gaussian quadrature integration routine. The application of such routines results in a significant reduction in the integration time. However a check for the convergence of the numerical results should always be included. This can be done by calculating the numerical sum for an increased number of integration points, as is done in an adaptive Kronrod extension of the Gaussian quadrature. We present the results for numerical integration with 20 and 1000 points in Figs. 3(a,b) and 3(c,d), respectively. Our Figs. 3(a) and 3(b), in which case the numerical integration did not converge, clearly exhibits the same characteristics as Figs. 2(a) and 4 of [1].

In conclusion, we have presented the correct results corresponding to those in error in the paper by Biss and Brown [1]. The error in their results is most likely caused by the lack of convergence in numerical integration. Therefore, the conclusions drawn from these results are incorrect. Close to the interface near-field effects contribute to the field distribution in the second medium, and we observe a smaller spot than expected by the diffraction limit [4]. However, when only far-field effects contribute to the field distribution, the FWHM of the focused spot is determined by the diffraction limit.