1. Introduction

A vector description for the strong focusing of light using high numerical apertures is essential, as all three mutually orthogonal electric field components appear in the focal region. The vector properties not only affect the local field direction but also the irradiance (time-averaged electric energy density) distribution in the focal region. This paper gives the results of calculating the irradiance in the focal region by the numerical integration of the Rayleigh-Sommerfeld diffraction integral of the first kind. The numerical integration method (2DSC method) is based upon a two-dimensional form of Simpson’s 1/3 rule [1]. In the 2DSC method, a two-dimensional diffraction integral is evaluated directly without the diffraction integral being reduced to a set of one-dimensional integrals. The only assumption is that the Kirchhoff boundary conditions are acceptable. The Rayleigh-Sommerfeld of the first kind is evaluated because it is a very convenient formulation for the diffraction integral, since it is consistent and it is only necessary to know the electric field within the aperture and not the gradient of the field. The 2DSC method is better than the method proposed by Richards and Wolf [2] as it is applicable when modeling low Fresnel number systems as well as systems with large Fresnel numbers.

The focusing of light beams with linear polarization have substantial disadvantages. The irradiance distribution in the focal plane for beams with linear polarization are not symmetrical and the interaction with matter at the focal spot depends upon the direction of the polarization. The asymmetry for linear polarized light is due mainly to a contribution to the irradiance from the Z component of the electric field directed along the optical axis as shown in Fig. 1(c).

 

Fig. 1. Irradiance distribution (dB) in the focal (XY) plane for a beam linearly polarized in the X direction. (a) X component, (b) Y component, (c) Z component and (d) total irradiance.

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A radially polarized beam (sometimes referred to as a cylindrical vector beam) produces a symmetrical irradiance distribution in the focal plane and gives maximum absorption in the focal region. Some of the advantages of a symmetric polarized beam include improved trapping and accelerating of particles, material processing, light propagation in hollow fibers and the focusing of light with high NA lens. Advantages of smaller spot sizes include the probing and manipulation of atoms or molecules in nano-scale quantum systems, microscopes, optical data storage, lithography and laser cutting [3–6].

One proposal to create radially polarized beams is to use half wave plates cut into four quadrants with linear polarization in the four quadrants directed outward. The addition of the contributions from each quadrant gives pseudoradial polarization [6–7]. The calculations for pseudoradial beams with circular and annular apertures are compared with calculations for a uniform profile linearly polarized beam, and pure radial polarization beams with uniform and doughnut profiles. The Z axis corresponded to the optical axis and the origin of the coordinate system is the center of the circular aperture. The parameters for the calculations were: radius of circular aperture, a=2.500 mm, inner radius of annular aperture, a _inner=0.900a, numerical aperture, NA=0.900, focal length, f=1.211 mm, wavelength, λ=632.8 nm. The polarization of a beam is given by a unit vector p=[p_x p_y p_z]: linear [1 0 0]; pure radial [cosβ sinβ 0]; pseudoradial [1 0 0] for β=-π/4 to π/4, [0 1 0] for β=π/4 to 3π/4, [-1 0 0] for β=3π/4 to 5π/4, [0–1 0] for β=5π/4 to 7π/4, where β is the aperture angle measured with respect to the X axis. The amplitude profile, T for the doughnut radial polarized beam is T=ρ exp[-ρ²/(1.5a)²)] where ρ is the distance of an aperture point from the origin.

2. Results

For a pseudoradial polarization beam, the electric field in the focal region is calculated by summing the contributions of the linearly polarized light from each of the four quadrants. Figure 2 shows the irradiance in the focal plane for the pseudoradial polarized light for circular and annular apertures. Figure 3 shows the irradiance W and its components W _x, W _y and W _z along the radial X direction (a); axial Z direction (b); and the irradiance W in decibels along one of the discontinuities at 45° to the X axis and along the X or Y axis (c). The focal spot is circular and is comparable in size to that of a pure radial polarized beam but the distribution is no longer symmetric in the focal plane and the pattern is distorted along the lines of discontinuities between adjacent quadrants and there are no definite zeros for the irradiance along the X and Y axes. There is a very significant decrease in the width of the central maximum in the focal plane when an annular aperture is used without significantly increasing the height of the side lobes.

 

Fig. 2. The irradiance (dB) in the focal plane for a pseudoradial polarized beam. (a) Circular aperture and (b) Annular aperture.

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Fig. 3. Uniform four quadrant pseudoradial polarizated beam. The total irradiance W and its X, Y and Z components, W _x, W _y and W _z in the radial X direction (a) and axial Z direction (b). The total irradiance W (dB) along a discontinuity (at 45° to the X axis) and along either the X or Y axis (c).

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The full width at half maximum (fwhm) along an axis is twice the distance measured in wavelengths from the maxiumum in the irradiance to the point where the irradiance is half the maximum value. The fwhm in the focal plane is the focal spot size defined in terms of the effective diameter of a circular focal spot along the contour representing a value of half the maximum irradiance around the geometrical focus. Figure 4 summarizes the calculations for the fwhm for the different beams for the circular and annular apertures. For a circular aperture, the focal spot size for the radial polarized beams is not smaller than the effective spot size of the linear polarized beam. However, Fig. 4 clearly shows the reduction in focal spot size for the annular aperture for all beams, and the smaller the spot size the stronger the electric field strength at the focus.

The focal spot size is very similar for the uniform radial, doughnut radial and pseudoradial polarized beams. However, if a surface could be covered with a special photosensitive surface that is only sensitive to the narrow longitudinal field component, even smaller spot sizes are possible [5, 8–10]. For the four quadrant pseudoradial polarized beam, the overall energy distribution is not radially symmetrical, but in close proximity to the geometrical focal point, it is approximately symmetrical with an almost circular focal spot similar to the radial polarization. Therefore, a pseudoradial polarized beam is a good approximation to a radial polarized beam in the region very close to the geometrical focal point.

 

Fig. 4. The full width at half maximum of the total average electric energy density and the longitudinal component (L) measured in wavelength units for circular and annular (A) apertures for the different beams.

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Figure 5 summarizes the strengths of the first side lobe measured in decibels for the circular and annular apertures for the different beams. The results summarized in Fig. 5 shows the significant enhancement of the side lobe strength in the focal plane for all beams, especially for the case of the linear polarized beam for the annular aperture. The annulus has a significant effect upon the depth of field along the optical axis, increasing the width of the central maximum along the optical axis by a factor of about 10 with very little effect upon the strength of the side lobes. For the linear polarization, with a high numerical aperture, the annular aperture does not produce a significantly smaller spot compared with the circular aperture because of the increase in strength of the side lobes in the focal plane.

 

Fig. 5. The strength of the first side lobes along the coordinate axes measured in decibels for the circular and annular apertures (A) for the different beams

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3. Conclusion

For the focusing of light by a lens in large numerical aperture systems, it is most important that the polarization of the incident beam be taken into account in describing the electric and magnetic fields and the resulting average energy density in the focal region. A beam which has a cylindrical symmetry, for example, a beam with radial polarization has more desirable characteristics than that for linearly polarized beams which produce non-symmetrical field distributions in the focal plane. A proposal to produce a radial polarized beam is to use a half wave plate cut into four quadrants that gives linear polarization in the four quadrants directed outwards (pseudoradial polarization). In the focal region, the average electric energy density distribution for the pseudoradial polarization is very similar to that of beams with uniform and doughnut profiles and radial polarization. An annular aperture compared with a circular aperture results in smaller focal spot sizes in the focal plane and a very significant increase in the depth of focus about the focal plane. For the radial polarization, the electric field at the geometrical focus is purely longitudinal and if one could detect only the longitudinal component, very the small focal spot sizes can be achieved with an annular aperture.