Cylindrical vector beams have been proposed and demonstrated for applications ranging from microscopy to high energy physics. In this paper, we analyze the three-dimensional field distributions of radial and azimuthal beams focused near a dielectric interface. We give particular attention to the classic problem of high numerical aperture focusing from an immersion lens to a glass-air interface and find that the use of radially and azimuthally polarized illumination for this type of imaging provides an impressive lateral confinement of the fields over a wide range of interface positions.
©2001 Optical Society of America
Cylindrical-vector beams are solutions of Maxwell’s equations which obey cylindrical symmetry both in field-amplitude and polarization. Such solutions have been the topic of numerous recent theoretical and experimental investigations [5–23]. It has been found, for example, that such beams may be produced by active or passive means [13–17]. The use of an interferometric polarization converter has recently allowed the application of focused cylindrical-vector beams to several different types of microscopy .
Meanwhile, the propagation and focusing properties of these beams remain of continued interest. Three recent papers [14,18,19], have analyzed the high numerical-aperture focusing properties of the beams in free space, and have shown that a highly inhomogeneous distribution of electric field directions exist near focus and that, in the case of a radially polarized beam, the non-propagating longitudinal component to the electric field may achieve a strength several times that of the propagating component.
Fields focused onto an interface have been previously investigated [2–4], these papers have only considered linearly polarized light. Recently Helseth has considered interfacial focusing with beams of arbitrary polarization and gave one example of a radially polarized beam . The strong longitudinal component created by radially polarized beams naturally raises the question of the distribution of fields at various interface positions. Recent investigations of optical focusing inside a dielectric have demonstrated that focusing a source of illumination over a cone of light which extends beyond the critical angle produces astigmatic effects due to the Goos-Hänchen phase shift experienced by plane waves undergoing total internal reflection.
In this paper, we analyze the through focus three-dimensional field distributions of a cylindrical vector beam incident on a dielectric interface, with special attention to using such beams for high numerical aperture illumination from inside a dielectric. This is of considerable interest because, as we will show, a cylindrical vector beam, by its very nature, possesses a plane wave spectrum which is either entirely s-like or entirely p-like. There is therefore no astigmatism and the lateral confinement of the fields near focus is impressive over a wide range of interface positions.
To find the reflected and transmitted fields of the focused field, an initial focused field is found using the theory described by Richards and Wolf  and applied to cylindrical vector beams by Youngworth and Brown .
An initial field, represented by an apodization amplitude function, ℓo (θ), is refracted by an aplanatic lens and focused near the interface as shown in figure 1. The forms of the azimuthally and radially polarized beams are found in the same fashion as Török et al  and Helseth .
2.1 Azimuthally Polarized Light
For azimuthally polarized light the procedure is similar to that of Helseth , the only differences being that the incoming light is completely s polarized. The initial field after refraction by the aplanatic lens is,
This field is propagated to the interface by the usual method of the angluar spectrum representation. Reflected and transmitted fields are assumed and Maxwell’s boundary conditions are applied to these fields. The results are,
The integration of the azimuthal angle can now be performed numerically. Care has to be taken when integrating if the entering angles exceed the critical angle of the system due to the fact that complex poles exist in the integrand for azimuthal angles at the critical angle.
2.2 Radially Polarized Light
The case of radially polarized light is similar and this polarization has been considered by Helseth . The result has the same integral form as azimuthal light, but with different , , and ,
3. Results and Discussion
So far the solutions for the reflected and transmitted fields have been left in a general form. No apodization function has been chosen. Here we will choose an apodization function originally derived by Jordan and Hall , which describes the Bessel-Gauss class of fields. The lowest order field is given by,
Here θ max is the maximum entering angle of the beam, and β is a ratio of the pupil radius to the entering beam radius. In these examples β=1, which indicates that the pupil fill is equal to 1. With this apodization function, eqs. (2)–(10) must be evaluated numerically. The follow results were obtained using a Gaussian quadrature integration method.
Figures 2a and 2b show the radial and longitudinal field intensities at an interface for two sets of parameters. Figure 2a considers an oil immersion lens focusing into air, and figure 2b examines focusing in air onto a silicon (n=3.55) substrate, such as might be required in semiconductor inspection.
Here we see that the radial field component is continuous across the interface, while the longitudinal component is discontinuous. This is expected according to Maxwell’s boundary conditions.
The fields for azimuthally polarized light can be obtained using a similar procedure. In this case we find only one vector component in the azimuthal direction and the field is everywhere s polarized. In Figures 3a and 3b we see two examples of azimuthally polarized light focused onto an interface. Figure 3a considers an oil immersion lens focusing in air, and figure 3b considers focusing from air onto a silicon substrate. We see again that the on axis null is preserved through the focus. Also since the field is transverse to the plane of the interface, there is no field discontinuity as the beam passes through the interface.
We now consider an immersion lens (NA=1.4) focusing a radial beam to a dielectric/air interface. Figure 4 shows the total intensity of a radial beam as the interface is moved through focus. As we adjust the position of the interface (which is equivalent to defocusing the beam) we see that the longitudinal field focal spot can extend beyond the interface over a considerable range of defocus positions, but is shifted from the origin. The deviation of the focal spot can be thought of, in a geometrical optics fashion, as a result of spherical aberration due to refraction at the interface. Using the full width at half maximum (FWHM) of the focal spot as a gauge, we may now compare the resolution of various polarizations when focused beyond the dielectric-air interface. Figure 5 shows the FWHM of the intensity of a focused, linearly polarized beam, a focused radial beam (total intensity), and the longitudinal component of a focused radial beam for different values of the interface position.
For radially polarized light, the FWHM of the focal spot is somewhat smaller than that of the linear field for a wide range of interface positions before the geometrical focus. However, if we consider only the longitudinal component of the radially polarized beam, it has the smallest FWHM of the other two beam polarizations, over a large range of interface positions before the geometrical focus. We propose, therefore, that linear or nonlinear mechanisms which detect the scattering of longitudinal fields can provide, in air, a resolution equivalent to what is possible with immersion objectives. A mechanism that is strongly responsive to just a longitudinally polarized field is surface harmonic generation (HG). In principle, a surface HG field could be produced at an interface to create an image of the interface, with better resolution than conventional linear or circularly polarized beams. This is likely to be of great importance, for example, in semiconductor mask and wafer inspection.
The three dimensional fields of cylindrical vector beams focused through a dielectric interface have been calculated and compared to circularly polarized light. Examples have been given both for air objectives focused on to a high index (e.g. semiconductor) substrate and a dielectric/air immersion system. For the case when the beam is focused through an immersion lens (NA=1.4), the radial beam is of considerable interest because of the rather tight confinement of the longitudinal field and its extent into the air. Such beams are likely to play important roles in a variety of surface imaging applications.
We would like to thank Kathleen Youngworth, Professor Lukas Novotny, and Professor Emil Wolf for helpful discussions. This work was supported in part by the Semiconductor Research Corporation under contract 776.001.
References and links
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