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Theory of high numerical aperture focusing

When you focus an X polarised beam with a lens, what is the polarisation in the focal spot? The answer to this questions will depend on the numerical aperture (NA) of the focusing lens. The NA of the lens on the other hand depends on the angle made by a marginal ray with the optic axis. Higher the value of this angle, higher is the NA. For a low NA lens the focal spot is primarily X polarised, however, for a high NA lens there will be a considerable amount of Z polarised light in the focal spot in addition to the X polarised light. Figure 1(a) shows the focal intensity distribution considering X polarised light only, due to the focusing of an X polarised laser beam by a high NA (i.e. NA=1.2) lens. Figure 1(b) shows the corresponding Z polarised focal intensity distribution. For the above mentioned NA, the maximum of the Z polarised focal intensity is about 15% of the maximum X polarised focal intensity. Due to the presence of this significant amount of Z polarised light, in the form of a dumbbell along the X axis, the resultant focal intensity distribution is not circularly symmetric. Figure 1(c) shows the resultant focal intensity distribution, where, both X polarised intensity as in figure 1(a) and Z polarised intesity as in figure 1(b) are considered. Thus the focal spot in the high NA case is no longer an Airy pattern. However, if the same X polarised laser beam is focused by a low NA lens, the focal spot is circularly symmetric and is closely represented by an Airy pattern. This is because, in the low NA case, the Z polarised focal intensity distribution is too small to be considered. Figure 1(d) shows the resultant focal spot of an X polarised beam focused by a 0.1 NA lens. It is to be mentioned here that for an X polarised beam, contribution from the Y polarised light in the focal spot is negligible both in the high and low NA cases.

Figure 1: Simulated images of focal spots using high and low NA lenses.

Amplitude and intensity distribution in the focal plane of a lens corresponding to the X, Y and Z polarisations can be computed using the vectorial diffraction theory of Richards and Wolf. This theory can be extended to compute the focal field of an arbitrarily polarised beam. It has been shown that such focal computation can be efficiently performed by using the commercially available fast Fourier transform (FFT) tools. More about focal field conputation of an arbitrarily polarised beam using FFT tools is found here.

The vectorial diffraction theory, mentioned above, can be used to compute the focal spots of some important beams with cylindrically symmetric polarisation profiles. One such beam is a radially polarised beam. A radially polarised beam has polarisations along the radius vectors. Thus if an axial section, containing the optic axis, of a radially polarised beam is considered, the beam is plane polarised. Such a beam when focused by a high NA lens, theory suggests, will result in focal spot that is mainly axially (Z) polarised. Figures 2(a) and 2(b) show the laterally (X+Y) polarised and axially (Z) polarised focal intensity distributions when a radially polarised beam is focused by a 1.2 NA lens. The radially polarised beams find apllications in number of areas such as in the study of molecular orientations, near field microscopy, second harmonic generation microscopy, etc.

Figure 2: Simulated images of laterally polarised and axially polarised focal spots of a radially polarised beam.