Abstract
We describe the energy and momentum flux in the case of an aberrated optical imaging system with a high numerical aperture (NA). The approach is based on the extended Nijboer-Zernike diffraction theory, that, in its high-NA version, yields an accurate analytic representation of the electromagnetic field vectors in the focal region of imaging systems that suffer from aberrations and/or transmission defects. In an earlier publication, we have derived the electromagnetic energy density from the field vectors. In this paper, we expand our analysis to the energy flow (Poynting vector) and to the quantities related to the linear and angular momentum of the radiation. Several examples of the energy and momentum flow are presented. In particular, we show how the linear and angular momentum distribution in the focal region depend on the initial polarization state and on the parameters describing the wavefront shape of the converging beam. For the angular momentum flow, we show how the separation between spin and orbital momentum is modified when going from the paraxial case to a high-NA focused beam.
Highlights
P U B L IC A T IO N S Optics Research Group, Faculty of Applied Sciences, Technical University Delft, Lorentzweg 1, NL2628 CJ Delft, The Netherlands Optics Research Group, Faculty of Applied Sciences, Technical University Delft, Lorentzweg 1, NL2628 CJ Delft, The Netherlands Philips Research Europe, HTC 36 / 4, NL-5656 AE Eindhoven, The Netherlands
The approach is based on the extended Nijboer-Zernike diffraction theory, that, in its high-numerical aperture (NA) version, yields an accurate analytic representation of the electromagnetic field vectors in the focal region of imaging systems that suffer from aberrations and/or transmission defects[1]
Special attention has been paid to the angular momentum representation because its description at high numerical aperture is the subject of a discussion on what would be the optimum representation of angular momentum, going from the paraxial to the high-aperture case
Summary
Expressions have been given for the Cartesian electric field components in the focal region of a high-NA system in the presence of a linear state of polarisation in the entrance pupil, Received August 28, 2007; published November 29, 2007. State of polarisation in the entrance pupil can be described by multiplying the x- and y-components in the entrance pupil with complex numbers a and b, respectively, and summing the vector components in the high-NA focal region with these weighting factors. The defocus parameter f is orientated in the opposite sense as compared to the real-space coordinate z and equals zero in where a and b are normalised complex factors with |a|2 + |b|2 = 1 that determine the incident state of polarisation and R|nm|(ρ) is a radial Zernike polynomial of radial degree n and azimuthal order m. Using Eq(1) the components of the magnetic induction vector are written as
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