Extended Nijboer-Zernike (ENZ) Analysis
Nijboer's results for the amplitude distribution in focus were extended by him for low-order, small amplitude aberrations to a small region out of the perfect focal plane, but his calculations relied on ad hoc procedures that become awkward when the effect of higher-order, general aberrations of larger amplitude must be considered further out of focus. A recent research result has opened the way to an analytic description of the focal field over a larger volume and for appreciable values of the aberration coefficients. The basic mathematics that allow this extension to the focal region have been described by [Janssen] and the analysis has been applied to a certain number of relevant optical problems in a paper by [Braat, Dirksen and Janssen], both published in 2002.
Forward and backward calculation: aberration retrieval
The extended Nijboer-Zernike analysis can be used for the forward calculation to obtain the complex amplitude and intensity in the focal region and it then competes with other numerical methods based on, e.g., the Fourier transform. The advantage of the extended Nijboer-Zernike analysis is its close connection to the existing practice in physical optics and its inherent accuracy due to the well-understood and non-compromising convergence behaviour of the series expansion for the focal field.
The backward calculation can be used to obtain the complex field in the exit pupil starting from intensity data in the focal volume. With this approach, also called aberration retrieval, an area of great practical importance is addressed because this can solve the problem of finding the lens imperfections from the aberrated intensity point-spread function of the optical system. Interesting applications are found in the field of high-resolution optical lithography where the optical defects of the projection lens can be derived from recorded point source images in a photoresist layer (see [Dirksen, Braat, Janssen and Juffermans]).
High numerical aperture extension
The original Extended Nijboer-Zernike diffraction analysis was limited to scalar optical fields, a good approximation for imaging systems with a numerical aperture smaller than 0.60. At larger numerical aperture, several complications are encountered that have been treated in the literature. The most important complication is the fact that the vector nature of the electromagnetic field has to be included (vector diffraction). The measurable quantity in the focal region, the intensity, is now obtained from the modulus squared of a vector quantity, the electric field. The exact state of polarization of the electric and magnetic field components at the entrance of the optical system also has to be included in the analysis. Some geometrical effects like the defocusing factor and the amplitude distribution on the exit pupil sphere have to be adapted. All this is needed because we now have to cope with opening angles of the focusing beams that approach half of the full solid angle of 4π.
With the foregoing extensions to the scalar case and some other minor ones we have obtained analytic expressions for the electric and magnetic field components in the focal region of a high-NA imaging system [Braat, Dirksen, Janssen, van de Nes]. The colorful picture on the website homepage, made according to algorithms from this publication, corresponds to the intensity distribution that is found at high numerical aperture; in this case, the NA was 0.85, a value at which vector effects become clearly visible (see also Applications: forward calculation).
The next step was to create a backward calculation scheme for the vector diffraction case so that the aberrations and transmission defects of a high NA optical system can be retrieved. Although the scheme, including the retrieval of the so-called ’polarization aberration’, comprises rather lengthy expressions, the retrieval at high numerical aperture has become feasible [bibliography] and is applied both to synthetic data [van Haver, Braat, Dirksen, Janssen 1] (also see: [van Haver, Braat, Dirksen, Janssen 2]) for testing purposes and to experimental data; in the latter case, only a special illumination with natural light was available. At this moment, our preferred applications are found in state-of-the-art lithography and in advanced microscopy.
Imaging of extended objects
The imaging of extended objects is made possible in the ENZ-approach by first calculating the complex field in the entrance pupil of an optical system. The field scattered by the object is obtained by a scalar calculation or by solving an electromagnetic boundary problem. The entrance pupil field then follows by calculating the far-field plane wave spectrum of the field scattered by the object. For not too large objects (up to an extent of typically twenty diffraction units), the plane wave spectrum can be very well represented by a complex Zernike expansion with coefficients of higher order than is customary for aberration calculations. Radial degree and azimuthal order up to 20 or 30 are used in such an expansion. The completeness and orthogonality of a Zernike expansion allow for a stable representation of such a relatively complicated scattered field in the entrance pupil. With the aid of the coefficients of the Zernike expansion, the field in the exit pupil is calculated, taking into account the amplitude and phase defects of the imaging system. Details of this approach are found in [van Haver, Braat, Janssen, Janssen, Pereira] and [van Haver, Janssen, Braat, Janssen, Urbach, Pereira]. The influence of the object illumination is taken into account by a sufficiently dense angular sampling over the incoherent source, imaged at infinity by a Köhler-type illumination system.
Imaging of an extended object in a stratified medium
The image space close to the focal region is often nonhomogeneous. The receiving surface can be covered by thin layers, for instance an anti-reflection coating or a sensitive medium like a photoresist layer. If these layers have interfaces that are parallel to each other and have their normal vectors pointing along the optical axis, the well-known thin-layer theory for light reflection at and transmission through these layers can be applied. For the high-NA case, see [van de Nes]. The incorporation of this thin-layer formalism in the ENZ-approach for high-NA imaging is described in [Braat, van Haver, Janssen, Pereira] and [van Haver]. In each layer of the stack, a forward and a backward propagating spectrum of plane waves is calculated. By a proper fitting of these plane wave spectra with four sets of complex Zernike coefficients, the focal field in a specific layer is obtained.
Energy and momentum flow in the focal region
Quantities that are quadratic in the electric and magnetic field components determine the electric and magnetic energy densities and the energy flow (Poynting vector). The density vectors of linear and angular momentum and the tensors that describe the flow of these momentum components are also obtained from products of electric and magnetic field components. The ENZ-theory enables an analytic treatment of these quadratic electromagnetic quantities in the focal region. This is especially useful in the case of a high-NA focus with arbitrary amplitude and phase of the lens transmission function and a general state of polarisation in the entrance pupil ([Braat, van Haver, Janssen, Dirksen] and [van Haver]). An example of the electric energy density and electromagnetic energy flow in a high-numerical-aperture focal region is given in the Figure below; the aberration is astigmatism.
Fig. 1. Upper row: cross-sections of the energy density (color-shading, units J/m3) in three image plane settings (from left to right: two focal depths in front of the nominal focal plane, in focus, and two focal depths beyond the nominal focal plane. In the same graphs, the (x, y)-components of the Poynting vector have been represented by arrows. The length of the arrows has been normalized to the largest transverse component in the picture. In the lower row, the z-component of the Poynting vector has been represented by means of color shading. The incident state of polarization is linear (horizontal direction), the numerical aperture of the focused beam is 0.95, astigmatic aberration.