Applications: A robust Bessel-Bessel expansion for arbitrary defocus values

The basic version of the ENZ-theory uses a so-called power-Bessel expansion of the field in the focal region. The axial excursion is limited in this case. A range typically from -15 to 15 focal depths is covered by this method which is largely sufficient for all well-corrected optical systems. For larger defocus ranges, a second version has been devised that is based on a Bessel expansion for both the radial and the axial coordinate (Janssen, Braat and Dirksen). In this case, numerically reliable results are obtained for defocus values of an unrestricted number of focal depths. In the figure below, on the left-hand side, a strongly defocused diffraction pattern and, on the righthand side, a radial cross-section of the intensity after averaging over the azimuthal coordinate. The defocusing amounts to approximately 25 focal depths from the nominal image plane. The system is supposed to be aberration-free and has an unobstructed circular pupil with uniform transmission. The radially symmetric diffraction 'shadow' of the circular pupil is located far in the Fresnel regime and shows a multitude of fine oscillations. 


The intensity distribution in the focal region at a large defocus value. Left-hand picture: measured two-dimensional spatial distribution; right-hand picture: azimuthally averaged measured intensity distribution as a function of the radial coordinate

Using the measured defocused intensity distribution as input data, we have fitted these data to our Bessel-Bessel analytic prediction. The defocus parameter and the radial scale were used as fit parameters. We have fitted by hand as well as by using a special, one-focus-plane ENZ-retrieval approach. In the latter case the (unknown) defocus and radial-scale parameters are incorporated in the retrieval procedure, yielding overall retrieved aberration energies that depend on the assumed values of these parameters. The parameters that yield the least overall aberration energy are taken as estimates for the actual parameter values: these values give the best agreement with the assumption that the system is aberration-free. In the figure below we present the experimental and the fitted intensity distribution.

A radial cross-section of the circularly symmetric diffraction pattern produced by a strongly defocused spherical wave (defocus approximately 25 focal depths). The experimental curve does not show all detail of the fitted analytic distribution but the general correspondence is good

Another example of the application of the Bessel-Bessel representation of the through-focus intensity distribution is found in the next figure. An effective defocusing has been produced here by means of a Fresnel-lens structure in the pupil of the imaging lens and the intensity maximum of the diffraction image has been axially shifted over a distance of 5 μm.

An axial cut (through-focus) of the intensity distribution of a strongly defocused light distribution. The defocus has been produced by a Fresnel-lens structure in the object plane that was illuminated by a coherent plane wave. The resulting defocusing in the image plane amounts to 5 μm

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