Applications: Structured and accurate forward calculation

A point source (spatial delta-function) emits a spherical wave that is imaged by the optical system. The coherent point source produces a field distribution in the nominal image plane, called the (complex) point-spread function. In the case of an incoherent quadratic detector, only the intensity distribution can be obtained in the image plane.

Schematic layout of an optical imaging system

The original Nijboer-Zernike theory produces the amplitude and intensity distribution in the focal plane in the presence of an aberrated imaging system. Both the aberration of the system and the defocusing of the image plane should remain relatively small in the Nijboer-Zernike theory. In the Extended Nijboer-Zernike theory, both the aberration and the  defocusing can assume larger values; as a result, we can use analytic formulas that describe the amplitude and intensity distribution in an image volume. Using the Extended Nijboer-Zernike formalism, we obtain explicit expressions for axial, radial or azimuthal cross-sections of the through-focus intensity distribution. This is advantageous with respect to computational burden. Other numerical schemes rely on point-by-point evaluation of numerical integrals or on Fourier transforms that only yield a solution for a single plane. In this sense, the numerical results from the Extended Nijboer-Zernike analysis can serve as a benchmark with respect to other purely numerical techniques in diffraction theory like Fourier transform methods.

An example of a forward calculation based on the ENZ-theory is the intensity pattern below. The pupil function was moderately aberrated (Strehl ratio typically 0.80) by contributions from relatively high-order Zernike polynomials. In the corresponding in-focus intensity pattern we observe a widely spread low-level intensity contribution that resembles a laser speckle pattern. Note that all higher-order diffracted energy is roughly contained within two circles given by v=nmin+3 and v=nmax+3 with nmin and nmax equal to the lowest and highest radial order of the Zernike polynomials in the aberration function; the central part of the diffraction image has retained its appearance of a classical Airy disc.  

An ENZ-calculated diffraction pattern derived from a pupil function with a relatively large content of high-order Zernike aberration terms (up to the 23rd radial order) that emulates the effect of low-angle scattering in the optical system. The contour levels in the figure have to be multiplied by a factor of 10-2 . The central peak intensity (Strehl intensity) is approximately 0.80 (just diffraction-limited). The lateral (x,y)-coordinates are in units of  v=2πr with the radial coordinate r expressed in the diffraction unit  λ/NA in the image plane

High-NA image formation

The ENZ-diffraction theory has been extended to high-NA imaging by executing the following steps:

  • introduction of the vector character of the electromagnetic field
  • inclusion of a non-uniform amplitude distribution on the exit pupil sphere due to the high-NA focusing
  • introduction of a refined defocusing phase factor in the diffraction integral
  • allowing a general state of polarization in the entrance pupil of the optical system
  • the derivation of new semi-analytic expressions for the vector diffraction integrals in the case of an aberrated focused beam

The colourful picture that was presented on the home page is an example of a high-NA forward diffraction calculation using ENZ-theory; the numerical aperture was 0.85 in this case. More field distributions (modulus of the total field in the focal region) are shown in the figures below, now at the extremely high numerical aperture of 0.95. The first two pictures present a hypothetical classical Airy disc distribution (absolute amplitude) and, at the same scale, the corresponding picture at high numerical aperture (NA=0.95). The incident light is linearly polarized along the horizontal direction. It is clearly seen that the high-NA amplitude distribution has become elliptical. In the horizontal cross-section, the full width at half maximum (FWHM) is smaller than for the hypothetical Airy disc; in the horizontal cross-section, a substantial broadening occurs.
In the next figure, high-NA plots have been produced of the modulus of the field amplitude in the case of a slightly astigmatic beam with the azimuth of the wavefront aberration at 30 degrees with the horizontal axis. The incident polarization was again along the horizontal axis. The left-hand and right-hand graphs correspond to the geometrical focal lines, the central graph pertains to the best focus position. The angle between the incident linear state of polarization and zero azimuth of the astigmatic aberration give rise to a complicated through-focus behaviour.


The modulus of the complex amplitude distribution in focus according to the scalar diffraction theory (left-hand figure: Airy disc) and the modulus of the total electric field in the case of a focused high-NA beam (normalized in size to the same scale as the Airy disc; NA=0.95). The incident light is linearly polarized along the horizontal direction


The modulus of the electric field in the case of an astigmatic beam when focused at high NA (0.95). The left-hand figure and right-hand figure correspond to the focal positions of the geometrical focal lines; the central figure is the best-focus position. The azimuth of the astigmatic aberration is at 30 degrees with the horizontal axis, the incident state of polarization is linear in the horizontal direction

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