On the computation of the Nijboer-Zernike aberration integrals at arbitrary defocus
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Citations
Digital in-line holography with an elliptical, astigmatic Gaussian beam : wide-angle reconstruction
Zernike representation and Strehl ratio of optical systems with variable numerical aperture
The Extended Nijboer-Zernike Diffraction Theory and its Applications
High-NA aberration retrieval with the Extended Nijboer-Zernike vector diffraction theory
Image formation in a multilayer using the extended Nijboer-Zernike theory
References
Handbook of Mathematical Functions
Principles of Optics
Principles of Optics
Orthogonal Polynomials
Related Papers (5)
Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System
Frequently Asked Questions (17)
Q2. What is the accuracy of the Bessel-Bessel method?
The accuracy of the so-called Bessel-Bessel method is guaranteed to relative intensity values down to machine precision due to the fact that the coefficients of the analytic series expansion are all positive and bounded to the interval [0, 1].
Q3. What is the main result of the FT-scheme?
In [1]-[2], the evaluation of the diffraction integral is carried out using numerical Fourier transform techniques, and the effect of the phase factor is mitigated by intentionally introducing a compensating quadratic phase factor that can be incorporated in the FT-scheme.
Q4. How is the electromagnetic field in the aperture matched?
The electromagnetic field in the aperture is matched by means of an expansion in multipole far-field radiation patterns using spherical harmonics and the field distribution in any defocused position is obtained by applying the known propagation effects to the multipole distributions, involving again well-converging spherical Bessel functions with the geometrical distance from the focus as argument.
Q5. What is the axial intensity distribution in the image plane?
In the image plane, the intensity distribution is strongly defocused and the sharp focus is only found at a distance 5 µm beyond the paraxial image plane.
Q6. What is the purpose of this paper?
In this paper the authors treat the strong defocus problem in the framework of the recently developed extension of the Nijboer-Zernike approach to the computation of optical point-spread functions of general aberrated optical systems [6]-[10].
Q7. How can the authors find finite series expressions for the products R02kR m ?
By using results of the relatively recent modern theory of orthogonal polynomials, as can be found in [17], Secs. 6.8 and 7.1, the authors are able to find finite series expressions, with favourable properties from a computational point of view, for the coefficients needed in the “linearization” of the products R02kR m n .
Q8. What is the phase factor in the diffraction integral?
the phase factor in the diffraction integral is proportional to the projection of the defocus distance onto the direction of the plane wave contribution in the integral.
Q9. How many axial depths can be handled?
the use of the series in (3) is limited to a range like |f | ≤ 5π, so that an axial range of the order of typically ten focal depths can be handled.
Q10. What is the largest modulus for l f?
The terms in the series V PBS00 have largest modulus for l ≈ f and r = 0 of the order ef/2f√2πf while V00 itself has a modulus of the order 1/f .
Q11. Why is the product in (31) small?
It is useful to note that, due to the occurrence of the min-operand in (32), (33), the product in (31) is small whenever one of the factors is small.
Q12. What is the contribution of the defocus parameter to the diffraction integral of a?
In the extension to the Nijboer-Zernike approach, power series expressions involving the defocus parameter f , with coefficients explicitly given in terms of Bessel functions and binomial coefficients, were given for the contribution to the diffraction integral of a single aberration term βnm R m n (ρ) cos mϑ with R m n (ρ) a Zernike polynomial, see [11], Sec. 9.2.
Q13. What is the smallest term on the second line of (44)?
In particular, the low-amplitude wrinkles on the decaying side of |2V00(r, f)|2 around v = 2πr = 200 can be identified as an interference of the (small) term on the second line of (44) and the (large) term on the first line.
Q14. What is the problem of writing products R02kR m n?
Then one is faced with the problem of writing products R02kR m n as a linear combination of Zernike polynomials with upper index m so as to be able to apply (11).
Q15. Why is the new method used to check numerical methods?
Because of its basic accuracy, the new method can also be used to check numerical methods (e.g. Fourier transform methods) with respect to the required sampling density for achieving a desired precision.
Q16. How many focal depths are required for the new method?
The authors have compared the new method with some of the existing ones and concluded that the convergence of former methods (see [6],[7] and [13], respectively) is appropriate for a total axial defocusing range of typically ten focal depths.
Q17. What is the radiometric effect in the vectorial treatment of the diffraction integral?
The radiometriceffect encountered in the vectorial treatment of the diffraction integral is integrated in the analysis by using a Fourier-Gegenbauer expansion in [3]; along the same lines, the inclusion of circularly symmetric aberrations in the diffraction integral is demonstrated in [4].