On the computation of the Nijboer-Zernike aberration integrals at arbitrary defocus
Summary (2 min read)
1 Introduction
- The computation of strongly defocused amplitude distributions has been considered by several authors.
- Their effort has been directed towards the stable evaluation of the diffraction integral in the presence of a strongly oscillating defocusing phase factor in the integrand of the basic diffraction integral.
- The approach has been extended further in [8] so as to cover the cases of high numerical aperture which requires computation of field components as well as inclusion of the radiometric effect and the state of polarization.
- The evaluation of the required Bessel functions is normally no problem since one can exploit recursion formulas as in [12], expression 9.1.27 on p. 361, when the efficient computation of the Bessel functions is not already available in the software environment of the user.
- The authors conclude this section with some comments on different approaches that can be found in the literature.
2 Main result
- See Subsec. 2.2 below, and its proof.the authors.
- For the basic definitions and properties of the Zernike polynomials Rmn the authors refer to [11].
2.1 Linearization of products of Zernike polynomials
- The non-negativity of all cl’s and the fact that the cl’s vanish outside the claimed l-range is an immediate consequence of [19], Corollary 5.3.
- They are therefore quite small, and computation of the cl’s according to (18)–(19) hardly suffers from loss of digits.
- The authors now present the main result of this paper.
- The resulting expression for Tm,m+2p is of the same complexity as what one gets by using (36).
4 Extension to high-numerical aperture sys-
- In [8], the Nijboer-Zernike approach has been further extended so as to include aberrated optical systems with high numerical aperture using a vectorial diffraction formalism à la Ignatowskyi/Richards-and-Wolf.
- Then the authors are in the situation of Sec. 2 and they can apply the main result directly.
5 Examples
- In this section the authors compare the methods embodied by formulas (3-5) and by the main result in Subsec.
- 2, respectively, for the computation of some Vnm’s with respect to the (r, f)-range for which they produce accurate results.
- Both formulas (3-5) and formulas (26-27) are converted straightforwardly into computer codes, especially when the available software environment allows accurate computation of high order Bessel functions.
- For the series in (3) this is an overkill as well: it follows from the analysis in [7].
- Figure 4 seems to confirm this statement.
5.2 Strongly defocused fields
- 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.
- In figure 5, left, a cross-section of the radially symmetric intensity pattern has been given that contains the z-axis.
- In figure 5, right, the axial intensity has been plotted as a function of the axial z-coordinate, showing the same focal shift phenomenon due to the presence of the zone plate structure.
- 3 Accuracy comparison for the Tnm-functions Jj+1(2πr) (2πr)j+1 . (47) Recalling the results from the preceding sections, the authors are allowed to take the values calculated according to the Bessel-Bessel method (BBS) as the absolute reference because of the guaranteed convergence of this method.
- The fact that curves A and B virtually coincide shows that the PBS- and BBS-methods produce an equal disparity with respect to the Cao-method.
6 Conclusion
- A new analytic calculation method has been devised that solves the convergence problem when computing strongly defocused aberrated diffraction patterns in the extended Nijboer-Zernike theory.
- The resulting analytic expression contains an expansion in terms of well-converging spherical Bessel functions for the axial defocusing parameter and an expansion in Bessel functions of the first kind for the lateral coordinate in the defocused plane.
- 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].
- 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.
- The range is only limited by the practical calculation time, not by the convergence of the calculation method.
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"On the computation of the Nijboer-Z..." refers background in this paper
...As said, the series expression in (3) yields accurate results when some 3f terms are included in the series over l. The evaluation of the required Bessel functions is normally no problem since one can exploit recursion formulas as in [ 12 ], expression 9.1.27 on p. 361, when the efficient computation of the Bessel functions is not already available in the software environment of the user....
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...These rules are somewhat conservative since one can use instead of [ 12 ], 9.1.62 on p. 362, the sharper but more complicated inequality 9.1.63 on p. 362....
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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].