Pupil filter design by using a Bessel functions basis at the image plane

Many applications can benefit from the use of pupil filters for controlling the light intensity distribution near the focus of an optical system. Most of the design methods for such filters are based on a secondorder expansion of the Point Spread Function (PSF). Here, we present a new procedure for designing radially-symmetric pupil filters. It is more precise than previous procedures as it considers the exact expression of the PSF, expanded as a function of first-order Bessel functions. Furthermore, this new method presents other advantages: the height of the side lobes can be easily controlled, it allows the design of amplitude-only, phase-only or hybrid filters, and the coefficients of the PSF expansion can be directly related to filter parameters. Finally, our procedure allows the design of filters with very different behaviours and optimal performance. ©2006 Optical Society of America OCIS codes: (100.6640) Superresolution, (220.1230) Apodization References and links 1. G. 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Introduction
Point Spread Function (PSF) engineering by pupil filters is a relevant technique in several applications, such as confocal microscopy [1][2][3][4], optical data storage [5,6], astronomy [7,8], and free space communications [9].The design of filters for PSF engineering basically consists in the definition of some figures of merit that describe the desired light distribution, and the calculation of the filter parameters from these figures of merit.Radial-symmetric filters have been usually preferred for the ease of their fabrication and analysis.A seminal study on radial-symmetric filters was performed by Toraldo [10].Sheppard [11] introduced the parabolic approximation of the PSF, which has been used in most methods for designing amplitude filters.Moreover, this parabolic approximation was generalized for general complex filters [12], allowing in the last years a fast development of phase-only and hybrid filter designs for different applications, such as three-dimensional superresolution [13], superresolving readout systems [14], systems with variable superresolution [15], or transverse superresolution [16,17], even generalized out of the focal plane [18].In spite of the success of these methods, the use of the parabolic approximation has some disadvantages, because an order 2 expansion can not take into account the side lobes, which are very important in many applications, and, evidently, lacks some precision.
In this work, we develop a new design procedure that is not based on the parabolic approximation.Instead, the PSF is expanded in a very interesting basis: the first order Bessel functions.Within this basis, the PSF must not be approximated for the filter design process, and, consequently, the results are more precise.Furthermore, figures of merit that take into account the PSF side lobes can be easily included.This procedure is applicable to annular filters, which can yield any desired performance, as they can be used as a basis for any radialsymmetric function.In fact, it has been shown that they can be converted to continuous filters [8].In this work, we will use annular binary filters with only two possible values of the phase: 0 and π (this restriction presents a relevant advantage and does not mean a performance limitation).Hence, the parameters that describe the filters are the zone radii and the corresponding transmittance of each zone.Nonetheless, it is possible to build amplitude filters, phase-only filters or hybrid ones by this procedure, and, thus, take advantage of their different performances [19,20].It must be noted that other works have also calculated the parameters of complex [20] or phase-only filters [9,21] without the parabolic approximation.However, in both case the analysis is very restricted.The first of these works did not pay attention to the axial behaviour, while the latter two did not take into account the focus displacement, and, thus, the behaviours are described not too precisely.
In some applications that take advantage of pupil filters, it is necessary to improve transverse resolution [7,9,20].These applications may also need to maintain a certain value of the PSF peak.In other applications, axial resolution is improved [15,22,23].There are applications where 3D superresolution is expected [13,14,24], or where the depth of focus must be increased [25], or even a diffraction pattern must be modified [26].Such a great variety of performances can be obtained with annular binary filters.
In this work, to show the additional contribution of this new technique, we will pay special attention to the attainment of transverse superresolution with moderate sidelobes and moderate energy loss at the PSF core, which is a common situation in many practical devices.We show that the designed filters can attain slightly better results than the best hybrid filters proposed in the scientific literature [20].In fact, we show that they allow a very high superresolution value (near 60% of the core width) with a core that stands out over its side lobes.
The scheme of this paper is as follows.First, the new procedure for filter design is described.For this task, the first-order Bessel functions basis is introduced, then, the figures of merit that characterize the PSF are defined, and, finally, the way the filter parameters are calculated is shown.Once, the new procedure has been described, the axial behaviour is analyzed.Then, we show some results obtained by simulation and, lastly, the main conclusions of our work are outlined.

Filter design procedure
This study shows a new procedure for the design of filters that allow us to control the light intensity distribution near the geometrical focus of an optical system.In this section, we first explain the basis that is used for the expansion of this distribution.As stated, this light intensity distribution is characterized by some figures of merit.Hence, the next step is to define the figures of merit and to show how the filter parameters can be obtained from the desired figures of merit using the new procedure.

The first-order Bessel-functions basis
We will work within the framework of the scalar diffraction theory.Let us consider a general complex pupil function P(ρ) = T(ρ)⋅exp[j⋅φ(ρ)], where ρ is the normalized radial coordinate over the pupil plane, T(ρ) is the transmittance function and φ(ρ) is the phase function of the filter.For a converging monochromatic spherical wave front passing through the center of the pupil, the amplitude U in the focal region may be written as [11] where v and u are radial and axial dimensionless optical coordinates with origin at the geometrical focus, given by λ and r and z are the usual radial and axial distances.The PSF can be obtained as the squared modulus of eq.( 1).In the parabolic approximation [11,12] the intensity distribution provided by the filter is expanded in series near the geometrical focus.However, in this case the PSF will not be approximated.Instead, let us consider that the pupil function is a filter formed by N annuli with different amplitude transmittances, t i , but only two alternative values of the phase, 0 and π: Of course, ρ 0 = 0 and ρ N = 1.In eq.( 2), the first zone of the filter has π phase, the second zone 0, the third π, etc.It must be noted that the behaviour of this filter is identical to that of the complimentary filter with 0 phase in the first zone, π in the second, etc [27].It is evident that these filters can be amplitude-only, phase-only or hybrid filters.It is important to note that the special phase values, 0 and π, are the only values that prevent a focus displacement, a fact that is often forgotten in other works (see appendix) producing confusing results.This choice of the filter shape does not mean a great lack of generality.In fact, any filter that yields a symmetric axial PSF, centred at the geometrical focus, can be expressed as a linear combination of such annuli.This case of a symmetric axial PSF is the most practical one, and, consequently, the behaviour required by most applications can be attained by the proposed filters.Thus, the transverse amplitude at the focal plane (u = 0) can be expressed as In order to understand why this expression is fundamental, let us recall that the contribution of a very thin annulus is given by the zero-order Bessel function [28], which can be expressed in terms of first order Bessel functions as If we introduce this expression in eq.( 3) we can recover eq.( 1) conveniently discretized.It is evident that any radial-symmetric function can be written as a linear combination of annuli, and, hence, can be analyzed with this formalism.It is also important to note that the coefficients of the J1 functions in the image plane are directly the parameters of the filter in the pupil plane (radii and transmittance).Thus, it is easy to relate both planes.

Figures of merit
As stated, there are several different objectives in practical applications, as transverse superresolution or apodization, axial superresolution, depth of focus improvement, etc.There can also be additional restrictions like moderate energy loss or reduced side lobes height.These characteristics of the light intensity distribution are described by figures of merit.Let us then describe the figures of merit that will be used in our procedure.In this section, we will only consider the transverse behaviour.The first figure of merit is the Strehl ratio.It is a relevant parameter for analyzing image quality, and is defined as the ratio of the intensity at the focal point to that corresponding to an unobstructed pupil: Here it is assumed that there is no focus displacement and the PSF core reaches its maximum value at u = 0.The next figure of merit gives a measure of the superresolution performance in the transverse direction.The transverse spot size, denoted by G, is the distance from the core to the first zero of the transverse PSF.These figures of merit are normalized so that for the unobstructed pupil they are equal to unity.A filter is superresolving if G is lower than unity and it is an apodizer when G is greater than unity.The last figure of merit for describing the transverse light distribution is the side lobe intensity ratio, Γ, defined as the ratio between the core intensity and the maximum intensity at the first side lobe: The position of the maximum of the side lobe can be easily obtained in the chosen basis.At such position, the derivative of the field, eq.( 3), must be null.From the properties of the Bessel functions it means that: This expression gives the first side lobe maximum and, thus it allows to calculate Γ from eq.( 6).The three figures of merit that we have just defined, allow a convenient description of the transverse behaviour of the light distribution at the focal plane.

Derivation of the filter parameters
Now, we have the whole background for describing the procedure that allows us to calculate the filter parameters, i.e. the transmittances and radii, from the desired figures of merit.As stated, this procedure is not an approximation as previous methods, [12] and is based on the use of the first-order Bessel-functions basis.The first step of the procedure is to decide the figures of merit that correspond to the desired distribution, S, G and Γ.Of course, it is not necessary to fix all of them.For example, if we are interested in a certain level of superresolution, we would have to fix G.In some applications it will be necessary to have a certain Strehl ratio too, though in some others this condition may not be required (for example if the light source is a laser that can yield as much energy as necessary).Analogously, some applications require a strict control of the side lobe height (as systems for data recording) while others (as confocal systems) do not require this restriction.Once these figures are chosen, the conditions are applied this way: Resolution condition: Side lobe height: where v 1zA = 3.83 is the position of the first zero of the Airy function and ε is a value low enough for obliging the PSF to reach a minimum value at v = v 1zA .On the other hand, if ε is extremely low it may be difficult to obtain the desired values of the other figures of merit, so that a compromise for the value of ε must be searched.S 0 and Γ 0 are the minimum acceptable values for the Strehl and the side lobe ratios.Finally, v 1s is the position of the maximum of the first side lobe.Its initial value is G ⋅ v 1sA , where v 1sA is the corresponding value for the Airy function.Using this initial value, the filter parameters are calculated from eq.( 8) by an optimization process.This first calculation usually provides a filter with a PSF close enough to the desired behaviour.Nonetheless, if it is not accurate enough, the position of the side lobe maximum can be obtained from the roots of eq.( 7), and this value can be used to solve again eq.( 8).This process can be repeated in an iterative procedure if necessary.Convergence is actually very fast.

Axial PSF
In many applications, the most relevant characteristic of an optical system is its transverse response.Nonetheless, even in these applications, it is interesting to understand the axial behaviour of the PSF, which is often forgotten in the scientific literature.The first important comment, already stated, is that special values for the phase have been chosen, 0 and π, because they are the only values that prevent a focus displacement (see appendix).Consequently, we assume that the focal plane is situated at u = 0, where the axial PSF presents a local maximum or minimum.However, it is interesting, and essential in some applications, to acquire more information about the axial PSF.As we have done with the transverse response, it is possible to obtain a natural basis for representing the radial-symmetric filters axial response.If we introduce eq.( 2) into eq.( 1) for v = 0, then the axial field distribution is: It is easy to reach this expression with the change of variable η =ρ 2 , because then the axial field distribution is proportional to the Fourier transform of the pupil function (expressed in terms of variable η instead of ρ).In the next section we show some results that we have obtained for different filters, and we will point out the relevant role of the axial PSF in some applications.

Results
In this section, we analyze the capacity of the new procedure for designing filters with relevant performances.Let us first centre our attention on a relevant problem pointed out by Narayan et al. [20].In some applications high values of transverse superresolution are interesting.However, both amplitude and phase filters are impractical for achieving a spot width, G, below 0.7.The latter ones produce large side lobe intensities, with a maximum Γ near to 2, while amplitude filters provide extremely low Strehl ratios (10 -3 -10 -6 ).For these reasons Narayan et al designed hybrid filters composed of a three-zone binary filter and a 12 thorder polynomial amplitude [20].These hybrid filters performed better than phase-only filters in terms of side lobe ratio, at the cost of a lower Strehl.From now on, we will design fivezone hybrid filters with our new procedure, in order to show that the new method can yield slightly better results than previous methods.Moreover, we will analyze the light distribution and we will point out some problems that were not taken into account in previous works.Let us consider an application where very high transverse superresolution, G = 0.64, moderate Strehl loss, S > 0.14 and moderate side lobes, Γ > 4, are required.If we apply the procedure described in section 2, a filter with five zones that fulfils such conditions can be found using a Mathcad 13 (Mathsoft Applications) built-in function named Find, which chooses between different methods for solving equations depending on whether the problem is linear or not.The parameters of this filter are radii (0, 0.103, 0.177, 0.352, 0.654) and transmittances (0.604, 0.683, 0.808, 0.843, 1).Fig 1 shows the transverse PSF, where it can be seen that it fulfils the conditions previously imposed.For comparison, the transverse PSF Transverse PSF v obtained in the work of Narayan et al is also shown.It can be seen that the new filter improves the Strehl ratio, with similar superresolution and side lobe height.The amplitude shape is simpler (i.e.easier to implement in most devices) and the conditions that must be imposed in the parameter searching process are simpler too.Here, it is worth noting that the election of a filter with five zones allows enough degrees of freedom for the PSF shape but at the same time does not make the computing process too slow.Hence, this new filter presents some advantages.
Fig. 1.Transverse PSF designed with our procedure to obtain a G value of 0.64 with S = 0.14 (solid curve) in comparison with the result found by Narayan et al. (dotted curve).For comparison the PSFs for an unobstructed pupil (dashed double-dotted curve) is also shown.v is normalized, so that the first zero of the PSF for the unobstructed pupil is unity.
It is very interesting to observe not only the transverse response, but also the axial one.Figure 2 shows the axial distribution of both the recently presented filter and that of Narayan et al.It can be seen that in both cases the axial PSF presents a local minimum, and not a local maximum at u = 0.This fact can be a problem in certain applications, so it is important to take it into account.Consequently, it is necessary to define a new figure of merit in order to force the axial PSF to have a maximum at u = 0.
A possible way is the most direct one, to impose the value of the PSF at u = 0 to be higher than the value at another value u 1 , where u 1 is a value around 12.6, the first zero for the unobstructed pupil PSF.When the design method is applied, it can be found that it is not possible to obtain a local maximum at u = 0 if G < 0.70, even with low Γ.Thus, in figure 3 a solution with this lower resolution, G = 0.70 is shown.The parameters of the filter are radii (0, 0.098, 0.222, 0.238 0.472 1) and transmittances (0.598, 0.712, 0.824, 0.955, 1).The PSF presents a dramatic increase in S and Γ = 4.In figure 4 we can see that the PSF has a high depth of focus, and the axial core is almost flat (due to the change from a local minimum to a local maximum at u = 0 as G increases near 0.7).This analysis shows that depending on the application it may be very difficult to find a practical superresolving filter that yields a spot width, G, below 0.7.
Finally, it is interesting to compare the filter behaviour with that of the well-known phase-only annular binary filters.As can be seen in fig 3 and 4 a 0-π two-zones phase-only filter [27]  hybrid filters.The side lobe ratio is slightly lower than that achieved by the hybrid filter while the Strehl ratio is slightly higher.Its axial behaviour is also similar to that found for hybrid filters, which leads to consideration of the fact that there is not a local maximum at u = 0 if G < 0.7 may be a general result for any kind of filter.
Fig. 2. Axial PSFs corresponding to the filters in figure 1: filter designed with our procedure (solid curve), the one designed by Narayan et al. (dotted curve) and the unobstructed pupil (dashed double-dotted curve).u is normalized, so that the first zero of the PSF for the unobstructed pupil is unity.Fig. 3. Transverse PSF designed with our procedure to obtain a G value of 0.7 (dotted-dashed curve).The result for a two-zones phase-only filter (dashed curve) is very similar.For comparison the PSFs for an unobstructed pupil (dashed double-dotted curve) and for the filters in figure 1, the filter designed with our procedure for G=0.64 (solid curve) and the one designed by  3: filter designed with our procedure to obtain a G value of 0.7 (dotted-dashed curve), the two-zones phase-only filter (dashed curve), an unobstructed pupil (dashed double-dotted curve), filter designed with our procedure for G=0.64 (solid curve) and the one designed by Narayan et al. (dotted curve).u is normalized, so that the first zero of the PSF for the unobstructed pupil is unity.

Conclusions
In this work a new procedure for designing pupil filters has been described.This new procedure is not based in a second-order expansion of the PSF as most previous methods, but it uses an exact representation of the PSF in terms of first-order Bessel functions.It allows the control at the same time of the height and width of the transverse PSF core and of the side lobes.Furthermore, we have shown how to take the axial response into account in this filter design procedure.Results obtained by simulation show that this design procedure allows us to obtain filters with performances slightly better than the best previous methods and, at the same time, with a better simultaneous control of all the PSF characteristics: energy loss, transverse response, axial response and side lobes.where * means complex conjugate.For annular pupil filters with N zones the integration can be easily performed: with a little amount of algebra it can be deduced that this expression vanishes if sin (φ i -φ j ) is null for every pair of annuli i,j.Thus, we demonstrate that focus displacement is avoided if, and only if, φ i -φ j = mπ j i, ∀ , where m is an integer.
with intermediate radius ρ 1 =0.437 can achieve a performance comparable to that of Narayan et al. (dotted curve), are also shown.v is normalized, so that the first zero of the PSF for the unobstructed pupil is unity.