Pareto optimality between width of central lobe and ... - OSA Publishing

Research Article

Vol. 54, No. 31 / November 1 2015 / Applied Optics

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Pareto optimality between width of central lobe and peak sidelobe intensity in the far-field pattern of lossless phase-only filters for enhancement of transverse resolution SOMPARNA MUKHOPADHYAY AND LAKSHMINARAYAN HAZRA* Department of Applied Optics & Photonics, University of Calcutta, JD-2, Sector-3, Salt Lake, Kolkata 700098, India *Corresponding author: [email protected] Received 22 July 2015; revised 21 September 2015; accepted 2 October 2015; posted 2 October 2015 (Doc. ID 246461); published 27 October 2015

Resolution capability of an optical imaging system can be enhanced by reducing the width of the central lobe of the point spread function. Attempts to achieve the same by pupil plane filtering give rise to a concomitant increase in sidelobe intensity. The mutual exclusivity between these two objectives may be considered as a multiobjective optimization problem that does not have a unique solution; rather, a class of trade-off solutions called Pareto optimal solutions may be generated. Pareto fronts in the synthesis of lossless phase-only pupil plane filters to achieve superresolution with prespecified lower limits for the Strehl ratio are explored by using the particle swarm optimization technique. © 2015 Optical Society of America OCIS codes: (100.6640) Superresolution; (100.5090) Phase-only filters; (180.1790) Confocal microscopy; (180.5810) Scanning microscopy; (350.4855) Optical tweezers or optical manipulation; (350.5730) Resolution. http://dx.doi.org/10.1364/AO.54.009205

1. INTRODUCTION Tailoring the point spread function of an optical system by pupil plane filtering continues to play significant roles in diverse applications, e.g., confocal microscopy, optical micromanipulation, free-space optical communication, etc. For applications in astronomy and spectroscopy, pupil plane filtering is commonly used for either suppression of sidelobes and/or narrowing down the central lobe of the point spread function, as and when required. Traditionally the latter processes are known as apodization and superresolution, respectively [1–4]. In both these cases the least resolvable distance is diffraction limited even for aberration-free systems. The effects are to enhance either the low-frequency components or the high-frequency components in the image at the cost of the other within the frequency passband stipulated by diffraction limits. The term “superresolution” is now often used to represent any attempt to transcend the limits of resolution set by diffraction effects in optical systems. Overcoming the diffraction limit of resolution constitutes one of the core challenges of modern optical engineering. The advent of novel light sources, detectors, and digital computers with phenomenal number crunching ability has opened up new frontiers to circumvent the limitations in resolution imposed by diffraction effects, and several ingenious methods, e.g., near-field microscopy, are being explored [5–9]. 1559-128X/15/319205-08$15/0$15.00 © 2015 Optical Society of America

Notwithstanding the major strides made by these techniques in the field of high-resolution imaging, investigations on the use of pupil plane filtering to obtain high and superresolution in the field of optical imaging continue unabated since 1952 when Toraldo di Francia first proposed the use of pupil filters for superresolution over a restricted field of view [10–12]. On the other hand, many nonreal time approaches for superresolution imaging or related applications call for energy-efficient techniques that can provide narrowing down the central lobe from its size in the Airy pattern, the point spread function in a diffraction-limited system. Narrow focal spots in scanning imagery and optical tweezers are obvious examples. In these applications, often narrowing of the transverse point spread function is obtained by using high numerical aperture optical systems without any pupil plane filtering. Theoretical analysis of system performance in such a case calls for an application of vector diffraction theory [13,14]. However, in applications using focusing objectives with numerical aperture less than 0.7, the far-field performance can be reliably modeled by the scalar diffraction theory, providing thereby an analytical formulation which is more amenable to further manipulation for analysis and synthesis. Most studies on Toraldo filters reported so far in the literature are based on the scalar diffraction theory, and the same is being adopted in our studies.

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Lossy, lossless, and leaky filters have been explored for this purpose. Each of them has also been studied for the two cases: continuous and piecewise continuous filters, obviously to alleviate the fabrication problem [15–46]. Any attempt for narrowing the central lobe of the point spread function (PSF) is accompanied by enhanced intensity of the neighboring sidelobes, of which the first sidelobe plays a dominant role in degrading the quality of the image. The overall effect is reduction of contrast in the final image. Also, the increase in sidelobe intensity is accompanied by a decrease of intensity in the central lobe; this decrease in central intensity from that for a uniform pupil is expressed by a ratio known as the “Strehl ratio.” The larger the value of the Strehl ratio, the more useful is the filter in practical applications. The inverse relationship between the width of the central lobe and the intensity in the sidelobe may be looked upon as a multiobjective (MO) optimization problem. The requirement for useful values of the Strehl ratio is the major constraint of this optimization problem. MO problems are difficult to solve compared to single objective problems, since they require a priori knowledge about the relative importance of the objectives. Instead of having a unique solution, an MO problem typically has a set of acceptable trade-off solutions called Pareto optimal [47,48] solutions forming a curve in the objective function space. This is commonly known as the Pareto front. A solution belongs to the Pareto set if there is no other solution that can improve one objective without degrading the other. In the problem concerned, the width of the central lobe of the transverse PSF and the peak intensity in the sidelobes are two such conflicting objectives to be optimized. In general, analytical methods for tackling the problem of synthesis of these filters in accordance with a set of prespecified requirements for the Strehl ratio, full width at half-maximum (FWHM, where the intensity drops down to 50% of its peak value), and peak intensity in the sidelobe do not exist, except for the trivial cases. Most of the researchers mentioned in references [15–46] used semianalytical approaches for tackling the problem of synthesis of pupil plane filters. Some of them formulated the problem in a manner suitable for application of the local optimization algorithm. The low value of the Strehl ratio, in the solutions so obtained, severely restricts their applicability in practice. Stochastic global optimization techniques hold better promise for providing useful solutions, and in a recent communication [49] we reported an implementation of evolutionary programming (EP) for the synthesis of phase-only Toraldo filters. Our formulation enabled us to look for useful solutions with reasonable values for the Strehl ratio. However, it is noted that success of any venture with EP depends, to a large extent, on the proper choice of genetic operators, namely the selection probability, the crossover probability, and the mutation probability. Unless suitable values are assigned to these parameters, the stochastic search is affected by slow convergence and undue stagnation problems. For a specific class of problems, it requires extensive numerical experiments to determine proper values for the genetic operators [50]. In order to circumvent this problem, we explore the possibility of using an implementation of the particle swarm optimization (PSO) technique [51–55] for determining Pareto

fronts of optimal phase filters. The PSO is a heuristic technique of search methodology, based on the notion of collaborative behavior or swarming in groups of biological organisms. The PSO is similar to techniques of evolutionary programming in the sense that both are random initial population-based approaches, and the members share information with their neighbors among the population to perform the search using a combination of deterministic and probabilistic rules. However, instead of using the three classical genetic operators, each candidate solution termed as “particle” in PSO, adjusts its trajectory in the objective space according to its own motion and as well as the motion of its companions. The absence of the genetic operators in PSO makes it relatively easier to implement in practice. Recently, the PSO algorithm has been used in the synthesis of axially superresolving binary phase filters, and Pareto optimality is shown between the axial FWHM of the central lobe and the sidelobe intensity [56]. PSO has also been experimented for global optimization of lens design [57]. In our investigations, globally or quasi-globally optimized superresolving phase filters with concentric, unequal area, annular zones of fixed phase are synthesized by the PSO technique. Using these filters we performed some numerical experiments and investigated their superresolving performance. For the point spread function on the transverse focal plane, the transverse FWHM of the central lobe and the maximum intensity in the first sidelobe are chosen as the performance parameters. The attempt for simultaneous reduction of these two factors with prespecified value for acceptable minimum value for the Strehl ratio provide sets of trade-off solutions approximating the Pareto front. The next section presents the mathematical derivation for the normalized transverse intensity distribution in the far field of a concentric multizone phase filter. Section 3 describes synthesis of optimal phase filters and the implementation of the PSO technique. Section 4 contains illustrative results for the Pareto fronts, followed by concluding remarks in Section 5. 2. MATHEMATICAL EXPRESSION FOR THE TRANSVERSE INTENSITY DISTRIBUTION IN THE FAR FIELD The complex amplitude distribution on the image plane in the far field of an axially symmetric imaging system, as shown in Fig. 1, is given by Z 1 F p f rJ 0 prrdr; (1) 0

where f r is the pupil function representing the complex amplitude over the exit pupil at E 0 . J N is the Bessel function of the first kind of order N. r is the normalized radial distance of a point A 0 on the exit pupil from the optical axis and is given by 0 , where ρ 0 is the radial distance of the point A 0 r ρ 0 ∕ρmax 0 from the center E 0 and ρmax is the radius of the exit pupil, and p is the reduced diffraction variable defined as 2π (2) p n 0 sin α 0 χ; λ where χ ( O 0 P 0 ) is the geometrical distance of the point P 0 on the image from the center of the diffraction pattern, α 0 is the

Research Article


F N p 2

M X m1

f m I m p;

where I m p is given by r J pr − r m−1 J 1 pr m−1 I m p m 1 m p J pr J pr r 2m 1 m − r 2m−1 1 m−1 : pr m pr m−1

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(8)

(9)

For an M zone phase filter having the same phase kαm over the m-th zone, Fig. 1. Exit pupil and image plane in the image space of an axially symmetric imaging system.

semiangular aperture, n 0 sin α 0 is the image space numerical aperture, and λ is the operating wavelength. For an arbitrary f r, the normalized complex amplitude F N (p) is given by Z 1 F p F N p 2 f rJ 0 prrdr: (3) F 0 0 0 F 0 0 is the amplitude at the center of the diffraction pattern for an Airy pupil, where f r 1 0

for 0 ≤ r ≤ 1 otherwise:

4

With a phase filter consisting of M concentric unequal annular zones the pupil function may be represented as f r

M X m1

f m W m r;

(5)

where f m is the fixed transmission over the m-th zone and W m r are zero-one functions, as shown in Fig. 2: W m r 1 for r m−1 ≤ r ≤ r m 0 otherwise:

6

2

M X m1

Z fm

0

rm r m−1

J 0 prrdr:

This may be represented as

Fig. 2. mth zero-one function W m r.

(10)

where k 2π∕λ is the propagation constant. Substituting Eq. (10) in Eq. (8), we can write F N p 2

M X

e ikαm I m p:

(11)

m1

Thus, the normalized transverse intensity distribution for an M zone filter on the paraxial image plane may be expressed as I N p jF N pj2 4

M X M X cos kαm − αn I m pI n p: m1 n1

(12) For an M zone phase filter, different combinations of r m and phases αm , over zones (m 1; …M ) give rise to different distributions for the normalized intensity I N p on the transverse plane. According to Eq. (12) the normalized central intensity on the image plane for an Airy pattern is unity. Hence, for a concentric annuli phase filter, the normalized central intensity I N 0, given by Eq. (12) is a direct measurement of the numerical value of the Strehl ratio S.

3. OPTIMUM PHASE FILTERS USING PSO A. Degrees of Freedom

The normalized far-field amplitude distribution is given by Z 1 M X fm W m rJ 0 prrdr F N p 2 m1

f m e ikαm ;

(7)

In the search for optimal phase filters consisting of concentric annular zones of unequal area, the phases in the annuli and the radii of the annuli are the available design parameters. All the design parameters are used in their normalized form. For such a filter with M zones the available number of degrees of freedom is [2M − 1]. The phase of the innermost zone is set to zero and the radius of the outermost zone or M -th zone is set to 1. Any of the remaining zones are allowed to have any phase, out of P finite discrete phase levels in the range (0, 2π). As a result, a particular zone can have any phase out of 0; 2π∕P; 4π∕P…; P − 12π∕P. The inner and outer radii of the m-th zone of an M zone filter are r m−1 and r m (m 1; …M ), respectively. Note that r 0 0 and r M 1. During stochastic search operation, r m is allowed to take 1, where any value within the range er m−1 ; e er m−1 r m−1 ∈;

(13)

e 1 1− ∈;

(14)

ϵ is a small number given by

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∈

30 : ρmax

(15)

with 0 ρmax ; (16) λ being equal to radius of the exit pupil in wavelengths. ϵ in Eqs. (13) and (14) above inhibits the occurrence of any zone width less than or equal to 30λ. Introduction of this minimum possible zone width retains validity of our analysis based on the scalar diffraction theory; it does also minimize microfabrication problems.

ρmax

B. Fitness Function

In our problem, the PSO optimization looks for the maximum value of the fitness function. The fitness function Φ in this case, is inversely related to the merit function ψ by 1 : (17) Φ 1ψ The formation of the merit function plays a critical role in the search for globally or quasi-globally optimum solutions. Constituents of the figures of merit are the FWHM of the central lobe of the transverse intensity distribution and the peak intensity in the sidelobes. The transverse FWHM is expressed as a dimensionless quantity 2ˆp, where 2π ˆ (18) pˆ n 0 sin α 0 ξ; λ 2ξˆ is the actual value of transverse FWHM. For a specific value of numerical aperture (n 0 sin α 0 ), the transverse FWHM in terms of fraction of wavelength can be readily obtained as 2ξˆ n 0 sin α 0 2ˆp: (19) λ 2π The merit function ψ has been formulated to incorporate prespecified values for 2ˆp and β, where β is the peak intensity in the sidelobes lying after the central lobe and within a prespecified value of p, say pmax. Note that in the Toraldo approach for superresolution pmax corresponds to the effective field of view. ψ is defined as a linear combination of the squared difference of the target and current values for 2ˆp and β, ψ ω1 ˆpT − pˆ C 2 ω2 βT − βC 2 ;

(20)

where ω1 and ω2 are the weighting factors. For the sake of simplicity, the factor 2 is merged in the factor ω1. The superscripts T and C indicate the target and the current values for the respective parameters. It should be noted that the Strehl ratio is not directly used in formation of the fitness function. Instead, a lower limit for the Strehl ratio is used as an acceptance criterion for a particular solution. Any solution providing value of the Strehl ratio larger than the stipulated minimum value is retained in the search procedure; but a solution providing a value of the Strehl ratio lower than the minimum value is rejected, and a new search is carried out. As the solution space is searched iteratively, the associated phase filters are generated. Using Eq. (12), the normalized transverse intensity is computed for a specific phase filter, at a set of equidistant values for p over (0, pmax ) at a particular

Research Article interval. For example, with pmax 10, the normalized intensity has been computed for p over 0.1 (0.1) 10. Although in the case of phase filters studied in the course of this investigation, the peak intensity in sidelobes outside the central lobe was observed to be the peak intensity in the first sidelobe; for the sake of generality, our algorithm searches for peak intensity in the sidelobes outside the central lobe from the onset of first sidelobe to pmax . The algorithm terminates either after a fixed number of epochs or after the set target is achieved by a “particle.” As the search ends, we have arrived at our best possible transverse phase filter. C. Multiobjective Optimization Using PSO

PSO is a heuristic global optimization method widely used for solving various computational optimization problems. The idea evolved from research on the behavioral patterns of bird and fish flock movements and was formally laid down by Kennedy and Eberhart in 1995 [51]. In basic PSO, an “n” particle swarm system is considered and each particle is a candidate solution in the representative search space. The particles, in the search space, change their position according to the law of inertia in such a manner that each individual particle tries to reach its most optimal position as well as the swarm converges toward its most optimal position. The system is initialized with a population of random candidate solutions with randomized velocities and positions in the search space. In contrast with genetic algorithm, the candidate solutions are real values and are not encoded binary strings; the absence of mutation and crossover operation makes such encoding unnecessary. The particles update their velocity and position in random direction, which is a crucial part of the algorithm. Each particle remembers the best fitness value achieved by it so far, and the associated solution is called “pbest;” the solution with the highest fitness across all particles is called “gbest.” At the end of an iteration step both “pbest” and “gbest” are updated if higher fitness is achieved. The velocity and the position are considered as multidimensional vectors and updates of them are governed by the following equations [53], v i;j t 1 ωv i;j t c 1 r 1;j x i;j0 t − x i;j t c 2 r 2;j g j t − x i;j t; x i;j t 1 x i;j t v i;j t 1;

(21) (22)

where v i;j t and x i;j t are the jth components of the velocity and the position vectors of the ith particle at time t, respectively. The term x i;j0 t is the jth component of “pbest” of ith particle and g j t is the jth component of “gbest” of the swarm at time t, respectively. The time increases by unity for successive iteration steps. The parameters ω0 ≤ ω ≤ 1.2, c 1 0 ≤ c 1 ≤ 2, c 2 0 ≤ c 2 ≤ 2 are user supplied coefficients, and r 1 and r 2 are random values in the range [0, 1] regenerated for each velocity update. The three terms in the right-hand side of Eq. (21) carry different significance. The first component ωvi;j t is known as the inertia component, and it keeps the particle moving in the same direction it was originally heading to. The inertia coefficient typically takes values between 0.8 to 1.2, with a lower value of ω encouraging faster convergence and higher value enforcing more thorough exploration of the search

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space. The second component c 1 r 1;j x i;j0 t − x i;j t is known as the cognitive component or the memory of the particle. It pushes the particle toward the region of the search space where it experienced a higher individual fitness or “pbest.” The cognitive coefficient c 1 is responsible for the strength of the push toward the “pbest.” The third and last component is the c 2 r 2;j g j t − x i;j t, commonly known as the social component, and causes the particle to move toward the overall best region found by the swarm so far. The social coefficient c 2 again affects the strength of the push toward the global optimum “gbest.” Typical values of c 1 and c 2 are close to 2.0. To prevent the particle from moving far beyond the search space in a single iteration step, the maximum velocity of each particle is clamped to a prespecified value v max , which may be considered as a representative of the resolution with which the search space is explored. Extremely high or extremely low values of v max forces the particle either to fly past the good solution or to get trapped between local optima. However, it is a general practice to limit the v max within the dynamic range of each variable in each dimension. In the present problem, a “particle” represents a phase filter comprised of phases and radii values. The associated intensity distribution is computed using Eq. (12). From this distribution transverse FWHM 2ˆp and sidelobe intensity m are determined. These values are then used to evaluate the merit function as well as the fitness of that particular filter using Eqs. (20) and (17). The best fitness value achieved by a particle is stored along with the radius and phase values (i.e., the solution). The solution is called “pbest” and the best fitness value among all particles of the swarm is considered to be the best fitness achieved by the swarm and the corresponding solution is stored as “gbest.” The velocity update takes values such that the particles converge toward their own best values as well as toward the swarm’s best solution. The position and velocity of a particle can both be considered as multidimensional vectors and each component of these vectors is a particular phase or radius value; e.g., for a three-zone filter available degrees of freedoms are two phase values φ1 and φ2 of the second and third zones, respectively, and two radii value r̴ 1 and r̴ 2 of the first and second zones (since the phase of the innermost zone is set to 0 and the radius of outermost zone is set to 1). In this case the velocity vector for a particle may be represented as V⃗ Δφ1 ; Δφ2 ; Δr̴ 1 ; Δr̴ 2 :

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Δr̴ 2 t 1 ωΔr̴ 2 t c 1 r 1 r̴ 20 t − r̴ 2 t c 2 r 2 g 4 t − r̴ 2 t:

(27)

The primed variables represent the solution corresponding to the “pbest” filter at time t; g represent the overall best solution “gbest” acquired by the swarm till time t. The updated values for the phases and radii may be expressed as φ10 t 1 φ1 t Δφ1 t 1;

(28)

φ20 t 1 φ2 t Δφ2 t 1;

(29)

r̴ 10 t 1 r̴ 1 t Δr̴ 1 t 1;

(30)

r̴ 20 t 1 r 2 t Δr̴ 2 t 1:

(31)

If the new fitness value of the particle is better than the fitness value corresponding to the solution stored in “pbest,” the current solution replaces the earlier “pbest” values. Once this updating process is completed for each particle in the swarm, the fittest particle is identified and if its fitness value is better than the fitness value for the solution in “gbest,” the values in “gbest” are updated with the values in “pbest” of this particle. The essential steps of the PSO algorithm are depicted in a flowchart in Fig. 3.

(23)

Here, Δ represents the step size of updating velocity. Using Eq. (22) the velocity updates for phases and radii in a single iteration are given by Δφ1 t 1 ωΔφ1 t c 1 r 1 φ10 t − φ1 t c 2 r 2 g 1 t − φ1 t;

(24)

Δφ2 t 1 ωΔφ2 t c 1 r 1 φ20 t − φ2 t c 2 r 2 g 2 t − φ2 t;

(25)

Δr̴ 1 t 1 ωΔr̴ 1 t c 1 r 1 r̴ 10 t − r̴ 1 t c 2 r 2 g 3 t − r̴ 1 t;

(26)

Fig. 3. Flow chart representation of the particle swarm optimization (PSO) algorithm.

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4. ILLUSTRATIVE RESULTS FOR THE PARETO FRONTS The PSO technique as elucidated above has been incorporated for the synthesis of concentric multizone unequal area phase filters to obtain a high transverse resolution at the focal plane of the imaging system. FWHM values for all PSFs are normalized by the FWHM of the Airy pattern and it is called normalized transverse FWHM Δ. Peak sidelobe intensity β is determined from values of normalized intensity I N p for a set of values of p over the interval (0, pmax ). Numerical results presented here correspond to pmax 10. The search was restricted to multizone filters with two, three, and four zones with various combinations of ω1 , ω2 , and prespecified values for the normalized transverse FWHM Δ and the peak intensity β in the sidelobes. A higher number of zones did not provide any tangible improvement. The search was carried out with a gradually increasing number of allowable phase levels from two to four and eight. Again it was noted that, in general, a further increase in the number of phase levels beyond certain values did not provide any significant advantage. Results for some phase filters with a number of zones more than two are found to have the same phases in contiguous rings; e.g., a four-zone filter with phases 0.0, 1.75, 1.75, 1.0 may occur. This three-zone filter may be considered as a special case of a four-zone filter having second and third zones merged. However, the search procedure is carried out for a series of prespecified target values for transverse FWHM 2ˆpT . Since the peak intensity in the sidelobe is usually desired to be as small as possible, a prespecified value for βT is always set to zero. Each search resulted in an optimal phase filter that provides a combination of 2ˆp and β values. Then the transverse FWHM 2ˆp is normalized by that of the Airy pattern. These sets of normalized transverse FWHM (Δ) and the corresponding sidelobe intensity (β) define a curve called the Pareto front, which is concave in nature. The normalized transverse FWHM Δ is plotted along the abscissa and the peak intensity of the sidelobe β is plotted along the ordinate. The Pareto fronts for two-zone filters with available phase steps 8 for each zone are shown in Fig. 4. Four curves highlight the fronts for four different prespecified lower limits of the Strehl ratio. Toward the left-hand side the Pareto fronts consist of narrower central lobes along with a relatively larger value of peak intensity in the sidelobe. Wider central lobes accompanied by negligible sidelobes appear toward the right-hand side of the Pareto front. The span of the front is large for the low cutoff value 0.1 for the Strehl ratio, and it decreases with an increase in the cutoff value for the Strehl ratio. Also, the Pareto fronts lose their concavity with the increase in the lower cutoff value of the Strehl ratio. As the cutoff limit of the Strehl ratio increases, the lowest attainable value for the FWHM increases. The Pareto front presented in Fig. 4 tends to unfold the intricate interrelationship between the FWHM, the peak sidelobe intensity, and the value of the Strehl ratio attainable by use of two-zone phase filters. It may be utilized at the stage of a preliminary feasibility study of two-zone phase filters for specific applications in practice. For example, if we need a normalized transverse FWHM 0.8 then it is clear from Fig. 4 that it cannot be obtained for higher cutoff values of the Strehl ratio,

Fig. 4. Pareto fronts for a two-zone filter at different lower cutoff values for the Strehl ratio.

because the vertical dashed line does not intercept the Pareto fronts for the higher cutoff values of the Strehl ratio 0.5 and 0.7. The possibility of obtaining FWHM 0.8 is increased as the Strehl ratio is sacrificed to lower value; the vertical dashed line intercepts the Pareto fronts for lower cutoff values of the Strehl ratio 0.1 and 0.3. But in both cases one has to put up with a larger intensity in the sidelobe. Moreover, the Pareto front can provide estimates for the maximum achievable value for the Strehl ratio and the peak intensity in the sidelobes for a target value for FWHM. The Pareto fronts for phase filters with two, three, and four concentric annular zones with eight available phase steps at each zone are shown in Fig. 5 below for two minimum acceptable values of the Strehl ratio.

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In the Pareto fronts shown in Fig. 5 above, the number of zones are varried from two to four, keeping the available phase level eight. It is clear that an increase in the number of annular zones does not influence the shape of the Pareto front significantly, but they lose their concavity with an increase in minimum acceptable value for the Strehl ratio. In Fig. 6, the effect of increasing the number of allowable phase levels in zones is studied. The Pareto fronts for two-zone phase filters are shown at different prespecified lower limits of the Strehl ratio for three different cases where two, four, and eight phase levels are allowed in each zone. It is observed that a change in the number of phase levels does not affect the shape of the Pareto front significantly. However, it may be noted that the concavity of the Pareto fronts is higher for a lower cutoff limit of the Strehl ratio in all cases. 5. CONCLUDING REMARKS

Fig. 5. Pareto fronts for two-zone, three-zone, and four-zone filters. Allowable phase level in each zone 8. Lower cutoff values for the Strehl ratio are 0.1 and 0.5.

This paper presents a new approach for global synthesis of Toraldo filters for superresolution. Optimal phase filters with concentric, unequal area annular zones of fixed phase are synthesized by the PSO technique. The FWHM of the central lobe of the transverse PSF and the peak intensity in the sidelobe over a predefined and restricted field of view are chosen as components of the figures of merit. This paper explores the Pareto optimal relationship between the two conflicting objectives, namely large value for FWHM and small value for peak sidelobe intensity, for different values of minimum allowable values for the Strehl ratio. Our investigation provided combinations of optimal values of two such conflicting objectives forming a curve known as the Pareto front in the objective space. The Pareto front is found to be concave in nature. The shape of the curve is independent of the number of zones or the number of allowable phase steps in each zone, but loses its concavity with higher cutoff values for the Strehl ratio. Numerical results presented in the earlier section use dimensionless variables. The latter are scaled/normalized in a manner that extends the usefulness of the same Pareto fronts for a large class of practical applications. Funding. Department of Science & Technology (DST), India; INSPIRE Fellow program “Assured Opportunity for Research Career” (AORC) (2010 [115]). REFERENCES

Fig. 6. Pareto fronts for two-zone filters with available phase levels two, four, eight; lower cutoff values for the Strehl ratio are 0.1 and 0.5.

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