Hybrid pupil filter design using Bessel series

Optics Communications 284 (2011) 4900–4902

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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m

Hybrid pupil filter design using Bessel series Noé Alcalá Ochoa a,⁎, J. García-Márquez a, A. González-Vega b a b

Centro de Investigaciones en Óptica, A.C., Loma del bosque 115, Col. Lomas del campestre, 37150 León, Mexico División de Ciencias e Ingeniería, Universidad de Guanajuato, Loma del bosque 103, Col. Lomas del campestre, 37150 León, Mexico

a r t i c l e

i n f o

Article history: Received 9 November 2010 Received in revised form 15 March 2011 Accepted 23 June 2011 Available online 7 July 2011 Keywords: Super-resolution Pupil engineering Bessel functions Complex pupil filters

a b s t r a c t We propose a simple method of designing pupil filters for transverse super-resolution. For this end we represent the amplitude Point Spread Function (PSF) as a series expansion, constructed from the Fourier transform of a basis of Bessel functions. With this representation we optimize the intensity PSF according to certain desired characteristics, such as a smaller disk diameter than the corresponding, clear aperture, Airy disk. It is proved that by using few basis functions, it is possible to design pupils with similar or better PSF characteristics than previously reported. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Super-resolution deals with the reduction of the PSF of a diffraction-limited optical system. This reduction may be transversal or longitudinal, although they are not independent [1]. The transverse section of a PSF of an axially symmetric pupil shows a central lobe surrounded by side lobes, generally of lower intensity. It has been shown with the design of different kinds of pupils, that the reduction in size of the central spot of the PSF has some detrimental effects, such as a reduction of the Strehl ratio and an increment in the brightness of its rings. Toraldo Di Francia [2] proposed pupils with equally spaced rings, and proved the possibility of reducing the size of the PSF as much as desired, displacing the most significant secondary lobes to almost any position. The price that had to be paid was a drastic reduction of the Strehl ratio. Other pupil designs based on filtering rings have been proposed in the last 12 years [1,3–5]. We can find experiments performed with these pupils in various disciplines. In astronomy, for example, they are used to optimize ground-based telescopes [6]; in laser ablation, the PSF diameter is reduced for femtosecond laser machining [7]; in microscopy, binary and phaseonly filters are employed in 4Pi confocal microscopes [8–10]; in high density data storage, super-resolving confocal readout systems are used to retrieve the data [11]; in near field microscopy, superresolving pupils can be synthesized by means of two spatial light modulators [12]. The aim of this work is to demonstrate the possibility of designing useful super-resolving pupils using Bessel series, by the simple ⁎ Corresponding author. Tel.: + 52 47 74 41 42 00; fax: + 52 47 74 41 42 09. E-mail addresses: [email protected] (N.A. Ochoa), [email protected] (J. García-Márquez), gonzart@fisica.ugto.mx (A. González-Vega). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.06.049

expedient of reducing to zero the intensity of the PSF at a predetermined radius, with the help of an optimization algorithm. For this radius, the Strehl ratio and the side-lobe intensities can be controlled with the number of basis functions. We present examples to show the capabilities of the proposed method; for example, using only three basis functions it is possible to get a PSF with similar characteristics than the methods described in Refs. [3,5]. Additionally we report pupils that produce a reduction factor up to 0.55, and moderate Strehl ratios. Finally, we also present results to show the limit of super-resolution that can be attained with this kind of filters. 2. Methodology We shall now describe the process to obtain the functions for the expansion of the PSF, and their relation to the pupil function. Let G(δ, v) be the normalized complex amplitude distribution of an axially symmetric, complex pupil function F(ρ): 1   2 Gðδ; vÞ = 2∫ F ðρÞ exp −i2πδρ Jo ðvρÞρdρ;

ð1Þ

0

where, ρ is the normalized radial coordinate and δ, v are the axial and the transverse dimensionless optical coordinates, respectively, defined by the equations ν = 2πrNA/λ and δ = ξ NA 2/2λ, where r and ξ = z − f are the radial and axial distances from the focus, respectively, NA is the numerical aperture, and λ is the wavelength. Let us suppose that we can describe the pupil function as K

F ðρÞ = ∑ Cn J0 ðαn ρÞ; n=0

ð2Þ

N.A. Ochoa et al. / Optics Communications 284 (2011) 4900–4902

4901

where J0(x) is the Bessel function of the first kind and zero order, αn are the roots of J1(x), Cn are the, possibly complex, coefficients to be calculated, and K + 1 the number of basis functions that we adopt.To calculate the transverse amplitude at F(ρ) we set δ = 0 in Eq. (1). Using now Eq. (2) in Eq. (1) we obtain K 1 GðvÞ = Gð0; vÞ = ∑ Cn J0 ðαn ρÞ2∫ J0 ðαn ρÞJo ðvρÞρdρ: n=0

ð3Þ

0

Evaluating the integrals as shown in Ref. [13], we can rewrite Eq. (3) as K

GðvÞ = ∑ Cn Gn ðvÞ; n=0

ð4Þ

with   2 2 Gn ðvÞ = 2J0 ðαn Þ v J1 ðvÞ = v −αn :

ð5Þ

In a 2-D plot, the functions Gn(v) look like a normalized Airy function with its main maximum shifted to the radial coordinate v = αn. In particular, G0 = 2J1(v)/v has its principal maximum at v = α0 = 0, with an amplitude J0(α0) = 1. Other properties of these functions can be found in Ref. [13]. Now we must find the coefficients Cn (Eq. (4)) that reduce the intensity of the PSF to zero at the predetermined radius. Once the coefficients Cn are found, these must be recalculated as Cn/max[|F(ρ)|], so that the condition |F(ρ)| ≤ 1 can be satisfied (Eq. (2)). Even if we assume that the coefficients are real, as we have done for simplicity in this work,F(ρ) may take negative as well as positive values, corresponding thus to a hybrid pupil filter; that is, a filter of variable amplitude and phase, albeit the phase variation is restricted to the values 0 and π in this case. In contrast, in amplitude-only pupil filters F(ρ) N 0, and in phase-only pupil filters |F(ρ)| = 1, which implies constant complex amplitude transmission. As in previous works [3,5], the spot size reduction of the central lobe ε, the Strehl ratio S, and the relative side-lobe intensity Γ are the parameters to be controlled. These parameters are defined as follows: ε=

D ; Dc

ð6:1Þ

S=

PSF ð0Þ ; PSFc ð0Þ

ð6:2Þ

Γ=

PSF ð0Þ ; PSF ðμ1 Þ

ð6:3Þ

Fig. 1. Normalized PSFs obtained with our method (continuous) and the method proposed by Canales and Cagigal [5] (dotted). In both cases a super-resolving gain parameter ε = 0.64 was employed.

In our optimization experiments we readily noticed that, for a constant value of K + 1, the Strehl ratio decreases and the side-lobe intensity increases, both rapidly, as soon as we start reducing the value of the predetermined radius εv1. This is a well known development in super-resolving PSFs. Increasing the value of K + 1, but keeping constant the value of εv1, reduces still more the Strehl ratio, but also reduces the side-lobe intensity for that particular radius, increasing the value of Γ, albeit only moderately.

3. Results We shall present now some optimization results, comparing them with those obtained in previous works. Finding the coefficients Cn to satisfy Eq. (7) with ε = 0.64 and K = 3 (only three coefficients), gives us a PSF with S = 0.0876 and Γ = 4.41 (Fig. 1). In this figure we can see our optimized PSF (continuous line) and the super-resolving PSF proposed by Canales and Cagigal [5] (dotted line), for the same value of ε. It can be noticed that we obtained a slightly larger value of Γ, but the Strehl value reported in Ref. [5] is higher than ours (0.147). The optimum coefficients Cn are (0.2959, −0.8860, 1.1571). Using these three coefficients in Eq. (2), we can easily reconstruct the pupil function. This is shown in Fig. 2. Our optimum pupil, therefore, is

where D means diameter of the central spot, the sub index “c” means clear aperture, and μ1 is the position of the first secondary maximum. We have been successful in using Eqs. (4) and (5) to make Iðεv1 Þ = jGðεv1 Þj

2

= 0;

ð7Þ

where I is the PSF intensity, v1 is the first zero of the Airy function, and ε b 1 is the super-resolving gain parameter. This means that the optimization was only carried out at the radial coordinate εv1. The Strehl ratio and the side-lobe intensities depend on the value of K + 1, the number of functions involved in the optimization process. The larger the value of K, the higher the reduction in S and the increment in Γ. We choose the Nelder–Mead fitting algorithm (MatLab) as optimization method. Since the optimization time and results depend heavily on the choice of initial values, we found through experience that if the coefficients Cn are initialized as Cn = 1/J0(αn), the algorithm converges quickly.

Fig. 2. Pupil function corresponding to the PSF in Fig. 1.

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N.A. Ochoa et al. / Optics Communications 284 (2011) 4900–4902

price in the Strehl ratio. The relative secondary-lobe height, on the other hand, does not increase rapidly, and at most doubles its intensity within the interval 0.9 N ε N 0.1. So, usable values would depend on the application at hand; for example, if we need ε = 0.55, we obtain S = 0.0132 and Γ = 2.22, the above-mentioned values. Thus S0 = S = 0.0132 and S1 = S0/Γ = 0.006, in Fig. 4. For ε = 0.47, we obtain S = 0.008 and thus S0 = S and S1 = S0/Γ = S0 = 0.008, which correspond to the point of intersection of the curves in Fig. 4. 4. Conclusions

Fig. 3. PSF intensity profile with a super-resolving gain parameter ε = 0.55.

hybrid, and can be seen as a phase, annular pupil, modulated by a positive continuous non-smooth function. It is in fact similar to the PSF reported by Gundu et al. [3], with practically the same value of Γ. Optimizing again for the same value of ε = 0.64, but choosing K = 6 and 10, we obtain S = 0.0258 and 0.01 and Γ = 5.52 and 6.12, respectively. It should be noticed here the inverse relationship between S and Γ, and the moderate increment of Γ with K. A further increment in resolution can be obtained, for example, setting ε = 0.55 and K = 6. We then obtain the PSF of Fig. 3, with values of Γ = 2.22 and S = 0.0132. The coefficients Cn in this case are (0.1151, −0.4756, 0.5652, −0.7026, 0.7838, −0.9014). Note that with this pupil design we are able to get tolerable Strehl ratios and Γ values for image processing applications. In order to know if there is a limit to the degree of super-resolution that can be attained with these filters, both in terms of a theoretical limit, regardless of practical considerations, and of a maximum usable limit, we elaborated Fig. 4. In it we plot the Strehl ratio of the central maximum and of the first side lobe, S0 (continuous line) and S1 (dashed line), respectively, as functions of the super-resolution gain parameter ε, for K = 6. It can be noticed that theoretically we can get almost any amount of super-resolution, at the expense of the Strehl ratio reduction of the central lobe, and the Strehl ratio increment of the secondary lobe. It is interesting to observe that, with this filter (K = 6), a modest increment in resolution (ε = 0.9) is paid at a considerable

Fig. 4. Strehl ratio curves for the central (continuous) and the first side lobe (dashed), as function of the super-resolution gain parameter, ε.

In conclusion, we have presented a simple procedure to design super-resolving pupils based on Bessel series expansions. With this procedure the central lobe reduction (ε) of the PSF can be reached independently of the number of basis functions that are used for this purpose (K + 1 ≥ 3). The Strehl ratio and the side-lobe intensity, on the other hand, depend inversely on each other, although, with the reduction of ε, the former decreases drastically, whereas the later, in comparison, remains fairly constant. Some curves are presented to appreciate this effect. We believe that the method proposed here accelerates the computation of super-resolving pupils, allowing their design in-situ. Acknowledgments Noé Alcala Ochoa would like to thank the support of the CONACyT through project 133495, and Arturo González-Vega the grant PROMEP/103.5/10/4684. We gratefully acknowledge the comments of the anonymous reviewers, that helped us to improve significantly the article, and also some suggestions from J. E. A. Landgrave. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

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