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Shaded-Mask Filtering for Extended Depth-of-Field Microscopy
Shaded-Mask Filtering for Extended Depth-of-Field Microscopy
Journal of information and communication convergence engineering. 2013. Jun, 11(2): 139-146
Copyright ©2013, The Korean Institute of Information and Commucation Engineering
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/li-censes/bync/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
  • Received : March 10, 2013
  • Accepted : April 15, 2013
  • Published : June 30, 2013
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About the Authors
Isabel Escobar
Department of Applied Physics, University of Castilla la Mancha, Cuenca E-16071, Spain
Genaro Saavedra
Department of Optics, University of Valencia, Burjassot E-46100, Spain
Manuel Martínez-Corral
Department of Optics, University of Valencia, Burjassot E-46100, Spain
manuel.martinez@uv.es
Arnau Calatayud
Centro de Tecnología Físicas, Polytechnic University of Valencia, Valencia E-46022, Spain
Ana Doblas
Department of Optics, University of Valencia, Burjassot E-46100, Spain

Abstract
This paper proposes a new spatial filtering approach for increasing the depth-of-field (DOF) of imaging systems, which is very useful for obtaining sharp images for a wide range of axial positions of the object. Many different techniques have been reported to increase the depth of field. However the main advantage in our method is its simplicity, since we propose the use of purely absorbing beam-shaping elements, which allows a high focal depth with a minimum modification of the optical architecture. In the filter design, we have used the analogy between the axial behavior of a system with spherical aberration and the transverse impulse response of a 1D defocused system. This allowed us the design of a ring-shaded filter. Finally, experimental verification of the theoretical statements is also provided.
Keywords
I. INTRODUCTION
Imaging systems with a high depth-of-field (DOF) are required in many applications across different fields [1 - 5] , such as microscopy [6 - 8] and communications [9] . However, most imaging systems described in the literature are very sensitive to defocusing. This means that small misalignments between the object and image plane impose great limitations on the imaging systems. In order to increase the DOF of these systems, numerous studies have been carried out recently along these lines [10 - 16] .
The trivial method for increasing the DOF is to reduce the numerical aperture (NA) of imaging systems. However, this provokes a dramatic decrease in the transverse resolution: there exists a compromise between the transverse resolution and DOF. Thus, considerable effort has been expended in attempting to increase the DOF of imaging systems without undermining their resolution. In a particular case of the extension of the DOF while maintaining the transverse resolution [8 , 13] , many pupil masks have been designed based on phase [6 , 7 , 13 - 15 , 17 , 18] and/or amplitude [19 - 21] . It is of interest that the advantages of phase masks are superior to those of amplitude masks, including the fact that the most recently developed amplitude masks are not sufficiently light efficient. Another technique is the utilization of a multifocal concept, which uses various lenses of different focuses [22] .
In this work, we analyze the enhancement of DOF in imaging systems. For this, we employ the analogy between the axial behavior of a system affected by spherical aberration and the transverse response of an imaging system. To study the increase in the DOF, we implement an amplitude filter in two optical systems. Similarly, any other filter designed to reduce the spherical aberration can be used to increase the DOF.
The paper is organized as follows. In Section II, we derive the equations that describe an optical system affected by spherical aberration. Also, as a specific case worth analyzing, the complex amplitude distribution is particularized to study the axial behavior of the imaging system. Section III is devoted to demonstrating the similarity between the axial response of an imaging system with spherical aberration and the transverse response of an optical system. Finally, in Section IV, we set up two different experiments. The first of them is implemented in an imaging system whose NA is low. For this case, we show both the numerical and experimental verification; the high agreement between them is clear. The second experiment is performed with conventional scanning microscopy. To sum up, in Section V, we conclude the main achievements of our research.
II. THE MISMATCH INDEX-INDUCED SPHERICAL ABERRATION
In this section, we study the aberration in a high-NA system when the wave field is focused through several media stacked perpendicularly to the optical axis.
Let us start by considering a high-NA objective lens illuminated by a monochromatic collimated beam with wavelength λ . The geometry of the objective is illustrated in Fig. 1 . Contrary to what happens in the paraxial approach, the objective is characterized by its principal surfaces, which are a planar surface, S 1 , and a spherical surface S 2 , with focus f and centered at the focal point, F. In most of the high-NA objectives reported in the literature, the aperture stop is located at the back-focal plane. Thus, if a monochromatic planar wave strikes the objective lens, the emerging wave field is a truncated spherical wavefront [23] . This wavefront is focused passing through a dielectric layer, the coverslip, whose thickness is t and refractive index n ’, immersed into a medium with a different refractive index.
The amplitude distribution at the neighborhood of the focal plane can be calculated according the scalar, nonparaxial Debye’s formulation [24] and assuming that the sine condition [25 , 26] holds. After straightforward maths, the complex amplitude distribution along the optical axis is given by [27]
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where w 40 and w 20 are, respectively, the well-known spherical-aberration coefficient and defocus coefficient, as measured in units of wavelength, and q ( ζ ) is the apodized amplitude transmittance of the aperture stop.
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Conceptual diagram to explain the focus process in a high numerical aperture objective into two media separated by a planar interface.
III. DESIGN OF BEAM-SHAPING ELEMENTS TO INCREASE THE DEPTH-OF-FIELD
Our aim here is to study beam-shaping elements that increase the DOF of optical systems. For that, let us consider a conventional two-dimensional (2D) imaging system, which basically consists of a telecentric arrangement, as shown in Fig. 2 .
The telecentricity provides two important properties to the system: the system is 2D linear and shift-invariant. Therefore, the 2D irradiance distribution at the image space can be expressed as the 2D convolution between a scaled version of the 2D object and a 2D function, which is called the intensity point-spread function (PSF) of the imaging system [28] ,
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where M = − f 2 / f 1 is the magnification of the imaging system and the intensity PSF is then obtained as the square modulus of the amplitude PSF
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where, p ( xp , yp ) represents the amplitude transmittance of the aperture stop ( Fig. 2 ). We have also omitted some irrelevant factors. In the particular case in which the pupil function is separable in Cartesian coordinates, the amplitude PSF can be rewritten as h '( x , y ; z ) = h'x ( x ; z ) h'y ( y ; z ), being
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Schematic of a two-dimensional telecentric imaging system. The light emanating from the object is collected by the objective (L1) and focused by the tube lens (L2).
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Note that the coordinates in the aperture stop plane have been normalized as
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(a) Shaded ring filter for reduction of the spherical aberration impact, (b) Cartesian version of the filter.
where r max is the radius of the circle in which the square pupil is inscribed.
It is interesting to note the similarity between Eqs. (1) and (4). This implies that the axial response of an imaging system affected by spherical aberration behaves similarly to the transverse response of an imaging system with a square pupil. This reasoning leads us to conclude that the amplitude profile family designed to reduce the spherical aberration may also be used to provide greater tolerance to defocusing in imaging systems.
The general case study can be particularized to the case of binary masks known as shaded ring (SR) filters. These filters are composed of three annular zones with two different transmittances, and each mask is uniquely specified by two construction parameters (μ, η) as defined in Fig. 3 . From Fig. 3 , it is trivial to realize that a square filter produces a significant loss of resolution in certain transverse directions because the entire pupil size is not used. Consequently, a corresponding radial version has been designed ( Fig. 4 ), where the transverse resolution is now the same in all directions. After a numerical optimization procedure [29] , we have selected the values μ = 0.4 and η = 0.7.
Obviously, to evaluate the PSF in this case, it is more convenient to rewrite Eq. (12) in cylindrical coordinates. Moreover, by employing the analogy with the axial response of an imaging system with spherical aberration, the amplitude PSF is
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where ρ = rp / r max and the defocus coefficient is defined as
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Structure of the optimized shaded ring filter, called the defocus tolerance filter, which increases the depth-of-field. The two construction parameters are μ = 0.4 and η = 0.7.
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Conceptual experiment to demonstrate the extended field of view in a low numerical aperture imaging system. The defocus tolerance filter is placed at the front focal plane of L1.
IV. EXPERIMENTAL VERIFICATION
To demonstrate the effects of defocusing in a low-NA imaging system apodized with both the clear aperture and an optimum SR filter, which is referred to as defocus tolerance (DT) filter, we prepared the experimental setup shown in Fig. 5 . The DT filter was fabricated with highcontrast photographic film (Kodak Technical Pan; Rochester, NY, USA). For the illumination of a USAF 1951 resolution chart, we employed the diffused light proceeding from a white source.
In the setup of Fig. 5 , the imaging system was operated in telecentric mode and was composed of two converging lenses whose focal lengths were f 1 = 400 mm and f 2 = 100 mm. According to this, the resolution test was placed at the front focal plane of L1 . To capture the images, we used a CCD camera (JAI/Pulnix TM-765E; Copenhagen, Denmark) composed of 765 × 578 square pixels of 11 μm on each side.
In our experiment we recorded a set of 2D images at different axial positions, z . For simplicity, it is convenient to mount the CCD on a micrometric translation stage to provide a high-precision at different axial positions. Specifically, we selected z = 0, 2.75, 5.50, and 8.25 mm, which, according to Eq. (16), correspond to defocus coefficients of w 20 = 0, -1, -2, and -3. Note that, theoretically, the PSF is symmetrical about the focal plane, so that we do not consider positive defocus parameters because we assume that we would obtain similar results.
In Figs. 6 and 7 we show the numerical and experimental results of the resolution target with the circular aperture and the DT filter. Clearly, the similarity between the experimental and calculated results is apparent. These figures also indicate that the greater the defocus coefficient, the smaller response of the clear aperture. Note that we cannot discern the low frequencies in the elements 1 and 2 of group 0 (1 and 2 LP/mm, respectively) for w 20 = -3. However, the response of the DT filter remains fairly stable and for w 20 = -3 it is possible to detect frequencies of 2.52 LP/mm corresponding to element 3 of group 1. Another fact to consider, in the case of a non-apodized system (left row of Fig. 7 ), is the contrast inversion in several elements of the test group 1 for w 20 = -2.
Finally, to demonstrate the experimental case of high-NA, we arranged the experimental setup schematized in Fig. 8 ; this arrangement corresponds to a conventional scanning microscope. For this experiment, the light emerging from a fiber coupled to a He-Ne laser ( λ = 632.8 nm) was collimated through a converging lens of focal length fL 1 = 200 mm. After passing through a relay system and a beamsplitter, the wave field was focused via a microscope objective, whose NA was 0.9, onto the sample.
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Numerically-evaluated results corresponding to (a) the clear aperture and (b) the defocus tolerance (DT) filter for different values of the defocus coefficient w20. As shown in the right column, the improvement with the depth-of-field is apparent.
The signal reflected by the sample was finally focused onto a pinhole of radius of 50 μm.
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Experimental results of a resolution chart corresponding to (a) the clear aperture and (b) the defocus tolerance (DT) filter. It is clear that the response of the DT filter remains fairly stable, and it is possible to detect higher frequencies with it.
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Schematic layout of practical implementation of a conventional scanning microscope. The special feature of such an arrangement is the insertion of a relay system, which makes possible the introduction of an apodized filter.
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Experimental verification of an extended depth-of-field for a high numerical aperture scanning microscope. For the measurement, we used the tracks on a CD as the object. (a) Clear aperture, (b) the defocus tolerance (DT) filter.
We have carefully chosen the radius of the pinhole given that the detection was not confocal [30 - 33] . The pinhole was placed in front of a detector; in our case, it was a photomultiplier tube. A small fragment of an original music CD was imaged. This object was composed of a collection of tracks recorded on the CD.
Our goal was to increase the DOF. This task can be accomplished by modifying the exit pupil with the use of a DT filter. For this purpose, we used a relay system set up from RL1 ( fRL 1 = 200 mm) and RL2 ( fRL 2 = 175 mm).
In Fig. 9 , we show the experimental results. In this case, an increase in one unit in w 20 value corresponds to an axial displacement of z = 1.12 μm. Again we can see that the DT filter provides very stable behavior as the defocus parameter increases. The DT filter significantly improves the image quality from values superior to -2.
V. CONCLUSIONS
In summary, in this work it has been shown that the DOF has been increased in an imaging system affected by spherical aberration. This result opens the way to reducing the defocus in an imaging system by using an amplitude filter designed firstly to reduce spherical aberration. The improvement of DOF has been checked with an optimized SR filter (called a DT filter), which has been implemented in two types of experimental architecture. It should be noted that in both experiments, the DT filter provides very stable behavior compared to the clear aperture.
Acknowledgements
This work was funded in part by the Ministerio deEconomia y Competitividad, Spain, under Grant DPI2012-32994, and also by Generalitat Valenciana under GrantPROMETEO2009-077. Furthermore, A. Doblas acknowledgesfunding from the University of Valencia through thepredoctoral fellowship program Atraccion de Talent.
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Zhou S. , Zhou C. 2004 “Discrete continuous-phase superresolvingfilters” Optics Letters 29 (23) 2746 - 2746    DOI : 10.1364/OL.29.002746
Yun M. , Liu L. , Sun J. , Liu D. 2005 “Three-dimensionalsuperresolution by three-zone complex pupil filters” Journal of the Optical Society of America A 22 (2) 272 - 277    DOI : 10.1364/JOSAA.22.000272
Gundu P. N. , Hack E. , Rastogi P. 2005 “Apodized superresolutionconcepts and simulations” Optics Communications 249 (1-3) 101 - 107    DOI : 10.1016/j.optcom.2005.01.025
Pereira S. F. , van de Nes A. S. 2004 “Superresolution by means ofpolarization, phase and amplitude pupil masks” Optics Communications 234 (1-6) 119 - 124    DOI : 10.1016/j.optcom.2004.02.020
Canales V. F. , de Juana D. M. , Cagigal M. P. 2004 “Superresolutionin compensated telescopes” Optics Letters 29 (9) 935 - 937    DOI : 10.1364/OL.29.000935
Caballero M. T. , Andres P. , Pons A. , Lancis J. , Martinez-Corral M. 2005 “Axial resolution in two-color excitation fluorescencemicroscopy by phase-only binary apodization” Optics Communications 246 (4-6) 313 - 321    DOI : 10.1016/j.optcom.2004.11.035
de Juana D. M. , Oti J. E. , Canales V. F. , Cagigal M. P. 2003 “Transverse or axial superresolution in a 4Pi-confocal microscopeby phase-only filters” Journal of the Optical Society of America A 20 (11) 2172 - 2178    DOI : 10.1364/JOSAA.20.002172
Liu L. , Liu C. , Howe W. C. , Sheppard C. J. R. , Chen N. 2007 “Binary-phase spatial filter for real-time swept-source opticalcoherence microscopy” Optics Letters 32 (16) 2375 - 2377    DOI : 10.1364/OL.32.002375
Jia J. , Zhou C. , Liu L. 2003 “Superresolution technology forreduction of the far-field diffraction spot size in the laser freespacecommunication system” Optics Communications 228 (4-6) 271 - 278    DOI : 10.1016/j.optcom.2003.10.011
Ojeda-Castaneda J. , Ledesma S. , Gomez-Sarabia C. M. 2013 “HyperGaussian windows with fractional wavefronts” Photonics Letters of Poland 5 (1) 23 - 25
Diaz F. , Goudail F. , Loiseaux B. , Huignard J. P. 2009 “Design of acomplex filter for depth of focus extension” Optics Letters 34 (8) 1171 - 1173    DOI : 10.1364/OL.34.001171
Golub M. A. , Shurman V. , Grossinger I. 2006 “Extended focusdiffractive optical element for Gaussian laser beams” Applied Optics 45 (1) 144 - 150    DOI : 10.1364/AO.45.000144
Liu L. , Diaz F. , Wang L. , Loiseaux B. , Huignard J. P. , Sheppard C. J. R. , Chen N. 2008 “Superresolution along extended depth offocus with binary-phase filter for the Gaussian beam” Journal of the Optical Society of America A 25 (8) 2095 - 2101    DOI : 10.1364/JOSAA.25.002095
Sheppard C. J. R. 2011 “Binary phase filter with a maximally-flatresponse” Optics Letters 36 (8) 1386 - 1388    DOI : 10.1364/OL.36.001386
Sheppard C. J. R. , Mehta S. 2012 “Three-level for increased depthof focus and Bessel beam generation” Optics Express 20 (25) 27212 - 27221    DOI : 10.1364/OE.20.25.027212
Sheppard C. J. R. 2013 “Pupil filter for generation of light sheets” Optics Express 21 (5) 6339 - 6345    DOI : 10.1364/OE.21.006339
Castro A. , Ojeda-Castaneda J. , Lohmann A.W. 2006 “Bow-tieeffect: differential operator” Applied Optics 45 (30) 7878 - 7884    DOI : 10.1364/AO.45.007878
Zalevsky Z. , Shemer A. , Zlotnik A. , Eliezer E. B. , Marom E. 2006 “All-optical axial super resolving imaging using a low-frequencybinary-phase mask” Optics Express 14 (7) 2631 - 2643    DOI : 10.1364/OE.14.002631
Ojeda-Castaneda J. , Tepichin E. , Diaz A. 1989 “Arbitrarily highfocal depth with a quasioptimum real and positive transmittanceapodizer” Applied Optics 28 (13) 2666 - 2670    DOI : 10.1364/AO.28.002666
Hammond C. M. Jr 2000 “Apparatus and method for reducing imagingerrors in imaging systems having an extended depth of field” US patent 6097856
Miller D. , Blanco E. 2003 “System and method for increasing thedepth of focus of the human eye” US patent 6554424
Ben-Eliezer E. , Marom E. , Konforti N. , Zalevsky Z. 2005 “Experimental realization of an imaging system with an extendeddepth of field” Applied Optics 44 (14) 2792 - 2798    DOI : 10.1364/AO.44.002792
Sheppard C. J. R. , Gu M. 1993 “Imaging by a high aperture opticalsystem” Journal of Modern Optics 40 (8) 1631 - 1651    DOI : 10.1080/09500349314551641
Debye P. 1909 “Der lichtdruck auf kugeln von beliebigem material” Annalen der Physik 335 (11) 57 - 136    DOI : 10.1002/andp.19093351103
Born M. , Wolf E. 1999 Principles of Optics 7th ed Cambridge University Press New York, NY
Sheppard C. J. R. , Matthews H. J. 1987 “Imaging in high-apertureoptical system” Journal of the Optical Society of America A 4 (8) 1354 - 1360    DOI : 10.1364/JOSAA.4.001354
Escobar I. , Saavedra G. , Martinez-Corral M. , Lancis J. 2006 “Reduction of the spherical aberration effect in high-numericalapertureoptical scanning instruments” Journal of the Optical Society of America A 23 (12) 3150 - 3155    DOI : 10.1364/JOSAA.23.003150
Martinez-Corral M. , Saavedra G. 2009 “The resolution challenge in3D optical microscopy” Progress in Optics 53 1 - 68
Escobar I. , Sanchez-Ortiga E. , Saavedra G. , Martinez-Corral M. 2011 “New analytical tools for evaluation of spherical aberrationin optical microscopy” Springer in Optical Fluorescence Microscopy: from the Spectral to the Nano Dimension Heidelberg, Germany 85 - 99
Wilson T. , Carlini A. R. 1988 “Three-dimensional imaging inconfocal imaging-systems with finite sized detectors” Journal of Microscopy 149 (1) 51 - 66    DOI : 10.1111/j.1365-2818.1988.tb04561.x
Kimura S. , Munakata C. 1990 “Dependence of 3-D optical transferfunctions in the pinhole radius in a fluorescent confocal opticalmicroscope” Applied Optics 29 (20) 3007 - 3011    DOI : 10.1364/AO.29.003007
Gu M. , Sheppard C. J. R. 1993 “Effects of a finite-sized pinhole on3D image formation in confocal two-photon fluorescencemicroscopy” Journal of Modern Optics 40 (10) 2009 - 2024    DOI : 10.1080/09500349314552011
Gauderon R. , Sheppard C. J. R. 1999 “Effect of a finite-size pinholeon noise performance in single-, two-, and three-photon confocalfluorescence microscopy” Applied Optics 38 (16) 3562 - 3565    DOI : 10.1364/AO.38.003562