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Image Formation in Structured Illumination Wide-Field Fluorescence Available online at www.sciencedirect.com Micron 39 (2008) 808–818 www.elsevier.com/locate/micron Image formation in structured illumination wide-field fluorescence microscopy Dejan Karadaglic´ a,*, Tony Wilson b a University of Liverpool, School of Biological Sciences, Biosciences Building, Crown Street, Liverpool L69 7ZB, UK b University of Oxford, Department of Engineering Science Holder Building, Parks Road, Oxford OX1 3PJ, UK Received 24 October 2007; received in revised form 19 January 2008; accepted 19 January 2008 Abstract We present a theoretical analysis of the image formation in structured illumination wide-field fluorescence microscopy (SIWFFM). We show that the optically sectioned images obtained with this approach possess the optical sectioning strengths comparable to those obtained with the confocal microscope. We further show that the transfer function behaviour is directly comparable to that of the true confocal instrument. The theoretical considerations are compared with and confirmed by experimental results. # 2008 Elsevier Ltd. All rights reserved. Keywords: Structured illumination; Optical sectioning; Optical microscopy; Fluorescence microscopy; Wide-field microscopy; Confocal microscopy 1. Introduction 1996; Wilson et al., 1996), whereas the second has attempted to modify the conventional optical microscope in such a way as to The confocal microscope possesses a number of advantages introduce optical sectioning (Neil et al., 1997). If we define over the conventional optical microscope, and the optical optical sectioning as the requirement that all spatial frequencies sectioning is arguably the most important one (Wilson and within the transfer function attenuate with defocus then it is Sheppard, 1984; Wilson, 1990; Pawley, 1995). The traditional clear that the conventional microscope does not possess this construction of a confocal microscope employs a point light property. However, if we consider for example the conventional source together with a point pinhole detector. It is the presence fluorescence case then we find that it is only the zero spatial of the pinhole aperture which is responsible for the optical frequency which does not attenuate with defocus; all the other sectioning since it physically blocks light from out-of-focus frequencies are imaged less strongly with increasing defocus. planes from reaching the photodetector and hence precludes This suggests that if we modify the illumination system in a this light (information) from contributing to the final image. A conventional microscope, so as to project a single spatial complete image of the object is built up by scanning. These frequency fringe pattern onto the object, the microscope will systems, which are usually built as an ‘add-on’ to a image those parts of the fringe pattern efficiently whose conventional microscope, are very powerful and have found projection onto the object lies within the focal region. We thus great application is a wide number of fields. They normally obtain an image of the object in which the focal optical section require laser light sources since standard microscope illumina- is labelled by a sharply focused grid of lines. Simple processing tion systems are usually insufficiently bright. In order to of three images, taken at three relative spatial positions of the overcome these problems a number of approaches have been fringe pattern, permits both an optically sectioned and a taken. Among those there are two developed at the University conventional image to be extracted in real-time. The approach of Oxford. The first one has concentrated on improving the has been demonstrated in both brightfield reflected light design of traditional pinhole-based systems (Jusˇkaitis et al., imaging (Neil et al., 1997; Wilson et al., 1996) as well as fluorescence (Neil et al., 1998, 2000; Lanni and Wilson, 2000). In this article we theoretically describe the fluorescence case * Corresponding author. Tel.: +44 151 7954508; fax: +44 151 7954431. using two methods of obtaining structured illumination: fringe E-mail addresses: [email protected] (D. Karadaglic´), projection and grid projection. In the fringe projection case the [email protected] (T. Wilson). structured pattern is formed by the interference of two coherent 0968-4328/$ – see front matter # 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.micron.2008.01.017 D. Karadaglic´, T. Wilson / Micron 39 (2008) 808–818 809 illumination beams, so as to create the appropriate pattern on aperture of the objective lens and l denotes the wavelength. the specimen. In the grid projection approach the structured Although the excitation light, used to create interference pat- pattern is formed by imaging a physical grid object on the tern, may be coherent in nature, the light emitted from the specimen. fluorescent specimen is completely incoherent, and given by There have been numerous attempts to exploit structured illumination in microscopy to achieve different goals. Iemissionðt1; w1Þ¼f1 þ m cosðnt1 þ fÞg f ðt1; w1Þ (2) Notably, Gustafsson has shown that using basically the same principle as described in this paper it is possible theoretically If we now assume that the objective lens has the amplitude to improve the lateral resolution of the optical microscope by point spread function h2, and since the final image recorded by a factor of two (Gustafsson, 2000). Further, exploiting the the detector in the image plane ðt; wÞ is equal to the convolution nonlinearity properties of fluorophore saturation theoretically of the emission light and intensity point spread function of the an unlimited improvement is possible (Gustafsson, 2005). lens (Wilson and Sheppard, 1984), we may write for the However, in the reality the improvement is limited by the intensity of that image signal-to-noise ratio and the photobleaching of imaged ZZ fluorophores. Bailey et al. (1993) introduce standing wave Iðt; wÞ¼ f1 þ m cosðnt þ fÞg f ðt ; w Þjh ðt þ t ; w excitation in the fluorescence microscopy, by which they 1 1 1 2 1 achieve improved axial resolution. Failla et al. (2003) name þ w Þj2dt dw (3) this type of microscopy spatially modulated illumination 1 1 1 microscopy and claim it can be used for the measurement of The excitation and the fluorescence emission wavelengths the structures with accuracy close to 1 nm. This approach also have also been taken to be equal for simplicity, although in the uses structured illumination, but unlike of the approach reality, the emission wavelength must be longer. described in the paper, its sinusoidal pattern is laid along the Eq. (3) may be written in the following form: illumination axis, which complicates the system setup, and limits its potential applications. m m The analysis in this article emphasises the optical sectioning Iðt; wÞ¼I0ðt; wÞþ expð jfÞInðt; wÞþ expð jfÞIÀnðt; wÞ 2 2 side of the approach, and presents experimental results which (4) confirm that the optically sectioned images of high quality can be achieved by this method. where ZZ 2. Fringe projection—imaging theory 2 I0ðt; wÞ¼ f ðt1; w1Þjh2ðt1 þ t; w1 þ wÞj dt1dw1 (5) Let us assume the fluorescence specimen, characterised by a spatial distribution of fluorescence f ðt1; w1Þ, Fig. 1,is represents a conventional fluorescence image and Inðt; wÞ is illuminated with a one-dimensional interference intensity given by pattern of the form ZZ 2 Iexcitationðt1; w1Þ¼1 þ m cosðnt1 þ fÞ (1) Inðt; wÞ¼ expð jnt1Þ f ðt1; w1Þjh2ðt1 þ t; w1 þ wÞj dt1dw1 where m denotes the modulation depth, n the spatial frequency (6) and f a spatial phase. We have elected to work in optical co- ordinates ðt; wÞ which are related to real co-ordinates (x, y) via à and we note that IÀnðt; wÞ¼In ðt; wÞ where the asterisk denotes ðt; wÞ¼ð2p=lÞðx; yÞn sin a where n sin a is the numerical complex conjugate. If we now introduce the spectrum of the specimen, F(m, n) of f ðt1; w1Þ, where m and n are spatial frequencies in the t1 and w1 directions, respectively, we may recast Eq. (6) as ZZ Inðt; wÞ¼expð jntÞ Fðm; nÞCðm þ n; nÞ exp½ jðmt þ nwÞdmdn (7) where the transfer function C(m, n) is given by the convolution of the pupil functions C(m, n)=P(m, n) P*(m, n)(Wilson and Sheppard, 1984). Since the function C(m, n) is also a function of axial distance, u, we will add that parameter as another argument in the function. The normalised defocus co-ordinate u is related to 2 Fig. 1. Schematic diagram of a fringe projection imaging system. the actual defocus, z, via u =(8p/l)z sin (a/2). 810 D. Karadaglic´, T. Wilson / Micron 39 (2008) 808–818 Eq. (7) now permits us to write an alternative expression for Now the question is—how to extract the image jInðt; w; uÞj the detected image as from the raw data? One method would be to record three raw ZZ images taken at three equally spaced positions of the fringe f f Iðt; w; uÞ¼ Fðm; nÞCðm; n; uÞ exp½ðmt þ nwÞdmdn pattern. We take the spatial phase = 0 (a constant) at the first ZZ position and take an image I1. We then move the fringe pattern m by one third of a period, f = f +2p/3, and take image I . þ expð jfÞ expð jntÞ Fðm; nÞCðm þ n; n; uÞ 0 2 2 Finally, image I3 is taken corresponding to f = f0 +4p/3. It is m then a simple matter to compute  exp½ jðmt þ nwÞdmdn þ expð jfÞ expð jntÞ 2 ZZ 3m I þ I eÀ j2p=3 þ I e j2p=3 ¼ expð jf ÞI ðt; w; uÞ (10)  Fðm; nÞCðm À n; n; uÞ 1 2 3 2 0 n  exp½ jðmt þ nwÞdmdn (8) and hence À j2p=3 j2p=3 We see that the raw image consists of three terms.
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