Fluorescent image formation in the fibre-optical confocal scanning microscope

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1 JOURNAL OF MODERN OPTICS, 1992, VOL. 39, NO. 4, Fluorescent image formation in the fibre-optical confocal scanning microscope X. GAN, MIN GU and C. J. R. SHEPPARD Department of Physical Optics, University of Sydney, New South Wales 2006, Australia (Received 3 July 1991 ; revision received 11 November 1991) Abstract. Reported in this paper are theoretical studies on two-dimensional and three-dimensional image formation with fluorescent objects in a fibre-optical confocal scanning microscope. An effective point spread function is introduced to derive the two- and three-dimensional optical transfer functions. It is found that, unlike confocal fluorescence microscopes with a finite circular detector, there is no missing cone of spatial frequencies, and no negative tail in the transfer function. The effect of the system parameters on the optical sectioning property is also investigated. 1. Introduction In biological applications, confocal microscopes are often used in a fluorescent mode, in which the microscope is used to localize in three dimensions fluorescent labels attached to cell constituents. In order to understand fully the image formed, and the limitations of the system, it is important to consider theoretically the image formation process. In the confocal fluorescence microscope, the spatial cut-off frequencies in both axial and transverse directions are twice as large as those in a conventional fluorescence microscope [1, 2]. Three-dimensional (3D) image formation becomes possible in the confocal scanning optical microscope ; this is very important since a practical specimen always has a finite thickness, that is it is a 3D object. The confocal fluorescence microscope is a fully incoherent system. The twodimensional (2D) and 3D optical transfer functions have been presented in order to describe image formation in the confocal fluorescence microscope with point source and detector [3, 4]. For all finite sizes of pinhole, a negative tail appears in the optical transfer function (OTF), which becomes most significant when the pinhole size is about equal to the central lobe of the Airy disc [5]. Recently a new implementation of the confocal scanning microscope, the fibreoptical confocal scanning microscope, has been proposed [6-8]. This makes the microscope very compact and will therefore find many practical applications. Some investigations on image formation in the fibre confocal microscope with nonfluorescent objects have been reported by Gu et al. [6, 7] (for the transfer function) and Gu and Sheppard [8] (for the signal level). As a result of using single-mode fibres to replace the pinholes of the conventional confocal microscope, the fibre confocal microscope behaves as a fully coherent system, which contrasts with the confocal microscope with finite source and detector, which behaves as a partially coherent system [9]. Another difference between these two systems is that the signal level in the fibre confocal microscope does not necessarily increase as the fibre spot size /92 $ Taylor & Francis Ltd.

2 826 X. Gan et al. increases, which means that in a practical system the property of coupling efficiency must be considered [8]. This paper reports on the fluorescent image formation process in the fibre-optical confocal scanning microscope, including the 2D and 3D transfer functions as an extension of previously presented imaging theory. Section 2 gives an expression for the point spread function for the fluorescent fibre confocal microscope. The 2D and 3D transfer functions and optical sectioning effects of the system are described in section 3. Then finally numerical results for the 2D and 3D transfer functions and optical sectioning effects with different fibre spot sizes are given in section The effective point spread function A schematic diagram of the fibre-optical confocal scanning microscope is shown in figure 1, where two single mode fibres F 1 and F 2 are used as an illumination source and a collection device respectively. The light emitted from the latter is collected by a large-area detector D. The light is focused to, and collected from, the object by an objective lens 0 1. The object is characterized by a fluorescent strength f. In practice the beam splitter can be replaced by a fibre coupler, in which case scanning is achieved by movement of the fibre tip. In order to analyse this system, we have divided the imaging process into two stages. First we consider the illumination from the fibre source on the specimen, the intensity distribution at a point r o just after the object plane being derived according to imaging theory [10] as I(ro,r s)= Jh i (r i + M )UI (r i )dr l f(rs-ro), (1) 1 where U1 is the 2D amplitude mode profile of fibre F 1, parameter M 1 =dlf denotes the magnification of the objective lens, where f is the focal length of the lens 0 1, and h 1 is the point spread function of the objective lens, which is the Fourier transform of its pupil function. The quantity f(rg -r o ) is the fluorescent strength function of the specimen and r s is the scan distance. Considering the second stage, although the fluorescent image is obtained by incoherent imaging processing, we start by assuming a point object on the object plane, which can be expressed as a delta function. The amplitude distribution U' after the collector lens is U'(r o ;r2)=h2( ro+ M 2I, (2 a) M 2), where h 2 is the point spread function of the collector lens, assumed to be different from h 1 for generality. According to the method of Gu et al. [6] for coupling with a fibre and collection by a large-area detector, the amplitude point spread function h e of the second stage can be expressed as r h ero) = J h 2 (r o+)u 2*(r 2 )dr 2, M2 where U2 is the amplitude profile of the single-mode fibre F 2, and parameter M2 = f/d denotes the magnification of the collector lens. So the corresponding effective intensity point spread function for a point r o =(xo, y o, z o ) is (2 b) 2 heff(ro) - ( 2 c)

3 Fluorescent image formation 827 fibre F t which can be written as x d L1 detector D Figure 1. The geometry of a fibre-optical confocal scanning microscope. Combining equations (1) and (2), and integrating over the whole specimen plane, we obtain the intensity collected by the detector given as a function of the scanning coordinates : I(r.)= I J J h2(ro+ M ) Uf(r 2 ) dr 2 2 C r1 +M / U1(r1) dr1 J hl 1 2 objective 0 object 2 f (r,- r o ) dr o, (3 a) I(r,)= Jg(r o )f(r8 ro )dro, (3) where g(ro) is the intensity point spread function of the system, given by g(ro)= J J J J m Ul(rl)Ur(ri)hl(rl+ ro)hr(r' Ml ra ~Uf(r 2 ) M l r' x U2(r2)h2 ro )hf ro + M2) dr 1 dri dr 2 dr2. (4) 3. The optical transfer function Image formation in a microscope can be described by using an OTF. In this section, we shall give both the 2D and the 3D transfer functions for the fibre microscope for fluorescent objects The 2D optical transfer function The periodic components of a fluorescent specimen are given by its spectrum, which can be derived as the Fourier transform of the object fluorescent function : F(p)=5 f(r)exp(-21cip r )dr (5) 7.0

4 828 X. Gan et al. where p = (m, n, s) denotes the spatial frequency in Fourier space, and F(p) denotes the spectrum components. The image intensity can be written as in which from equations (4)-(6), we have which is the OTF of the system. Our discussion here is limited to the case of circular single-mode fibres and lenses with circular pupil functions. So, in the in-focus case, by applying cylindrical symmetry the OTF can be expressed as C(l) _ [ U 1(M1l)P1(2f l )] [ Ur(M1l)Pr(Afl )] [Ui(M1l)P2(tfl)] [U2(M1l)Pi(2fl )] (8) where (&& denote the 2D convolution operation, the superscript asterisks denote the complex conjugate, and l=(m 2 +n 2 ) 1 / 2 represents the radial spatial frequency. U 1 and U2 are the Fourier transforms of the amplitude mode profiles of the fibres, which represent the angular spectra of the light emitted by the fibres. Here we have assumed that the wavelength of the fluorescent light is equal to that of the illumination light. For the object in focus, we also limit our discussion to the simplest case when both fibres have the same spot size under the Gaussian approximation [11 ] for the amplitude profile of the single-mode fibres, and the pupil functions of the objective and collector lenses are the same. We have for the pupil functions, assuming them to be aberration free, 1, r<a o, PI(r) = P2 (r) _ (9 a) 0, r > a o, where a o is the aperture of the lenses, and for the fibre amplitude profiles _ -0 U1(r)=U 2 (r)=exp -2 r 2 (9b) 0 where ro is the radius of the fibre core, so that the corresponding Fourier transforms are U 1 (r) = U2 (r) = 2irro exp [ - i (2t[lr o) 2 ]. (9 c) By putting equations (9) into equation (8), in this case the first 2D convolution operation is the coherent transfer function (CTF) for the fibre microscope with a non-fluorescent object, which has already been derived by Gu et al. [6]. The last 2D convolution is the complex conjugate of the first convolution which in the present case is also equal to the first convolution because the CTF is a real function [6]. Then the OTF of the fibre microscope for a fluorescent object is given by the CTF convolved with itself, that is f f I(rs) = J C(p)F(p) exp (2itip - rs ) dp (6) C(P) = J m n/2 n/2 C(l) = K exp [-4A(m 2 + n 2 )] exp {- :'A [(m - l) 2 +n 2 ] f 0 g(ro) exp (-2nip ro) dro (7) x [1-exp (-Apo)](1-exp (-Apo )] db' db dm dn, (10)

5 Fluorescent image formation 829 where K is a normalization coefficient, and Po = - 2 (m 2 +nz)1/2lcos 01 + [1 -*(m 2 +n 2 ) sine 0]1/2 ' (10a) PO= - i[(m - l) 2 +n2] 1/2 Icos0'1+{1 - ā[(m - l)2 +n 2 ]sin 2 0'} 1/2 A= 2naoro z Ad (lob) (10c) is a dimensionless parameter, which represents the size of the objective pupil relative to that of the beam size at a distance d. If A=0, then C(l) becomes the OTF with point source and point detector The 3D optical transfer function For the defocused case, the pupil function including the defocus effect for the reflection geometry can be given as p1 (r, e p [2 (ao)2]' r< a o, (11 a) 0, r > a o, (11 b) P 2 (r, u)=p 1 (r, u). (11 C) Here r = (x2 +y2)1/2 represents the transverse optical coordinate and u represents the axial optical coordinate, which is related to the defocus distance z by u = -z sin (12) 2 (2 ) in which a is the semiangle of the objective lens. By using the method of Sheppard et al. [10] and Gu et al. L6,7], the 3D OTF given by equation (7) can be derived as C(l,s)=1 : [ U1(M1l)P1(Afl,u)] [U*(M1l)Pr(Afl, u)] [U2(M1l)P2(tfl,n)] [U2(M1l)P2(tfl,u)]exp( - isu)du. (13) As we found in the last section for the 2D transfer function, the 3D OTF of the fluorescent fibre microscope is the 3D CTF [6] of a transmission-mode fibre microscope convolved with itself : ff exp(-a{ h(m 1-2 +n 2 )+[s'/(m 2 +n 2 ) 1 J2] 2 }) C(l, s) _ (m2 + n2)112a erf [Re ( yo ) A'I'] exp[ - Aa[(m- 1)2 +n 2 ] + {(s' - 211/2121 s)i [(m-l)2 X [(m -1 ) 2 +n2] "2 x erf [Re (yo) A"2] dm do ds', (14) where Re indicates the real part of its argument, y o and yo are given by C ((M2 +.n 2)1/2 I sl )2]1/2, Yo= (m 2 +n 2 )1/2(15)

6 830 X. Gan et al. Cl-([(m-1)2+n2]l/Z + Is-s'J )2]1/2 yo= (16) 2 and erf is the error function [12] defined as [(m-1)2+n2]1/2 x erf(x)=- exp(-t 2 )dt. (17) Optical sectioning The strong optical sectioning property of the fibre confocal scanning microscope allows the formation of the 3D image of a thick object. Optical sectioning can be investigated by considering the axial image of a thin fluorescent sheet. Since we have calculated the 3D transfer function, the simplest way to calculate the axial image of a thin fluorescent sheet is by just taking the one-dimensional (1D) Fourier transform of the section of the 3D OTF C(1=0,s) [5]. 4. Numerical results The 2D OTF, normalized by C(l=0), of the fibre confocal scanning microscope for fluorescent specimens is shown in figure 2. The cut-off frequency is four, which is twice that of a conventional fluorescence microscope and the same as that for the confocal fluorescent microscope with a finite source and detector [2-9]. When A increases from zero, the central lobe of the transfer function becomes narrower, which means that the effective cut-off frequency is reduced. As we can see from the plot, there are no negative values appearing for any value of A in the whole region within the cut-off frequency. For the confocal fluorescent microscope with a finitesized source and detector, however, for a finite size of pinhole, negative values do appear in the OTF [5]. These are recognized as resulting from the negative parts of the Fourier transform of the detector distribution and hence are absent for the assumed Gaussian beam profile. The fact that the fibre microscope exhibits no negative tail in the OTF avoids imaging artefacts and is advantageous for image restoration l Figure 2. In-focus optical transfer functions C(l) for different dimensionless fibre spot sizes A.

7 Fluorescent image formation 831 The 3D OTFs, normalized by C(1=0, s=0), for the fluorescent fibre-optical confocal scanning microscope are displayed in figure 3 for different values of the dimensionless parameter A, corresponding to different fibre spot sizes. The cut-off frequencies in the transverse and axial directions are four and one respectively, which correspond to twice those of the conventional fluorescence microscope. They are both identical with those of the confocal scanning microscope with a finite source and detector [2-9]. When A increases from zero, as the fibre spot size increases, the central part of the transfer function reduces in extent and the value near the cut-off -4 (b) Figure 3. 3D OTF C(l,s) : (a) A=1 ; (b) A=5.

8 8 3 2 X. Gan et al. C(l, 0) ` A=10 A=7 A=5 A=3 A= `` A=1 A= Figure 4. Cross-section C(1, 0) of the OTF in the transverse direction for different dimensionless fibre spot sizes A. l S Figure 5. Cross-section C(0, s) of the OTF in the axial direction for different dimensionless fibre spot sizes A Figure 6. Variation in the detected intensity with the axial optical coordinate u in the case of a perfect fluorescent planar object. U

9 Fluorescent image formation A Figure 7. The half-width of the curves in figure 6 as a function of the dimensionless fibre spot size A. becomes very small, so that the effective cut-off frequency is reduced. As we mentioned for the 2D transfer function, there is also no negative tail for any size of fibre spot size. Another advantage of the fibre microscope which we can observe from the 3D transfer function is that there is no spatial frequency missing cone appearing for any value of A. By comparison, the 3D transfer function for the microscope with a finite-sized detector exhibits a missing cone of spatial frequencies, in which region the information in the objective is lost, and which becomes more significant with increasing size of pinhole [5]. When A increases, this means that the fibre spot sizes for the illuminating and collecting fibres increase together, so that the combined effect of both pupil functions, corresponding to the objective and collective lenses, on the 3D transfer function is to reduce the missing cone. Again this improves image formation and reconstruction. Figures 4 and 5 show the cross-sections C(l, 0) and C(0,s) of the 3D OTF. The axial response for a perfectly fluorescent sheet, normalized by I(u=0), for the fibre microscope is numerically displayed in figure 6. It illustrates the strong optical sectioning property of the fibre microscope, which allows the possibility of 3D image formation. The central lobe becomes broader as A increases, showing that the axial discrimination becomes poorer when the fibre spot size increases. Figure 7 shows how the half-width of the axial response varies as a function of parameter A corresponding to different fibre spot sizes. It is seen that A must be less than about two for the axial resolution to be within 10% of its optimum value. 5. Conclusions We have investigated image formation in the fluorescent fibre-optical confocal scanning microscope. By using fibres as both source and detector, the system retains similar effective cut-off spatial frequencies as for those in the fluorescent microscope with a finite-sized detector, but has the advantages of no spatial frequency missing cone nor negative tail in the OTF. References [1] SHEPPARD, C. J. R., 1986, Optik, 72, 131. [2] WILSON, T., 1989, J. Microsc., 154,143.

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