Image formation in a fiber-optical confocal scanning microscope

Transcription

1 Gu et al. Vol. 8, No. 11/November 1991/J. Opt. Soc. Am. A 1755 Image formation in a fiber-optical confocal scanning microscope Min Gu, C. J. R. Sheppard, and X. Gan Department of Physical Optics, University of Sydney, New South Wales, Australia Received July 30, 1990; second revised manuscript received May 16, 1991; accepted May 17, 1991 Theoretical studies on image formation in a confocal scanning microscope with optical fibers as the transmission medium are reported. Theoretical analyses show that this new kind of microscope can be considered a coherent imaging system, even for finite fiber spot size. Based on these studies the coherent transfer functions in both in-focus and defocused cases are derived and calculated. The axial coherent transfer functions are also obtained, and, furthermore, the optical-sectioning property of the microscope system is investigated with the consideration of the image formation of a perfect-reflection planar object and a point object. 1. INTRODUCTION The confocal scanning microscope (CSM) has been extensively studied and applied to many practical fields, such as wafer metrology, industrial inspection, and biological microscopy, because it has important advantages over conventional microscope systems.1-8 For example, the CSM gives superior resolution owing to the fact that the cutoff spatial frequency is twice as high as that of the conventional coherent system.1 7 Furthermore, it has a strong optical-sectioning effect' because the out-of-focus information is detected much less strongly, whereas in a conventional system it is merely blurred. The opticalsectioning effect permits three-dimensional image formation. 9 -" Besides, making use of a point detector in the CSM will reduce the amount of flare and scattered light. 4 The fiber-optical CSM, which uses optical-fiber components that may include optical fibers and gradient-indexrod lenses rather than the conventional optical lenses as the transmission medium, is a new development of the CSM. These components make the system compact, and it will thus have numerous uses in material processing and biomedical industries. In this new kind of the microscope, the light source is not a point source but the tip of an optical fiber, and the signal is collected by another optical fiber that delivers the signal to a detector. Because of the introduction of the optical fibers that have finite sizes of cross section, the properties of the image formation of the system will be different from those of the CSM using a point source and a point detector or a finite-sized detector. We will consider in this paper the image formation in a reflection-mode fiber-optical CSM. The paper is organized as follows. First, the theoretical analysis of the imaging formation in the fiber-optical CSM is described in Section. Following this description, we study the transverse transfer functions of the system in both in-focus and defocused cases, as well as the axial coherent transfer function for the latter case (Section 3). In Section 4, the optical-sectioning property is investigated. A discussion is finally put forward in Section 5.. FIBER-OPTICAL CONFOCAL SCANNING MICROSCOPE The fiber-optical CSM that we study here is schematically shown in Fig. 1. In this reflection-mode microscope the light source is a tip of an optical fiber Fl. Light is collimated by lens L, onto the objective 01, with pupil function P1. An object with the amplitude reflectivity rf is placed on the focal plane of the objective. The signal from the scanned point of the object is collected by the objective and focused by lens L onto the tip of another optical fiber F, which delivers the signal to the center of a detector D whose size is much larger in comparison with the diameter of the fiber. In practice the same fiber can be used for illumination and collection with the use of either a beam splitter at the far end of the fiber or a fourport fiber coupler. To analyze the imaging formation in this system, we first restrict ourselves to the one-dimensional case for simplicity. Assume that x denotes the variable on the plane perpendicular to the optical axis and that z is the axial variable. The practical microscopic system is cylindrically symmetrical because the lenses and fibers are circular, so that the dimension x may be considered as the radial direction by our replacing x with r = (X + y)11. Furthermore, both fibers F, and F are assumed as singlemode fibers. Let U,(xo) be the amplitude distribution of the electric field on the output end of fiber Fl. After passing through the imaging system, the field amplitude at a point x on the input end of fiber F can be expressed as, according to the imaging theory, 3 U(x., x ) = ff Ui(xo)hi(x + xi)rf(xs-x,) X h X + X)dxdxl, (1) where x5 represents the scan position at the object and h, and h are the amplitude point-spread functions 3 of the /91/ $05.00 i 1991 Optical Society of America

2 1756 J. Opt. Soc. Am. A/Vol. 8, No. 11/November 1991 Gu et al. fiber F 1 d L objective 01 U 3 (x, X3) = f1(x 3 )exp(ip3,zo) f fl*(x )U (X., x )dx - (5) I fi(x) dx *... where zo is the length of optical fiber F. Therefore the object intensity measured by the large area detector D, which has a uniform intensity sensitivity, is,(x.) = U3(x 3, x3)1 dx3. (6) Fig. 1. Geometry of (LI, L, lenses). fiber F 6 detector D a reflection-mode fiber-optical CSM objective and the collector, corresponding to the Fourier transform of the pupil functions P and P, respectively. (For generality of the derivation, we have used P and P to express the pupil functions of the objective and the collector, respectively.) Parameters M = f/d and M = d/f denote the magnifications of the lenses, where f and d represent the focal length of the objective and the distance between the lens and the tip of the optical fibers, respectively. In order to obtain the field amplitude on the output end of fiber F, we now consider the propagation of field U (x 8, x ) through optical fiber F. In terms of Snyder and Love1 an arbitrary electric field U(x, z) can be expanded as a superposition of a complete set of orthonormal modes in an optical fiber, viz., U(x,z) = aj~f(x)exp(i,8jz), () where f(x)exp(i,3jz) denotes a complete set of the orthonormal modes in a particular fiber that is a bound solution and propagates along the z direction without attenuation,,3g is the propagation constant, and aj is called the modal amplitude and can be determined by orthonormal relations. Functions f&(x) are profiles of the fiber modes on the plane perpendicular to the z axis. If the optical fiber is a single-mode fiber, the summation overj includes only the first term withj = 1, and thus Eq. () becomes where U(x,z) = a1f,(x)exp(i131z), (3) a, = I f1*(x)u(x, z = O)dx : d f. I fi(x)l dx Applying Eqs. (3) and (4) to our problem, we can derive the field amplitude U 3 (x 8, X3) at a point X 3 on the output end of the second fiber F as Substituting Eqs. (5) and (6) and using Eq. (1) give 1(X 3 = JJ3J ifl *(X )Ui(xo)hl(xo + M )rf(xs - X,) x h (XI + X)f(x )Ui*(xo')hl*(Xot + M )r*(x - x,') X h *(X,, + M )dodxo'dxdxldx dx ', where the condition that f I (x ) dx 3 is constant has been used because the profile fa(x3) is fixed for a particular fiber. Equation (7) can be written as,(x.) =I g1(x1)g1*(xl')rf(xs - x)rf*(xs - x')dxldxl' (7) = lg,(x 8 ) 0 rf(xs)i, (8) where 0 denotes the convolution operation. Equation (8) shows that the microscope discussed above behaves as a coherent microscope with an effective point-spread function given by g,(x) =f U(xo)hi(xo + M )fi*(x )h (x + X)dxdx = [U1(x/M1) 0 h,(x/m1)][f,*(x/m,) 0 h (x)], (9) where we have used the fact that M = 1/M1. Therefore the introduction of the single-mode optical fibers in the CSM does not change the coherence of the image system. In this respect it behaves quite differently from a finitesized incoherent detector TRANSFER FUNCTIONS OF THE IMAGE SYSTEM The image formation of a microscope can be described by the transfer function that gives the strength of a periodic component in the intensity image and the cutoff spatial frequency of the microscope. In this section we derive the transfer functions of our imaging system in both in-focus (Subsection 3.A) and defocused (Subsection 3.B) cases. A. In-Focus Transfer Functions The periodic components of the object are given by its spectrum, which can be derived by the Fourier transform of the object reflectivity rf, viz., Rf(m) f rf(x)exp(-7rixm)dx, (10)

3 Gu etal. so that rf(x, - x) = f Rf(m)exp[ri(xs - xl)m]dm, (11a) rf*(xs - xi') = Rf*(p)exp[-7ri(x, - xl')p]dp. (b) Substituting Eqs. (11) into Eq. (8) and comparing with I(X.) = ifc(m; p)rf(m)rf*(p)exp[7ri(m - we have C(m; p) = JJ g(xi)g,*(xi') p)x,]dmdp, (1) x exp(-7rimx + pxl')dxldxl', (13) where m and p in C(m; p) are the spatial frequencies in the transverse direction. C(m; p) is called the transfer function for our given system and describes the strength of a periodic component in the intensity image resulting from the presence of two spatial frequencies in the object amplitude reflectivity rf. Since our system is a coherent image system, a transverse coherent transfer function c(m) can be introduced so that C(m; p) can be rewritten as where C(m;p) = c(m)c*(p), (14) c(m) = fg 1 (xi)exp(-7rimxi)dxi, (15a) Vol. 8, No. 11/November 1991/J. Opt. Soc. Am. A 1757 is, Ul(r) = f(r). Under the condition of the Gaussian approximation for the circular, linear-polarized fiber, the amplitude profile of the single-mode fiber with a spot size r is f,(r) = exp[-(1/)(r/ro) ], so that the corresponding Fourier transform is (19) i(l) = 7rro' exp[-(1/) ( 7rlro)']. (0) Therefore, if the objective is aberration free and has the pupil function with a radius of ao, Eq. (18) can be derived, with the use of Eq. (0) and the method of Sheppard et al. 14,1 5 as where c(l) = exp exp 7r4l - exp(-ai)] A J~r x [1 - exp(-ap0)]dol, (1) Po = -[(I cos 0)/] + [1 + 1 sin (O/4)]1/ A = (7raoro/Ad). () (3) Equation (1) and 1 have been normalized by c(l = 0) and by the cutoff spatial frequency of the conventional microscope, lo = ao/af, respectively. A is a dimensionless parameter, denoting the effects of the fiber spot sizes, the objective pupil radius ao, and the distance d. If A = 0, then c(l) becomes the coherent transfer function of the CSM with a point source and a point detector, given by c*(p) = f gi*(x')exp(7ripxi')dxi', (15b) which are the Fourier transforms of the effective spread functions gl(x,) and gl*(xi), respectively. The image intensity is thus expressed as x ~~~~~~~~~~ I(xJ) c(m)rf(m)exp(-rrimx,)dm (16) of the CSM. Substituting Eq. (9) for g,(xi) and using a combination of the shift and similarity theorems, 13 we obtain c(m) = [1(M m)p(afm)] 0 [f1(m m)p (Afm)], (17) c(l) = (/vr){cos'1(1/) - (1/)[1 - (1/)]1/} (4) The numerical calculations of Eq. (1) are shown in Fig., compared with the coherent transfer function of the CSM with a point source and a point detector, corresponding to the parameter A = 0 [see Eq. (4)]. It can be shown that the cutoff spatial frequency is still twice that of the conventional microscope, which is the same as that Interestingly, we find that the plots of c(l) for the fiber CSM corresponding to A 0 cross the plot for the CSM (A = 0) at a certain spatial frequency. The larger the value of the parameter A, the closer the crossing point is to the vertical axis. For a given value of A, when where U0(m) and f,(m) are the Fourier transforms of U,(x) and fi(x), respectively. In the case of circular single-mode fibers and circular pupil functions of the lenses Eq. (17) can be easily generalized to the cylindrically symmetric two-dimensional case: c(l) = [U,(Mll)P1(Afl)] 00 [A(Ml)P (Afl)], (18) where 0 represents the two-dimensional convolution operation and I = ( + n )"1 denotes the radial spatial fre- quency in the transverse direction. It is apparent that, if U, and f, are delta functions that correspond to a point source and a point detector, Eq. (8) then reduces to the coherent transfer function of the CSM. 3 In the following derivation we therefore assume that both fibers F, and F have the same amplitude profile; that 8 c( Fig.. In-focus transverse coherent transfer functions c(l) for different dimensionless fiber spot sizes A.

4 1758 J. Opt. Soc. Am. A/Vol. 8, No. 11/November 1991 Gu et al. Refc(l, u)} 1 r-- Im{c(l, u)} u=0 (a) for different values of the parameter A and the defocused distance u. In the case of defocusing the coherent transfer functions include a real part and an imaginary part, which have negative components when u increases. For = 0 the real part of c(o, u) decreases by a factor compared with c(l = 0, u = 0) of the in-focused case or even becomes negative as u increases. However, it always gives a positive value when A is large [see Fig. 4(b)]. To study this phenomenon further, we can analytically derive the normalized expression of c(0, u) as c0(0,u) -A[ - exp(-a + i)] (9) (A - iu) [1 - exp(-a)] 8 or, for the real and imaginary parts, 6 Re(c(l, u)} 1 *raw_ 8 I 6-4 =0 (b) Fig. 3. Defocused transverse coherent transfer functions c(l, u) for A = 1: (a) real part, (b) imaginary part. - B_.5 I the spatial frequency I is less than the crossing point, c(l) for A 0 is larger than that for A = 0 but decreases more quickly than that of the CSM if 1 is larger than the crossing point. Eventually, the plots of c(l) become narrow when A becomes a large value. For nonzero A the transfer function always has 0 slope for = -O.: Im(c(l, u)} IT 8- (a) B. Defocused Transfer Functions For the defocused case, the pupil function including the defocusing effect can be expressed as 4," [a factor exp(-ikz) is suppressed here] Pl(r, u) = exp[(iu/) (r/ao) ] P,(r, ) = 0 (r < a), (5a) (r > ao), (5b) where u is an axial optical coordinate, related to the real axial distance from the focal plane z by u = (87r/A)z sin (a/), (6) U 6 l - - l He 5 1 (b) Fig. 4. Defocused transverse coherent transfer functions c(l, u) for A = 5: (a) real part, (b) imaginary part. I and sin a = a/f is the semiangular aperture. The defocused transverse coherent transfer function c(l, ) in the reflection mode is thus, in terms of Eq. (18), c(l, ) = [i(mjl)p,(afl, )] [ (Mjl)P,(Afl, u)], (7) which can be evaluated as A [-(A- i)11 c(lu, ) = [1 - exp(-a)](a - i) exp 4 {[/ x 1- exp[-(a - iu)p 0 ]Ido 0 (8) where c(l, i) has been normalized by c(l = 0, u = 0). The defocused transfer functions are plotted in Figs. 3 and U 14 A=1 A=5 A=1 Fig. 5. Variations of c(0, u) with the axial optical coordinate u. The dashed curves represent the imaginary parts of c(0, u), while the solid curves represent the real parts of c(o, u).

5 Gu et al. c(s) Vol. 8, No. 11/November 1991/J. Opt. Soc. Am. A 1759 object. Its Fourier transform is then Rf(1) = 5(l). The detected intensity varying with the axial optical coordinate u is thus, in terms of Eqs. (16) and (8), I(u) = Ic(l = 0, U) A{1 - exp[-(a - iu)]} [1 - exp(-a)](a - iu)l = Cr + Ci, (3) which becomes S Fig. 6. Axial coherent transfer functions c(s) for different values of A. { l= - exp(-a)cos u] + exp(-a)u sin u}a CR(O U) (A + )[1 -ep-) (30a) c 1 (0, ) =u[1 - exp(-a)cos u] - exp(-a)a sin u}a (A + Ut )[1 - exp(-a)] (30b) U)=(u/) sin u/ for the CSM (A = 0). The numerical calculations of Eqs. (31) are displayed in Fig. 7(a). When A increases, the plots of I(u) become broad. This effect confirms that the strength of the optical-sectioning effect decays with increasing A. This result is emphasized in Fig. 7(b) in which we plot the halfwidth U1/ of the curves I(u) as a function of the dimensionless fiber spot size A. It is shown that the half-width I(u) (33) They reduce to, in the case of the CSM (A = 0), CR(O, U) = sin u/u, (31a) c 1 (0, a) = (1 - cos u)/. (31b) Equations (30) are numerically calculated and shown in Fig. 5. We find that, when A is small, both the real and the imaginary parts of c(o, u) are oscillating functions along the u axis, but the plots of cr(o, u) become broad, and the side peaks gradually disappear with increasing values of the parameter A. This property implies that the optical-sectioning effect decreases with increasing A, which is discussed further in Section 4. With the help of c(l, u), a three-dimensional coherent transfer c(l, s) with an axial spatial frequency s may be obtained from the Fourier transform of c(l, u) performed with respect to u. The axial coherent transfer functions c(s) = c(l = 0, s) are shown in Fig. 6, where s has been normalized by a /Af. It can be seen that the axial transfer functions c(s) are cut off at s = 1. For A = 0, c(s) is a square function that can be analytically derived by the Fourier transform of Eqs. (31). As A increases, c(s) decreases with s and eventually becomes a delta function as A -* X. It should be pointed out that the axial coherent transfer functions shown in Fig. 6 have a constant axial spatial frequency shift" so given by. U 1 / U so = 1/[ sin (a/)]. 4. OPTICAL-SECTIONING EFFECTS As mentioned in Section 1, one of the main advantages of the CSM is the optical-sectioning effect that makes it possible to form a three-dimensional image of a thick object. To investigate the strength of optical sectioning in our system, we will first consider a uniform planar object that is scanned through the focus. In this case the amplitude reflectivity rf = 1, corresponding to a perfect-reflector Fig. 7. (a) Variations of the detected intensity with the axial optical coordinate u in the case of a perfect reflection planar object. (b) Half-width of the curves of (a) as a function of the dimensionless fiber spot size A.

6 1760 J. Opt. Soc. Am. A/Vol. 8, No. 11/November (u) 0. Gu et al. behaves as a partially coherent system, 6 and of course in any real system the pinhole must be of finite size. This point can be made even clearer by the following comparison. For a CSM with a finite-sized detector D(x), U in Eq. (1) is still true in front of the detector, so that the detected intensity is 0. 1(x 3 ) = U (x., x )1 D(x)dx. (37) 0 ~-I I Fig. 8. Variations of the detected intensity with the axial optical coordinate u in the case of a point object. does not change significantly if A < 1. For A > 5 the half-width is close to its geometrical-optics limit of U 1 / = A. This occurs because the Gaussian variation is by then truncated little by the edge of the pupil. In contrast, for a system with a finite-sized detector, the behavior approaches the geometrical-optics limit slowly. The perfect-reflector planar object includes only a periodic component with zero spatial frequency. Next we turn to an alternative way of studying the opticalsectioning effect of the point object, which includes all periodic components with the same amplitude. -If the object is a single point, we have Rfl) = 1. The intensity distribution is then, according to the generalization of Eq. (16) for cylindrical coordinates, 3> ~~~~ I(v, ) f c(l, u)jo(vl)ldl (34) where the integration is a Fourier-Hankel transform of the defocused transverse coherent transfer function c(l, u), Jo is a Bessel function of the first kind of order zero, and v is the transverse optical coordinate, related to the radial distance r, given by v = (7r/A)r sin a. (35) For investigating the optical-sectioning effect, we set v = 0 in Eq. (34), substitute Eq. (8) into Eq. (34), and calculate the axial intensity distribution I(0, u). The plots of I(0, ) normalized by I(v = 0, u = 0) are shown in Fig. 8, which includes the plot of (0, u) analytically given by' 5 I(0, i) = sin 4 (u/4) (36) (U/4) 4 for the CSM (A = 0). When A < 1, the plots of I(0, u) are almost superposed together, which implies again that the optical-sectioning effect does not change appreciably in this case. 5. DISCUSSION 5 It has been shown that the fiber-optical confocal scanning microscope behaves as a coherent imaging system, even for finite values of fiber spot size. In this respect it differs fundamentally from a confocal microscope with a finite-sized pinhole placed in front of the detector, which However for a fiber-optical CSM the detected intensity is, in terms of Eq. (6), 1(xJ) = f U (x 3, x )fl*(x)1 dx - (38) It is therefore seen that, in the former case, intensity is integrated over the detector aperture, while in the latter, field amplitude is integrated over the fiber profile. Hence, the fiber-optical CSM is coherent because it is linear in amplitude. The coherent nature of imaging in the fiber-optical CSM may prove advantageous when used for quantitative microscopy or for confocal interference microscopy. Further, because it behaves in a coherent fashion, it is possible to describe the imaging performance completely by a coherent transfer function, as has been calculated in this paper. This description contrasts with that for the microscope with a finite-sized detector, 6 which must be described by a partially coherent transfer function or a transmission cross coefficient, and for this reason theoretical images for objects only of simple form, such as a single point and a plane, have been described in the literature. Under the Gaussian approximation, the fiber spot size ro can be expressed,' for a single-mode fiber with a step refractive-index profile, as r = p( n V)". (39) where p, the core radius of the fiber, is in the range of -5 pm and V is the fiber parameter. For a single-mode fiber the maximum value (p. 343 of Ref. 1) of V is.59 so that r is in the range of /.tm. From Eq. (3) and with the wavelength of illuminating light 633 nm and the radius ao of the pupil 3 mm, it can be seen that A can be readily fixed in the range by appropriate choice of the distance d. For values of A at the lower end of this range the image performance, both for lateral resolution and optical sectioning, are substantially the same as in the ideal confocal microscope. Finally, it should be pointed out that the performance of the fiber-optical CSM in fluorescence is also different from that in the conventional confocal fluorescent microscope in that the optical transfer function of the former does not show the spatial-frequency-missing cone and negative tails that are two characteristics in the latter case.' 6 REFERENCES 1. C. J. R. Sheppard, "Scanning optical microscopy," in Advances in Optical and Electron Microscopy, R. Barer and V. E. Cosslett, eds. (Academic, London, 1987), Vol. 10, pp C. J. R. Sheppard and A. Choudhury, "Image formation in the scanning microscope," Opt. Acta 4, (1977).

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