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  168 West, M. l. Slomianka, L and Gundersen, H. l G. 1991): Unbiased stereo ogical estimation of the total number of neurons in the subdivisions of he rat hippocampus using the optical fractionator. Anatomic. Record. 23 : 482-497. Optical Fibre Probe Microscopy C lR Sheppard and H. Fatemi Department of Physical Optics, School o Physics University of Sydney, NSW 2006, Australia ABS1RACT The optical fibre probe microscope is a compact system which allows investigation of surface topography. The effects of surface height, tilt and refractive index on the image were investigated both eoretically and experimentally. 1 Introduction In the optical fibre probe microscope Cerre et aI., 1991), light is launched into a single mode optical fibre and is incident on the object Fig. 1-1). The reflected light is coupled back into the fibre, and after passing through a fibre coupler its intensity measured with a photodiode detector. An image is built up by scanning the fibre tip relative to the object surface. This can be achieved by mechanically moving either the fibre tip or the object itself The strength of the light coupled back into the fibre depends on the reflectivity ofthe sample, but also its distance from the fibre i.e. the surface height) and the tilt of he surface. In practice there is also a reflection oflight from the fibre tip, which results in interference fringes being formed as the object is scanned in depth. The interference pattern consists of fringes within an envelope defined by upper and lower parts. Investigation of he details of he nterference allows us to extract the various properties of he surface at the particular scan point. In this way it is very similar to the confocal surface profiling methods, except that in the present case there is no peak in intensity because the radiation from the fibre is divergent rather than convergent. The light emerging from a single mode optical fibre is closely approximated by a Gaussian beam. The variation in intensity along the axis is proportional to 1 z / Z0>2 , where z is the distance from the beam waist, located at the fibre tip, and Zo is the confocal parameter. 169  170 Fig.1 1. A schematic diagram of the optical fibre probe microscope. 2 Experimental Investigation o Image Formation fthere is a sample in front of the illuminating fibre tip, a fraction of the light can be reflected from the sample surface back towards the fibre. Part of his light couples back into the fibre, whilst the rest is reflected by the fibre tip back to the sample. Thus multiple reflections are set up, similar to a Fabry-Perot resonator. The fibre tip is viewed and positioned relative to the sample surface Fig. 1-2), and can be tracked from side and top views through an image grabbing system. For a given surface tilt the axial position is scanned using a feedback-controlled piezoactuator, driven by a sawtooth generator. The signal is digitally accumulated in a storage oscilloscope and processed using a computer. For the particular case when the normal to the sample surface is parallel to the axis, i.e. when y=0 in Fig. 1-2, the experimentally recorded intensity is shown in Fig. 1-3 Fatemi and Sheppard, 1993; Sheppard et aI. 1995). t is possible to measure the geometrical parameters of the light emitted from the fibre, e.g. zo by observations using a plane mirror as sample. f 171 In the present study, we examine the effects when the surface is tilted, so that y;e ). We assume that the value of Zo and thus the far-field angle of divergence, e and also the amplitude reflection coefficient ro of he fibre tip, are known. The aim is to measure the amplitude reflection coefficient of he sample, and the local tilt of he sample surface. The method is valid when the illuminated surface can be assumed locally plane. Spatial resolution is thus limited to about the fibre spot size, which is around 3 Ilm in our experiments. In order to investigate the behaviour of the signal with different sample parameters, we used a tilted mirror as sample. Fig.1-4 shows an example of an experimental plot. The figure covers the lower part of the envelope of he interference fringes, showing a zero in intensity the position of which can be used to extract the reflectivity and tilt of he sample. The existence of his zero limits such measurements to a value of greater than 0.2. In practice this covers a wide range of ndustrial and biological samples ã single mode optical fibre Fig.1 2. A close up of the fibre tip and sample. cco f\ cam~a V  172 :i: <0 c: ID E 10llm 20llm Dis1ance Fig. 1-3. Experimentally recorded axial response from an untilted mirror. ~e I 111-1 ,11. '.Ill,,' ' a I 11' 1,i JJO,1t 1 l i I,:l;~~- _lira ,.1,\ [ : I , I numI ~== _____ M_= ===ii=== u u Fig 1-4. An example of an interference pattern from a tilted mirror as sample. he lower part of he envelope is shown, depicting a zero in the envelope. 3 Theory We start by investigating focusing by a lens with a Gaussian weighting whose defocused pupil function can be assumed (Born and Wolf, 1989, p.440). .. 173 pep) = eXP(ikz)ex p( ~iUp2 )ex p( ~ p2) (13.1) where P is a nonnalized radial coordinate in the pupil plane in the range 0 ~ P < CfJ, and u the nonnalized defocus coordinate u =4kzsin2(a 12) (13.2) where a is the effective aperture of he system, corresponding to the value p =1. The field in the focal region can be calculated in the scalar paraxial approximation by the Kirchhoff diffraction integral U v,U) = exp(ikz) I x p( ~iUp2 )ex p( ~ p2 Yo Vp)PdP (13.3) where v is the transverse optical coordinate v= krsina (13.4) Evaluating the integral in Eq.(13.3) 1 V2 ) v,u) = . xp(ikz)exp . 1 + U 2(1 + U) (13.5) which can be also written in the fonn 1 ( iuv 2 ) ( v,u)=--. exp(ikz)exp 2 exp l+lU 2 1+u ) v 2 ) 2 1 +u 2) (13.6) These expressions can be seen to be equivalent to the usual expressions for a Gaussian beam (Yariv, 1975) with v=J rlw o (13.7) where Wo is the waist ofthe beam, u =zl Zo (13.8) where Zo is the confocal parameter zo=kw~/2 (13.9)  174 The intensity in the focal region can be written in the simple form 1 (v2) V,U) = exp (1+u 2) 1+u 2 13.10) n the far-field we can put \u\» 1 so that U v, u) = exp ikz) ex p iV 2 ) exp -~) lU 2u 2u ~13.11 and the intensity is 1 V2) J v,u) = -;;ex p ;; 13.12) By analogy with diffraction by a circular aperture we can associate v = u with the shadow edge which is here of course not a hard edge). Thus Z2 ine 2 =~exp -  . Z2 sma 13.13) where sine r 1 13.14) and sina = fi 1 kw o 13.15) can be identified as the numerical aperture of he beam. Here we should stress the distinction between the numerical aperture of he beam and the numerical aperture of a fibre, which for the single-mode case can be very different. Again by analogy with imaging by a circular aperture, the coherent transfer function CTF), is equal to the defocused pupil function 175 c /,u) = eXP ikz)exp  ~iUI2 p  ~/2) 13.16) where I is a normalized transverse spatial frequency, equal to the true spatial frequency normalized by sina lA = 11 .finwo) . For confocal reflection the defocused CTF is given by 2x , { (P)} l,u)=eXP (2ikz) Iex p -(l+iu) P2+4 pdpdlj> 13.17) or after normalization c l,u)=--. exp(-/2/4)exp -iu- exp 2ikz) ZZ l+lU 4 13.18) The 3-D CTF can be calculated by Fourier transformation ofEq. 13.20), or by a line integral over the product of he defocused pupil functions. Using the latter method or, after normalizing c l,s) = 2n ex p {- p2 )r(S-2kz+ p P c l,s) = exp(-s), 2 s -~so 4 where the offset So is given by So = 2kzo = 2W~ = 1/2sin2(a 12) The coherent transfer function for confocal reflection is shown in Fig. 1-5. 13.19) 13.20) 13.21)
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