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 :
482497.
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. 11). 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 FabryPerot resonator. The fibre tip is viewed and positioned relative to the sample surface Fig. 12), 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 feedbackcontrolled 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. 12, the experimentally recorded intensity is shown in Fig. 13 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 farfield 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.14 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. 13. Experimentally recorded axial response from an untilted mirror.
~e
I
1111 ,11.
'.Ill,,' '
a
I
11'
1,i JJO,1t
1
l i I,:l;~~
_lira
,.1,\
[
:
I
, I
numI ~==
_____
M_= ===ii===
u
u
Fig 14. 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 farfield
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 singlemode 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 3D 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(S2kz+
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. 15. 13.19) 13.20) 13.21)