INTRODUCTION
Recently a variety of optical systems have been reported in literature for
different sensing applications (Yuan et al., 2009;
Babchenko and Maryles, 2007; Baptista
et al., 2006; Peng et al., 2005;
Donlagic and Cibula, 2005; Qiu et
al., 2004). Some of the successful optical designs are reported, which
are based on the intensity modulation technique and used fibers as waveguide
in sensing operations (Golnabi, 2002; Golnabi
and Jafari, 2006; Golnabi et al., 2007; Golnabi,
2000a; Golnabi and Azimi, 2007). The ease of operation,
cost factor and precise performance requirements has led to the design of the
plastic fibers in a doublefiber probe (Golnabi and Azimi,
2008). The double fiber design in spite of small probe size can be used
for local and remote noncontact distance measurements that are required in
many applications. The reported simulation results are based on a design that
uses the intensity modulation of reflected light from a surface and it can be
used in different applications. Considering the importance of such a double
fiber system in this article we report simulation results obtained by variation
of different important parameters controlling the system performance and merits.
The goal of this study is to provide some simulation result, which leads to
the optimum situation in using such an optical fiber sensor. The reported simulation
results can offer some advantages concerning the optimization of crucial parameter
which can be helpful in design and construction of similar displacement sensors.
MATERIALS AND METHODS
Development of the reported method and related computations are performed for
the period of 20089. The doublefiber designs used in this modeling consist
of a pair of similar plastic optical fibers (Weinert, 1999).
As can be seen in Fig. 1, a double fiber assembly is used
in this modeling in which in one end; two fibers are separated and they are
close to each other at the other end. At this end two fibers are attached sidebyside
together to form a single double fiber. The double end is mounted in a fixed
holder assembly in line with the reflecting target. The separated fiber ends
can be connected to the light source and photodetector accordingly. More experimental
details about double fiber usage can be found in other references (Golnabi
and Azimi, 2008). All the double fiber assemblies shown in Fig.
1ac include similar plastic optical stepindex fibers.
The overall diameter of each fiber is estimated to be about 2.2 mm and the typical
core diameter considered to be about 860 μm in order to be compatible with
the practical values (Golnabi and Azimi, 2008).
As shown in Fig. 1a the first doublefiber probe denoted
as EF has an equal size for the transmitter and receiver fibers.

Fig. 1: 
Double fiber arrangements. (a) Equal fibers design (EF), (b)
Transmitter fiber Shorter (TS) and (c) Receiver fiber Shorter (RS) 
The second arrangement shown in Fig. 1b is denoted as TS
in which the transmitter fiber is shorter than the receiver fiber by amount
of w. The third arrangement shown in Fig. 1c is denoted as
RS in which the receiver fiber is shorter than the transmitter fiber by the
same amount of w. From now on we refer to these cases as EF, TS and RS designs,
respectively.
The reflection concept of light from a surface for this case based on the geometrical
optic can be determined from Fig. 2a. The fibertofiber distance
indicated in Fig. 2a by t, which is one of the variables of
the parametric studies. The distance t for the probe can be varied from 03
mm. The core radius of each fiber is another parameter of this study which can
be varied from 100500 μm. The NA of the fiber in double fiber assembly
is also varied and simulation results are reported for such variations. Similar
parameter studies are performed for TS and RS designs. In addition for such
designs length difference denoted by W in Fig. 1b and c
are varied and the computed results are compared.
Operation of the modeled doublefiber sensor designs is based on the intensity
modulation of the light reflected off the target. In Fig. 2a
the image of the transmitted light on the entrance plane (image plane) of the
receiving fiber is displayed for EF case. Let us consider the beam cross section
on the target plane to be circular in the XYplane, where Zaxis shows the direction
of the propagation of beam. If r_{1} and r_{2} represent the
transmitter (Tfiber) and receiver fibers (Rfiber) core radius, respectively,
then the overlap of two circles determines the amount of light received by the
receiver fiber. For our case in Fig. 2a, we assume r_{1}
= r_{2} for similar transmitter and receiver fibers.

Fig. 2: 
Light reflection diagram for the case of EF design. (a) Spot
light on the image plane or receiving fiber. (b) Cross sectional overlap
of the spot light image and receiving fiber 
Four distinct cases can be considered for such an analysis, as shown in Fig. 2b. Considering first case (a) when there is no crossing of two circles then no specular light is received by the Rfiber (dead region). Second case where there is a possible crossing in two different situations. Diameter AB is either above crossing line CD, case (b) or below the crossing line CD of two circles, case (c). In the last case (d) the small circle can be completely inside the spot light circle. Actually the amount of received light depends on the overlapping area between these two circles. Using geometrical relation one can compute overlapping cross section ΔS for the four described cases. For case a with no overlap one can write:
For case b with small overlap where AC line is above CD we have:
where, R is the radius of the spot light on the target plane (Fig. 2b). Parameters x, R and y are defined as followings:
For case c where AD is below the interception line CD we write:
where, x and R are as above and y is given as:
For the case of d when small circle is inside the spot light circle we can write:
Thus ΔS can vary from zero from Eq. 1 to a maximum value given by Eq. 8. The received power of the light on the target plane for the case a to c is equal to:
where, I_{0} is the launching intensity by the Tfiber at the target plane. The reflected intensity from the target plane to be received by the Rfiber is obtained by:
where, ρ shows the reflection coefficient for the target plane. Theoretical result indicates that by increasing axial distance the collected intensity is increased and at a certain distance reaches a maximum. For the case d intensity variation with distance is given by:
where, Z is the axial distance and Eq. 11 is just the regular intensity decreasing rule. As can be seen the light intensity entered in the receiving fiber varies by the axial distance and as a result in simulation the reflected intensity is plotted as a function of this variable.
SIMULATION RESULTS
All the reported results are computations based on the developed formulas accomplished
by different written programs using the MATLAB software. In the first study
simulation results for the case of EF design are reported. Figure
3 shows the computed reflected intensity for a target as of function of
the axial distance Z for different fibertofiber distance values, t. The assumed
parameters in this computation are NA = 0.2 and r = 430 μm. Intensity variation
is considered for the case of t = 1, 2 and 3 mm in the computations and variation
are plotted for Z variation of about 24 mm. For all curves shown in Fig.
3, the respond curve starts from a near zero value and reaches a maximum
at a particular axial distance and then drops slowly to a minimum value. As
can be seen in Fig. 3, by increasing the t value three major
changes occur in the intensity plot. First the dead region distance is increased
by increasing the t value. For example for t = 1 mm the dead region starts from
zero to about 2.5 mm while for the tvalue of 3 mm the dead region is from zero
to about 7 mm. Second point and perhaps the most important difference, is the
shift of the peak intensity to longer distances by increasing the tvalue. For
instance, for t equal to 1 mm the peak is located to about 4 mm while it is
shifted to about 9 mm for the tvalue of 3 mm.

Fig. 3: 
Reflection intensity as a function of axial distance for different
t values for EF design 

Fig. 4: 
Reflection intensity as a function of axial distance for different
radius r values for EF design 
Third notable points of this study is the variation the bandwidth (FWHM) values
for different respond curves. By increasing the t value as can be seen in Fig.
3, the amount of the intensity peak remain constant while the curve bandwidth
(FWHM) is considerably increased by increasing the t value. Since all curves
obey the same behavior for the falling parts, therefore the observed increase
in bandwidth is as a result of the slowing variation of the raising part of
the intensity curve with respect to this parameter.
In Fig. 4 the computed reflected intensity as a function of axial distance Z is presented for different r values for EF probe. For this case the assumed parameters in computation are t = 1 mm and NA = 0.2. Intensity variation is considered for the case of r = 100, 200, 300, 400 and 500 μm in the computations and variation are plotted for Z variation of about 24 mm. For all curves shown in Fig. 4, the respond curve starts from a near zero value and reaches a maximum at a particular axial distance and then drops slowly to a minimum vale. As can be seen in Fig. 4, by increasing the fiber core radius value, r, some observations can be made in the intensity plots. First the dead region distance remains constant by increasing the r value.
Second point and perhaps the most important difference, is the shift of the
peak intensity to longer distances by increasing the r value. For instance,
for r equal to 100 μm the peak is located to about 3 mm while it is shifted
to about 5 mm for the r value of 500 μm. Third point and also important
one, is the increase of the peak intensity as a result of the r changes. For
example, for r equal to 100 μm the maximum peak is about 0.02 on the arbitrary
scale while it is increased to about 0.78 on the same scale for the r value
of 500 μm. Another notable point of this study is the variation of the
bandwidths (FWHM) values for different respond curves.

Fig. 5: 
Reflection intensity versus axial distance for different NA
values for EF design 
By increasing the r value as can be seen in Fig. 4, the amount
of the curve bandwidth (FWHM) is increased accordingly. Since all the intensity
curves obey the same behavior for the falling part, therefore the observed increase
in the bandwidth is due to the slower variation of the raising part of the intensity
curve with respect to r.
In Fig. 5 the computed reflected signal intensity for a target
as a function of the scan distance Z for different fiber numerical aperture,
NA, values for the same EF design are presented. The assumed parameters in this
computation are t = 1 mm and r = 430 μm. Intensity variation is considered
for the case of NA = 0.2, 0.3 and 0.4 in the computations and variations are
plotted for Z variation of about 24 mm. As shown in Fig. 5,
the respond curves start from a near zero value and reach a maximum at a particular
axial distance and then drop slowly to a minimum vale. As can be seen in Fig.
5, by increasing the numerical aperture value three major changes occur
in the intensity plot. First the dead region distance is decreased by increasing
the NA value. For example for NA = 0.2 the dead region starts from zero to about
5 mm while for the NA value of 0.4 the dead region is for zero to 2.5 mm. Second
point and perhaps the most important difference, is the shift of the peak intensity
to longer distances by decreasing the NA value. For instance, for NA equal to
0.4 the peak is located to about 3 mm while it is shifted to about 7 mm for
the NA value of 0.2. Other notable points of this study are the variation of
the maximum and the bandwidths (FWHM) values for different respond curves. By
increasing the NA value as can be seen in Fig. 5, the amount
of the intensity peak is slightly reduced while the curve bandwidth (FWHM) is
considerably reduced by increasing the NA value.

Fig. 6: 
Reflection intensity as a function of axial distance for different
t values for TS design 
Since both curves obey the same behavior for the falling part, therefore the
observed decrease in bandwidth is as a result of the sharpening of the raising
part of the intensity curve.
In the next study simulation results for the TS probe are given. Figure
6 shows the computed reflected intensity for a target as a function of the
axial distance Z for different fibertofiber t values. For this case the assumed
parameters in computation are NA = 0.2, r = 430 μm w = 2 mm. Intensity
variation is considered for the case of t = 0, 1, 2 and 3 mm in the computations
for TS design and variation are plotted for Z variation of about 24 mm. For
all curves shown in Fig. 6, each respond curve starts from
a near zero value and reaches a maximum at a particular axial distance and then
drops slowly to a minimum vale. As can be seen in Fig. 6,
by increasing the t value three major changes occur in the intensity plot. First
the dead region distance is increased by increasing the t value (compare to
Fig. 3 for EF design). For example for t = 1 mm the dead region
starts from zero to about 3.5 mm while for the t value of 3 mm the dead region
is from zero to about 8 mm. Second point and perhaps the most important difference,
is the shift of the peak intensity to longer distances by increasing the t value.
For instance, for t equal to 1 mm the peak is located to about 6 mm while it
is shifted to about 11 mm for the t value of 3 mm. Third notable points of this
study is the variation the bandwidth (FWHM) values for different respond curves.
By increasing the t value as can be seen in Fig. 6, the amount
of the intensity peak remain almost constant while the curve bandwidth (FWHM)
is considerably increased by increasing the t value. Since all curves obey the
same behavior for the falling parts, therefore the observed increase in bandwidth
is as a result of the slowing variation of the raising part of the intensity
curve with respect to this parameter.

Fig. 7: 
Reflection intensity as a function of axial distance for different
w values for TS design 
In Fig. 7 the computed reflected intensity for a target as a function of the scan distance Z for different fibers length difference, w, values for the similar TS design is shown. For this study the assumed parameters in computation are t = 1 mm, NA = 0.2, r = 430 μm. In this study role of length difference in computation results are investigated. Intensity variation is considered for the case of w = 1, 2, 3, 4 and 5 mm in the computations and variation are plotted for Z variation of about 24 mm. For all curves shown in Fig. 7, the respond curve starts from a near zero value and reaches a maximum at a particular axial distance and then drops slowly to a minimum vale. As can be seen in Fig. 7, by increasing the length difference, w, value three major changes occur in the intensity curves. First, the dead region distance is increased by increasing the w value. For example for w = 1 mm the dead region starts from zero to about 3 mm while for the w value of 5 mm the dead region is from zero to about 5.0 mm. Second point and perhaps the most important difference, is the shift of the peak intensity to longer distances by increasing the w value. For instance, for w equal to 1 mm the peak is located to about 5 mm while it is shifted to about 7 mm for the w value of 2 mm. Other notable points of this study are the variation of the maximum and the bandwidths (FWHM) values for different respond curves. By increasing the w value as can be seen in Fig. 7, the amount of the intensity peaks remain constant while the curve bandwidth (FWHM) is nearly constant as can be seen in Fig. 7.
The simulation results for the RS probe design are presented. Figure
8 displays the computed reflected intensity for a target as a function of
the scan distance, Z, for different fibertofiber, t values.

Fig. 8: 
Reflection intensity versus axial distance for different t
values for RS design 
For this case the assumed parameters in computation are NA = 0.2, r = 430 μm
and w = 2 mm. Intensity variation is considered for the case of t = 0, 1, 2
and 3 mm in the computations for RS design and variation are plotted for Z variation
of about 24 mm. For all curves shown in Fig. 8, each respond
curve starts from a near zero value and reaches a maximum at a particular axial
distance and then drops slowly to a minimum vale. As can be seen in Fig.
8, by increasing the t value as stated before three major changes occur
in the intensity plots. First the dead region distance is increased by increasing
the t value (compare to Fig. 3 for EF and Fig.
6 for TS design). For example for t = 1 mm the dead region starts from zero
to about 1.5 mm while for the t value of 3 mm the dead region is from zero to
about 6 mm. Second point and perhaps the most important difference, is the shift
of the peak intensity to longer distances by increasing the t value. For instance,
for t equal to 1 mm the peak is located to about 3 mm while it is shifted to
about 8 mm for the t value of 3 mm. Third notable points of this study is the
variation the bandwidth (FWHM) values for different respond curves. By increasing
the t value as can be seen in Fig. 8, the amount of the intensity
peak remain almost constant while the curve bandwidth (FWHM) is considerably
increased by increasing the t value. Since all curves obey the same behavior
for the falling parts, therefore the observed increase in bandwidth is as a
result of the slowing variation of the raising part of the intensity curve with
respect to this parameter.
In Fig. 9 the computed reflected intensity for a target as
a function of the axial distance, Z, for different fiber length differences,
w, values for the same RS design is displayed. For this investigation the assumed
parameters are NA = 0.2, r = 430 μm and t = 1 mm. In this study role of
length difference in computation results are investigated for the RS probe design.

Fig. 9: 
Reflection intensity versus axial distance for different w
values for RS design 
Intensity variation is considered for the case of w = 1, 2, 3, 4 and 5 mm in
the computations and variation are plotted for Z variation of about 24 mm. For
all curves shown in Fig. 9, the respond curve starts from
a near zero value and reaches a maximum at a particular axial distance and then
drops slowly to a minimum of zero value. As can be seen in Fig.
9, by increasing the length difference, w, value three major changes occur
in the intensity curves. First the dead region distance is considerably decreased
by increasing the w value. For example for w = 1 mm the dead region starts from
zero to about 2 mm while for the w value of 5 mm the dead region is very short
close to near zero (compare results with that of Fig. 7).
Second point and perhaps the most important difference, is the shift of the
peak intensity to shorter distances by increasing the w value (The results for
Fig. 7). For instance, for w equal to 1 mm the peak is located
to about 4 mm while it is reduced to about 2 mm for the w value of 5 mm. Other
notable points of this study are the variation of the maximum and the bandwidths
(FWHM) values for different respond curves. By increasing the w value as can
be seen in Fig. 9, similar to the case of TS design, the amount
of the intensity peaks remain constant while the curve bandwidth (FWHM) is nearly
constant as can be seen in Fig. 9.
In the last study simulation results for three types probes are compared. Figure
10 shows this comparison of the reflection intensities as a function of
axial distance Z for designated EF, TS and RS designs. For all cases the assumed
common parameters in computation are t = 1 mm, NA = 0.2 and r = 430 μm
and w = 2 mm for TS and RS designs. It is easier to see the role of length difference
in this comparison.

Fig. 10: 
Comparison of the reflection intensities for different EF,
TS and RS designs 
As usual intensity variation is plotted for three probes versus axial distance,
Z, for a range of about 24 mm. For all curves shown in Fig. 10,
each respond curve starts from a near zero value and reaches a maximum at a
particular axial distance and then drops slowly to a minimum of near zero value.
From this comparison three points can be concluded.
First, the dead region distance is smaller for the RS, which is about 2 mm, next for EF is about 3 mm and for the TS is the highest about 4 mm. Second point and perhaps the most important one, is the difference of the position of the peak intensity curves. For RS the peak position is at about 3 mm, for EF about 4.5 mm and finally for TS is at about 5.5 mm. The third notable points of this comparison are the variation of the intensity maximum values and the bandwidths (FWHM) values for different respond curves. As can be seen in Fig. 10, the amount of the intensity peaks remain almost constant (0.58) while the curve bandwidth (FWHM) is smallest for RS (2.4 mm), next for EF (2.7 mm) and finally for TS is the highest value (3.1 mm) for the assumed input values.
DISCUSSION
Simulation results reveal some points that are discussed in more detail in this section. As can be seen in results, the respond curve includes three parts, the first and second region for increasing intensity and the third region for decrease in reflected light intensity as a function of the axial distance. From simulation results four general cases can be recognized. First, when there is no crossing between the two light cones (dead region) and the reflected intensity is zero. When there is a crossing between the two coins two cases can happen (either AB, above the horizontal cross section line CD; or AB below the cross section line) and the crossing area can increase from zero to a maximum value of Tfiber core area. When the circle is totally inside the spot light circle, then the amount of cross section ΔS is always a constant but the intensity of the received light in this area is decreased by increasing the axial distance.
For example result shown in Fig. 3 displays the respond curves
of the EF doublefiber design computed from the Eq. 2 and
6 for the first rising region of the intensity curve and for
the second region from Eq. 11. Two points can be concluded
from the theoretical results shown here for different t values. First point
is that the peak of the respond curve is shifted towards a larger distance as
t increases. Second point is that for the first increasing part of the intensity
curve, the probe with the smaller fibertofiber t distance shows a sharper
increase in comparison with that of the large fibertofiber distance. As can
seen in Fig. 3, the decay for the second region of the respond
curve is about the same for all cases and the overall curve bandwidth for larger
t is a little wider than for the smaller t values.
Fibers core and overall size and as a result the fibertofiber distance has a great impact on the probe sensitivity and dynamic range. Fiber core radius and fiber overall size play also an important role in the maximum intensity and the bandwidth of the respond curve of such a probe. The length difference in the transmitting and receiving fibers is crucial and shifts the place of the intensity maximums in the respond curves. For TS and RS probe designs in respect to the EF case, the length difference shifts the maximum with respect to the EF for the TS design towards higher distance while for the RS design moves the place of maximum towards a shorter axial distance.
For all designs, dead region is controlled by parameters t, NA, but r has little effect in this respect. Maximum place of the intensity curve is governed mainly by t and NA. Maximum peak value of the intensity curve is governed mainly by r and NA. The bandwidth (FWHM) of the respond curve is controlled by t, NA and r values. It must be pointed out that since the core and cladding are the same for all three arrangements, thus the noted difference in Fig. 10 is only due to the length difference. The parameter w denoted for the length difference mainly controls the dead region, the place of intensity maximum and the bandwidth (FWHM) of the intensity curve.
Finally, the simulated respond curves are very similar to the experimental
respond curve observed in the previous experimental studies. In general the
developed theoretical formulas and algorithms effectively describe the performance
of doublefiber probes suitable for reflection/displacement sensors. A similar
behavior is noted in experimental measured intensity as shown in Fig.
3 of Reference by Golnabi and Azimi (2008) for the
sensor deign using the EF fiber probe. Given simulation results are in support
of the previous ones and there is no contradiction in results. From practical
point of view all the three arrangements simulated here are very easy to fabricate
and offer a good size probe for sensing purposes depending on the application
requirements. The EF arrangement provides good results for the sensing operation
such as reflection or displacement measurements. However, the RS probe geometry
offers a better design with the less dead region in comparison with the EF and
TS designs as shown in Fig. 10. It must be pointed out that
other geometries can be used for displacement sensing. More details about such
designs in term of performance evaluation can be found in literature (Golnabi,
1999, 2000b).
CONCLUSIONS
This article has described the simulation results for operation of three different double fiber probes. It was shown that presented information can be useful in design and operation of such noncontact sensor systems that can be used in a variety of applications. The reported simulation results reveal the following conclusions:
• 
Effect of t: This parameter controls the dead region
distance and it is increased by increasing the t value. Controls the position
of the maximum in intensity curve. Any increase of t increases the place
of the peak. By increasing the t value the amount of the intensity peak
remains nearly constant while the curve bandwidth (FWHM) is increased 
• 
Effect of r: Controls the position of the peak in intensity
respond curve. By increasing the r value the peak is shifted towards higher
axial distance values. Also important point is the increase of the peak
intensity as a result of the increase in r. By increasing the r value the
intensity curve bandwidth (FWHM) is increased accordingly. The parameter
r has little effect on the dead region range and by changing the fiber core
radius the dead region distance remains almost constant 
• 
Effect of NA: By increasing the NA value three major
changes occur in the intensity plots. First, the dead region distance is
decreased by increasing the NA value. Second point and the most important
effect, is the shift of the peak intensity to shorter distances by an increase
in the NA value. By increasing the NA value the amount of the intensity
peak is slightly reduced while the curve bandwidth (FWHM) is considerably
reduced by increasing the NA value 
• 
Effect of w: First, For TS and RS designs the dead
region distance is increased by increasing the w value. Second, the dead
region distance is smaller for the RS, next for EF (w = 0) and for the TS
is the highest. Third point and perhaps the most important ones, is the
difference of the position of the peaks in intensity curves. For RS the
peak position occurs at shorter axial distance, next for EF and finally
for TS occurs at larger distances. The fourth notable points of this comparison
are the variation of the intensity maximum values and the bandwidths (FWHM)
values for different respond curves. The amount of the intensity peaks remain
almost constant for three designs while the curve bandwidth (FWHM) is smallest
for RS, next for EF and finally the TS design has the largest value 
ACKNOWLEDGMENT
This study was supported in part by the Sharif University of Technology Research program. The authors gratefully acknowledge the grant money devoted to this research (Grant No. 3104).