INTRODUCTION
Light guides are transparent devices that conduct the flow of light from a
source to the point of interest and optical fibers are one of the most effective
links in this respect. In recent years a variety of experimental and theoretical
investigations have been made to study the bending loss mechanisms in different
fibers and fiber cable transmission lines (Baptista et
al., 2006; Babchenko and Maryles 2007; Senior,
1985). A Plastic Optical Fiber (POF) has a transparent inner core and a
thin exterior cladding and a protective sleeve. Typically, POF has a continuous
operating range of 55°C to 70°C and can withstand up to 100°C for
a short period of time less than 1 min. Silica glass optical fiber has a better
light transmission characteristic (less loss) than POF and can tolerate higher
temperatures than plastic fibers. However, POFs are more flexible, less problems
to fabricate into special assemblies and lower in cost than glass fibers (Zubia
and Arrue, 2001; Levi, 1980). The typical white light
attenuation in POF is 0.2 to 0.25 dB mm^{1}. Light guides and scanners
often using the POFs for the beam shaping (Asadpour and
Golnabi, 2008, 2010) and light transmission are
reported (Jafari and Golnabi, 2008; Haghighatzadeh
et al., 2009; Entezari and Golnabi, 2011;
Khorramnazari and Golnabi, 2011). POFs can be used for
sensing operations as well (Golnabi and Jafari, 2006;
Jafari and Golnabi, 2010).
The main objective of this study was to introduce a simple and sensitive optomechanical system in order to investigate the fiber bending effects. For this purpose in the first section of this report the fiber bending losses are described.
FIBER BENDING LOSSES
A general theory, which considers the solution of the wave equation propagating
in the optical fibers is given (Van Etten and Plaats, 1991).
Optical fibers suffer certain losses at bends or curves on their propagation
path. Bending losses in fibers can be considered as microbend (tightdiameter
bend) and macrobend (largediameter bend) losses and can result losses of the
transmitted power. Microbend losses are associated with the small perturbations
of the fiber for example induced by factors such as uneven coating application
or cablinginduced stresses (centimeter in radius). Microbend losses may be
a function of temperature and installation stresses or pressure on the fiber.
Macrobending affects the lesserconfined modes due to fiber bending in the fiber
waveguide.

Fig. 1: 
The geometrical arrangement for a bend fiber 
Due to bending the evanescent fields reach into the cladding and thus can be
affected by the distortion of the cladding as well as core. The result of the
microbend perturbation is to cause the coupling of the propagating modes in
the fiber by changing the optical path length and the multimode fiber loss from
random bends is given by Allard (1990):
where, M_{L} is multimode loss; N is number of bumps per unit length; h is average height of the bump; b is fiber diameter; a is fiber core radius; E_{F} is elastic modules of the fiber; E is elastic modulus of the surrounding medium and Δ is index difference between the core and cladding. From Eq. 1 increasing the refractive index of the core decreases the fiber’s sensitivity to such a bending loss. Increasing overall fiber diameter also decreases sensitivity while increasing core diameter increases the sensitivity. The loss of the higher modes causes a gradual increase in multimode fiber attenuation.
Consider a fiber with the cross sectional core radius of r_{0}, which is bended to a circular form with a radius of r_{b} where the center of the bend is point O as shown in Fig. 1. The optical axis passing through the center of the core circle as shown in Fig. 1 is point O. The first reflection angle at the first bend point B in Fig. 1 is given by:
where, angle
is the angle between the incident ray and horizontal axis. The optical path
by the light from the point A and the first total reflection point B (line AB)
in the bend fiber is given by:
where, angle β is the angle of the line OB with respect to line OA and the ratio of the beam path with the related length of the fiber is:
If we define:
then we have:
The critical bending ratio is then given by:
where n_{1} is the refractive index of the core material and n_{2} for cladding and we assume average index of the refraction to be:
and the relative index of refraction difference is given by:
One can say that by bending the fiber, Numerical Aperture (NA) is reduced to the value given by (virtual numerical aperture):
and it is equivalence to an increase in the refractive index value of the cladding n_{2} by a factor of:
and whenever this factor is less than ratio of n_{1}/n_{2} then the relating beam incident into the optical axis escapes from the fiber due to such a bending effect. From Eq. 6 the critical bending radius for ray launched on optical axis can be obtained from the following relation:
where all the quantities are defined before. As can be seen in Eq.
12 where the core radius is increased the critical bending radius is increased
accordingly and by decreasing the index of refractive difference factor the
critical bending radius is also increased. For a radius given equal or smaller
than the value given by Eq. 12 the incident ray is escaping
from the fiber wave guide, which causes the bending loss effect.
According to Fig. 1, the given formulas are for the case of a ray parallel to horizontal axis on the optical axis of the figure just incident on the center part of the fiber core section. For a parallel ray with the height of h from such optical axis the governing relation mentioned as Eq. 2 is replaced by:
where, we have r_{0}<h<r_{0} and the rest of equations
described remain unchanged. From given discussions it is obvious that for small
index difference even a small bend in the fiber causes that particular beam
to escape from the guiding fiber. Thus for such conditions for small microbend
situation of the fiber from straight line position will cause loss of such beams.
For the case that there are many of such microbending points in the fiber length
this effect will cause a considerable loss in transmitted beam power (Allard,
1990; Crisp and Elliott, 2005).
Wavelength of the incident beam also affects the macrobend losses in a fiber. In general such a bend loss can be considered as the radiation attenuation coefficient, which is given by:
where, r_{b} is the radius of curvature of the fiber bend and parameters
c_{1} and c_{2} are constants (Senior, 1985;
Allard, 1990). Large or macrobending losses tend to occur
in multimode fibers at a critical radius of curvature (r_{b})_{c},
which may be obtained from:
where, n_{1} and n_{2} are the refractive index of the core and cladding materials, respectively. As can be seen in Eq. 15, shorter wavelength light implies a lower critical bend radius and also smaller index difference is more suitable for less bending loss effect as described in the previous section.
EXPERIMENTAL METHOD
The basic experimental arrangement used in this research is shown in Fig.
2a, which consists of a light source, a transmission fiber, a mechanical
device for fiber deformation generation and a light power meter for power monitoring.
Figure 2a shows the principle of experiment where a multimode
plastic fiber is used for the output power modulation experiment. The light
source can be either a coherent laser light, white or color LED, or a white
filament lamp. The light sources used here are either a white LED or a color
light LED all operating at a supply voltage of 5 V for a better comparison.
The Plastic Optical Fibers (POFs) as described can operate successfully at visible
wavelength range and thus such light sources used in this experiment (Zubia
and Arrue, 2001; Mohanty and Kuan, 2011).
Since, the cross section of the fiber is large enough, therefore, the source
light is directly coupled to the fiber. The POF used here are effective but,
better fibers and a better treatment of the fiber ends can improve the coupling
efficiency of the fiber to source and finally to photodetector. The overall
diameter of this multimode fiber is about 2.2 mm with a cladding diameter of
about 1 mm and the core diameter of about 0.860 mm. The fiber length used here
is about 7 m. One end of the transmission fiber is connected to the LED light
source and the other end is connected to the power meter for power monitoring.

Fig. 2: 
Block diagram of the experiment (a) and (b) cylinder for the
rolling effect 
In this experiment to roll the fiber a cylindrical piece as shown in Fig.
2b is used and the fiber is rolled tightly around its surface. By using
different cylinders with different diameters it is possible to generate different
bending diameters for the measurement. Since the fiber length is fixed by reducing
the cylinder diameter the number of turns are increased, respectively. For this
experiment four different cylinders are used to provide the bending diameters
of 18, 15, 9.82 and 7.33 cm for the rolled fiber.
EXPERIMENTAL AND THEORETICAL RESULTS
For a more consistent operation of the LED light source, in the initial experiment the variation of the output power as a function of the supply voltage is investigated. Fig. 3 shows the variation of the light source power as a function of the supply voltage. As can be seen the power behavior is different for the supply voltage range of 33.5 V but for the range of 45 volts the response is linear with a smooth increase. Thus supply voltage of 5 V is used through all the reported experiments for the LED light sources. All the experiments for the given light sources are performed at the similar sourcefiber coupling arrangement and for uniform illumination intensity at such a fixed power supply voltage.
In the first study for the initial reference the power transmission for the fiber is measured where there is no bending deformation along the fiber length. As can be seen in Fig. 4, experiments are performed for the green, orange, white and red LED light color sources. For comparison the supply voltage for the LEDs are kept similar at 5 V level. Vertical axis shows the transmitted power for the tested plastic fiber for the different light colors and for a better comparison the values are shown on the graph bars in Fig. 4. For the tested LEDs the green LED shows the highest transmitted power (7456.0 nW) while the red LED shows the lowest transmitted power (5317.2 nW).
In the next experiments a cylinder as shown in Fig. 2b is
used to roll the same fiber. By using different cylinders with different diameters
it is possible to generate different bending diameters for the measurements.
Since the fiber length is fixed by reducing the cylinder diameter the number
of rolled turns is increased, accordingly. For this experiment results for the
bending diameters of 18, 15, 9.82 and 7.33 cm are reported. Power modulation
for the rolled fiber with bending diameter of 18 cm (number of turn equal to
10) for different LED light colors is shown in Fig. 5. As
indicated the vertical axis shows the transmitted power for the tested plastic
fiber for the different light colors and for a better comparison the power values
are shown on the graph bars in Fig. 5.

Fig. 3: 
Variation of the light source as a function of supply voltage 

Fig. 4: 
Output powers of a fiber with no bending effect for different
LED light colors 

Fig. 5: 
Power modulation of a rolled fiber with bending effect (n
= 10 turn, d = 18 cm) for different LED light colors 

Fig. 6: 
Power modulation of a rolled fiber with bending effect (n
= 12 turn, d = 15 cm) for different LED light colors 

Fig. 7: 
Power modulation of a rolled fiber with bending effect (n
= 18 turn, d = 9.82 cm) for different LED light colors 
For the tested LEDs the green LED shows the highest transmitted power (7349.8
nW) while the red LED shows the lowest transmitted power (5249.0 nW). In the
next investigation results of the power modulation for the plastic fiber with
bending diameter of 15 cm and a number of turns equal to 12 for different LED
light colors are shown in Fig. 6. For example, for the tested
LEDs the green LED shows the highest transmitted power (7291.4 nW) while the
red LED shows the lowest transmitted power (5188.8 nW).
Next experiment considers the results of power modulation for the plastic fiber
with the bending diameter of 9.82 cm (number of turns equal to 18) for different
LED light colors as shown in Fig. 7.

Fig. 8: 
Power modulation of a rolled fiber with bending effect (n
= 24 turn, d = 7.33 cm) for different LED light colors 
The transmitted powers for the tested plastic fiber for the different light
colors are shown. For example, for the tested LEDs the green LED shows the highest
transmitted power (6687.6 nW) while the red LED shows the lowest transmitted
power (4641.6 nW). Last experiment considers results of the power modulation
for the plastic fiber with bending diameter of 7.33 cm (number of turns equal
to 24) for different LED light colors. As usual as shown in Fig.
8, the vertical axis shows the transmitted power for the tested plastic
fiber for the different light colors and for a better comparison the values
are indicated on the graph bars in Fig. 8. For instance, for
the tested LEDs the green LED shows the highest transmitted power (5863.3 nW)
while the red LED shows the lowest amount of the transmitted power of 4179.4
nW.
Now it is constructive to study theoretically the bending process for the rolled
fiber. Thus, in the last study the effect of the macrobending is investigated.
Based on Eq. 12 the critical bending radius is computed by
a Matlab program as a function of the core index of refraction for a fixed value
of the cladding index of refraction. As can be seen in Fig. 9,
the critical bending radius depends on the fiber core radius and such computations
are performed for three different fiber core sizes. By decreasing the core radius
the critical bend radius is decreased, accordingly. For example for 430 μm
the critical bending radius is about 12 cm, for r = 200 μm is reduced to
5.5 cm and finally for the r = 50 μm is about 1.8 cm. This means that for
bending radius less than the given critical values, the fiber experiences power
losses under macrobending condition. As can be seen in Fig. 9,
for the fixed refractive index of the cladding (1.43), the critical bending
radius is decreased by increasing the core refractive index.

Fig. 9: 
Critical bending radius as a function of the core index of
refraction for a fixed value of the cladding index of refraction for different
core radii 
However, for the assumed values such drop in radius strongly depends on the
refractive index difference and for Fig. 9, a sharp drop is
noted for the core refractive index of 1.44 to 1.47. When the difference in
the index of refractions is very large the variation is very slow and the critical
bending radius is almost constant. In term of the critical optical bending radius
(case that total reflection fails), for the case of core refractive index of
n_{1} = 1.5, cladding index of n_{2} = 1.48, from Eq.
12 the critical bending radius is 6.4 cm. For the fixed core radius of 430
m, when the index difference term is 0.01 (1.441.43) the critical bending radius
is 12 cm, while for index difference of 0.02 (1.51.48) the critical bending
radius is reduced to 6.4 cm.
DISCUSSION
Figure 48 are good bases for the comparison
purposes and three major points can be discussed from the given results. First,
for the no bending situation and all the bending cases the trend of the power
transmission variation is the same. For all the measurements the green LED shows
the highest transmitted power while the red LED shows the lowest output power.
Second and perhaps the more important one, when there is no perturbing bending
force the amount of the transmitted power as expected is the highest value for
all the illumination source light colors. For a given source light, by reducing
the bending diameter the power loss is increased, accordingly. For example,
for the green LED for the case of forcefree the transmitted power is 7456.0
nW, while for bending 18 cm diameter is 7349.8 nW, for 15 cm bending diameter
is 7291.4 nW, for 9.82 cm bending diameter is 6687.6 nW and finally for the
bending diameter of 7.33 cm transmitted power is decreased to 5863.6 nW. Third
point is that at first look the power loss comparison for different launching
light color shows the green transmitted light is higher than for the other colors.
However, this does not mean that there is a lower attenuation for the green
light as a result of the bending forces for the fiber. For this case the fact
is that the green LED has a higher output and as a result the fiber input power
for this source is higher than those of the other sources. A more precise analysis
for this case requires different color LEDs with the compatible output powers
for a better comparison.
Now it is useful to consider the theoretical results in the following discussions. As stated, the critical bending radius depends on the fiber core radius. By decreasing the core radius the critical bend radius is decreased, accordingly. For example for 430 μm the critical bending radius is about 12 cm, for r = 200 μm is reduced to 5.5 cm and finally for the r = 50 μm is about 1.8 cm. For the fixed refractive index of the cladding (1.43), the critical bending radius is decreased by increasing the core refractive index. Such decrease in radius strongly depends on the refractive index difference and a sharp drop is noted for the core refractive index of 1.44 to 1.47. When the difference in the index of refractions is very large the variation is very slow and the critical bending radius is almost constant. In term of the critical optical bending radius (case that total reflection fails), for the case of core refractive index of n_{1} = 1.5, cladding index of n_{2} = 1.48, the critical bending radius is 6.4 cm.
For the fixed core radius of 430 m, when the index difference term is 0.01 (1.441.43) the critical bending radius is 12 cm, while for index difference of 0.02 (1.51.48) the critical bending radius is reduced to 6.4 cm. The critical bending radius depends on the fiber core radius, where by decreasing the core radius the critical bend radius is decreased, accordingly. For the core radius of 430 μm the critical bending radius is about 12 cm, while for the r = 50 μm is reduced to only 1.8 cm. For a fixed refractive index of the cladding (1.43), the critical bending radius is decreased by increasing the core refractive index.
As a result of this study when the difference in the index of refractions is
very large the critical bending radius variation is very slow and the critical
bending radius is almost constant. For a fiber with the low value of the critical
bending radius it means that such a fiber can be bended to a small radius at
the expense of a lower power loss, while for the high value of the critical
bending radius power loss occurs even for the high bending cases. As can be
seen in theoretical results by decreasing the bending radius below the critical
value the numbers of the escaped rays at the interfaces are increased and as
a result the output power is reduced. When the bending radius is smaller than
the critical bending radius the escape of the incident rays at the interfaces
is enhanced and more power is lost at the interfaces. This is in agreement with
the experimental results for the fiber output measurements. When the bending
radius is decreased the amount of the power loss is increased and as a result
as shown in Fig. 58 the measured transmitted
power is decreased accordingly.
As described in literature many attempts have been made to reduce the fiber
bending losses. HoleAssisted Fiber (HAF) is an attractive lightguide method
with small bending losses. Such a design of fiber is unique because of the air
holes in the cladding part of the fiber materials. Its characteristics are realized
simply by introducing several air holes into a conventional fiber with a specific
refractive index profile (Ieda et al., 2008).
Design and characteristics of singlemode and low bending loss HAF are investigated
both numerically and experimentally. They introduced an air filling fraction
S and clarified the S dependence of the bending loss and the cutoff wavelength.
They showed that HAFs with the desired parameter values can be designed roughly
by taking account of the S dependence and the relative index difference dependence.
Their results reveal that HAF with a loss of less than 0.1 dB/turn at 1625 nm
and with a 5 mm bending radius can be achieved while maintaining transmission
characteristics comparable to those in conventional single mode fibers.
In another study characteristics of bending loss optimized hole assisted fiber
is described (Nakajima et al., 2010). That paper
describes the transmission characteristics of a HAF, which has a conventional
germaniumdoped core surrounded by several air holes. They reported the design
principle of the HAF in terms of both bending loss and modefield diameter characteristics.
Their results show that bending loss optimized HAF will be useful for constructing
a flexible optical network. Our experimental results revealed that the reported
system here can be used to measure the fiber bending loss for the conventional
fiber arrangement and for the fabricated HAFs. As indicated by many articles
designed singlemode and low bending loss HAFs are beneficial for constructing
future networks and the bending loss measurement is an important issue.
CONCLUSIONS
It was shown that intensity modulation technique can be used in a perturbing
design in order to obtain information about bending losses in plastic fibers.
The following main points can be concluded from our results. First, the fiber
dimension and materials are important in bending loss mechanisms. Second, the
roll effect is more pronounced in the fiber with smaller bending diameter. Third,
the light wavelength and the nature of the deformation of the fiber play important
roles in the mechanism and as a result in power loss. The experimental arrangement
used here effectively shows the phenomenon of the bending losses. Experimental
results verify that the reported arrangements can be effectively used to monitor
the modulated transmitted intensity caused by such mechanical perturbations
in the fiber waveguide.
Considering the theoretical results, for a fixed value of the fiber refractive index of the cladding, the critical bending radius is decreased by increasing the core refractive index. When the difference in the index of refractions (În = n_{1}n_{2}) is very large the critical bending radius variation is very slow and the critical bending radius is almost constant. By decreasing the bending radius the numbers of the escaped rays at the interfaces are increased and as a result the output power is reduced. This is in agreement with the experimental results for the fiber output measurements. When the bending radius is decreased the amount of the power loss is increased and as a result the measured transmitted power is decreased, accordingly.
Using the reported techniques one can extract information concerning the nature and amount of power losses in optical fibers with a good precision. Obtained results show that the reported system provides a simple and accurate means for the bending and rolling loss investigations of the plastic optical fibers even for largediameter bending cases with the minor deformation losses. In order to compare and verify the experimental results a theory for the critical bending radius is developed and there is a good agreement between the two results. As a result, the reported method can be used for the bending loss monitoring for both the single and multimode fibers. The results reported here are for the multimode plastic fibers but can be used for the conventional or modified singlemode fibers of other fabricated fibers with different materials and corecladding arrangements.