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A Series Journal of the Chinese Physical Society
Distributed by IOP Publishing
Online: http://iopscience.iop.org/cpl
http://cpl.iphy.ac.cn
CHINESE
PH Y SIC A L S OCIET Y
ISSN: 0256
-
307X
中国物理快报
Chinese
Physics
Letters
Volume 29 Number 11 November 2012
CHIN. PHYS. LETT. Vol. 29, No. 11 (2012) 114217
Theoretical and Experimental Study of a Numerical Aperture for Multimode
PCS Fiber Optics Using an Imaging Technique
Saman Q. Mawlud**, Nahlah Q. Muhamad
Department of Physics, Education Collage, Scientific Department, Salahaddin University, Erbil, Kurdistan Region, Iraq
(Received 5 August 2012)
We study theoretically and experimentally the properties of numerical aperture (𝑁𝐴)of multimode graded-index
plastic core silica (PCS) fibers by using an image technique. A He-Ne laser at wavelength 632.8 nm and output
power 1 mW is used as the transmitter light source. The output beam images and intensity profiles of an optical
fiber are investigated by using an imaging technique. The laser beam profiles captured by a sensitive digital Nikon
camera are processed and analyzed by using a Gaussian intensity distribution in a 2D graph. A MathCAD 14
program is used for converting the image of the laser output beam into data. The theoretical and experimental
values of the numerical aperture for the used optical fiber in this study are found to be 0.5 and 0.4924, respectively.
The theoretical value of 𝑉-number is also calculated to be approximately 2482.
PACS: 42.55.Wd, 42.81.−i, 42.55.−fDOI: 10.1088/0256-307X/29/11/114217
In general, a large 𝑁𝐴means a smaller clad-core
ratio or a smaller fiber core. Smaller cores and core-
clad ratios lead to less material expense incurring and
more flexible fibers.[1]The 𝑁𝐴is intended as a mea-
sure of the light capturing ability of the fiber. Numer-
ical aperture determines the light gathering ability of
the fiber. It is a measure of the amount of light that
can be accepted by a fiber.[2,3]
In the past all silica fibers were restricted to nu-
merical apertures of 0.22 or below but nowadays a
pure silica core and doped silica clad fibers of this 𝑁𝐴
are not very thermally stable for large diameter sizes.
The thermal problems are related to the interface be-
tween the doped and undoped silica, so that today
0.22 𝑁𝐴fibers with cores much greater than 1 mm are
available with suitable thermal stability. The value of
𝑁𝐴ranges from 0.13 to 0.50, a large 𝑁𝐴implies that
a fiber will accept a large amount of light.[4]Multi-
mode optical fibers with wide light-acceptance angles
and high trapping efficiency are essential for a variety
of applications. In recent years, they have become in-
creasingly important in such fields as the detection of
charged particles and ionizing radiation.[4]
In graded-index multimode fibers, the core is com-
posed of many different layers of glass, chosen with in-
dices of refraction to produce an index profile approx-
imating a parabola, where from the center of the core
the index of refraction gets lower toward the cladding.
Since light travels faster in the lower index of refrac-
tion glass, the light will travel faster as it approaches
the outside of the core. Likewise, the light travel-
ing closest to the core center will travel the slowest.[5]
For a single-mode fiber, the attenuation reduces for
increasing distance of transmission whereas for a mul-
timode fiber, the attenuation increases when the dis-
tance of transmission increases.[6]A graded index fiber
has the same dimensions as a step index fiber. The re-
fractive index of the core changes slowly between the
fiber axis and the cladding. This is achieved by using a
varying level of dopant across the diameter of the core.
The gradations are not linear. They follow a parabolic
index profile. It is important to realize that graded
index fibers are relatively difficult to make and are
therefore significantly more expensive than step index
fibers.[3]Plastic optical fibers are large core step-index
multimode fibers, although graded-index plastic fibers
are under development, because plastic fibers have a
large diameter and can be cut with simple tools, they
are easy to work with and can use low-cost connec-
tors. Compared to glass fibers, PCS fibers are much
easier to connect because of their large diameters, and
the coupling of light from a source is also very efficient
due to large 𝑁𝐴and large core diameter.[2]
The numerical aperture 𝑁𝐴is a measurement of
the ability of an optical fiber to capture light. All
fibers have acceptance angles. The sine of half the ac-
ceptance angle of a fiber is known as the 𝑁𝐴. The 𝑁𝐴
of the fiber, and also the acceptance angle, is deter-
mined by the ratio of refractive indices of the optical
fiber core and its cladding. Rays entering the fiber
at an angle greater than the 𝑁𝐴will not be reflected
internally.
We can define the numerical aperture 𝑁𝐴depend-
ing of basic geometrical optics of the fiber by the
equation[6]
𝑁𝐴=𝑛2
1−𝑛2
2= sin 𝜃𝐴,(1)
where 𝑛1is the refractive index of the core, 𝑛2is the
refractive index of the clad, and 𝜃𝐴is half the ac-
ceptance angle. The multimode optical fiber used in
this study has the specific characteristics 𝑛1= 1.485
**Corresponding author. Email: samanqadir@gmail.com
©2012 Chinese Physical Society and IOP Publishing Ltd
114217-1
CHIN. PHYS. LETT. Vol. 29, No. 11 (2012) 114217
and 𝑛2= 1.40. Using Eq. (1), we calculate the theo-
retical value of the numerical aperture of the optical
fiber used and it is approximately equal to 0.5. The
number of the modes allowed in a given fiber is deter-
mined by a relationship between the wavelength of the
light passing through the fiber, the core diameter of
the fiber and the material of the fiber. This relation-
ship is known as the normalized frequency parameter,
or 𝑉-number. The mathematical description of the
𝑉-number is given by[6]
𝑉=2𝜋𝑁𝐴𝑎
𝜆,(2)
where 𝑎is the core radius of the optical fiber and it
is equal to 500 µm,[7]𝜆is the wavelength of the used
laser light and it is 632.8 nm. We find the theoret-
ical value of 𝑉-number by substituting the values in
Eq. (2) and it is approximately equal to 2482. A multi-
mode fiber has a 𝑉-number that is greater than 2.4045
for most optical wavelengths, and therefore will prop-
agate in many paths through the fiber. The relation-
ship between 𝑉-number and the numerical aperture is
shown in Fig. 1.
2
2.5
3
3.5
4
4.5
5
0.4 0.5 0.6 0.7 0.8 0.9 1
V
-
number X103
Numerical aperture
Fig. 1. Relation between 𝑉-number and numerical aper-
ture of the PCS multimode fiber.
8
6
4
2
0
-2
-4
-6
-8
-8-10 -6 -4 -2 0 2 4 6 8 10 12 14
54.2O
4..5O
37.4O
33.5O
31.4O
29.1O
26.9O
27.7O
28.1O
27.7O
28O
28.8O
29O
29.3O
29.4O
Fig. 2. Measurement of the acceptance angle for MMF.
We drew several concentric circles of increasing di-
ameter from 2.8 cm to 16.9cm on a small paper screen
as shown in Fig. 2. The screen is positioned in the far
field so that the axis of the fiber, at the output end,
passes perpendicularly through the center of these cir-
cles on the screen.
Multimode fiber
Z
imD
Fig. 3. Measurement of the diameter 𝐷of the spot.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.00 0.50 1.00 1 .50 2.00 2.50 3.00 3 .50 4.00
Output Power (mW)
x
-
axis direction (mm)
Fig. 4. Gaussian distribution of He-Ne laser light output.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 1 2 3 4 5 6 7 8
Output power (mW)
x
-
axis direction (mm)
Fig. 5. Gaussian distribution of PCS multimode fiber out-
put.
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
0 5 10 15 20
Numerical aperture
Beam diameter (mm)
Fig. 6. Relation between beam diameter and the numer-
ical aperture.
The fiber end, which is mounted on a 𝑥𝑦𝑧-
positioner, is moved slightly toward or away from the
screen so that one of the circles there just circum-
scribes the far-field radiation spot. The distance 𝑧
between the fiber end and the screen increases from
1 cm to 15cm, and the diameter 𝐷of the coinciding
circle is measured accurately as shown in Fig. 3. The
114217-2
CHIN. PHYS. LETT. Vol. 29, No. 11 (2012) 114217
𝑁𝐴is calculated using the equation[6]
𝑁𝐴= sin 𝜃𝐴= sin[tan−1(𝐷/2𝑍)].(3)
Figure 4shows the Gaussian beam of the output
power of the He-Ne laser light at wavelength 632.8 nm.
The 𝑥-axis represents the distance of the photodiode
moving across the horizontal displacement of the pho-
todiode, and the 𝑦-axis represents the output power
of laser light, at 90% of the output power from the
Gaussian beam giving the diameter of the laser light.
From the figure, the diameter is found to be 0.4 mm.
The same procedure is repeated for the output from
the fiber. Figure 5shows the Gaussian beam of the
output power from the MMGI optical fiber, the dis-
tance between the fiber and the photodiode is 10 mm,
and then the angle between the fiber and the distance
gives the acceptance angle.[8]The acceptance angle of
the PCS fiber is equal to 𝜃𝐴= 29.5, at 10% of the
output power from the Gaussian beam giving half the
acceptance angle of the laser light. Using Eq. (1) and
from the figure, we can find the numerical aperture to
be 0.4924, i.e.
(𝑁𝐴)pcs = sin 𝜃𝐴= 0.4924.
The relationship between the beam diameter 𝑑or
the mode field diameter and the numerical aperture
is depicted in Fig. 6; the value of numerical aperture
increases when the beam diameter is reduced. The
following equation shows the relation between the di-
ameter of the beam and 𝑁𝐴:[9]
𝑑=2
𝜋𝑀2𝜆
𝑁𝐴
,(4)
where 𝑀2is the beam quality.
0
4
8
12
16
0 5 10 15
Beam diameter (mm)
Distance (cm)
Fig. 7. Relation between the distance from the output
end of a multimode fiber and the beam diameter.
The beam diameter is defined as the distance
across the center of the beam for which the irradi-
ance equals 1/𝑒2of the maximum irradiance (1/𝑒2=
0.135). Figure 7shows the linear relationship between
the beam diameter and the distance from fiber illu-
mination end at 632.8 nm. We can observe that the
beam diameter increases with the increasing distance
between the output of the fiber ends and the screen.[10]
The fundamental-mode field distribution of the
multimode fiber can be well approximated by a Gaus-
sian function that can be written in the form[10]
𝑃(𝑟) = 𝐴exp(−𝑟2/𝑤2),(5)
where 𝑤is referred to as the spot size of the mode
field pattern. The quantity 𝑑= 2𝑤is usually referred
to as the mode field diameter (MFD).
A digital camera is used to take an image of the
output beam of the He-Ne laser directly and output
laser from the ends of the fiber. Figure 8shows the
image of the output spot of the He-Ne laser at near
field 10 cm from the source and its Gaussian profile
distribution by using the MathCAD 14 package. The
real image of the spot in Fig. 8can be converted to
the greyscale image, and the logical analysis is per-
formed to study the change in the characteristics of
the numerical aperture of the laser beam. The oc-
curred distortion in the output beam image is due to
the diffraction from the output aperture. The speckle
pattern occurs because of the inter-modal interference
that cannot be predicted and formulated easily. We
can reduce the speckle and the noise distortion of an
output coherent source by using optical diffusers in
the beam path.[11,12]
0
50
100
150
200
250
0 500 1000 1500 2000
Intensity (arb. units)
x
-
direction(arb. units)
Fig. 8. The image of the output laser and the Gaussian
intensity distribution profile as a function of 𝑥.
0
50
100
150
200
250
0 300 600 900
Intensity (arb.units)
x
-
direction (arb. units)
Fig. 9. The image of the output laser from the ends of
the fiber and the Gaussian intensity distribution profile as
a function of 𝑥.
This process was repeated for the output laser spot
from the MMGI fiber optics. The intensity profile is
very similar to a Gaussian distribution for a laser out-
put and for the MM fiber output. The image gets
closer to the hat-top distribution.[13]Figure 9shows
the intensity distribution of the laser output as a func-
tion of 𝑥for the output laser from the ends of fiber
optics.
114217-3
CHIN. PHYS. LETT. Vol. 29, No. 11 (2012) 114217
In conclusion, we have studied the numerical aper-
ture at far field and near field radiation for a multi-
mode graded index optical fiber. The theoretical value
of numerical aperture for the multimode fiber is calcu-
lated to be 0.5, the experimental value of the numeri-
cal aperture is also studied and obtained to be 0.49247,
which gives a good agreement with the theoretical re-
sults. The data from both lasers and fiber output
are used to draw a Gaussian profile distribution ap-
proximation of the peak of the far field and near field
radiation. The intensity of the light fluctuates espe-
cially when the intensity is higher. The fluctuation is
affected by many factors, such as the reflection process
of laser light. In order to see if the reflection light in-
fluences the numerical aperture of the optical fiber, we
use a reflective glass to reflect the light back directly
to the laser to decrease the fluctuation of the laser
light. From the work we can conclude that the image
technique to the laser beam profile and its energy dis-
tribution can be computerized in a very short time,
and the developed technique is compact and accurate
in measuring the laser beam characteristics instead of
using the traditional system. The proposed method
of analysis provides a better view of the laser beam
pattern showing even minor deviation in the unifor-
mity, and provides accurate quantitative analysis that
is helpful to reduce operator error and misalignment.
References
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[6] Subrahmanyam N, Brij L and Avadhanulu M N 2009 A Text
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and Imaging (New York: Optic Industries)
114217-4
Chinese Physics Letters
Volume 29 Number 11 November 2012
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114216 High Current Transfer Ratio Organic Optocoupler Based on Tandem Organic Light-Emitting
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114217 Theoretical and Experimental Study of a Numerical Aperture for Multimode PCS Fiber
Optics Using an Imaging Technique
Saman Q. Mawlud, Nahlah Q. Muhamad
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Locally Resonant Elements
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118501 High Sensitivity Magnetic Field Sensors Based on Nano-Polysilicon Thin-Film Transistors
ZHAO Xiao-Feng, WEN Dian-Zhong, ZHUANG Cui-Cui, LIU Gang, WANG Zhi-Qiang
118502 Structural Design of a Compact in-Plane Nano-Grating Accelerometer
YAO Bao-Yin, ZHOU Zhen, FENG Li-Shuang, WANG Wen-Pu, WANG Xiao
118503 Ultra Low Dark Current, High Responsivity and Thin Multiplication Region in InGaAs/InP
Avalanche Photodiodes
LI Bin, YANG Huai-Wei, GUI Qiang, YANG Xiao-Hong, WANG Jie, WANG Xiu-Ping, LIU Shao-Qing,
HAN Qin
118701 Preliminary Study on Neutron Radiography with Several Hundred keV Fast Neutrons
LI Hang, ZOU Yu-Bin, LU Yuan-Rong, GUO Zhi-Yu, TANG Guo-You
118901 Promotion of Cooperation in a Spatial Public Goods Game with Long Range Learning and
Mobility
XIAO Yao, HUA Da-Yin
