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A comparative study of spliced optical fibers
ARTICLE IN PRESS
Optics and Lasers in Engineering 41 (2004) 277–287
A comparative study of spliced optical fibers H. El Ghandoora, I. Nasserb,*, Afaf Abdel-Hadyc, A. Al-Shukrib b
a Physics Department, Girls College, Dammam, Saudi Arabia Physics Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia c KFUPM, P.O. Box 1994, Dhahran 31261, Saudi Arabia
Received 20 October 2002; accepted 25 November 2002
Abstract A fusion-spliced optical fiber is examined with a laser sheet of light. A CCD camera is used to record the transverse interference pattern from the fiber. The buckling on the fiber material in one direction of the spliced point is distinct inside the transverse interference pattern. The refractive index profile inside the fiber core, obtained at different illumination directions, is calculated using a new method showing the change in the refractive index due to fusion splicing of the fiber. A simple theoretical model is introduced to simulate the anomalous behavior in the transverse interference fringes due to a slight change of the optical parameters. r 2003 Elsevier Science Ltd. All rights reserved. PACS: 42.25.H; 78.20.C; 42.81.D; 42.87, 81.70, 07.60.L Keywords: Fiber optics; Interferometry; Laser sheet
1. Introduction The optical fiber [1–4] used in communication systems needs to be spliced mechanically or by fusion splicing techniques. Splicing in the region of connections (of two fibers or more) causes buckling of the fiber material. This buckling causes an appreciable change (fluctuation) in the refractive index of the fiber material, which in turn leads to some loss in the transmitted information. The most significant parameters contributing to this loss are the differences between: the core diameters (geometric imperfection); the numerical apertures; the profile parameters; and the refractive indices of the conjoined fibers. *Corresponding author. E-mail address: [email protected] (I. Nasser). 0143-8166/04/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0143-8166(02)00200-2
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Theoretical investigations of fiber losses have been described by several authors, see Ref. [5] for details. Also several experimental verifications have been reported, without comparative studies, between spliced and non-spliced fibers. The power loss in a graded index multimode optical fiber is more sensitive to fusion or mechanical splicing than a step-index multimode fiber with the same core diameter. Optical fiber is characterized by different parameters; one of them is the refractive index. The refractive index [6] is a crucial parameter that is used to explore the optical properties of the material. Many optical methods [6–10] have been widely used for measuring the refractive index of optical fibers and transparent materials. A recent study [9] of the refractive index profile of highly oriented fibers used the variable wavelength interferometric technique. This technique is used for the determination of the mean refractive indices and the birefringence of highly birefringent fibers. A comparative study [10] of interferometric techniques (using multiple-beam Fizeau fringes, a two-beam interference Pluta microscope, and the automatic variable wavelength interference technique) measured the optical properties of fiber. Unfortunately, all these techniques have drawbacks. Consequently, a new experimental method [11,12], using a laser sheet of light has been developed to examine the effect of fusion splicing buckling on the refractive index profile of an optical graded index fiber. This method is non-destructive, non-contact, well suited for computer-aided measurements, needs no complicated experimental set-up, and is sensitive to very small variations in the refractive index. It has a precision of 0.000001, and an error of not more than 106. This paper is organized as follows: In Section 2, the experimental set-up will be demonstrated. Section 3, being an improvement on the assumptions made in our recently published theoretical model [11,12], utilizes a different approach to ray tracing [13]. The model is based on the geometrical-optics approximation and is independent of the particular shape of the fiber. Finally, the experimental results will be analyzed and discussed in Section 4, followed by a conclusion in Section 5.
2. Experimental set-up The experimental set-up used in this work is shown in Fig. 1, where a He–Ne laser of wavelength 632.8 nm is expanded using a diverging lens L1. A collimating lens L2 is used to produce parallel beams of laser light. These beams are passed through a cylindrical lens L3 that produces a sheet of laser light illuminating the optical fiber being investigated. The optical fiber is thus totally immersed in the laser sheet of light. The sheet of light formed has a thickness of 1 mm, covering the fiber, which has a total diameter of 200 mm. Thus we obtain two segments of the light sheet: one segment passing outside the fiber; the other segment, of thickness 200 mm, passing through the fiber. Due to the gradient in the refractive index of the fiber core, multiple reflections and refractions of light waves occur throughout the cladding and the fiber core. The interference of these waves with the waves passing outside the fiber leads to constructive or destructive interference, producing a transverse interference fringe
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Screen
L3 L2
Optical fiber
PC
L1 He-Ne laser
f
CCD camera
Transverse interference fringe pattern
Fig. 1. The schematic diagram of the experimental set-up used to image fringe patterns, see text for details.
pattern (TIP). This fringe pattern displays the refractive index information of the fiber core. The produced TIP is projected on the screen at a distance ‘‘f ’’ from the optical fiber. The fringe patterns were imaged using a CCD camera connected to a computer for conducting the data acquisition and analysis.
3. Theoretical model The ray tracing inside and outside the optical fiber is shown in Fig. 2. In Fig. 2, we will consider three sets of rays: the first set, Ray (1), passes only through the cladding and travels the distance FH; the second set of rays, Ray (2), passes through the cladding distance AB; the core distance BD; then the cladding (once again) distance DE; the third set, Ray (3), is the direct rays, which are not incident on the fiber, but propagate above and below it. As an improvement to the assumptions made in our recently published model [11,12], a different theoretical approach to ray tracing will be followed [13]. Consider a cross section of a GRIN optical fiber, Fig. 2, having a cladding of refractive index ncl and radius Rou ; and a core of refractive index nc ðrÞ with radius Rin ; where r ¼ ðX 2 þ Y 2 Þ1=2 is the distance measured from the core center ‘‘O’’. The optical path length for the direct ray, Ray (3), is given by OPL3 ¼ 2Rou þ f
ð1Þ
and the optical path length for Ray (1), through the cladding, has the form: OPL1 ¼ ðRou XF Þ þ FHncl þ HP;
ð2Þ
where P is the intersection point of Ray (1) with the screen. For Ray (2), consider a sheet of laser light incident at point A with an incident angle yA ; where Rin Rin arcsin ncl oyA oarcsin ncl : Rou Rou
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Optical fiber y-axis
Ray (1)
F
H A
θA
ϕA
B
C
Ray (2)
D O
x-axis E
Ray (3)
2 Rou
f
Fig. 2. The ray tracing inside the optical fiber and in the surrounding medium, where AðXA ; YA Þ; BðXB ; YB Þ; DðXD ; YD Þ; F ðXF ; YF Þ; and HðXH ; YH Þ:
At A; the Cartesian coordinates (XA ; YA ) are: YA ¼ Rou sin yA ; XA ¼ Rou cos yA ; and Snell’s law is represented as na sin yA ¼ ncl sin jA ;
ð3Þ
where na ¼ 1:000275 is the refractive index of the surrounding medium (air in our case) and jA is the refractive angle at the point A: The equation of the straight line that joins the points A and B is given by Y ¼ m 1 X þ b1 ;
ð4Þ
with m1 ¼ tan cA ; b1 ¼ ðYA m1 XA Þ and cA ¼ yA jA ; and the Cartesian coordinates of point B are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m1 b1 7 R2in þ m21 R2in b21 XB ¼ ; ð5Þ 1 þ m21 YB ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2in X12
In Eq. (5), we will take the negative sign because XB is negative.
ð6Þ
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The optical path length for Ray (2) is OPL2 ¼ ðRou XA Þ þ ABncl pffiffiffiffiffiffiffiffiffiffiffiffi Z rB ¼ X 2 þY 2 B B þ pffiffiffiffiffiffiffiffiffiffiffiffi nc ðrÞ dr þ DEncl þ EQ; rA ¼
ð7Þ
XA2 þYA2
where Q is the intersection point of Ray (2) with the screen. As ABEDE; approximately equal to the clad thickness, and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi BD ¼ ðXB XA Þ2 þ ðYB YA Þ2 EðXB XA ÞE2BC;
ð8Þ
where C is the intercept point with the y-axis, Eq. (7) reduces to OPL2 E ðRou XA Þ þ 2ABncl Z XA þ2 nc ðX ; Y Þ dX þ EQ:
ð9Þ
0
Eq. (9) is applied to calculate the theoretical optical path length of Ray (2) leading to the formation of the interference fringe pattern. Because of the optical path difference (OPD) between Rays (2) and (3), there will be a phase difference leading to constructive and destructive interference, i.e. the obtained TIP. In our discussions and also in the graphs, we will use R as the ratio between the cladding diameter and the core diameter; OPD is in arbitrary units; the refractive index of the fiber cladding ncl is 1.4597 and its radius (Rou ) is 62 mm; and that of the core center n0 is 1.4777 and its radius (Rin ) is 25 mm. For a qualitative calculation, nc in the third term of Eq. (9) plays a crucial rule in predicting the fringe pattern. Many empirical relationships [2] can be used in calculating OPL2; one of them is nc ðrÞ ¼ n0 ½1 Dðr=Rin Þa ; ðn20
n2cl Þ=2n20 ;
0prpRin ;
ð10Þ
where D ¼ and a is an adjustable parameter characterizing the shape of the index profile. It is also known that a plays an important role in calculating the optical path length for the GRIN optical fiber. To calculate the refractive index from the transverse interference fringes of the fiber, we implemented a method based on the well-known formula of Abel’s inversion. The method relates the deflection angle, e; of light passing through the fiber and the refractive index gradient of the fiber. The deflection angle (defined as the angle between the plane of the original laser sheet and the heights of the observed constructive (or destructive) interference fringes on the screen) is measured experimentally for the fringes and used as a data function using Abel’s integral to obtain the refractive index distribution function. Symbolically, the refractive index nðrÞ at a position r along the fiber cross-sections is given by the formula [2,11]: Z nRin N e pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dY : nðrÞ ¼ nRin ð11Þ 2 p r Y r2
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To use Eq. (11) in actual calculation, one has to capture the resultant transverse interference pattern by a CCD camera, as in Figs. 3(a) and (b), and store them in a computer for digital processing. These images were digitally processed using special software that was developed to conduct a search for the maximum and minimum brightness of the fringes and to report them as functions of position (v and u). Here, v represents the heights of the fringes from the original laser sheet plane to the positions of the fringes appearing on the perpendicular screen and u is the coordinate of the fringe parallel to the plane of the fiber. As a result, highly contrasted
Fig. 3. Raw images (upper half) of the transverse interference fringe pattern from a spliced optical fiber: (a) away from the spliced region, (b) in the spliced region, where the discontinuity of the fringe pattern can be seen in the middle.
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constructive and destructive interference fringes could be exactly defined and subsequently it was possible to make accurate measurements. By measuring the deflection angles (e) one can determine the refractive index of the fiber using Eq. (11) after it expands in series form as [11]: ni nRin ¼
M nRin X em Si;m ; p m¼i
ð12Þ
with an initial value of e0 ¼ 0:0: In Eq. (12), we divided the fiber into M equal intervals starting from the center, and the values of Si;m are [11]: S0;1 ¼ 2 log 2;
ð13Þ
S0;m ¼ ðm þ 1Þlogðm þ 1Þ 2m log m þ ðm 1Þlogðm 1Þ for mX2;
ð14Þ
Si;m ¼ ðm þ 1Þlogf½ðm þ 1Þ þ ð2m þ 1Þ1=2 =mg ð2m þ 1Þ1=2
for m ¼ iX1;
ð15Þ
Si;m ¼ ðm þ 1Þlogf½ðm þ 1Þ þ ðm þ 1Þ2 i2 1=2 g 2m logfm þ ðm2 i2 Þ1=2 g þ ðm 1Þ logfðm 1Þ þ ½ðm 1Þ2 i2 1=2 g ½ðm þ 1Þ2 i2 1=2 þ 2ðm2 i2 Þ1=2 ½ðm 1Þ2 i2 1=2
for mXi þ 1X2:
ð16Þ
4. Experimental results and discussion The experimental TIP from a spliced optical fiber is shown in Figs. 3(a) and (b). The differences between the two patterns are in the fringe spacing and visibility. In Fig. 3(a), the fringes are photographed away from the spliced region. They are close to each other and have a sharp bell shape. The fringes in the spliced region, Fig. 3(b), are much more separated and irregular in shape. The buckling in the spliced fiber is responsible for the irregular behavior of the fringe pattern. This irregularity is due to a slight change in the optical parameters, such as the differences in core diameter (geometric imperfection) and the differences in the numerical aperture of the fiber. In the spliced region of the fiber, the refractive index distribution, calculated using Eq. (12), is represented in Fig. 4. The variation (fluctuation) in the refractive index at the spliced region (asymmetric region) is obvious. It is clear from the refractive index distribution, Fig. 4, that there is a remarkable change in the fiber refractive index due to the fusion splicing, which is due to buckling at the two spliced ends of the fiber core. This buckling changes the fiber’s diameter, leading to another change in its numerical aperture. Of course, this will lead to another change in the entrance angle
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1.45
Refractive Index
1.44
1.43
1.42
1.41
1.40 0
20
40
60
80
100
120
140
160
180
200
220
Position from the centre (Arbitrary units)
Fig. 4. Refractive index profile along the spliced optical fiber diameter, where the profile is not asymmetrical in the spliced region due to buckling.
through the fiber, which will cause a decrease in the efficiency of the optical communication system transmitting information Most of the experimental work [2,3] in this field assumes a to be in the range (1.8–2) and R ranging from 1.5 to 2.5. The manufacturer usually provides these values. In this work we have two parts. The first is the experimental part considering the same values of a and R as given previously. The second is the theoretical model where we cover another range for a and R to test the prediction sensitivity of the fringe patterns proposed by the model. The theoretical simulation of the interference fringe pattern for the case where R ¼ 2:5 as a function of the incident angle yA (in radians) is shown in Fig. 5 for different values of a: a solid curve for a ¼ 0; a dotted curve for a ¼ 1:5; and a dashed-dotted curve for a ¼ 2:5: In Fig. 5, as a decreases, the fringes become flatter, but the width stays the same. Also, as a goes to 0, the fringe pattern tends to be bell shaped with no dip. Fig. 6 is used to study the effect of increasing R on the shape of the fringe pattern. The theoretical simulation of the interference fringe pattern, for the case where a ¼ 2:0; as a function of the incident angle yA (in radians) is plotted in Fig. 6 for different values of R: a solid curve for R ¼ 2:0; a dotted curve for R ¼ 2:5; and a dashed-dotted curve for R ¼ 4:0: It is interesting to note that as R increases the fringes become sharper and their width decreases. The minimum of the central dip,
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Fig. 5. The theoretical interference fringe pattern for the case where R ¼ 2:5 as a function of the incident angle yA (in radians) for different values of a: a solid curve for a ¼ 0; a dotted curve for a ¼ 1:5; and a dashed-dotted curve for a ¼ 2:5:
however, stays the same. Physically, as R increases, yA decreases and consequently the number of rays (2) that passes through the core decreases. This leads to sharp fringes.
5. Conclusions In this paper we have presented a simple technique to examine spliced and nonspliced fibers in order to explain the loss in efficiency of a communication system using optical fiber cables. In this method the transverse interference pattern obtained in the transmission of a sheet of laser light through an optical fiber is imaged with a CCD camera. The fringes are analyzed using a modified software package. The method is non-destructive and relatively insensitive to vibration compared to other optical interferometric methods. Due to the fact that the fringe pattern can be seen
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Fig. 6. As Fig. 5 but with a ¼ 2:0 and at different values of R: a solid curve for the R ¼ 2:0; a dotted curve for R ¼ 2:5; and a dashed-dotted curve for R ¼ 4:0:
by the naked eye (by transmission or reflection on the fiber, as carried out with a laser sheet), we believe that the described technique can be easily manipulated in many different areas of scientific research and industry, such as optical testing and reflection remote sensing.
Acknowledgements The authors gratefully acknowledge KFUPM’s support in carrying out this research, and the assistance of the University Editing Board.
References [1] Ghatak K, Thyagrajan K. Optical electronics. Cambridge: Cambridge University Press, 1989. [2] Marcusen D. Principles of optical fiber measurement. New York: Academic Press, 1981.
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[3] Barakat N, Hamza AA. Interferometry of fibrous materials. Bristol: Adam Hilger, 1990. [4] Beck Th, Reng N, Weber H. Optical fibers for material processing lasers. Opt Laser Eng 2000;34(6):255–591. [5] Barakat N, El-Henawi HA, Medhat M, Sobie M, El-Diasti F. Interferometric examination of spliced optical fibers. Appl Opt 1986; 25(19): 3466–8 and the references therein. [6] Shakher C, Nirala AK. A review on refractive-index and temperature profile measurements using laser-based interferometric techniques. Opt Laser Eng 1999;31(6):455–591. [7] Wang YR, Qu XM, Cai LZ, Ma BM, Sun DL. Simultaneous determination of the refractive index and wedge angle of an optical wedge plate using a photorefractive holographic interferometer. J Mod Opt 1999;46:1369–76. [8] Le Menn M, Lotrian J. Refraction index measurement by a laser-cube-capillary technique. J Phys D: Appl Phys 2001;34:1256–65. [9] Sokar TZN, Bakary MA. The refractive index profile of highly oriented fibers. J Phys D: Appl Phys 2001;34:373–8. [10] Hamza AA, Sokar TZN, El-Farahaty KA, El-Dessouky HM. Comparative study on interferometric techniques for measurements of the optical properties of a fiber. J Opt A: Pure Appl Opt 1999;1:41–7. [11] El-Ghandoor H, Abd El Ghafar EA, Hassan R. Refractive index profiling of a GRIN optical fiber using a modulated speckle sheet of light. Opt Laser Technol 1999;31(7):481–8. [12] El-Ghandoor H, Nasser I, Abd-El Rahman M, Hassan R. Theoretical model for the transverse interference pattern of GRIN optical fiber using a laser sheet of light. Opt Laser Technol 2000;32(4):281–6. [13] Synovec RE. Refractive index effects in cylindrical detector cell designs for microbore highperformance liquid chromatography of liquid using laser beam displacement. Anal Chem 1987;59:2877–84.
Optics and Lasers in Engineering 41 (2004) 277–287
A comparative study of spliced optical fibers H. El Ghandoora, I. Nasserb,*, Afaf Abdel-Hadyc, A. Al-Shukrib b
a Physics Department, Girls College, Dammam, Saudi Arabia Physics Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia c KFUPM, P.O. Box 1994, Dhahran 31261, Saudi Arabia
Received 20 October 2002; accepted 25 November 2002
Abstract A fusion-spliced optical fiber is examined with a laser sheet of light. A CCD camera is used to record the transverse interference pattern from the fiber. The buckling on the fiber material in one direction of the spliced point is distinct inside the transverse interference pattern. The refractive index profile inside the fiber core, obtained at different illumination directions, is calculated using a new method showing the change in the refractive index due to fusion splicing of the fiber. A simple theoretical model is introduced to simulate the anomalous behavior in the transverse interference fringes due to a slight change of the optical parameters. r 2003 Elsevier Science Ltd. All rights reserved. PACS: 42.25.H; 78.20.C; 42.81.D; 42.87, 81.70, 07.60.L Keywords: Fiber optics; Interferometry; Laser sheet
1. Introduction The optical fiber [1–4] used in communication systems needs to be spliced mechanically or by fusion splicing techniques. Splicing in the region of connections (of two fibers or more) causes buckling of the fiber material. This buckling causes an appreciable change (fluctuation) in the refractive index of the fiber material, which in turn leads to some loss in the transmitted information. The most significant parameters contributing to this loss are the differences between: the core diameters (geometric imperfection); the numerical apertures; the profile parameters; and the refractive indices of the conjoined fibers. *Corresponding author. E-mail address: [email protected] (I. Nasser). 0143-8166/04/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0143-8166(02)00200-2
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Theoretical investigations of fiber losses have been described by several authors, see Ref. [5] for details. Also several experimental verifications have been reported, without comparative studies, between spliced and non-spliced fibers. The power loss in a graded index multimode optical fiber is more sensitive to fusion or mechanical splicing than a step-index multimode fiber with the same core diameter. Optical fiber is characterized by different parameters; one of them is the refractive index. The refractive index [6] is a crucial parameter that is used to explore the optical properties of the material. Many optical methods [6–10] have been widely used for measuring the refractive index of optical fibers and transparent materials. A recent study [9] of the refractive index profile of highly oriented fibers used the variable wavelength interferometric technique. This technique is used for the determination of the mean refractive indices and the birefringence of highly birefringent fibers. A comparative study [10] of interferometric techniques (using multiple-beam Fizeau fringes, a two-beam interference Pluta microscope, and the automatic variable wavelength interference technique) measured the optical properties of fiber. Unfortunately, all these techniques have drawbacks. Consequently, a new experimental method [11,12], using a laser sheet of light has been developed to examine the effect of fusion splicing buckling on the refractive index profile of an optical graded index fiber. This method is non-destructive, non-contact, well suited for computer-aided measurements, needs no complicated experimental set-up, and is sensitive to very small variations in the refractive index. It has a precision of 0.000001, and an error of not more than 106. This paper is organized as follows: In Section 2, the experimental set-up will be demonstrated. Section 3, being an improvement on the assumptions made in our recently published theoretical model [11,12], utilizes a different approach to ray tracing [13]. The model is based on the geometrical-optics approximation and is independent of the particular shape of the fiber. Finally, the experimental results will be analyzed and discussed in Section 4, followed by a conclusion in Section 5.
2. Experimental set-up The experimental set-up used in this work is shown in Fig. 1, where a He–Ne laser of wavelength 632.8 nm is expanded using a diverging lens L1. A collimating lens L2 is used to produce parallel beams of laser light. These beams are passed through a cylindrical lens L3 that produces a sheet of laser light illuminating the optical fiber being investigated. The optical fiber is thus totally immersed in the laser sheet of light. The sheet of light formed has a thickness of 1 mm, covering the fiber, which has a total diameter of 200 mm. Thus we obtain two segments of the light sheet: one segment passing outside the fiber; the other segment, of thickness 200 mm, passing through the fiber. Due to the gradient in the refractive index of the fiber core, multiple reflections and refractions of light waves occur throughout the cladding and the fiber core. The interference of these waves with the waves passing outside the fiber leads to constructive or destructive interference, producing a transverse interference fringe
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279
Screen
L3 L2
Optical fiber
PC
L1 He-Ne laser
f
CCD camera
Transverse interference fringe pattern
Fig. 1. The schematic diagram of the experimental set-up used to image fringe patterns, see text for details.
pattern (TIP). This fringe pattern displays the refractive index information of the fiber core. The produced TIP is projected on the screen at a distance ‘‘f ’’ from the optical fiber. The fringe patterns were imaged using a CCD camera connected to a computer for conducting the data acquisition and analysis.
3. Theoretical model The ray tracing inside and outside the optical fiber is shown in Fig. 2. In Fig. 2, we will consider three sets of rays: the first set, Ray (1), passes only through the cladding and travels the distance FH; the second set of rays, Ray (2), passes through the cladding distance AB; the core distance BD; then the cladding (once again) distance DE; the third set, Ray (3), is the direct rays, which are not incident on the fiber, but propagate above and below it. As an improvement to the assumptions made in our recently published model [11,12], a different theoretical approach to ray tracing will be followed [13]. Consider a cross section of a GRIN optical fiber, Fig. 2, having a cladding of refractive index ncl and radius Rou ; and a core of refractive index nc ðrÞ with radius Rin ; where r ¼ ðX 2 þ Y 2 Þ1=2 is the distance measured from the core center ‘‘O’’. The optical path length for the direct ray, Ray (3), is given by OPL3 ¼ 2Rou þ f
ð1Þ
and the optical path length for Ray (1), through the cladding, has the form: OPL1 ¼ ðRou XF Þ þ FHncl þ HP;
ð2Þ
where P is the intersection point of Ray (1) with the screen. For Ray (2), consider a sheet of laser light incident at point A with an incident angle yA ; where Rin Rin arcsin ncl oyA oarcsin ncl : Rou Rou
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Optical fiber y-axis
Ray (1)
F
H A
θA
ϕA
B
C
Ray (2)
D O
x-axis E
Ray (3)
2 Rou
f
Fig. 2. The ray tracing inside the optical fiber and in the surrounding medium, where AðXA ; YA Þ; BðXB ; YB Þ; DðXD ; YD Þ; F ðXF ; YF Þ; and HðXH ; YH Þ:
At A; the Cartesian coordinates (XA ; YA ) are: YA ¼ Rou sin yA ; XA ¼ Rou cos yA ; and Snell’s law is represented as na sin yA ¼ ncl sin jA ;
ð3Þ
where na ¼ 1:000275 is the refractive index of the surrounding medium (air in our case) and jA is the refractive angle at the point A: The equation of the straight line that joins the points A and B is given by Y ¼ m 1 X þ b1 ;
ð4Þ
with m1 ¼ tan cA ; b1 ¼ ðYA m1 XA Þ and cA ¼ yA jA ; and the Cartesian coordinates of point B are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m1 b1 7 R2in þ m21 R2in b21 XB ¼ ; ð5Þ 1 þ m21 YB ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2in X12
In Eq. (5), we will take the negative sign because XB is negative.
ð6Þ
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The optical path length for Ray (2) is OPL2 ¼ ðRou XA Þ þ ABncl pffiffiffiffiffiffiffiffiffiffiffiffi Z rB ¼ X 2 þY 2 B B þ pffiffiffiffiffiffiffiffiffiffiffiffi nc ðrÞ dr þ DEncl þ EQ; rA ¼
ð7Þ
XA2 þYA2
where Q is the intersection point of Ray (2) with the screen. As ABEDE; approximately equal to the clad thickness, and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi BD ¼ ðXB XA Þ2 þ ðYB YA Þ2 EðXB XA ÞE2BC;
ð8Þ
where C is the intercept point with the y-axis, Eq. (7) reduces to OPL2 E ðRou XA Þ þ 2ABncl Z XA þ2 nc ðX ; Y Þ dX þ EQ:
ð9Þ
0
Eq. (9) is applied to calculate the theoretical optical path length of Ray (2) leading to the formation of the interference fringe pattern. Because of the optical path difference (OPD) between Rays (2) and (3), there will be a phase difference leading to constructive and destructive interference, i.e. the obtained TIP. In our discussions and also in the graphs, we will use R as the ratio between the cladding diameter and the core diameter; OPD is in arbitrary units; the refractive index of the fiber cladding ncl is 1.4597 and its radius (Rou ) is 62 mm; and that of the core center n0 is 1.4777 and its radius (Rin ) is 25 mm. For a qualitative calculation, nc in the third term of Eq. (9) plays a crucial rule in predicting the fringe pattern. Many empirical relationships [2] can be used in calculating OPL2; one of them is nc ðrÞ ¼ n0 ½1 Dðr=Rin Þa ; ðn20
n2cl Þ=2n20 ;
0prpRin ;
ð10Þ
where D ¼ and a is an adjustable parameter characterizing the shape of the index profile. It is also known that a plays an important role in calculating the optical path length for the GRIN optical fiber. To calculate the refractive index from the transverse interference fringes of the fiber, we implemented a method based on the well-known formula of Abel’s inversion. The method relates the deflection angle, e; of light passing through the fiber and the refractive index gradient of the fiber. The deflection angle (defined as the angle between the plane of the original laser sheet and the heights of the observed constructive (or destructive) interference fringes on the screen) is measured experimentally for the fringes and used as a data function using Abel’s integral to obtain the refractive index distribution function. Symbolically, the refractive index nðrÞ at a position r along the fiber cross-sections is given by the formula [2,11]: Z nRin N e pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dY : nðrÞ ¼ nRin ð11Þ 2 p r Y r2
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To use Eq. (11) in actual calculation, one has to capture the resultant transverse interference pattern by a CCD camera, as in Figs. 3(a) and (b), and store them in a computer for digital processing. These images were digitally processed using special software that was developed to conduct a search for the maximum and minimum brightness of the fringes and to report them as functions of position (v and u). Here, v represents the heights of the fringes from the original laser sheet plane to the positions of the fringes appearing on the perpendicular screen and u is the coordinate of the fringe parallel to the plane of the fiber. As a result, highly contrasted
Fig. 3. Raw images (upper half) of the transverse interference fringe pattern from a spliced optical fiber: (a) away from the spliced region, (b) in the spliced region, where the discontinuity of the fringe pattern can be seen in the middle.
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constructive and destructive interference fringes could be exactly defined and subsequently it was possible to make accurate measurements. By measuring the deflection angles (e) one can determine the refractive index of the fiber using Eq. (11) after it expands in series form as [11]: ni nRin ¼
M nRin X em Si;m ; p m¼i
ð12Þ
with an initial value of e0 ¼ 0:0: In Eq. (12), we divided the fiber into M equal intervals starting from the center, and the values of Si;m are [11]: S0;1 ¼ 2 log 2;
ð13Þ
S0;m ¼ ðm þ 1Þlogðm þ 1Þ 2m log m þ ðm 1Þlogðm 1Þ for mX2;
ð14Þ
Si;m ¼ ðm þ 1Þlogf½ðm þ 1Þ þ ð2m þ 1Þ1=2 =mg ð2m þ 1Þ1=2
for m ¼ iX1;
ð15Þ
Si;m ¼ ðm þ 1Þlogf½ðm þ 1Þ þ ðm þ 1Þ2 i2 1=2 g 2m logfm þ ðm2 i2 Þ1=2 g þ ðm 1Þ logfðm 1Þ þ ½ðm 1Þ2 i2 1=2 g ½ðm þ 1Þ2 i2 1=2 þ 2ðm2 i2 Þ1=2 ½ðm 1Þ2 i2 1=2
for mXi þ 1X2:
ð16Þ
4. Experimental results and discussion The experimental TIP from a spliced optical fiber is shown in Figs. 3(a) and (b). The differences between the two patterns are in the fringe spacing and visibility. In Fig. 3(a), the fringes are photographed away from the spliced region. They are close to each other and have a sharp bell shape. The fringes in the spliced region, Fig. 3(b), are much more separated and irregular in shape. The buckling in the spliced fiber is responsible for the irregular behavior of the fringe pattern. This irregularity is due to a slight change in the optical parameters, such as the differences in core diameter (geometric imperfection) and the differences in the numerical aperture of the fiber. In the spliced region of the fiber, the refractive index distribution, calculated using Eq. (12), is represented in Fig. 4. The variation (fluctuation) in the refractive index at the spliced region (asymmetric region) is obvious. It is clear from the refractive index distribution, Fig. 4, that there is a remarkable change in the fiber refractive index due to the fusion splicing, which is due to buckling at the two spliced ends of the fiber core. This buckling changes the fiber’s diameter, leading to another change in its numerical aperture. Of course, this will lead to another change in the entrance angle
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1.45
Refractive Index
1.44
1.43
1.42
1.41
1.40 0
20
40
60
80
100
120
140
160
180
200
220
Position from the centre (Arbitrary units)
Fig. 4. Refractive index profile along the spliced optical fiber diameter, where the profile is not asymmetrical in the spliced region due to buckling.
through the fiber, which will cause a decrease in the efficiency of the optical communication system transmitting information Most of the experimental work [2,3] in this field assumes a to be in the range (1.8–2) and R ranging from 1.5 to 2.5. The manufacturer usually provides these values. In this work we have two parts. The first is the experimental part considering the same values of a and R as given previously. The second is the theoretical model where we cover another range for a and R to test the prediction sensitivity of the fringe patterns proposed by the model. The theoretical simulation of the interference fringe pattern for the case where R ¼ 2:5 as a function of the incident angle yA (in radians) is shown in Fig. 5 for different values of a: a solid curve for a ¼ 0; a dotted curve for a ¼ 1:5; and a dashed-dotted curve for a ¼ 2:5: In Fig. 5, as a decreases, the fringes become flatter, but the width stays the same. Also, as a goes to 0, the fringe pattern tends to be bell shaped with no dip. Fig. 6 is used to study the effect of increasing R on the shape of the fringe pattern. The theoretical simulation of the interference fringe pattern, for the case where a ¼ 2:0; as a function of the incident angle yA (in radians) is plotted in Fig. 6 for different values of R: a solid curve for R ¼ 2:0; a dotted curve for R ¼ 2:5; and a dashed-dotted curve for R ¼ 4:0: It is interesting to note that as R increases the fringes become sharper and their width decreases. The minimum of the central dip,
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Fig. 5. The theoretical interference fringe pattern for the case where R ¼ 2:5 as a function of the incident angle yA (in radians) for different values of a: a solid curve for a ¼ 0; a dotted curve for a ¼ 1:5; and a dashed-dotted curve for a ¼ 2:5:
however, stays the same. Physically, as R increases, yA decreases and consequently the number of rays (2) that passes through the core decreases. This leads to sharp fringes.
5. Conclusions In this paper we have presented a simple technique to examine spliced and nonspliced fibers in order to explain the loss in efficiency of a communication system using optical fiber cables. In this method the transverse interference pattern obtained in the transmission of a sheet of laser light through an optical fiber is imaged with a CCD camera. The fringes are analyzed using a modified software package. The method is non-destructive and relatively insensitive to vibration compared to other optical interferometric methods. Due to the fact that the fringe pattern can be seen
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Fig. 6. As Fig. 5 but with a ¼ 2:0 and at different values of R: a solid curve for the R ¼ 2:0; a dotted curve for R ¼ 2:5; and a dashed-dotted curve for R ¼ 4:0:
by the naked eye (by transmission or reflection on the fiber, as carried out with a laser sheet), we believe that the described technique can be easily manipulated in many different areas of scientific research and industry, such as optical testing and reflection remote sensing.
Acknowledgements The authors gratefully acknowledge KFUPM’s support in carrying out this research, and the assistance of the University Editing Board.
References [1] Ghatak K, Thyagrajan K. Optical electronics. Cambridge: Cambridge University Press, 1989. [2] Marcusen D. Principles of optical fiber measurement. New York: Academic Press, 1981.
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[3] Barakat N, Hamza AA. Interferometry of fibrous materials. Bristol: Adam Hilger, 1990. [4] Beck Th, Reng N, Weber H. Optical fibers for material processing lasers. Opt Laser Eng 2000;34(6):255–591. [5] Barakat N, El-Henawi HA, Medhat M, Sobie M, El-Diasti F. Interferometric examination of spliced optical fibers. Appl Opt 1986; 25(19): 3466–8 and the references therein. [6] Shakher C, Nirala AK. A review on refractive-index and temperature profile measurements using laser-based interferometric techniques. Opt Laser Eng 1999;31(6):455–591. [7] Wang YR, Qu XM, Cai LZ, Ma BM, Sun DL. Simultaneous determination of the refractive index and wedge angle of an optical wedge plate using a photorefractive holographic interferometer. J Mod Opt 1999;46:1369–76. [8] Le Menn M, Lotrian J. Refraction index measurement by a laser-cube-capillary technique. J Phys D: Appl Phys 2001;34:1256–65. [9] Sokar TZN, Bakary MA. The refractive index profile of highly oriented fibers. J Phys D: Appl Phys 2001;34:373–8. [10] Hamza AA, Sokar TZN, El-Farahaty KA, El-Dessouky HM. Comparative study on interferometric techniques for measurements of the optical properties of a fiber. J Opt A: Pure Appl Opt 1999;1:41–7. [11] El-Ghandoor H, Abd El Ghafar EA, Hassan R. Refractive index profiling of a GRIN optical fiber using a modulated speckle sheet of light. Opt Laser Technol 1999;31(7):481–8. [12] El-Ghandoor H, Nasser I, Abd-El Rahman M, Hassan R. Theoretical model for the transverse interference pattern of GRIN optical fiber using a laser sheet of light. Opt Laser Technol 2000;32(4):281–6. [13] Synovec RE. Refractive index effects in cylindrical detector cell designs for microbore highperformance liquid chromatography of liquid using laser beam displacement. Anal Chem 1987;59:2877–84.