Development of single mode fiber coupling coefficient using kinetic model

Optik 121 (2010) 2240–2244

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Optik journal homepage: www.elsevier.de/ijleo

Development of single mode fiber coupling coefficient using kinetic model T. Phattaraworamet a, T. Saktioto b, J. Ali c, M. Fadhali d, P.P. Yupapin e,, J. Zainal c, S. Mitatha a a

Hybrid Computing Research Laboratory, Faculty of Engineering, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand Physics Department, Math and Science Faculty, University of Riau, Pekanbaru, Indonesia Institute of Advanced Photonics Sciences, Science Faculty, Universiti Teknologi Malaysia, 81310 Skudai, Johor Bahru, Malaysia d Physics Department, Science Faculty, Ibb University, Yemen e Advanced Research Center for Photonics, Applied Physics Department, Science Faculty, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand b c

a r t i c l e in f o

a b s t r a c t

Article history: Received 3 April 2009 Accepted 10 September 2009

We propose a simple kinetic model that can be used to improve the coupling coefficient value of a single mode fiber coupler in the fabrication process. The proposed model is time independent, where the internal and external parametric functions are included. The simulation is integrated over the coupling ratio range for the various fiber separations. The coupling coefficient value of the device is examined by using the coupling ratio range from 1% to 75%. The result obtained is compared with the experimental results, where it is noted that the separation of fiber cores significantly affects the coupling coefficient, exhibiting exponential behavior. We also found that the coupling coefficient gradient is significantly changed with respect to the coupling ratio. This model can be used to determine power losses of the fiber coupler at the coupled region, while the fabrication of the fiber coupler is operated. & 2009 Elsevier GmbH. All rights reserved.

Keywords: Single mode fiber Fiber coupler Coupling coefficient Kinetic model

1. Introduction Over the past two decades, a single mode fiber (SMF) coupler has been one of the most important fiber optic devices in telecommunications. Various applications have demonstrated the use of such fiber coupler as an optical fiber junction, where it is used to combine and split the optical signal/power for optical switch, tunable filter and modulator [1–3]. In production, a 1  2 SMF-28es fiber coupler was successfully fabricated by using a slightly unstable torch flame within the range 800–1350 1C, where the injection of the hydrogen gas at a pressure of 1 bar was employed as shown in Fig. 1. The fibers were pulled by using a vacuum pump in the range 7500–9500 mm with a velocity E100 mm/s [4], where two adjacent fibers were twisted at once to obtain the tightly coupled length. In the process shown in Fig. 1, a laser diode with 1 mW peak power at wavelength of 1310 nm was launched into the straight fiber. At the coupling region, light was split and detected by a photo-detector. By using this operation, the coupling ratio of the fused coupler within the range 1–75% is obtained. In order to determine the power losses over the coupling region at the junction of the fiber coupler, it is important to examine the correct coupling coefficient and fiber core separation. As the fiber core’s radius and normalized

frequency are two major parameters of the coupling coefficients [7,8]. We should be more careful with the approximation and coupling coefficient calculations [9,10]. Generally, the calculation of the coupling coefficient for coupled fibers is also very complicated, especially, when the core and cladding refractive indices are nearly the same value. In this work, in order to calculate these parameters under various conditions theoretically, a simple kinetic model has been developed and used, whereas the modeling is approached by using an empirical calculation with time independence. We found that the results obtained from this model can be used to compare with the experimental results and treat them by imposing several coupling ratios (CR) and separations. The modeling can also be used to provide the inputs in spectroscopic investigation of the fiber coupler and microstructure optical fiber [11,12]. In practice, fiber coupler fabrication has resulted in good coupling ratio (CR) and coupling coefficient for both transmission and coupling power [5,6]. However, the effect of heating on coating materials, fiber geometry and structure may cause the change in refractive indices.

2. Kinetic model of fiber coupling coefficient  Corresponding author.

E-mail addresses: [email protected] (T. Phattaraworamet), [email protected]. th (P.P. Yupapin). 0030-4026/$ - see front matter & 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2009.12.005

The proposed kinetic model used is based on the fact that the two fibers are coupled experimentally. The two coupled fibers are completely subjected to the coupling ratio setup. Since the coupling

T. Phattaraworamet et al. / Optik 121 (2010) 2240–2244

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with fraction power of Pb can then be defined by [14] CRjPb ¼

ratio of the fiber coupler can be varied, the coupling coefficient can be affected. By using the kinetic model, we can develop the small range into micro- and nano-scale in determining the value of the coupling coefficient as a result of coupling ratio. This model shows how the light slightly penetrates both the core and cladding during fusion in the fabrication process. Power transfer of the fiber coupler at the coupling region is affected by the coupling coefficient, whereas the coefficient is represented by the cosine or sine function of power. The percentage of power transfer to a receiving output port with respect to the total power of all output ports is called the coupling ratio. To describe the fiber coupling coefficient as a function of coupling ratio, the kinetic model is proposed using the continuity equation with parametric vectors, which is written as ð1Þ

where k is the coupling coefficient. S is the source, n and t are velocity and time, respectively. Since k is time independent, and n is constant ! r k ¼ S !

The notation of r is operated along a function of a coupling ratio (CR), and hence r  CR, or

ð3Þ

k ¼ ðp=2ÞðOd=aÞexp½ðA þBd þ cd2 Þ

ð4Þ

2

where A=5.2789 3.663 V+0.3841 V , B= 7769+1.2252 V–0.012 V2, C= 0.0175 0.0064V 0.0009V2, d =(n21  n22)/n21 and d ¼ d=a. The initial values of core and cladding refractive indices are nco =1.4677 and ncl = 1.4624, respectively, where a is the fiber core radius, and d is the separation between two fiber cores. For a SMF-28es, the core and cladding diameters are 8.2 and 125 mm, respectively, The dominant mode is LP01, with a normalized frequency, V= (2pa/l) (n21 n22)1/2 [15,16] is equal to 2.405. Using Eqs. (3) and (4), the first term in Eq. (2) is then obtained by ( ! ! 2 @k 0 p pffiffiffi @ ½expððAþ Bd þCd ÞÞ ¼ d @ðCRÞ @ðCRÞ 2a pffiffiffi !  2 d @ 1 exp½ðA þ Bd þ Cd Þ þp @ðCRÞ a 2 ! pffiffiffi ) 2 p @ þ exp½ðA þ Bd þ Cd Þ ð5Þ ð dÞ @ðCRÞ 2a For simplicity, Eq. (5) can be divided into three separated terms: ! @k 0 ! ! ! ¼ k 01 þ k 02 þ k 03 ð6Þ @ðCRÞ

!

The external parametric function of the source S is temperature (T), which is involved in the heating of fibers, whereas the internal parametric function is the source wavelength (l) and power (P). k0 represents the coupling coefficient due to the fiber geometrical effects. Therefore, S can be expressed as

k o1 ¼

ð2Þ

In order to obtain a complete solution, each term in Eq. (2) is evaluated thoroughly. The coupling ratio in terms of power splitting fraction at the fiber junction is defined [13] Pb ðzÞ Pa ðzÞ þPb ðzÞ

where Pa and Pb are transmission and coupling power, respectively. If the source power (P0) has no loss along the fiber length and junction, then P0 = Pa(z)+ Pb(z). However, at the coupling region, power transfer is affected by the fiber’s geometrical and structural behaviors. Taking this into account, the coupling ratio

2

("

d expððA þ Bd þ Cd ÞÞ

2a "

3:663 þ 2ð0:3841VÞ

! # @V @ðCRÞ

! # @V @ðCRÞ " ! #) 2 @V þ ð0:00642ð0:0009VÞÞd @ðCRÞ !

with f0 is the CR first-order derivation. Then the equation becomes

p pffiffiffi

þ ð1:22522ð0:0152ÞÞd

! !! ! ! @k 0 þf 0 ð l ; P Þ þ f 0 ð T Þ S ¼ @ðCRÞ

CRð%Þ ¼

sin2 ½ðk2 þ d Þ1=2 z x100%

where

! ! @k ¼ S @ðCRÞ

! ! !! ! @k @k o ¼ þf 0 ð l ; P Þ þ f 0 ð T Þ @ðCRÞ @ðCRÞ

2

k

Here, d is the phase mismatch factor defined as (b1  b2)/2, where b1 and b2 represent the phase constant of fiber 1(straight fiber) and 2(bending fiber) as shown in Fig. 1. Assuming d = 0, the phase velocities in the two modes are equal such that the complete power transfer occurs over the coupling length, z = Lc, or Lc = p/(2k). If d a0 then the maximum fraction of power is k2 =ðk2 þ d2 Þ. A simple empirical relationship is then used to calculate the coupling coefficient, where [13]

Fig. 1. A schematic of a SMF-28es coupler fabrication process [4].

! ! @k ! þ r  ðv k Þ ¼ S @t

k2 2 þ d2

(

pffiffiffi

k 02 ¼ p

!

k 03 ¼

(

p

2a

d

2a

2

exp½ðA þBd þ Cd Þ 2

2

"

exp½ðA þ Bd þ Cd Þ

! @a @ðCRÞ

n2 1 22 n1

!

! ! !#) n22 @n1 n2 @n 2 þ n1 @ðCRÞ n31 @ðCRÞ

The second term in Eq. (2) shows that the power (P) and source wavelength (l) are also affected by the fiber coupler’s geometry and structure. Since they are constant at the input and output ports, the total power does not change, except partial changes along the coupling region. Unless the propagation constant remains, the wavelength will be affected. Hence, 2 0 13 " !! ! !! ! !# @ k @ l @k @P 0 ! ! @ A5 þ f ðl ; P Þ¼4 ð7Þ @l @P @ðCRÞ @ðCRÞ Consider the case when the power and wavelength do not depend on each other. After the fiber was perturbed by heating

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with no changes toward the coupling ratio range, it is inferred that the derivation only depends on the coupling coefficient. Experimentally, heat is imposed into the fiber coupling region, which is not a constant but depends on the position. Assuming that at the highest temperature position, the coupling ratio can be achieved even in a slightly unstable torch flame, the third term in Eq. (2), T, can be expressed as " !! ! !# @k @T 0 ! f ðT Þ¼ ð8Þ @T @ðCRÞ Both the coupling coefficient and temperature are varied over the source flame and coupling ratio, respectively. A higher coupling coefficient is obtained when the heating duration is longer. However, since the timer controls the pre-set of the coupling ratio at the earliest and optimum conditions, the coupling coefficient can be time independent. This means that the coupling coefficient is experimentally controlled by the coupling ratio itself rather than the flame source. The second part of the derivation shows that the temperature is not constant. This is experimentally true as the torch flame flow produces the temperature variations with respect to the fiber position in all cases. Fiber surfaces are not only heated at the center of coupling region or coupling length but also in the range of the coupling region. Therefore, this term can be determined from the experimental results. Combining Eqs. (6)–(8) and substitute into Eq. (2), we have ðIÞ

j

2 0 13 ! !! ! @k @ k @ l ! ! ! @ A5 ¼ ð k 01 þ k 02 þ k 03 Þ þ 4 @ðCRÞ @l @ðCRÞ j

j

!! ! !# " !! ! !# @k @P @k @T þ @P @T @ðCRÞ @ðCRÞ j

j

j

þ

j

"

ðIIIÞ

j j

ðIIÞ

length which requires wider distance when the coupling ratio increases. Theoretically, it is not expected for the coupling length to be wider as long as the coupling power has been coupled and has reached the pre-set value of the coupling ratio resulting in linear trend lines. It is affected by the time and pulling of fibers during fusion. In fact, the coupling ratio can be reached with no coupling length dependence or a constant value. Again, this occurs due to the effect of fabrication in order to obtain a certain coupling length, refractive index changes mainly to achieve the separation of fiber axis of two cores in the shortest time. The range of coupling length has the distance possibility of the two cores so that the power can transfer to another fiber within a much shorter time as compared with the constant coupling range. In Fig. 3, the coupling coefficient is also nearly linear in the range of 0.55–0.85 mm  1, and the trend lines gradient is 0.0024. The curve describes various values of d, which is shown by the dot that decreased with increase in the coupling ratio. However, d variations are not very clear. A complete coupling coefficient has been achieved, while the minimum d to obtain the splitting power during fusion cannot be detected. Therefore, to determine a minimum d for each coupling ratio, we find the radial and axial part of fiber changes are almost comparable, then the d gradient is nearly proportional to the coupling length gradient with 0.0134. Since the fibers are heated with a slightly unstable torch flame, both figures cannot reach a complete range of 100% coupling ratio. The higher coupling coefficient indicates that more power can be split to another fiber. The coupling coefficient increases over the coupling ratio in order to reach the nearest separation between the cores. Although the coupling coefficient does not affect the coupling ratio, it depends on the distance between two cores. Therefore in both figures, the coupling length and coefficient are not implicitly affected by the provided coupling ratio. The simulation result obtained when the kinetic model is used as shown in Fig. 4. It is found that the coupling coefficient of four different separation fibers of two cores increases from 0.024 to 0.054 over the coupling ratio range 0–100%. The positive gradient occurs due to increase in coupling

ð9Þ

Again, when the vectors are used to describe the separations of fiber axes (d) as a row vector, the distribution of the d along the coupling ratio can be visualized.

3. Integration and comparison of coupling coefficient Experimentally, the coupling coefficient implicitly depends upon time. However, since the coupling ratio is also actually the coupling power the coupling coefficient is affected by fiber dimension. It is purely controlled by the coupling ratio. Fig. 2 depicts the coupling

Coupling Coefficient /mm

2242

1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 5 10 15 20 25 30 35 40 45 50 55 60 0 y = 0.0024x + 0.6065, Coupling Ratio, x/(100-x); x = Fiber 1 R = 0.2993

65

70

75

80

Fig. 3. Graph of the coupling coefficient after fusion [4].

1.8 Coupling Length (mm)

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

5

10

15

y = 0.0134x + 0.4232 R = 0.8264

20

25

30

35

40

45

50

55

60

65

70

75

80

Coupling Ratio (x/100-x); x = Fiber 1

Fig. 2. Graph of the coupling length to reach coupling ratio [4].

Fig. 4. Plot of the coupling coefficient with d1 = 9.5  10  6 m; d2 = 10  10  6 m; d3 =10.5  10  6 m; d4 = 11  10  6 m (dV/dr= 0.5; da/dr=1044.3864  10  6; dn2/ dr=2.05  10  6; dn1/dr= 1.05  10  6) [17].

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length and the separation fiber axes along the coupling ratio. Comparing it with the experimental result, the gradient difference is obtained to be a factor of 10. This means that the fiber coupling is very much affected by pulling and heating although theoretically. It has no source of power imposed to obtain the coupling coefficient, which is described by Eq. (4). In Fig. 5, the separation difference of two cores d in a row vector which is shown from the highest to lowest coupling coefficient, it is set respectively as follows: d ¼ ½6:5; 7; 7:5; 8; 8:5; 9; 9:5; 10; 10:5; 11; 11:5; 12  106 m: In Fig. 5, the 12th value of d is expected as the highest possible distance between two cores. This vector describes how the distances of the two cores move closer and perpendicular to the propagation direction of the wave as the coupling ratio increases. Simulation results indicate that the higher the coupling coefficient, the smaller the separation between the cores. Additionally, the coupling coefficient increases exponentially more towards the vertical rather than the horizontal direction. If the coupling coefficient is increased significantly, then integration over the coupling ratio results in a large gradient range of the coupling coefficient. Fig. 6 depicts the coupling coefficient integration difference from the final point to initial point. Calculation was carried out using the Runge Kutta method in the range of CR for each fiber separation. These show that the coupling coefficient decreases 10  2/CR or by a factor of 10  4 and nearly linearly for fiber separation which is greater than 7.8  10  6 m. Below this value, it will exhibit a slightly exponential form. The power tends to move significantly and rapidly due to the effect of the surrounded refractive index material, which is actually associated with two claddings. Fig. 7 shows the coupling coefficient as a function of fiber separation. The two curves almost superimpose each other as a result of the small difference between the final and initial integration of the coupling coefficient. From Eq. (9), it is found that the change of k01 in term of I is significantly changed. This means that it contributes to the

Fig. 7. Graph of the coupling coefficient changes for CR= 1–100%.

coupling coefficient by a factor of 10  2 in the range 0.0539– 0.00245, while for k02 and k03 by a factor of 10  18 and 10  4, respectively. The k is much more affected by the exponential 2 factor of ½ðA þ Bd þcd Þ, where it implicitly explains the normalized frequency gradient, which is the dominant factor compared to the derivation of both d and a. On the other hand, term II and term III are weak disturbances in terms of the coupling coefficient. The terms II and III are very small and absorbed in the functions of k01 and k03. The inequality of ð@k=@lÞ 4 4ð@l=@ðCRÞÞ and ð@k=@PÞ 4ð@P=@ðCRÞÞ describes the gradient results of two equations on the left-hand side, which is higher than that of the wavelength and power gradient. The wavelength and partial power over the coupling ratio are of orders 10  9 and 10  3, respectively, and thus they are much less compared to both sides on the left of the derivatives. Within the coupling coefficient range of 10  3, both the derivative equations lead to the first equation which is 106 times higher than the second equation. Quantitatively, the value of ð@k=@lÞ is not affected by coupling ratio fraction. On the other hand, it is affected by ð@k=@PÞ which is in the order of 1. The temperature derivatives obtained is ð@k=@TÞ o ð@T=@ðCRÞÞ. The left and right hand sides are different by orders 10  3 and 102, respectively. Multiplying each part of term II and term III the order of 10  1 per thousands of the coupling coefficient is produced. This shows that the two perturbation parts contribute to the coupling coefficient over the coupling ratio effectively at the coupling region.

4. Conclusion

Fig. 5. Graph of the coupling coefficient for several separation fiber axis (d).

We have developed a simple kinetic model which is used to determine the coupling coefficient of fiber optic coupler over the range of coupling ratio 1–100%, which is found to be in good agreement with the experimental results. Results obtained have shown that the coupling coefficient has a slightly higher change over coupling ratio than that of the experimental result, whereas the separation between fiber cores is varied by a difference of 0.0106. This discrepancy occurrs due to the power and coupling length imposed by the coupling ratio for a constant separation fiber between two cores. While in the model, power and temperature have an insignificant influence. However, both coupling coefficients do not affect the overall performance of the SMF28es coupler. This model can also be applied for the multimode fibers.

Acknowledgments

Fig. 6. Graph of the coupling coefficient difference for CR = 1–100%.

We would like to acknowledge the Government of Malaysia, Universiti Teknologi Malaysia (UTM), Islamic Development Bank (IDB) and University of Riau, Indonesia, for supporting this research project.

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References [1] J.M. Senior, Optical Fiber Communications, Principles and Practice, 2nd ed, Prentice Hall of India, New Delhi, 1996. [2] I. Yokohama, J. Noda, K. Okamoto, Fiber-coupler fabrication with automatic fusion-elongation processes for low excess loss and high coupling-ratio accuracy, IEEE J. Lightwave Technol. 5 (7) (1987) 910–915. [3] L.B. Jeunhomme, M. Dekker, Single Mode Fiber Optics. Principles and Applications, Marcel Dekker Inc., New York, 1990. [4] Saktioto A. Jalil, R.A. Rosly, M. Fadhali, Z. Jasman, Coupling ratio and power transmission to core and cladding structure for a fused single mode fiber, J. Komunikasi Fisika Indonesia 5 (11) (2007) 209–212. [5] K. Akira, Coupling coefficients and coupled power equations describing the crosstalk in an image fiber, IEICE, Transactions on Electronics, The Institute of Electronics, Informatics and Communication Engineers E79-C(2) (1996) 243–248. [6] A.B. Grudinin, A.N. Guryanov, E.M. Dianov, S.V. Ignatev, S.I. Miroshnichenko, Experimental investigation of cross talk in two-channel fiber waveguides, Sov. J. Quantum Electron. 16 (1986) 238–240. [7] S.C. Tsang, K.S. Chiang, K.W. Chow, Soliton interaction in a two core optical fiber, Opt. Commun. 229 (12) (2004) 431–439. [8] A. Kazutoshi, N. Kyohei, S. Shin’ya, Effect of coupling coefficient dispersion on nonlinear fiber coupler, IEIC Technical Report, Institute of Electronics, Informatics and Communication Engineers 101 (334) (2001) 43–48.

[9] C. Dengpeng, Q. Jingren, Calculation of coupling coefficient between core mode and cladding modes of fiber grating, IEEE J. Lightwave Technol. 2 (1999) 18–22. [10] K. Akira, Coupling coefficients and random geometrical imperfections of an image fiber, IEICE Transactions on Electronics, The Institute of Electronics, Informatics and Communication Engineers E80-C(5) (1997) 717–719. [11] A. Sharma, J. Kompella, P.K. Mishra, Analysis of fiber directional couplers and coupler half-block using a new simple model for single-mode fiber, IEEE J. Lightwave Technol. 8 (2) (1990) 143–151. [12] X. Yu, M. Liu, Y. Chung, M. Yan, P. Shum, Coupling coefficient of two-core microstructured optical fiber, Opt. Commun. 260 (1) (2005) 164–169. [13] R.P. Khare, Fiber Optics and Optoelectronics, Oxford University Press, New Delhi, India, 2004. [14] A. Yariv, P. Yeh, Optical Waves in Crystals, Propagation and Control of Laser Radiation, John Wiley and Sons, Hoboken, New Jersey, 2003. [15] B. Ortega, L. Dong, Selective fused couplers consisting of a mismatched twincore fiber and a standard optical fiber, IEEE J. Lightwave Technol. 17 (1) (1999) 123–128. [16] N. Kashima, Passive Optical Components for Optical Fiber Transmission: British Library Cataloging in – Publication Dat., Artech House Inc., London, 1995. [17] Saktioto, J. Ali, M. Fadhali, and J. Zainal, Normalized frequency gradient of coupled fibers as a function of coupling ratio, International Proceeding of the Second International Conference on Optics and Laser Applications, ICOLA2007, Yogyakarta, Indonesia, 2007.