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Methodological Approach To Control the Refractive Index Profile of Graded-Index Polymer Optical Fiber Makoto Asai,*,† Yoshiki Mukawa,‡,§ Satoshi Takahashi,§ and Yasuhiro Koike§,|| Faculty of Science and Technology, ‡School of Integrated Design Engineering, Graduate School, and Keio Photonics Research Institute, Faculty of Science and Technology, Keio University, Yokohama, Kanagawa, 223-8522, Japan § Exploratory Research for Advanced Technology and Solution Oriented Research for Science and Technology (ERATO-SORST), Japan Science and Technology Agency ( JST), Kawasaki, Kanagawa, 212-0032, Japan
)
†
ABSTRACT: The graded-index polymer optical fiber (GI-POF) is a promising candidate for high-speed communication medium for very short reach networks such as home networks and office local area networks (LANs). It is well-known that there is an optimum refractive index profile to maximize bandwidth properties of GI-POF when the profile is expressed by the power law equation. The interfacial-gel polymerization technique, one of the major conventional fabrication methods of GI-POF, has many processes, and the formation mechanism of refractive index profiles intricately depends on many parameters of fabrication conditions. Thus is very difficult to control and form an optimum radial refractive index profile in the GI-POF. Recently, the coextrusion process has been proposed for mass production of GI-POF. In this process, the forming process of refractive index profiles can be expressed by a simple advection-diffusion equation. Therefore, we investigated the general methodology to control refractive index profiles by numerically solving the advection-diffusion equation. On the basis of our data, it was determined that refractive index profiles could be controlled by adjusting diffusion temperature and core ratio.
’ INTRODUCTION Today the number of services providing mass information, such as high-resolution and high-quality movies, is drastically increasing. This change was made possible by the fact that singlemode glass optical fiber (SM-GOF) networks have been constructed all over the world, and ultrahigh-speed data transmission has been achieved. However, the core diameter of SM-GOF is very small (5-10 μm), and an extremely precise technique is required for connecting the fiber to signal receiving devices. Therefore, it is difficult to install SM-GOF for very short reach networks (VSRN) such as home-networks, office local area networks (LANs), and in-vehicle networks. To overcome the difficulty, graded-index polymer optical fiber (GI-POF), which is known as easy-to-handle copper cables, has been proposed as a solution to the problem.1 GI-POF has sufficient flexibility to be used in VSRN. It also has a large core diameter (100-600 μm) which allows an easy connection with other devices using inexpensive connectors. Therefore, it can be installed at a fairly low cost. Moreover, the bandwidth of GI-POF can be enhanced by forming an optimum refractive index profile beyond tens of gigahertz for 100 m long fiber. Nuccio et al. (2008) reported that perfluorinated polymer-based GI-POF achieved over 40.0 Gbps for 100 m long fiber using a 1.55 μm laser.2a The most typical fabrication method is the interfacial gel polymerization technique.2b This method, however, is a batch process which requires many fabrication processes, resulting in high costs. It is also difficult to fabricate fiber continuously, resulting in low yields. Thus, the interfacial gel polymerization method is unsuitable for mass production of GI-POF. The coextrusion process has been proposed as the method which allows continuous fabrication of GI-POF.3-7 The coextrusion apparatus is shown in Figure 1. The fabrication process of this method is described as r 2011 American Chemical Society
follows: the core and cladding materials are melted in feeder 1 and feeder 2, respectively; the core material is composed of the base polymer and high refractive index dopant; the cladding material is composed of only the base polymer; the melting materials are extruded to the die section by nitrogen pressure and combined concentrically; and then, the high refractive index dopant within the core material diffuses radially in the diffusion section which forms a radial dopant concentration profile; after the diffusion section, the two-layer molten polymer with a graded-index profile is drawn to become fiber, “GI-POF”. In the coextrusion process, controlling dopant diffusion phenomena in the diffusion section is of significant importance in fabricating a high-bandwidth GI-POF with an optimum refractive index profile. Therefore, we investigated how to use computer simulation to fabricate a high-bandwidth GI-POF with an optimum refractive index profile by the coextrusion process.
’ THEORETICAL -3 DB BANDWIDTH Refractive index profiles of the GI-POF are generally approximated by the power law as shown in " g #1=2 r r e R0 nðrÞ ¼ n1 1 - 2Δ ð1Þ R0 ¼ n2 R0 e r Here, n1 is the refractive index of the core center, n2 is the refractive index of the cladding, r is the radius, R0 is the core Received: September 12, 2010 Accepted: February 3, 2011 Revised: January 24, 2011 Published: February 24, 2011 3895
dx.doi.org/10.1021/ie101888y | Ind. Eng. Chem. Res. 2011, 50, 3895–3899
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Figure 2. g-dependence of the theoretical -3 dB bandwidth of PMMA-DPS-based GI-POF. Spectral width σs = 1.0 nm. The wavelength of the light source was 650 nm. The fiber length was 100 m.
Figure 1. Schematic diagram of the (a) coextrusion process and (b) diffusion section.
radius, and g is the refractive index exponent. Δ is the relative index difference and is defined as Δ¼
n1 - n2 n1 - n2 = 2n1 2 n1 2
2
ð2Þ
The bandwidth of GI-POF has been analyzed by the WentzelKramers-Brillouin (WKB) method.8,9 We calculated the relationship between g and dispersion by the Olshansky and Keck approximation based on the WKB method.10 The modal dispersion σmod., material dispersion σmat., and total dispersion σtotal are given as follows: !1=2 " LnΔ g gþ2 4S1 S2 Δðg þ 1Þ S1 2 þ σmod: ¼ 2c g þ 1 3g þ 2 2g þ 1
σ mat:
1=2 4S2 2 Δ2 ð2g þ 2Þ2 þ ð3Þ ð5g þ 2Þð3g þ 2Þ 2 ! !2 2 2 Lσ s 4 2g 2 d n1 2 d n1 -λ ¼ - 2λ ðnΔÞS1 2g þ 2 λ dλ2 dλ2
þ ðnΔÞ
2
!2 #1=2 g-2-ε 2g gþ2 3g þ 2 S1 ¼
ð4Þ
g-2-ε gþ2
3g - 2 - 2ε S2 ¼ 2ðg þ 2Þ -2n1 λ dΔ ε¼ n Δ dλ dn1 n ¼ n1 - λ dλ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 σ total ¼ σ mod: þ σ mat: 2
Here, σs is the spectral width of the input pulse, n is the group refractive index, c is the speed of light, λ is the wavelength of light, and L is the transmission distance. The relationship between σtotal and theoretical -3 dB bandwidth is given by rffiffiffiffiffiffiffiffi ln 2 1 0:188 ð6Þ f-3dB ¼ ¼ 2 2π σtotal σ total Here, the theoretical -3 dB bandwidth was calculated with an assumption that the output pulse waveform was approximated by a Gaussian shape. Figure 2 shows the relationship between g and the theoretical -3 dB bandwidth in poly(methyl methacrylate)diphenyl sulfide-based (PMMA-DPS-based) GI-POF, when σs is set to 2.0 nm. On the basis of these data, the bandwidth reached 4.5 GHz for 100 m long fiber at approximately g = 2.4.
’ ADVECTION-DIFFUSION EQUATION The diffusion phenomena in the diffusion section can be described by the advection-diffusion equation. Dc ¼ Dm r2 c - u rc Dt
ð7Þ
Here, c is the dopant concentration, t is time, Dm is the mutual diffusion coefficient, and u is the velocity field. Note that the relation between n and c is known to be linear. To simplify calculation, we assumed the following: (1) The flow distribution within the diffusion section is constant and can be regarded as a laminar flow. (2) The melted polymer is a Newtonian fluid. (3) The dopant diffuses only in the radial direction. On the basis of these assumptions, eq 7 can be transformed as follows. Dc 1 Dc Dc Dc ¼ Dm r ð8Þ -u Dt r Dr Dr Dz In addition, we have known that Dm depends exponentially on dopant concentration.6 Dm ðcÞ ¼ D0 expðRcÞ
ð5Þ
ð9Þ
Here, D0 is the diffusion coefficient at infinite dilution, and R is the concentration dependence index. For instance, when there was a system in which PMMA was doped with DPS, we obtained D0 = 0.53 10-7 cm2/s and R = 3.73 103 cm3/ mol at T = 210 °C and weight average molecular weight of 3896
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Figure 3. (a) L-dependence of g. Calculation conditions: R1 = 2.50 mm; R0 = 0.75 mm; R = 3.73 103 cm3/mol; c0 = 6.97 mol/cm3; D0 = (0) 0.53 10-7, (4) 1.07 10-7, ()) 2.67 10-7, and (O) 5.33 10-7 cm2/s. (b) D0-dependence of gc.
Figure 4. (a) L-dependence of g. Calculation conditions: R1 = 2.50 mm; R0 = 0.75 mm; D0 = 0.53 10-7 cm2/s; c0 = 6.97 mol/cm3; R = (4) 1.76 103, (0) 3.73 103, ()) 5.70 103, and (O) 7.67 103 cm3/mol. Inset: magnification of range in which g sufficiently converges. (b) R-dependence of gc.
5.0 104, 11 In addition, u can be generally expressed by an exponential model as follows.12 ( ðd þ 2Þ=d ) r uðrÞ ¼ u0 1 ð10Þ R1 Here, u0 is the maximum velocity, uav is the average velocity, R1 is the inner radius of the diffusion tube, and d is the parameter of the exponential model. Then, uav can be described as dþ1 uav ¼ ð11Þ u0 3d þ 1 Since we assumed that the melted polymer was Newtonian fluid, d was set to 1.0. Moreover, uav was calculated through measurements of the average discharged rate of polymer from the diffusion section. Therefore, u0 was estimated by eq 11. In a system where PMMA was doped with DPS at T = 210 °C and Mw = 5.0 104, we obtained u0 = 0.17 cm/s.
’ REFRACTIVE INDEX PROFILES It was our interest to establish a reproducible method for forming an optimum refractive index profile in the coextrusion process. Thus, it was necessary to investigate the characteristics of the solution of eq 8. The solution of eq 8 (refractive index profile shape) is thought to be strongly affected by Dm expressed by eq 9. Consequently, our studies were conducted to determine the effects of D0 and R, which characterizes Dm, on the refractive
Figure 5. L-dependence of g. Calculation conditions: R1 = 2.50 mm; R0 = 0.75 mm; D0 = 0.53 10-7 cm2/s; R = 4.20 103 cm3/mol; c0 = 6.97 mol/cm3. The gray region represents a range of g = 2.4-2.5.
index profile (specifically the g value). Fixing R, we investigated effects of D0 on g. The results are shown in Figure 3. Figure 3a shows changes in g values for the corresponding diffusion tube length L. First, it was apparent that, for all D0, g values rapidly decreased in the early phase of diffusion, and the values converged to a certain g value after a certain period of time passed by (this g value was designated as gc). These results were very encouraging for fabrication of GI-POF. The reason is because the results imply that GI-POF with identical refractive profile can always be fabricated very steadily by controlling gc. Moreover, it was clearly understood that the greater the D0, the shorter the 3897
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Figure 6. (a) L-dependence of g. Calculation conditions: R1 = 2.50 mm; R0 = 0.75 mm; D0 = 0.53 10-7 cm2/s; R = 4.20 103 cm3/mol; c0 = (O) 6.97, (4) 8.23, and ()) 9.48 mol/cm3. Inset: c0-dependence of gc. (b) L-dependence of NA. The broken lines represent the diffusion length for forming the optimum refractive index profile. c0 = (O) 6.97, (4) 8.23, and ()) 9.48 mol/cm3. Inset: c0-dependence of NA.
time for g to converge to gc. The relationship between D0 and gc is shown in Figure 3b. It turned out to be that gc was hardly dependent upon D0. Fixing D0, we further investigated effects of R on g. Figure 4a shows changes in g values inside the diffusion tube. Similar to earlier results, it was apparent that, for any R, g values rapidly decreased in the early phase of diffusion and the values converged to a certain g value after a certain period of time had passed by. Also the greater R became, the shorter the time required for g to converge to gc. It was thought that the reason was that the diffusion coefficient became significantly greater in high concentration range when R became larger. In addition, Figure 4b shows the relationship between R and gc. From these data, it became apparent that gc became larger as R became greater. These results could be explained as follows: the difference between a diffusion coefficient in a high-concentration range and that in a low-concentration range became significantly greater; as a result, the progression velocity of the diffusion front was significantly slow since the diffusion coefficient of dopant in the vicinity of the core-cladding boundary area was significantly small compared to that in the core center, even though dopant in vicinity of the core center diffused rapidly in radial direction; consequently, dopant diffusing from the core center ended up accumulating in the corecladding boundary phase, refractive index profile was unlikely to form a smooth configuration, and gc values thereby became large. On the basis of these data, it was thought that the refractive index profile shape was mainly determined by R, and D0 directly determined the time necessary to form refractive index profile shape. Most importantly, it was suggested that it was possible to fabricate GI-POF with desired g by controlling R. We have already determined both experimentally and theoretically that R becomes smaller when diffusion temperature is high and becomes larger when temperature is low.11 Although, in general, D0 displays Arrhenius type of temperature dependence, it may be possible to converge g to desired gc without relying on changes in D0 associated with temperature control because gc practically depends on R. When R was approximated to purposely obtain gc = 2.4 using data from Figure 4b, the result turned out to be R = 4.20 103 cm3/mol. The data obtained by solving eq 8 using this R are shown in Figure 5. As expected, g converged within the range of 2.4-2.5, and it could be recognized that the refractive index profile formed with optimal distribution very steadily. Therefore, it was demonstrated that it was possible to control R by adjusting temperature, and thereby to fabricate, with high precision, GI-POF with desired g.
Numerical aperture (NA), an important parameter for the refractive index profile of a fiber, was investigated. NA determines bending loss characteristics and connection loss characteristics between fibers and is defined as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð12Þ NA ¼ n1 2 - n2 2 A conventional GI-POF has exhibited sufficient characteristics by means of imparting NA of approximately 0.2. At this point, however, a problem has arisen. That is, there is a possibility that NA decreases every time diffusion takes place. For instance, when the derived optimal refractive index profile shape is made to form by controlling R using temperature control as described earlier, the problem that NA may excessively decrease might occur. Thus, the effects on a refractive index profile were investigated when only the added amount c0 of dopant was varied. Figure 6a shows the changes in g values for diffusion tube lengths when a given amount of each dopant was added. It is particularly worth noting that g values converged to nearly identical gc regardless of the amount of dopant added. As stated above, the reason was thought to be attributed to the fact that a refractive index profile (specifically g value) was highly dependent upon R. Figure 6b shows changes in NA for diffusion tube lengths. NA is indicated by a broken line at the point where it could be determined that the g value commenced to converge. Also NA at those times are shown in the inset. It was clearly shown that when the added amount of dopant increased, NA was maintained at high levels. All the data above demonstrated that it was possible to control g values and NA independently. Furthermore, there is one more issue that needs to be discussed. D0 and R are substance-specific physical properties. The temperature dependence is also substance-specific. Namely, it is highly conceivable that the aimed value that was calculated in order to form an optimal refractive index profile may not be achieved by means of controlling temperature. For examples, the material itself could thermally decompose because of excessively high temperature, or the extrusion of material becomes impossible due to the low viscosity of the polymer which is caused by extremely low temperature. In such cases, the problem that remains to be solved is how the optimal refractive index profile should be formed. Namely, the method to control refractive index profile without controlling material-specific physical values also becomes an important factor. Varying the core diameter R0 in the early phase refractive index of the prediffusion stage, we investigated whether or not we were able to change the obtained refractive index shape. Figure 7a shows the changes in g values for 3898
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Figure 7. (a) L-dependence of g. Calculation conditions: R1 = 2.50 mm; D0 = 0.53 10-7 cm2/s; R = 4.20 103 cm3/mol; c0 = 6.97 mol/cm3; R0 = (O) 0.75, (4) 0.50, and ()) 0.25 mm. (b) R0-dependence of gc.
diffusion tube length L with each core diameter R0. It is evident that the smaller the R0 was, the faster the g values converged. It was thought that g values converged fast because it was possible to reduce the amount of substance transported per unit area necessary to diffuse through by reducing core diameter. In addition, Figure 7b shows relationship between core diameter R0 and its corresponding gc. On the basis of these data, it became evident that the smaller the diameter, the smaller the gc values became, and there was a correlation between core R0 and gc. This means that it is also possible to make an arbitrary refractive index profile form by manipulating R0. Since R0 is a parameter in structural design, it can be easily manipulated regardless of the characteristics of the substance (it can be easily realized through manipulating the added nitrogen pressure when the core and cladding materials are coextruded). Through the investigations above, we were able to obtain a general methodology to control refractive index profile in the extrusion process. With regard to the conventional fabrication of GI-POF, there have been no studies that accurately discuss general methodology to control refractive index. It is because the formation mechanism of the refractive index shape in the interfacial polymerization technique (the conventional fabrication method) is extremely complicated, and thus an equation that can predict the refractive index profile did not exist. For that reason, the refractive index profile has been controlled through various methods by trial and error. Control methods obtained in this manner have not been used as a general methodology because they were not effective in most cases if different materials were used (each method is material-specific). Consequently, in spite of the fact that high bandwidth is one of the most significant characteristics of GI-POF, it is the current status that most of the conventional GI-POFs possess neither an optimal refractive index profile nor a feasible maximum transmission band. In this respect, we believe that the refractive index control method presented in this study is a very general method, and it will become an extremely essential technology in research and development of the GI-POF.
’ SUMMARY The formation process of refractive index profile in the coextrusion process was simulated through numerical calculation of the advection-diffusion equation. In conclusion, in this study, we were able to demonstrate that it was possible to fabricate GI-POF with an arbitrary g not only in the method to control diffusion of the dopant itself through the means of controlling the temperature but also in the method to control diameter of the core.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT We thank Dr. T. Sassa for his technical advice in writing this paper. This work was partially supported by a Grant-in-Aid for JSPS Fellows. ’ REFERENCES (1) Koike, Y.; Asai, M. The Future of Plastic Optical Fiber. NPG Asia Mat. 2009, 1, 22. (2) (a) Koike, Y. High-Bandwidth Grade-Index Polymer Optical Fibre. Polymer 1991, 32, 1737. (b) Nuccio, S. R. et al., Proc. ECOC (2008). (3) Koike Y.; Narutomi M. Graded-Refractive-Index-Optical Material and Method for Its Production. U.S. patent 5,783,636, 1998. (4) Sohn, I.-S.; Park, C.-W. Diffusion-Assisted Coextrusion Process for the Fabrication of Graded-Index Plastic Optical Fibers. Ind. Eng. Chem. Res. 2001, 40, 3740. (5) Sohn, I.-S.; Park, C.-W. Preparation of Graded-Index Plastic Optical Fibers by the Diffusion-Assisted Coextrusion Process. Ind. Eng. Chem. Res. 2002, 41, 2418. (6) Asai, M.; Hirose, R.; Kondo, A.; Koike, Y. High-Bandwidth Graded-Index Plastic Optical Fiber by the Dopant Diffusion Coextrusion Proces. J. Lightwave Technol. 2007, 25, 3062. (7) Hirose, R.; Asai, M.; Kondo, A.; Koike, Y. Graded-Index Plastic Optical Fiber Prepared by the Coextrusion Process. Appl. Opt. 2008, 47, 4177. (8) Fleming, J. W. Material and Mode Dispersion in GeO2 3 B2O3 3 SiC2 Glasses. J. Am. Ceram. Soc. 1976, 59, 503. (9) Ishigure, T.; Nihei, E.; Koike, Y. Optimum Refractive-Index Profile of The Graded-Index Polymer Optical Fiber, toward Gigabit Data Links. Appl. Opt. 1996, 35, 2048. (10) Olshansky, R.; Keck, D. B. Pulse Broadening in Graded-Index Optical Fibers. Appl. Opt. 1976, 15, 483. (11) Asai, M.; Awata, M.; Koike, Y. Effects of Rotation of Benzene Rings on Diffusion of Solvents in Polymer Melts. Ind. Eng. Chem. Res. 2011in press. (12) McKelvey, J. M. Polymer Processing; John Wiley & Sons: 1962; p 69.
’ NOTE ADDED AFTER ASAP PUBLICATION Because of a production error, ref 2 and related citations are inaccurate in the version of this paper that was published online February 24, 2011. The corrected version was reposted to the Web February 28, 2011. 3899
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