Optimum refractive-index profileof the graded-index polymeroptical fiber, toward gigabit data links

Takaaki Ishigure, Eisuke Nihei, and Yasuhiro Koike

The optimum refractive-index distribution of the high-bandwidth graded-index polymer optical fiber1POF2 was clarified for the first time by consideration of both modal and material dispersions. Theultimate bandwidth achieved by the POF is investigated by a quantitative estimation of the materialdispersion as well as the modal dispersion. The results indicate that even if the refractive-indexdistribution is tightly controlled, the bandwidth of the graded-index POF is dominated by the materialdispersion when the required bit rate becomes larger than a few gigabits per second. It is alsoconfirmed that the material dispersion strongly depends on the matrix polymer and that the use of afluorinated polymer whose material dispersion 320.078 ns@1nm km24 is lower than that of poly1methylmethacrylate2 320.305 ns@1nm km24 allows for a 10-Gb@s signal transmission.Key words: Modal dispersion, material dispersion, graded index, polymer optical fiber 1POF2, high

bandwidth, WKBmethod. r 1996 Optical Society of America

1. Introduction

Recent interests and developments in fiber-optictechnology have permitted a high-speed multimedianetwork concept. However, the use of a single-mode glass fiber system is not necessarily suitablebecause of the requirement of precise handling andconnection and thus the high cost in a broadbandresidential area network, i.e., asynchronous-transfer-mode–local area network, interconnection, and soon. In contrast, polymer optical fiber 1POF2, whichhas more than 500 µm of core, is a promisingcandidate to solve such problems. Because all com-mercially available POF’s have been of the step-index 1SI2 type, the modal dispersion limits thepossible bit rate of POF links to less than 100megabits per second 1Mb@s2. Therefore we proposeda large-core, low-loss, and high-bandwidth graded-index polymer optical fiber 1GI POF2,1,2 and weconfirmed that a 2.5-Gb@s signal transmission for a100-m distance was possible with the GI POF.2,3

The authors are with the Faculty of Science and Technology,Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama 223,Japan, and with the Kanagawa Academy of Science and Technol-ogy, 1-1-1, Fukuura, Kanazawa-ku, Yokahama 236, Japan.Received 6 September 1995.0003-6935@96@122048-06$06.00@0r 1996 Optical Society of America

2048 APPLIED OPTICS @ Vol. 35, No. 12 @ 20 April 1996

It is well known that the dispersion of the opticalfiber is divided into three parts: modal dispersion,material dispersion, and waveguide dispersion.In the case of the SI POF, the modal dispersion is solarge that the other two dispersions can be approxi-mated to be almost zero. However, the quadraticrefractive-index distribution in the GI POF candecrease the modal dispersion dramatically, and wesucceeded in controlling the refractive-index profileof the GI POF to be an almost quadratic distributionby the interfacial-gel polymerization technique.2Therefore, in order to analyze the ultimate band-width characteristics of the GI POF, we clarify theoptimum refractive-index profile, in which not onlythe modal dispersion but also the material disper-sion is taken into account.

2. Experiment

A. Sample Preparation

The material dispersion that is caused by the wave-length dependence of the refractive index was ap-proximated by direct measurement of the refractiveindices of polymer at several wavelengths. Thesamples of poly1methyl methacrylate2 1PMMA2 andpoly1hexafluoro isopropyl 2-fluoroacrylate2 1PHFIP2-FA2 bulks for the measurement were polymerizedfrom their monomer. PHFIP 2-FA is one of thefluorinated polymers whose monomer unit contains

only three carbon-hydrogen bonds. As dopants tocontrol the refractive-index profile, benzyl benzoate1BEN2 for PMMA and dibutyl phthalate 1DBP2 forPHFIP 2-FA were used. The dopant feed ratio tothe monomer 1monomer@dopant2 was 5@1 1wt.@wt.2,which was the same ratio as that when the core ofthe GI POF was prepared. The monomer and dop-ant mixture with initiator and chain transfer agentwas placed in a glass ampule with an inner diameterof 18 mm, and it was polymerized at 90 °C for 24 h.Themethod of polymerization of the polymer rod wasdescribed in detail in Refs. 4 and 5.

B. Refractive Index

The refractive indices of the polymer bulks weremeasured by the use of Abbe’s refractometer. As away to measure the refractive indices at severalwavelengths, an interference filter was inserted inAbbe’s refractometer. Two or three measurementswere made at each wavelength, and an average ofthose was employed. Usually, index measurementswere fit to a three-term Sellmeier dispersion relationof the form6

n2 2 1 5 oi51

3 Ail2

l2 2 li2, 112

where n is the refractive index of polymer sample, Aiis the oscillator strength, li is the oscillator wave-length, and l is the wavelength of light.

3. Results and Discussion

A. Refractive Index

Average refractive indices of the PMMA, BEN-dopedPMMA, PHFIP 2-FA, and DBP-doped PHFIP 2-FAare summarized in Fig. 1 for different wavelengths,

Fig. 1. Refractive-index dependence of the polymers on wave-length: j, BEN-doped PMMA; c, PMMA;U, DBP-doped PHFIP2-FA; m, PHFIP 2-FA.

in which the solid curves are the best-fitted curvescomputed by a least-squares method. The coeffi-cients resulting from the Sellmeier fitting are givenin Table 1. It should be noted that the slopes of therefractive index versus wavelength curves indicatethat the PHFIP 2-FA has a smaller dispersion of therefractive index with respect to wavelength thanPMMAdoes.

B. Material Dispersion

The material delay distortion Dmat for propagation ofa pulse in the SI-type and the single-mode-types ofoptical fiber can be obtained from6

Dmat 5 21ldl

c 21d2n

dl22 L, 122

where s1 is the root-mean-square spectral width ofthe light source, l is the wavelength of transmittedlight, c is the velocity of light, d2n@dl2 is the second-order dispersion, and L is length of the fiber.Several measurements methods to investigate the

material dispersion of the optical fiber have beenalready reported.7,8 In addition, thematerial disper-sion of the glass fiber could be measured directly bymeasurement of the delay time of the light pulse atseveral wavelengths through the fiber. However,this method cannot be applied to POF, because thehigher attenuation of POF than glass optical fiberlimits the transmission distance of the light pulses,and a high accuracy is required to detect the pulsedelay time. Therefore dispersion measurements inthe bulk samples of the polymer were adopted.The material dispersions of PMMA, BEN-doped

PMMA, PHFIP 2-FA, and DBP-doped PHFIP 2-FAare indicated in Fig. 2; they were calculated from theresults of Sellmeier fitting, as shown Eqs. 112 and 122.The material dispersion of the silica obtained fromRef. 6 is also shown in curve 1C2 of Fig. 2. Thematerial dispersion of the PMMA is larger than thatof the silica from the visible to near infrared region.Furthermore, the signal wavelength for the PMMA-based GI POF is located in an optical window oftransmission loss around a 650-nm wavelength.Therefore the material dispersion of the silica-basedfiber at 1.3 µm or 1.55 µm, which are signal wave-lengths, is much smaller than that of PMMA at a650-nm wavelength.At a wavelength of 650 nm, the material disper-

sion of PHFIP 2-FA is 0.136 ns@1nm km2, whereas it is

Table 1. Coefficients of the Sellmeier Equation

Polymer A1 l1 A2 l2 A3 l3

PMMA 0.4963 71.80 0.6965 117.4 0.3223 9237PMMA-BEN 0.4855 104.3 0.7555 114.7 0.4252 49340PHFIP 2-FA 0.4200 58.74 0.0461 87.85 0.3484 92.71PHFIP 2-FA-DBP

0.2680 79.13 0.3513 83.81 0.2498 106.2

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0.305 ns@1nm km2 in the case of PMMA. In the caseof the fluorinated polymer matrix, a small amount ofan additive such as DBP or BEN, which has a largematerial dispersion, can increase thematerial disper-sion of the doped polymer. Therefore a fluorinateddopant should be used for fluorinated polymer inorder to keep the material dispersion lower.

C. Refractive-index Profile Consideration

The material dispersion mentioned above indicatesthe pulse spread caused by the variance of therefractive index on the wavelength of light if asingle-mode fiber or a SI fiber was fabricated by thepolymer. In the case of the GI fiber, the effect of therefractive-index profile should be considered withthe material dispersion.The refractive-index profile was approximated by

the conventional power law:

n1r2 5 n131 2 1ra2a

D4 , 132

D 5n12 2 n22

2n12<n1 2 n2n1

. 142

Here n1 and n2 are refractive indices at the centeraxis and the cladding of the fiber, respectively, a isthe radius of the core, a is the index exponent, and Dis the relative difference of the refractive index.The output pulse width from the GI POF was

calculated by the solution of the WKB method,9 inwhich both modal and material dispersions weretaken into account as shown in Eqs. 152–172. Heresintermodal, sintramodal, and stotal signify the root-mean-square pulse width caused by the modal dispersion,

Fig. 2. Comparison of the material dispersion among PMMA,partially fluorinated polymer, and silica: 1A2, BEN-doped PMMA;1B2, PMMA; 1C2, silica; 1D2, DBP-doped PHFIP 2-FA; 1E2, PHFIP2-FA.

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the intramodal 1material2 dispersion, and both disper-sions, respectively.

sintermodal 5LN1D

2c

a

a 1 1 1a 1 2

3a 1 221@2

3 3C12 1

4C1C2D1a 1 12

2a 1 1

14D2C2

212a 1 222

15a 1 2213a 1 2241@2

, 152

sintramodal 5ssL

l 312l2d2n1dl2 2

2

2 2l2d2n1dl2

1N1D2

3 C11 2a

2a 1 22 1 1N1D221a 2 2 2 e

a 1 2 22

32a

3a 1 241@2

, 162

where

C1 5a 2 2 2 e

a 1 2,

C2 53a 2 2 2 2e

21a 1 22,

e 522n1N1

l

D

dD

dl,

N1 5 n1 2 ldn1dl

.

Here ss is the root-mean-square spectral width of thelight source 1in nanometers2 and L is the fiber length.

stotal 5 31sintermodal22 1 1sintramodal2

241@2. 172

Figures 3 and 4 show the calculated stotal versusindex exponent a for PMMA-based GI POF andPHFIP 2-FA-based GI POF, respectively. Here itwas assumed that the fiber length and the spectralwidth of the light source 1laser diode2were 100 m and2 nm, respectively. Curves 1A2 in Figs. 3 and 4indicate the pulse width calculated only by themodal dispersion; i.e., parameters e and sintramodal arezero and N1 equals n1 in Eqs. 152–172. In this casestotal was minimized when a equaled 1.97 in bothPMMA-based and PHFIP 2-FA-based GI POF’s.In contrast, when the material dispersion at the650-nm wavelength was taken into account in Fig. 3,the total pulse width 1stotal2 becomes ten times larger,and the index exponent giving the smallest pulsewidth 1stotal2 is shifted to 2.33. It is also noteworthythat the effect of the material dispersion on the pulsewidth in PHFIP 2-FA-based GI POF in Fig. 4 issmaller than that in PMMA-based GI POF in Fig. 3.From the results shown in Figs. 3 and 4, we

estimated the possible bit rate in the GI POF linkwhen the rms spectral width of the light source wasassumed to be 2 nm. It was reported by Person-ick10,11 that the power penalty of the receiver reaches1 dB when the pulse width exceeds one fourth of thebit period 31@B 1in seconds2, where B is the bit rate

Fig. 3. Pulse width 1stotal2 versus index exponent of PMMA-basedGI POF, assuming equal power in all modes and a light sourcehaving a rms spectral width of 2 nm: 1A2, onlymodal dispersion isconsidered; 1B2, both modal and material dispersions at a 780-nmwavelength are considered; 1C2, both modal and material disper-sions at a 650-nm wavelength are considered.

Fig. 4. Pulse width 1stotal2 versus index exponent of PHFIP2-FA-based GI POF, assuming equal power in all modes and alight source having a rms spectral width of 2 nm: 1A2–1C2 are thesame as in Fig. 3.

1bit@s24. Therefore, possible bit rate Bp was definedby

Bp 51

4stotal

. 182

The calculated possible bit rate in PMMA-basedand PHFIP 2-FA-based GI POF links is shown in Fig.5. The results indicate that the maximum bit ratesof 25.7 Gb@s and 5.48 Gb@s can be achieved at a780-nm wavelength in PHFIP 2-FA-based GI POFand PMMA-based GI POF links, respectively. Inaddition, the maximum bit rates at the 780-nmwavelength are higher than those at the 650-nmwavelength in both PHFIP 2-FA-based and PMMA-based GI POF links, because the material dispersionbecomes smaller with the increase in wavelength oflight, as shown in Fig. 2. The attenuation spectra oflight transmission for both PMMA-based and PHFIP2-FA-based GI POF’s are shown in Fig. 6. As mostcarbon-hydrogen bondings in the PMMA-based GIPOF are substituted for carbon-fluorine bondings inthe PHFIP 2-FA-based POF, the absorption loss inthe near-infrared-to-infrared region caused by theharmonic generation of carbon-hydrogen stretchingvibration can be eliminated. Consequently, the to-tal attenuation at the near-infrared region of thePHFIP 2-FA-based GI POF is lower than that of thePMMA-based GI POF as shown in Fig. 6. Forinstance, the attenuation of the PHFIP 2-FA-basedGI POF at the 780-nm wavelength is 135 dB@km,whereas it is 840.5 dB@km in the PMMA-based GIPOF. This low attenuation at the near-infraredregion of the PHFIP 2-FA-based GI POF permits theuse of an inexpensive laser diode as the light source

Fig. 5. Relation between the bit rate and index exponent of100-m-length GI POF: 1A2, PHFIP 2-FA-based GI POF at a780-nm wavelength; 1B2, PHFIP 2-FA-based GI POF at a 650-nmwavelength; 1C2, PMMA-based GI POF at a 780-nm wavelength;1D2, PMMA-based GI POF at a 650-nm wavelength.

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for a compact disc. Therefore the fluorinated poly-mer-based GI POF is advantageous because of thelow attenuation at the near-infrared region andbecause of the low material dispersion.12In the POF link, it has been proposed that an

inexpensive LED can be utilized as the lightsource.13–15 However, because the LED has a largerspectral width than the laser diode, the effect of thespectral width on the GI POF link by the LED shouldbe investigated. The possible bit rate in a 100-mPMMA-based GI POF link is shown in Figs. 7 and 8,where the spectral widths of the light source areassumed to be 0, 1, 2, 5, and 20 nm. The wave-

Fig. 6. Total attenuation spectra of the GI POF: 1A2, PMMAbased; 1B2, PHFIP 2-FAbased.

Fig. 7. Relation between the bit rate in the 100-m PMMA-basedGI POF link and the index exponent at a 650-nm wavelengthwhen the light source has a finite spectral width as follows: 1A2, 0nm; 1B2, 1 nm; 1C2, 2 nm; 1D2, 5 nm; 1E2, 20 nm.

2052 APPLIED OPTICS @ Vol. 35, No. 12 @ 20 April 1996

lengths of the light sources in Figs. 7 and 8 are 650nm and 780 nm, respectively.In the case of the PMMA-based GI POF, the

maximum bit rate dramatically decreases with anincrease in the source spectral width, as shown inFig. 7. When the source width is 20 nm, the bit ratebecomes almost independent of the index exponent,as shown in curve 1E2 in Fig. 7. It was confirmedthat it is impossible to cover more than 1 Gb@s of bitrate unless a light source with less than a 5-nmspectral width is used and the refractive-index pro-file is tightly controlled to the order of 2 to 3 of indexexponent a.On the other hand, in the case of the PHFIP

2-FA-based GI POF, the low material dispersionmakes it possible to transmit more than 1 Gb@s, evenif the spectral width of the light source is more than10 nm.

4. Conclusion

The optimum refractive-index profile of the GI POFwas clarified with both modal and material disper-sions taken into account. The material dispersionseriously affects the bit rate performance of the GIPOF link when the signal spectral width is largerthan the order of nanometers. For a higher bit rateof more than 1 Gb@s in a 100-m POF link to beachieved, the source width should be less than 5 nmand the index profile of the PMMA-based GI POFshould be controlled to the order of 2 to 3 of indexexponent a.The PHFIP 2-FA-based GI POF can transmit a

higher bit rate than the PMMA-based GI POF, evenif the source width is as broad as 20 nm, because the

Fig. 8. Relationship between the bit rate in the 100-m PHFIP2-FA-based GI POF link and the index exponent at a 780-nmwavelength when the light source has a finite spectral width asfollows: 1A2, 0 nm; 1B2, 1 nm; 1C2, 2 nm; 1D2, 5 nm; 1E2, 20 nm.

fluorinated polymer has low material dispersion.Furthermore, the low absorption loss of the PHFIP2-FA-based GI POF at the near-infrared regionpermits the use of a longer wavelength in the POFlink. It is concluded that the fluorinated polymer-based GI POF is superior to the PMMA-based GIPOF in both the attenuation of light transmissionand the total bit rate at the near-infrared region,where several inexpensive LED’s and laser diodesexist.

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