Studies on the light-focusing plastic rod. 18: Control ofrefractive-index distribution of plastic radialgradient-index rod by photocopolymerization

Yasuji Ohtsuka and Yasuhiro Koike

The refractive-index distribution patterns obtainable by the photocopolymerization of ternary monomersystems were predicted with the help of the mechanism for forming a radial gradient index reported byOhtsuka and Koike. On the basis of this prediction, we successfully fabricated a convex lens, a concave lens,and a fiber with the W-shaped index distribution expected of an optical fiber with low modal dispersion. Itwas theoretically and experimentally confirmed that the index distribution of the rod by the photocopolymer-ization of a ternary monomer system could be widely controlled.

I. Introduction

Plastic radial gradient-index (GRIN) rods have beenmade by two-step copolymerization1l 2 and by photoco-polymerization. 3- 5 No other processes in a plasticGRIN rod have been reported, with the exception ofseveral patents.6 The GRIN rod prepared by photo-copolymerization can be easily heat-drawn into a plas-tic GRIN fiber.7 In addition, its preparation methodis simple compared to the former.

In Ref. 4, we proposed a new theory of the mecha-nism for forming the radial-gradient index in the pho-tocopolymerization of multiple monomer systems. InRef. 5, according to this, the radial GRIN rod with aquadratic-index distribution up to the periphery,which had a good imaging property, was successfullyobtained in the photocopolymerization of ternarymonomer systems for the first time.

The above theory suggested the other possibility ofcontrolling the index distribution into several differ-ent distributions from the quadratic one. In this pa-per, we demonstrate plastic radial GRIN rods with afew index-distribution patterns such as a concave lens

The authors are with Keio University, Faculty of Science & Tech-nology, Department of Applied Chemistry, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223, Japan.

Received 1 October 1984.0003-6935/85/244316-05$02.00/0.© 1985 Optical Society of America.

or a W-shaped distribution, which is expected as anoptical fiber, as well as a convex lens. It was confirmedthat the index distribution of a plastic radial GRIN rodcould be widely controlled in a ternary monomer sys-tem.

11. Preparation of the GRIN Rod

Full details of the photocopolymerization processwere already reported in Ref. 5. An outline of theprocedure is described here. A mixture of three mono-mers (M1, M2, and M3) containing an initiator is placedin a glass tube. Rotating the tube on its axis, it isexposed to UV light (Toshiba high-pressure mercuryvapor lamp H400P) passing between the two shades.Meanwhile the UV source is moved upward at a con-stant velocity. When the conversion P (from mono-mer to polymer) reaches Pc, a gel phase (copolymerphase) with polymer content Pu is initially formedfrom the inner wall of the glass tube. The copolymerphase thickens with irradiation time. When the con-tent of the glass tube is solidified to the center axis, theconversion reaches Pf. The temperature in the cham-ber is kept at 250C. After the UV irradiation thecopolymer rod with a glass tube was heat-treated at80'C for one day to complete polymerization.

The copolymer located in the region near the periph-ery is formed in the early stage of polymerization, andthe copolymer near the center axis is formed when theconversion P approaches Pf. Therefore, when thecomposition of the copolymer formed at conversion Pgradually changes to Pc < P < Pf, the radial distribu-tion of the copolymer composition is obtained. If therefractive indices of three homopolymers are different,a certain kind of radial GRIN must be formed.

4316 APPLIED OPTICS / Vol. 24, No. 24 / 15 December 1985

Ill. Change of Copolymer Composition duringPolymerization

Equation (1) shows the propagation reactions of co-polymerization between Mi and Mj monomers:

-Mi. + Mi kii MAM,.

-mi. + Mi -k MiM-(Polymer radical) (Monomer) (Polymer radical)

where kii and kij are the propagation rate constants.Here the monomer reactivity ratio rij is defined as

k /i 1,2,... n\rij = -' t -1,2 no (2)

The copolymer composition is a function of the mono-mer reactivity ratio and monomer feed ratio, and wecan calculate the change in the composition of thecopolymer formed during the copolymerizationcourse, details of which were described in Ref. 4.

The result calculated for the benzyl methacrylate(BzMA)-acrylonitrile (AN)-vinyl acetate (VAc)monomer system as the M1-M2-M3 system is shown inFig. 1. A double circle [BzMA/AN/VAc = 3/3/2(wt./wt./wt.)] expresses the monomer feed composi-tion. For each 5-wt. % rise in conversion P, the re-maining monomer composition is expressed by the tailend of each arrow (solid circle), and the composition ofthe terpolymer formed at each monomer compositionis expressed by the respective arrowheads. Therefore,two curves connecting each arrowhead and each arrowtail indicate changes in the instantaneous copolymercomposition and the remaining monomer composition,respectively, with conversion P. The compositions ofeach arrowhead Yk and of each arrow tail xk are formu-lated as

mkd(Mk)Yk ' (3)

Emkd(Mk)k= 1

XkO -I YkdP

Xk (4)

where d(Mk) is the differential of the amount (mole) ofthe Mk monomer unit in the copolymer formed at anyinstant, mk is the molecular weight of the Mk monomer,and Xko denotes the weight fraction of Mk in the initialmonomer mixture. The composition Yk of the formedcopolymer was converted into the refractive index withthe modified Lorentz-Lorenz equation,4 which as-sumed the additivity of the molar volumes of structur-al units. Then the refractive index of the copolymerformed at conversion P during copolymerization wascalculated.

IV. Control of the Refractive-index Distribution

As a result of carrying out the simulation explainedabove for many monomer systems, it was confirmedthat the radial-index distribution of the rod could bewidely controlled by the photocopolymerization of a

0 s0 0100VAc ( WT )

Fig. 1. Changes in the remaining monomer composition (arrowtail) and composition of the copolymer formed at each monomercomposition (arrowhead) with P = 5 wt. % ( = 1,2,...) in theBzMA-AN-VAc system. A double circle expresses the monomer

feed composition.

In

IIZ

zII:

Type I II

(M ) (M ) ( )

III IV V

(ti <n un,) (fl<l<fl3 ) (e,<fl <I)

Reactivity

VI

(cI <Ci

Fig. 2. Classification of ternary monomer system.

ternary monomer system. The number of the mono-mer reactivity ratio rij(Nrij) is

Nrij = n(n - 1). (5)

So, in a ternary monomer system, there are six kinds ofrij, which gives more freedom to control the indexdistribution than in the case of a binary monomersystem.

To comprehend the index-distribution patterns ob-tainable in the ternary monomer system, we selectedthe monomer system satisfying Eq. (6):

r12 > 1 r13 > 1 r 23 > 1,

r 2l < 1 r3 l < 1 r 32 < 1.(6)

From Eq. (2), it is expected that aM1 richer terpolymeris formed at the initial stage of copolymerization, M2richer terpolymer at the intermediate stage, and M3richer terpolymer at the final stage. Since the copoly-mer phase is formed from the inner wall of the glasstube, the following situation would be expected: M1richer terpolymer is placed in the peripheral region ofthe GRIN rod; M2 richer terpolymer is in the interme-diate region; and M3 richer terpolymer is in the centerregion.

Concerning the order of the refractive index for therespective homopolymer, the monomer system wasclassified into six types as shown in Fig. 2. Eachtriplet of points in Fig. 2 indicates M1 , M2 , and M 3homopolymers from the left side, respectively. Theordinate expresses the refractive index of each homo-polymer, and the abscissa indicates the reactivity ofcopolymerization, where the M1 monomer is first pref-erentially polymerized according to Eq. (6).

First, about thirty monomers which are suitable forphotocopolymerization were picked up. Then rij be-

15 December 1985 / Vol. 24, No. 24 / APPLIED OPTICS 4317

(1)

Table 1. Ternary Monomer System

Type Ml M2 M 3

I MMA Acrylonitrile (AN) Vinyl benzoate (VB)(1.49)a (1.52) (1.58)

II MMA VB Vinyl phenylacetate (VPAc)(1.49) (1.58) (1.567)

III MMA N-vinyl carbazole Vinyl acetate (VAc)(1.49) (1.68) (1.47)

IV Phenyl methacrylate Isopropyl methacrylate VB(1.57) (1.47) (1.58)

V Benzyl methacrylate VAc VPAc(1.57) (1.47) (1.567)

VI Methyl atropate MMA Ethyl acrylate(1.56) (1.49) (1.47)

a The value in parentheses represents the refractive index of each homopolymer.

Table 11. Values of 0, e, and r of Ternary Monomer Systems In Table I

Ml M2 M.,Q e Q e Q e r12 r2l r13 r3l r23 r32

I 0.74 0.40 0.60 1.20 0.061 -0.55 1.70 0.31 8.30 0.049 1.20 0.039II 0.74 0.40 0.06 -0.55 0.018 -1.08 8.30 0.05 22.76 0.005 4.53 0.167III 0.74 0.40 0.41 -1.40 0.026 -0.22 2.70 0.07 20.0 0.015 2.68 0.126IV 1.49 0.76 0.85 0.34 0.061 -0.55 1.32 0.65 9.60 0.020 10.30 0.043V 0.70 0.42 0.03 -0.22 0.018 -1.08 20.58 0.03 20.73 0.005 1.75 0.275VI 4.78 1.2 0.74 0.40 0.52 0.22 2.47 0.21 2.84 0.135 1.32 0.731

tween Mi and Mj monomers for all combinationsamong them were estimated by computer using Eq.(7)8:

rij = Q- exp[-e(ei - ej)],

rji = Qj exp[-e(ej - e 1)],

(7)

where Qi (or Qj) is the reactivity of the monomer Mi (orMj), and ej (or ej) is the electrostatic interaction of thepermanent charges on the substituents in polarizingthe vinyl group of monomer Mi (or Mj). Then, accord-ing to the definition of Fig. 2, all the ternary monomersystems satisfying Eq. (6) were classified into six types.Consequently, hundreds of kinds of ternary monomersystems in each type were selected with the help of acomputer. So, an advantage of the ternary monomersystem is that it can widely select the monomer system.The representative ternary monomer system in eachtype is listed in Table I. The values in parenthesesrepresent the refractive indices nD of the correspond-ing homopolymers. The values of Q, e, and rij of eachternary monomer system in Table I are listed in TableII.

Figure 3 shows the refractive index of the copolymerformed at conversion P during the copolymerization ofthe MMA-AN-vinyl benzoate (VB) monomer systemin Type I. With an increase in P, the copolymer re-fractive index gradually increases. As the copolymeris formed from the inner wall of the glass tube, thecopolymer formed at low P would be located in theregion near the periphery of the GRIN rod, and the

1.60

1.55

1.50

a 50

P (wt.%)

100

Fig. 3. Refractive index of MMA-AN-VB (Type I) copolymerformed at conversion P. MMA/AN/VB (wt./wt./wt.): A, 1/0/1; B,

1/1/3; C, 1/1.5/3; D, 1/2/3.

copolymer formed at higher P would be near the centeraxis of the rod. Therefore, from Type I, the parabolic-index distribution giving a convex lens would be ex-pected, details of which were discussed in Ref. 4. Itwas theoretically and experimentally confirmed, in thepreparation condition in Refs. 4 and 5, that the indexdistribution could not be controlled into a parabolicone in a binary monomer system. In MMA-AN-VB,and MMA-benzyl acrylate (BzA)-VB, which is also inType I, a good convex lens with parabolic-index distri-bution from the center to periphery was successfullyfabricated in Ref. 5. From Type II [MMA-VB-vinyl

4318 APPLIED OPTICS / Vol. 24, No. 24 / 15 December 1985

1.540

C

1.535

P (wt ) 1. 5u0 0.5 1.0

Fig. 4. Refractive index of BzMA-VAc-VPAc copolymer (Type V) r/ Rp

formed at conversion P. BMA/VAc/VPAc (wt./wt./wt: A, 1/1/1; Fig. 6. Calculated index distribution of BzMA-VAc-VPAc GRINB, 3/1/1; C, 7/1/1. rod (Type V). BzMA/VAc/VPAc = 2/1/1 (wt./wt./wt.). Pc = 0.25,

1 _____ Pu = 0.60, Pf = 0.80, and d = 2.

0 50

50 100

0.03

0.02

0

100

P (wt.%)

Fig. 5. Refractive index of MAt-MMA-EA copolymer (Type VI)formed at conversion P. MAt/MMA/EA (wt./wt./wt.): A, 1/1/1; B,

1/2/1; C, 1/2/2.

phenylacetate (VPAc)], the good convex lens was alsoexperimentally obtained in suitable conditions.

Figure 4 shows the calculated refractive index of acopolymer formed at conversion P for Type V (BzMA--VAc-VPAc). For example, curve A, with an increasein conversion P, the refractive index of formed copoly-mer steeply decreased from P = 30 wt. % and graduallyincreased from P = 50 wt. %. Therefore, in the result-ing GRIN rod, with an increase in distance r from thecenter axis, the refractive index would gradually de-crease in the center region and would increase in theregion near the periphery. From Type V, the W-shaped index distribution would be expected.

Figure 5 also shows the n-P curve for Type VI[methyl atropate (MAt)-MMA-ethyl acrylate (EA)].With an increase in conversion P, the refractive indexof copolymer gradually decreases. Therefore, fromTable VI, a concave lens would be expected.

According to the theory4 of forming a radial gradi-ent-index in the rod, we could predict the radial-indexdistribution of the GRIN rod in each type in Fig. 2.4For example, Fig. 6 shows the calculated index distri-bution of the GRIN rod in Type V [BzMA/VAc/VPAc= 2/1/1 (wt./wt./wt.)], where Pc = 0.25, Pu = 0.6, Pf =

0.01

0 0.5 1.0

( r/R,)2

Fig. 7. Calculated-index distributions of GRIN rods (Type VI): A,BzMA/AN/VAc = 3/3/2 (wt./wt./wt.); B, MAt/MMA/EA = 1/1/1

(Wt./Wt./WQ.)

0.8, and = 2. Here a is a parameter of the conversionof the remaining monomer mixture, and Rp is theradius of the rod. As crudely predicted in Fig. 4, theW-shaped index distribution could be expected fromType V.

Figure 7 also shows the calculated index distributionof the rod in Type VI, where Pc, Pu, Pf, and a are thesame as those in Fig. 6, and no is the refractive index atthe center axis. Curve A is for BzMA/AN/VAc =3/3/2, and curve B is for MAt/MMA/EA = 1/1/1(wt./wt./wt.). From Fig. 7, a concave lens could beexpected.

V. Fabrication of Radial GRIN Rods

In the ternary monomer systems picked up by theabove calculation, the GRIN rods were experimentallyobtained, and the index distributions of these were

15 December 1985 / Vol. 24, No. 24 / APPLIED OPTICS 4319

1.60

C1.55

1.50

01 an

0o

0.002

0

C

0

-0.002

-0.004

0 0.5 1.0

r/RpFig. 8. Index distributions of the GRIN rods fabricated in eachtype. Type I, MMA-AN-VB; Type III, MMA-N-VCa-VAc; TypeV, BzMA-VAc-VPAc; Type VI, BzMA-AN-VAc; A, BzMA-

MMA-VAc-VPAc.

measured by our nondestructive partial splittingmethod9 using Interphako' 0 interference microscopy.The representative index distributions of the GRINrods fabricated in each type were shown in Fig. 8.

The parabolic-index distributions giving the convexand concave lenses were experimentally obtained fromTypes I (MMA-AN-VB) and VI (BzMA-AN-VAc),respectively. In Type III (MMA-N-vinyl carbazole--VAc), the inversely W-shaped index distribution wasobtained. In Type V (BzMA-VAc-VPAc) and in thetetradic monomer system (BzMA-MMA-VAc-VPAc)(curve A), the W-shaped index distribution was ex-perimentally obtained. The 3 mm-diam. rod of curveA was continuously heat drawn into the fiber whichwould be expected as a new type optical waveguidewith low modal dispersion. It was experimentally con-firmed that the index distributions of the GRIN rodsby photocopolymerization were widely controlled.

VI. Conclusion

According to the mechanism of forming the radialgradient index, we comprehended the index distribu-tion patterns obtainable in the photocopolymerizationof ternary monomer systems by classifying the mono-mer systems among some thirty monomers into sixtypes by Eq. (6). Consequently, there were hundredsof kinds of ternary monomer system in each type. Wesucceeded in experimentally obtaining a convex lens, aconcave lens, and a fiber with the W-shaped indexdistribution which would be expected as an opticalfiber. It was theoretically and experimentally con-firmed that the index distribution of the rod preparedby the photocopolymerization of ternary monomer sy-sems was tightly controlled.

This paper is based on one presented at the FifthTopical Meeting on Gradient-Index Optical ImagingSystems, held in Monterey, Calif., 19-20 Apr. 1984.

References

1. Y. Ohtsuka and Y. Terao, "Studies on the Light-Focusing Plas-tic Rod. IX. Chemical Composition of the Copolymer Rod-Diethylene Glycol Bis(allyl Carbonate) with 2,2,3,3-Tetrafluo-ropropyl Methacrylate," J. Appl. Polym. Sci. 26, 2907 (1981).

2. Y. Ohtsuka and T. Sugano, "Studies on the Light-FocusingPlastic Rod. 14: GRIN rod of CR-39-trifluoroethyl Methacry-late Copolymer by a Vapor-Phase Transfer Process," Appl. Opt.22, 413 (1983).

3. Y. Koike, Y. Kimoto, and Y. Ohtsuka, "Studies on the Light-Focusing Plastic Rod. XIII. Photocopolymerization of MethylMethacrylate-Vinyl Esters of Aromatic Carboxylic Acid," J.Appl. Polym. Sci. 27, 3253 (1982).

4. Y. Ohtsuka and Y. Koike, "Studies on the Light-Focusing Plas-tic Rod. 16: Mechanism of Gradient-Index Formation in Pho-tocopolymerization of Multiple Monomer Systems," Appl. Opt.23, 1774 (1984).

5. Y. Koike, H. Hatanaka, and Y. Ohtsuka, "Studies on Light-Focusing Plastic Rod. 17: Plastic GRIN Rod Lens Preparedby Photocopolymerization of a Ternary Monomer Systems,"Appl. Opt. 23, 1779 (1984).

6. Japanese Patents (Kokal Tokkyo Koho), 75 83, 045; 78 21, 937;82 20, 601.

7. Y. Koike, Y. Kimoto, and Y. Ohtsuka, "Studies on the Light-Focusing Plastic Rod. 12: The GRIN Fiber Lens of MethylMethacrylate-vinyl Phenylacetate Copolymer," Appl. Opt 21,1057 (1982).

8. J. Brandrup and E. H. Immergut, Polymer Handbook (Wiley-Interscience, New York, 1975), p. 11-387.

9. Y. Ohtsuka and Y. Koike, "Determination of the Refractive-Index Profile of Light-Focusing Rods: Accuracy of a MethodUsing Interphako Interference Microscopy," Appl. Opt. 19,2866(1980).

10. Registered trade name Carl Zeiss, Jena, East Germany.

4320 APPLIED OPTICS / Vol. 24, No. 24 / 15 December 1985