Studies on the light-focusing plastic rod. 1 1: Preparationof a light-defocusing plastic rod

Yasuji Ohtsuka and Koichi Maeda

A light-defocusing plastic rod (LDR) with an antiparabolic refractive-index distribution was prepared bythe two-step copolymerization of diethylene glycol bis(allyl carbonate) (CR-39) with phenyl methacrylateor vinyl benzoate. A concave lens function of the representative LDR was demonstrated with the value ofchromatic aberration factor.

1. Introduction

A gradient-index cylindrical medium with a parabolicrefractive-index distribution"2 is currently manufac-tured by ion exchange of a cesium-based glass rod3 4

(Selfoc) and has been used for several imaging andlight-focusing devices.

On the other hand, a cylindrical medium with anindex distribution expressed as Eq. (1) was pointed outearlier to be of a concave lens function theoretically,5but any preparation of such a medium has not beenreported.

n(r) = n(0)(1 + 1/2 Br 2 ), (1)

where n(0) is the refractive index at the center axis, n(r)is the refractive index at a distance r from the centeraxis, and B is a constant of the refractive-index distri-bution.

In this paper, the following preparation procedure oflight-defocusing plastic rods (LDR) is reported: di-ethylene glycol bis(allyl carbonate (CR-39) was copo-lymerized with styrene (St), phenyl methacrylate(PhMA), or vinyl benzoate (VB) applying the "two-stepcopolymerization technique" for the GRIN plasticrod.6-8 A few optical characteristics of the resultingrods are also presented.

II. Experimental

A. Materials

CR-39 (P.P.G. Industries) was distilled at 130-140°C/0.01-0.03 mm Hg. Styrene (Junsei ChemicalCo.) was purified by distillation. PhMA (bp 64-660 C/2mm Hg) was synthesized from methacryloyl chlorideand sodium salt of phenol according to the procedureof Patai et al.9 VB (bp 69-710 C/3.5 mm Hg) was pre-pared according to the vinylation of benzoic acid withacetylene using mercuric acetate as the catalyst.10

Benzoyl peroxide (BPO) as an initiator was recrystal-lized from methanol.

B. Preparation Procedure

CR-39 (M, monomer) containing BPO (0.99 wt. %)was filled in a polyethylene casting tube with a 10-mmi.d. and heated at 79.6°C for 131 min to yield a pre-polymer gel rod (GR). The GR taken apart from thecasting tube was immersed in an M2 monomer (St,PhMA, or VB) at T2 (C) for t 2 (min) under nitrogen.After draining off the M2 monomer liquid, the rod washeat-treated at 79.60C for 19 hr under nitrogen in orderto copolymerize the remaining monomers.

C. Measurement of Optical Characteristics of the Rod

As shown in Appendix A, the reduction rate of theerect virtual image formed through the LDR is ex-pressed as

y = n(0)i/B 11 sinh(vBZ) + cosh(v"BZ),

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

Received 20 April 1981.0003-6935/81/203567-04$00.50/0.© 1981 Optical Society of America.

(2)

where 11 is the distance from the object to the entranceface of the rod with length Z.

In the resulting rod (radius Rp) the index distributionobeys Eq. (1) only in the region near the center axis(radius Rc). When 11 = 0, the reduction rate (yo) issimplified as

yo = cosh(xfBZ), (3)

3562 APPLIED OPTICS / Vol. 20, No. 20 / 15 October 1981

Table I. Selected Physical Properties of the Homopolymers

Abbe'sRefractive numbera Solubility

indexa nD -1 parameterbMaterial nD nF - nc Density 6(Vha/cm3)

CR-39 1.5001 58.8 1.32 10.7 polymerPSt 1.5907 30.8 1.04-1.065a 9.1PPhMA 1.5706 35 1.249c 10.2PVB 1.5775 30.7 1.224c 10.2

a Ref. 11.b Calculated by Hoy's attraction constants.12c Measured in our laboratory.

and in addition a radius of the object observed in theimage without distortion is equal to Rc. Therefore, wecan readily estimate a pair of B and Rc/Rp from imagingwhen 11 = 0.

As a measure of chromatic aberration the value of (BF- Bc)/BD was calculated, where subscripts F, C, and Drefer to the values based on F light, C light, and D light,respectively. The respective B values were derivedfrom the corresponding yo values observed under thecorresponding light (using suitable filters or a sodiumlamp).

111. Results and Discussion

The fractional solvent extraction of the used GR in-dicates the following composition of the GR: networkpolymer part (acetone-insoluble fraction), 20.8 wt. %;linear polymer part (acetone-soluble and methanol-insoluble fraction), 5.9 wt. %; CR-39 monomer part(methanol-soluble fraction), 73.3 wt. %. It should benoted that a large amount of CR-39 monomer remainsin the GR.

For preparation of a LDR, the refractive index of anM2 homopolymer (N2) must be higher than that of anMl (CR-39) homopolymer. It is clear from Table I thatall the examined monomer pairs fulfill this criterion.

A. CR-39-St System

Comparing the values of (N2 - NI) in the examinedmonomer systems, this system seems to be most favor-able, whereas a large difference in the solubility pa-rameter between a CR-39 polymer and PSt predictspoor compatibility of a CR-39 unit with a St unit. Infact the resulting rod from this system was found to betranslucent or opaque. This system was not successful.It should be noted that compatibility between the Mland M2 units is important with index difference (N2 -N1).

B. CR-39-PhMA System

This system yielded the rods with clarity as expectedfrom the small difference in solubility parameter inTable I. The immersion process was carried out in therange of T2 = 30-601C and t 2 = 80-280 min.

Dependences of the values of B and Rc/Rp on T2 andt2 are shown in Figs. 1 and 2. The distribution constantB increased with the immersion time t 2 to a maximumvalue, where diffusing PhMA reached the center axis

5

4

3

2

50 100 150 200 250 300

t2 (min)

Fig. 1. Dependence of the distribution constant B on the immersiontemperature T2. The CR-39-PhMA system. T2 (0): 0, 30.0; A,40.0; ,50.0; , 60.0. The CR-39-VB system. T 2 (C): -v-v-,

60.0.

40

30

'.420

10

50 100 150 200 250 300

t2 ( min )

Fig. 2. Dependence of Rc/Rp on the immersion temperature T 2.The CR-39-PhMA system. T2 (°C): 0, 30.0; A, 40.0; o, 50.0; ,

60.0. The CR-39-VB system. T 2 (°C): -v-v-, 60.0.

in the immersion process. A further elongation of t 2brings a reduction in the radial gradient of PhMAconcentration, which results in a decrease in B. Theimmersion time at the maximum point shifted towardshorter t 2 as the immersion temperature T2 was ele-vated. This is attributed to an increase in the diffusionrate of PhMA in the GR. The value of RcIRp was al-most constant on a level of 40% for T2 = 30-50°C andt2 > 120 min. On the other hand, a higher immersion

15 October 1981 / Vol. 20, No. 20 / APPLIED OPTICS 3563

lmm A B

Fig. 3. Virtual images of a checker pattern (1.6 mm2 squares) incontact with an end face of the LDR: (A) the CR-39-PhMA rod lenswith 4.4-mm diam, 20.9-mm length, and B = 5.54 X 10-3 mm- 2 ; (B)the CR-39-VB rod lens with 9.2-mm diam, 16.5-mm length, and B

= 1.66 X 10-3 mm-2

.

p

Fig. 4. Rays traveling through the LDR.

(r3, r )

20.9-mm length prepared in the condition of T2 =40.00 C and t2 = 160 min was ground down to 4.4-mmdiam by means of a centerless grinder. The B value was5.54 X 10-3 mm- 2 . (B) CR-39-VB rod lens. TheCR-39-VB LDR was prepared by a vapor-phasetransfer technique, whose details will be presented inthe near future. The GR was prepared by heatingCR-39 containing 0.50-wt. % BPO at 90.00C for 88 min.This GR was placed in the atmosphere composed ofnitrogen (150 mm Hg) and VB vapor (saturated vaporat 79.60C -9.0 mm Hg) at 79.6"C for 12 h, followed byheat-treatment under nitrogen at 79.60C for 19 h. Theresulting LDR with 10-mm diam and 16.5-mm lengthwas ground down to 9.2-mm diam. The B value was1.66 X 10-3 mm- 2 .

Figure 3 shows the concave lens function of theserods, where the values of the reduction rate (yo) were2.40 and 1.22, respectively.

The values of (BF - BC)/BD for the above two rodswere measured, where Eq. (B3) in Appendix B indicatesthat the estimable level is -1 X 10-2. The observedvalues were 5 X 10-2 for CR-39-PhMA and 9 X 10-2 forCR-39-VB, which are reasonable in comparison withthe respective B value (0.10 and 0.13) in Appendix Band also indicate high chromatic aberration of bothrods.

Unlike the light-focusing plastic rod, a suitablemonomer pair for a LDR with low chromatic aberrationwas not found before now.

IV. Conclusion

The LDRs were fabricated by the two-step copoly-merization of CR-39 with PhMA or with VB. The re-sulting rod lens was of high chromatic aberration. Itshould be stated that not only a convex rod lens (a LFR)but also a concave rod lens (a LDR) become availablenow. New light-focusing devices composed of both rodlenses could be fabricated, which will be presented inthe near future.

temperature such as 60°C brought lower values of bothRc/Rp and B, which are presumably due to a fast lev-eling off in the radial concentration of the PhMA unitbrought by a higher diffusion rate of PhMA in theGR.

C. CR-39-VB System

As predicted from the respective solubility parameterin Table I, a transparent rod lens was obtained from thismonomer system. As shown in Figs. 1 and 2, both Band Rc/Rp in this system had low values, which will beattributed to low polymerizability of the VB monomer.It should be noted that the CR-39-PhMA system issuperior to the CR-39-VB system for the two-step co-polymerization process.

D. Imaging Property and Chromatic Aberration of theRepresentative Rod

The representative rod lenses without image distor-tion were fabricated as follows: (A) CR-39-PhMA rodlens. The CR-39-PhMA LDR with 10-mm diam and

Appendix A: Imaging Characteristic of the LDR

The refractive-index distribution of the LDR withradius R and length Z is expressed as in Eq. (1). As-suming a meridional ray in a paraxial condition, tracingof rays traveling through the rod were carried out. Asshown in Fig. 4, a meridional ray emerging from pointP(l1,r0 ) with ray slope r travels through the rod andattains point Q(12,r3) with ray slope r3. The followingray matrix is obtained:

Ir I= M NIIr I where

K coshq + n(0)v'BI2 sinhq,L 11 coshq + n(0)i,/B 1112 sinhq

+ 1 sinhq + 12 coshq,

M n(0)VB sinhq,

N n(0)-\/B1 sinhq + coshq.

(Al)

(A2)

3564 APPLIED OPTICS / Vol. 20, No. 20 / 15 October 1981

2

0.5 1.0 1.5 2.0

q ( ByZ)

Fig. 5. Effect of q on the shift factor of imaging (fs) and the reduc-tion rate at 11 = 0(,yo): fs - /2)(sinhq)q/[1 + (sinhq)2].

The distance (s) from the focal point to the end face, thefocal length (f), and the distance (h) from the end faceto the principal plane are expressed as

1cothq

f (0)A/ sinhq (A3)

1h f-s = tanh(q/2),n(0)V~/_

where q -BZ.With the aid of image forming condition L = 0, re-

duction rates of the image (y) and the image plane (ii)are expressed as

1 = K = N = n(O)/ 11 sinhq + coshq, (2)

=i - 1 sinhq + n(0h)/ll coshq (A4)n()vKB coshq + n(0)\/Bl1 sinhq

When 11 =0

-yo = coshq. (3)

Appendix B: Chromatic Aberration Factor of the LDR

It is assumed that the object of radius R in contactwith the entrance face forms the erect virtual image ofradius y. Dependence of image radius y on wavelengthX is expressed as

y/R = cosh(x/BZ)]- 1 ,

(dy/dX) 1 (sinhq)q (dB/dX)R 2 (coshq) 2 B4

Then the value of (BF - BC)/BD is calculated from Eq.(B2):

BF-Bc (YF-Yc)BD fs-R

where subscripts F, C, and D refer to the respectivevalues based on F light, C light, and D light, and shiftfactor of imaging fs - (1/2) (sinhq) q/[1 + (sinhq)2].

As shown in Fig. 5, fs reaches its maximum value of0.289 at qmax = 1.463, where yo = 2.275. Consideringthe accuracy of measuring the image size 0.01 mm,I (BF - Bc)/BD I should be noted to be estimable onlyto a level of (fs R)-' X 10-2. Assuming that the spe-cific refraction and the specific volume of the copolymerare additive for (the corresponding specific quantity)X (weight fraction of the copolymer component), thetheoretical value of [(BF - BC)/BD] (B) is representedas Eq. (B3):

OB = (ND)2 |f( N (NF - Nc 1[(ND)2-(ND)1I, (B3)1(. N 12 1 D J J

where subscripts 1 and 2 refer to the CR-39 and M2homopolymers, respectively. Using the values of NDand Abbe's number for the respective homopolymer inTable I, OB values of the CR-39-PhMA and CR-39-VBsystems are 0.10 and 0.13, respectively.

References1. F. P. Kapron, J. Opt. Soc. Am. 60, 1433 (1970).2. E. G. Rawson, D. R. Herriott, and J. McKenna, Appl. Opt. 9,753

(1970).3. I. Kitano, K. Koizumi, H. Matsumura, T. Uchida, and M. Furu-

kawa, Suppl. J. Jpn. Soc. Appl. Phys. 39, 63 (1970).4. T. Miyazawa, K. Okada, T. Kubo, K. Nishizawa, I. Kitano, and

K. Iga, Appl. Opt. 19, 1113 (1980).5. H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).6. Y. Ohtsuka, Appl. Phys. Lett. 23, 247 (1973).7. K. Iga and N. Yamamoto, Appl. Opt. 16, 1305 (1977).8. Y. Ohtsuka, T. Sugano, and Y. Terao, Appl. Opt. 20, 2319

(1981).9. S. Patai, M. Bentor, and M. E. Reichmann, J. Am. Chem. Soc. 74,

845 (1952).10. S. R. Sandler, J. Chem. Eng. Data 18, 445 (1973).11. M. Kline, Analytical Chemistry of Polymers Part 3, Identifica-

tion Procedures and Chemical Analysis (Wiley-Interscience,New York, 1962), p. 46.

12. J. Brandrup and E. H. Immergut, Polymer Handbook (Wiley-Interscience, New York, 1975), Chap. 4, p. 339.

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0.3

0.25

0.2U,

4-4

0.15

0.1

(Bi)

(B2)