Print version ISSN 0103-9733
Braz. J. Phys. vol.31 no.1 São Paulo Mar. 2001
Soda-lime glass with gradient of refraction index (GRIN)
J. R. B. Paião, F. Pereira, S. Watanabe
Instituto de Física, Universidade de São Paulo,
CP 66318, CEP 05315-970, São Paulo, SP, Brazil
Received on 18 August, 2000
In a soda-lime glass provided by Companhia Vidraria Santa Marina, subsidiary of French Saint-Gobain, we induced a gradient of refraction of index (GRIN) by exchange of Na+ in glass by Li+ at 550oC for 12, 24, 36 and 48 hours and varying bath temperature at 525oC, 550oC, 575oC and 600oC for fixed time of 48 hours. Dn = 0.0107 ± 0.0005 and GRIN depth of about 2.4mm were obtained for the ion exchange at 550oC for 48 hours. The GRIN profile was fitted with er fc(x) function and then obtained the diffusion coefficient D = (1.6 ± 0.3) · 10-6 mm2/s.
The so called optical glasses have been traditionally homogeneous. During last thirty years, glasses having index of refraction varying with spatial coordinates have been introduced. Ion exchange in glasses was discussed by Garfinkel  and Doremus  in 1968 and 1969, respectively. Problems concerning optics of GRIN glasses have been treated by Sands [3, 4], Moore and Sands , and Kapron  in the early 70's. Several techniques have been used for manufacturing GRIN glasses such as neutron irradiation, chemical vapor deposition, polymerization techniques, ion exchange, ion stuffing and sol-gel process. CVD technique has been used to produce GRIN fibers for telecommunications . Ion exchange is probably the most widely used technique due to its relatively simple one compared to other processes. Ion stuffing consists in using a special glass that when heated, its phases separate . One of these phases is dissolved in an acid leaving a porous glass, then it is stuffed with ions or molecules in such a way that a gradient composition is produced. Sol gel process can be used in fabricating GRIN rods with large geometry . The gradient in the refractive index can be produced in the radial, axial or spherically symmetrical direction. Such GRIN elements have been used in copiers, facsimiles, endoscopes, etc. Several kinds of base glasses have been investigated. For exchange, Li+, Na+ K+, Tl+, Ag+ ions are frequently used .
II Experiments and results
Samples of soda-lime glass fabricated by Comp. Vidraria Santa Marina in S. Paulo, was used in the present work. Its composition is: 72SiO2 - 9CaO -13Na2O -4MgO - Al2O3 with small amounts of Fe2O3, TiO2 and K2O (these three compounds totaling about 1% weight are also found). The refractive index of this glass was found to be no = 1.5213 ± 0.0005. The ion exchange method was used for inducing the gradient of refractive index. Initially, we used a bath of molten of LiCl for the exchange Li+ « Na+ of the glass. The melting point of LiCl is 600oC, and since for soda-lime glass Tg ~ 610oC, immersing the glass into LiCl melt the glass devitrified strongly. We then followed Kindred et al. experiments , in which to lower the bath temperature, CaCl2 was mixed. They used an eutetic mixture of 60CaCl2 - 40LiCl with melting temperature of 496oC. In the present work the samples were kept immersed in such a bath at 550oC and varying time of ion exchange, Fig. 1, and varying bath temperature for
|Figure 1. GRIN profiles for different times of ion exchange (550oC).|
fixed 48 hours exchange time, Fig. 2. After ion exchange process the samples were cooled at room temperature. The glass presented crystallization characterized by milky colour. A surface layer was removed to eliminate devitrified portion. X-ray diffractogram indicates that the remaining part of the glass is amorphous. The measurement of variation of the refractive index n with depth was carried out removing successive layers and measuring n each time.
|Figure 2. GRIN profiles for different temperatures for fixed time of exchange (48 h).|
III Ionic diffusion and GRIN
Fick's laws describe the ions diffusion in a solid. Denoting by C(x, t), the concentration of diffusing species at depth x and at instant t, by D(T) the diffusion constant at temperature T, the solution of Fick's laws for D(T) constant at T is given by:
The refractive index of a GRIN glass is then dependent on C(x, t) :
Fig. 3 presents the GRIN profile for 48 hours and 550oC with depth of about 1.95 ± 0.05 mm.
The full curve in Fig. 3 is the best fit of experimental data, using the equation 3. The refractive index at the surface n0 = 1.5210 ± 0.0005, the maximum variation of n, Dn = 0.0107 ± 0.0005 and the diffusion coefficient D = (1.6 ± 0.3)·10-6mm2/s were obtained from this best fit.
|Figure 3. Axial GRIN profile produced by Li+ for Na+ exchange in a soda-lime glass for 48 hours and 550Co.|
IV Ray tracing
In Fig. 4, a laser beam is incident on a slab of GRIN glass with thickness L. a being the angle of incidence of laser beam and qo the angle of refraction at the entrance surface, after reaching GRIN region, at any point x, according to Snell's law:
|Figure 4. Laser beam incident on a slab of GRIN glass.|
where x is measured from beginning of GRIN and perpendicular to slab surface. The ray emerges at an angle a with respect to the normal to glass surface. Since in an homogeneous glass with refractive index n0, the ray entering under angle a emerges also forming an angle a, the effect of GRIN is to produce a displacement Dg of emergent ray compared to emergent ray from a glass without GRIN. A numerical calculation using the equation 4 produced for GRIN glass obtained here, a value of about Dg ~ 0.00202mm for a ~ 57o. Experimentally, this is a value too small to be measured due to the laser beam diameter. The table 1 lists values of Dg for a between 30 and 80o. The Fig. 5 shows the variation of Dg as function of a.
|Table 1. Dg as function of the incidence angle a.|
|Figure 5. Variation of Dg as function of a.|
V Discussion and Conclusion
According to Kindred , in many cases the refraction index is linearly dependent on the concentration. This is the present case. With the exchange Li+ « Na+, Haun et al.  did not fit the the GRIN profile with erfc(x) as we did. Fig. 1 has shown that both the depth of exchange and the variation Dn of the index of refraction increase with the time of exchange, however, the result shown in Fig. 2 indicated that for fixed time of 48 hours, the depth of exchange and Dn did not change with temperature in the range 525oC and 600oC. This last result is due to the increase in the thickness of crystallized layer. The maximum variation Dn and largest depth of exchange were obtained for time between 48 and 60 hours at 550oC, as well as for 48 hours at temperature between 550oC and 575oC. For 48 hours at 550oC, Dn = 0.0107 ± 0.0005 and a depth of 1.95 ± 0.05 mm were obtained. At this temperature a coefficient of diffusion of Li+ with value D = (1.6 ± 0.3)·10-6mm2/s was obtained. Meyer-Arendt  suggested the use of GRIN glass to eliminate spherical aberration. Grinding and polishing the GRIN glass region a preliminary result showed a lens without spherical aberration.
The authors acknowledge CNPq and FAPESP for financial support. Thanks are due to Eng. Marcos Gibin, Head, Centro Técnico de Elaboração do Vidro, Companhia Santa Marina, for providing us with the samples used in this paper.
 M. Garfinkel, J. Phys. Chem. 72, 4175 (1968). [ Links ]
 R. H. Doremus and J. Marinsky, in Ion Exchange in Glasses (Marcel Dekker, New York, 1969) p. 5. [ Links ]
 J. P. Sands, J. Opt. Soc. Am. 60, 1436 (1970). [ Links ]
 J. P. Sands, J. Opt. Soc. Am. 61, 1086 (1971). [ Links ]
 D. T. Moore and J. P. Sands, J. Opt. Soc. Am. 61, 1191 (1971). [ Links ]
 F. P. Kapron, J. Opt. Soc. Am. 60, 1433 (1970). [ Links ]
 M. A. Pickering, R. L. Taylor and D. T. Moore, Appl. Opt. 25, 3364 (1986). [ Links ]
 R. K. Morh, J. A. Wilder, P. B. Macedo and P. K. Gupta, in Digest of Topical Meeting on Gradient Index Optical Imaging Systens (Opt. Soc. Am.), Washigton, D. C. (1979). [ Links ]
 M. Yamane and S. Noda, J. Ceram. Soc. Japan, 101, 11 (1993) [ Links ]
 D. S. Kindred, J. Bentley and D. T. Moore, Appl. Opt. 29, 4036 (1990). [ Links ]
 D. S. Kindred, Appl. Opt. 29, 4051 (1990). [ Links ]
 N. Haun, D. S. Kindred and D. T. Moore, Appl. Opt. 29, 4056 (1990). [ Links ]
 J. R. Meyer-Arendt, in Introduction to Classical and Modern Optics (Prentice Hall, New Jersey, 1995) p. 123. [ Links ]