Buried Concentration Profiles for Optical Waveguides by One-Step Electro-Diffusion Process: Theoretical Analysis
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BURIED CONCENTRATION PROFILES FOR OPTICAL WAVEGUIDES BY ONE-STEP ELECTRO-DIFFUSION PROCESS: THEORETICAL ANALYSIS XIAOMING LI* AND PAUL F. JOHNSON**
R&D, Norton Company, 1 New Bond Street, Worcester, MA 01606 ** New York State College of Ceramics at Alfred University, Alfred, NY 14802 *
INTRODUCTION In recent years, a two-step electric field assisted diffusion or ion-exchange technique has been extensively studied for producing buried concentration profiles in glass [1,2,3], polymer [4,5], electrooptic -and semiconducting [6,7] substrate materials to fabricate buried optical waveguide devices. The technique contains two separate diffusion processes and is quite complicated, cost and time consuming. In addition, theoretical analysis of the technique is too complicated to be used for calculating and tailoring concentration profiles. Many efforts have been made to pursue a one-step process for producing buried profiles. Very recently, a novel one-step technique [8] involving electric field assisted diffusion of silver ions into glass from molten AgNO 3 bath with decaying silver concentration has been developed to produce buried Ag+ concentration profiles in glass substrate. The new technique is, from the practical operation point of view, relatively simple and is a significant improvement over the conventional two-step process. In this paper, we present the theoretical analysis of the new process using analytical solution on one dimensional linear field assisted diffusion equation with exponentially decaying boundary conditions. The solution theoretically proves for the first time that a buried cation concentration profile with nearly symmetric distribution can be achieved by a single electrodiffusion process. Potential applications of the theory in design and fabrication of buried optical waveguides are discussed. Effect of various processing parameters on the concentration distribution are theoretically analyzed. THEORETICAL SOLUTION Field-enhanced diffusion process can be analyzed by either linear [1,2] or non-linear [9] theories. For the purpose of simplification, we consider only linear analysis and treat electromigration of external cations into substrate as the diffusion of mobile ions in a semiinfinite medium with non-blocking electrodes under a dc field. Assuming that incoming ions, Ag+, and indigenous ions, Na+, have equal mobility, D and E are constants, the electric field assisted diffusion process could be described by linear diffusion equation
c D a2C D--
at
ax2
uE
ac ax
(1)
where C is the concentration of the diffusing ion, D is the interdiffusion constant, E is the applied electric field, andu is the ion mobility which is connected to D by the Einstein relation: .u = ezD =fkT
(2)
where e is the unit charge, z is the valence of the ions, k is the Boltzman constant, f is the correlation factor (0 < f < 1) and T is the absolute temperature in Kelvin. The diffusion process is defined by initial and boundary conditions C(x, 0) =0, C(oo, t) =0,
X>0 t>0
Mat. Res. Soc. Symp. Proc. Vol. 228. ©1992 M
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