Ion exchange in glasses is one method for producing optical waveguides and gradient index lenses. Relatively high concentrations of dopant ions are required to produce a gradient index lens and the dopant presence alters the diffusivity in the substrate. Derive an expression for the steady-state concentration profile within a planar substrate assuming the diffusivity varies linearly with dopant concentration and the glass substrate acts as a stagnant medium. Refer to Figure P4.34. How many total moles of ions are stored in the waveguide?

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FIGURE P4.34 y=0 y=L = X ao Dah = D(1+ax) ab Xa=0