In the above equation, there are two sets of key terms, 1) hyperbolic term 2) polynomial term. The hyperbolic term is expected to be the dominant term, as it can be seen from the overall phase shift seen across the interface (Fig ). In order to maintain the hyperbolic nature of this equation, we must apply the constraint n>3. As evident from the above analyses, going from equation (2) to equation ( ), the exponent of the hyperbolic term has decreased in denominator. We can also see this effect when comparing the dopant concentration in fig 2, which depreciates very quickly, as we increase z (dopant gradient ~ 8nm), and the holographic phase shift in fig , which depreciates much more slowly with increased z (holographic phase shift gradient ~ 80 nm). Based on the above approximate values of the gradients, we can write the expression equating the equal decay of dopant concentration and holographic phase shift, with increasing z: