The methods of piecewise constant and piecewise linear approximation of complicated inhomogeneous layers are investigated. The expressions for transmission matrices for linearly inhomogeneous and exponentially inhomogeneous layers are obtained. The accurate and approximate values of reflection coefficient against frequency for the exponential and linear matching layers are also obtained. Piecewise constant and piecewise linear approximation methods are compared according to the criteria of calculating time consumption and the accuracy of the result.

*The reflection coefficient of piecewise constant structure with 5 layers and the reflection coefficient of piecewise linear structure with 2 layers, that approximate the exponential inhomogeneous layer.*

*The accuracy of the reflection coefficients calculation.*

These methods allow us to obtain approximate values of the reflection coefficient of complex heterogeneous layers. The piecewise linear approximation method is more suitable in the case where the permittivity is a rapidly varying nonmonotonic function. If the number of layers is increasing, the error of the reflection coefficient for the piecewise linear approximation decreases faster than that for the piecewise constant approximation. With increasing frequency, the error of piecewise linear and piecewise constant approximation increases. An equal number of layers, the error of piecewise linear method is lower than the error of piecewise constant for all of the frequency range.

**Languages:** Matlab, Mathematica

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― Subrahmanyan Chandrasekhar
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“The black holes of nature are the most perfect macroscopic objects there are in the universe: the only elements in their construction are our concepts of space and time.”