Transfer Matrix Method

Transfer Matrix Method

The transfer matrix method is a widely used method for calculations light-matter interaction in a stack of parallel dielectric interfaces. The method treats the Fresnel reflection and transmission at the interface of two media as a matrix, and the propagation of the light in a particular medium as another matrix. This allows for the multiplication of a stack of matrices to obtain the transmission and reflection by a stack of dielectric layers. Since the mathematical operation is a multiplication of matrices, the method lends itself easily to numerical analysis using programming languages such as Matlab, Python, or any of a large number of other programming languages. Below, we describe the principles of the transfer matrix method as implemented for the measurements described in the manuscript.

Figure 1 shows a schematic of a multi-layer dielectric structure with alternating high and low index layers. and are the amplitudes of the forward and backward propagating wave in medium m. m=0 is the medium where the light is incident, and the subscript s represents the substrate.

Figure 1. A schematic of the forward and backward propagating fields in a multi-layer dielectric structure.

For the above setup, the fields are related in the following manner:

(1)

where the matrix M describes the behavior of all the intervening layers and is:

(2)

where Q is the number of dielectric layers. For the specific implementation, Q=2N, where N is the number of pairs of layers of alternating high and low refractive index media.

And,

(3)

(4)

where is the normal projection of the wave-vector in medium m, is the thickness of layer m, and is the angle if light travelling in medium m. Note that all the information necessary to compute matrix M is given by the material parameters of the layers, assuming that the incident angle is known.

Assuming:

since there is no backward propagating wave in the medium into which the light exits, and setting the amplitude of the incident wave gives the final version of equation (1) as:

(5)

allowing one to solve for and as a simple linear equation once the matrix M is solved for.

For the purpose of the computations described in the manuscript, the incident angle is assumed to be normal, i.e. (and consequently all ); thus both the transverse electric (TE) and transverse magnetic (TM) polarizations are degenerate and give the same result.

Matlab Implementation

The above was implemented in Matlab as a function. The assumptions were:

For all layer where m = 2, 4, 6, . . . Q, the refractive index was set to nL=1.34, and the thickness was set to dL.

For all layers where m = 1, 3, 5, . . . Q-1 the refractive index was set to nH (a relatively high refractive index), and the thickness was set to dH.

The residual of the difference between the spectrum resulting from initial values of N, nH, dL, and dH and the measured spectrum was minimized by varying nH, dL, and dH using Matlab’s Least Squares curve fitting routine lsqcurvefit for a range of values of N from 2 to 11. Subsequently, the spectrum corresponding to the N with the smallest residual was considered to be the final fit, and the corresponding fitting parameters recorded. This was done because N can have only integer values, and the lsqcurvefit routine cannot handle variables with only integral values.