The sequence of two layers have been placed above the perfect electric conductor (PEC) sheet for designing of electromagnetic wave absorber. The value of reflection coefficient at first layer is of great importance. The lesser the value of reflection coefficient at air-absorber interface, better will be the absorption performance [14]. The amplitudes of incident, reflected and transmitted powers through absorber structures have been calculated and various reflection coefficient graphs have been plotted at both normal and oblique incident angles to verify absorber performance.

The electromagnetic wave model of proposed double-layer absorber structure has been discussed for transverse electric (TE) mode here. In the model, the air-absorber interfacing layer and matched-lossy layer are assumed to be extremely thin resistive sheets of negligible thickness of conductance G1 and G2 respectively as shown in Fig. 1. The general equation of electric field intensity \(\overrightarrow{E}\) and magnetic field intensity \(\overrightarrow{H}\) for any arbitrary layer, *x*, having admittance Y can be expressed as follows [15]:

$$\overrightarrow{E}= {P}_{i}{e}^{-jk(x\text{cos}{\theta }-\text{y}\text{sin}{\theta })}+ {P}_{r}{e}^{jk(x\text{cos}{\theta }+\text{y}\text{sin}{\theta })}$$

1

…

$$\overrightarrow{H}= Y\left\{{P}_{i}{e}^{-jk(x\text{cos}{\theta }-\text{y}\text{sin}{\theta })}+ {P}_{r}{e}^{jk(x\text{cos}{\theta }+\text{y}\text{sin}{\theta })}\right\}$$

2

…

Where Pi and Pr are the amplitudes of the incident and reflected propagated waves respectively.

The boundary conditions on the tangential \(\overrightarrow{E}\) fields which have to be satisfied at matching-absorbing layer interface (G1) are as follows:

*G* *1* \(\overrightarrow{E}\) *+* *= G**1*\(\overrightarrow{E}\)*−* *=*\(\overrightarrow{J}\)

\(\overrightarrow{H}\) *+* *=* \(\overrightarrow{H}\)*−* *=* \(\overrightarrow{J}\)

*Also K* *p* *sin θ**p* *= K**q* *sin θ**q*

Here + and – signs show the electric fields at opposite sides on matched lossy layer interface and \(\overrightarrow{J}\) is the current density in sheet. Further in Fig. 1, let p and q are the matching layer and lossy layer respectively.

Further electric field for p and q layer can be expressed as:

$${\overrightarrow{E}}_{p}= {P}_{ip}{e}^{-jkp({x}_{p}\text{cos}{{\theta }}_{ip}+\text{y}\text{sin}{{\theta }}_{ip})}+ {P}_{rp}{e}^{-jkp({x}_{p}\text{cos}{{\theta }}_{rp}+\text{y}\text{sin}{{\theta }}_{rp})}$$

3

…

$${\overrightarrow{E}}_{q}= {P}_{iq}{e}^{-jkq({x}_{q}\text{cos}{{\theta }}_{iq}+\text{y}\text{sin}{{\theta }}_{iq})}+ {P}_{rq}{e}^{-jkq({x}_{q}\text{cos}{{\theta }}_{rq}+\text{y}\text{sin}{{\theta }}_{rq})}$$

4

…

Starting from perfect electric conductor (PEC) layer at plane x = 0 where Pi =1 and Pr = -1 passing through lossy absorbing layer and air-impedance matched layer with linear boundary transformation [15]

$${P}_{i}= \frac{{Z}_{o}{e}^{j{k}_{0}{x}_{p}cos{\theta }_{o}}}{2cos{\theta }_{0}}\left[{P}_{ip}\left(\frac{{cos\theta }_{o}}{{Z}_{o}}+\frac{{cos\theta }_{ip}}{{Z}_{p}}+{G}_{2}\right){e}^{-jkp{x}_{p}\text{cos}{{\theta }}_{ip}}+ {P}_{rp}\left(\frac{{cos\theta }_{o}}{{Z}_{o}}-\frac{{cos\theta }_{rp}}{{Z}_{p}}+{ G}_{2}\right){e}^{jkp{x}_{p}\text{cos}{{\theta }}_{rp}}\right]$$

… (5)

$${P}_{r}= \frac{{Z}_{o}{e}^{-j{k}_{0}{x}_{p}cos{\theta }_{o}}}{2cos{\theta }_{0}}\left[{P}_{ip}\left(\frac{{cos\theta }_{o}}{{Z}_{o}}-\frac{{cos\theta }_{ip}}{{Z}_{p}}-{G}_{2}\right){e}^{-jkp{x}_{p}\text{cos}{{\theta }}_{ip}}+ {P}_{rp}\left(\frac{{cos\theta }_{o}}{{Z}_{o}}+\frac{{cos\theta }_{rp}}{{Z}_{p}}-{G}_{2}\right){e}^{jkp{x}_{p}\text{cos}{{\theta }}_{rp}}\right]$$

… (6)

Where xp is the distance from the PEC layer to air-absorber interface and xq is the distance from PEC layer to matched-lossy interface.

\({k}_{0}= {\omega }_{0}\sqrt{{\epsilon }_{0}{\mu }_{0}}\) ; \({k}_{p}= {k}_{0}\sqrt{{\epsilon }_{rp}{\mu }_{rp}}\) and \({k}_{q}= {k}_{0}\sqrt{{\epsilon }_{rq}{\mu }_{rq}}\) are the wave numbers of free space, layer p and layer q respectively.

$${Z}_{0}= \frac{1}{{Y}_{0}}=377\varOmega$$

;

$${Z}_{p}= \frac{1}{{Y}_{p}}= {Z}_{0}\sqrt{\frac{{\mu }_{p}}{{\epsilon }_{p}}}$$

7

;…

$${Z}_{q}= \frac{1}{{Y}_{q}}= {Z}_{0}\sqrt{\frac{{\mu }_{q}}{{\epsilon }_{q}}}$$

8

…

where \({Z}_{0}, {Z}_{p}, {Z}_{q}\)are the impedance of free space, p layer and q layer respectively.

Reflection Coefficient of air-absorber interface is:

$$\left|\varGamma \right|= -20{{log}}_{10}\left|\frac{{P}_{r}}{{P}_{i}}\right|$$

9

…

In the current work, both layers (matching layer and lossy layer) are designed from epoxy resin loaded with reduced graphene particles but with different weight ratio. According to transmission line theory, the impedance of first layer i.e. matching layer must be approximately equal to free space impedance (377 ohm) [15]. This condition ensures the minimum reflection and maximum transmission of incident wave at air-absorber interface. To satisfy above, relative permittivity of the material should be equal to its relative permeability as seen from Eq. 7. Therefore a matching layer has been designed above absorbing layer, in which graphene oxide particles are blended in epoxy material with 5% weight ratio. This layer has 9.5 relative permittivity and 0.15 dielectric loss tangent particularly at 10 GHz frequency. Further the lossy/absorbing layer (next to matching layer) must possess high losses so as to absorb electromagnetic waves in terms of heat. For designing lossy layer, nanocomposites of graphene oxide particles blended in epoxy resin material with 15% weight ratio and 8 micrometer particle size has been proposed. This combination increases the relative permittivity of materials upto 18.5 and the dielectric loss tangent increased to 0.18 induces high losses to encountered electromagnetic waves [16]. Thus, most of the waves reaching to lossy layer dissipate and rest of the electromagnetic waves which manage to pass through lossy layer get fully reflected by perfect electric conductor layer which is placed at the back-end of absorber structure to ensure zero transmission.

The thickness of both the layers has been calculated on basis of quarter wavelength principle and it should be the odd multiple of quarter wavelength incident wave [17–18]. Therefore, for operating proposed electromagnetic absorber structure in X band, thickness of matching layer should be 2 mm and lossy layer is 0.5 mm backed by a perfect electric conductor sheet of negligible thickness.

To verify whether proposed absorber is insensitive to polarization, it has been illuminated with transverse electric (TE) and transverse magnetic (TM) polarization modes at different incident angles as depicted in Fig. 2 and Fig. 3.