The propose geometry of the presented problem of plane wave scattering from nihility sphere coated with DPS, DNG, MNG or ENG material is depicted in Fig. 1. Where *b* and *a* represents the radius of sphere with coating and without coating respectively. The outer medium i.e. *ρ* ≥ *b* is free space having wave number \( {k}_0=\omega \sqrt{\epsilon_0{\mu}_0} \) is represented by region 0. The coating medium *a* < *ρ* < *b* with wave number \( {k}_1=\omega \sqrt{\epsilon_1{\mu}_1} \) is termed as region 1. Region *ρ* < *a* with wavenumber \( {k}_2=\omega \sqrt{\epsilon_2{\mu}_2} \) is represented by region 2.

In spherical coordinate system (*r*, *θ*, *φ*) the spherical wave vector functions are given as.

$$ {\boldsymbol{M}}_{\sigma mn\gamma}^{(l)}=\nabla \times \left[\boldsymbol{r}{Y}_{\sigma mn}\left(\theta, \varphi \right){R}_n^l\left({k}_{\gamma} r\right)\right] $$

(3)

$$ {\boldsymbol{N}}_{\sigma mn\gamma}^{(l)} = \frac{1}{k_{\gamma}}\left[\nabla \times {\boldsymbol{M}}_{\sigma mn\gamma}^{(l)}\right] $$

(4)

Where *Y*
_{
σmn
}(*θ*, *φ*) represents spherical harmonic, the most significant property of spherical harmonics is its parity. Here *σ* represents the parity of spherical harmonics, which is even when *σ* = *e* and odd when *σ* = *o*. The radial function \( {R}_n^l\left({k}_{\gamma} r\right) \) transform to the spherical Bessel *J*
_{
n
}(*k*
_{
γ
}
*r*), Spherical Neumann function *n*
_{
n
}(*k*
_{
γ
}
*r*) and spherical Hankel function *h*
_{
n
}(*k*
_{
γ
}
*r*) corresponding to *l* = 1, 2, 3 respectively. Where subscript *γ* represents the appropriate wave number i.e., *γ* = 0, 1, 2 Represents the region 0 (free space) wave number and region 1 wave number *k*
_{1} and so on.

A Plane wave traveling in + *z* direction with its electric field polarized in the positive *x* direction is incident on metamaterial coated nihility sphere. The incident electromagnetic field in terms of spherical vector wave functions is given as.

$$ {E}_{\mathrm{inc}}(r)={E}_0{\displaystyle {\sum}_{n=1}^{\infty }{i}^n}\frac{\left(2 n+1\right)}{n\left( n+1\right)}\left({\boldsymbol{M}}_{\mathrm{oln}}^{(1)}- i{\boldsymbol{N}}_{\mathrm{eln}}^{(1)}\right) $$

(5)

$$ {H}_{\mathrm{inc}}(r)=-\frac{k_0}{\omega {\mu}_0}{E}_0{\displaystyle {\sum}_{n=1}^{\infty }{i}^n}\frac{\left(2 n+1\right)}{n\left( n+1\right)}\left({\boldsymbol{M}}_{\mathrm{eln}}^{(1)}+ i{\boldsymbol{N}}_{\mathrm{oln}}^{(1)}\right) $$

(6)

Where *k*
_{0} and *μ*
_{0} is the wavenumber and permeability of free space.

The scattered field can be written as

$$ {E}_{\mathrm{sc}}(r)={E}_0{\displaystyle {\sum}_{n=1}^{\infty }{i}^n}\frac{\left(2 n+1\right)}{n\left( n+1\right)}\left( i{a}_n{\boldsymbol{N}}_{\mathrm{eln}}^{(3)}-{b}_n{\boldsymbol{M}}_{\mathrm{oln}}^{(3)}\right) $$

(7)

$$ {H}_{\mathrm{sc}}(r)=\frac{k_0}{{\upomega \upmu}_0}{E}_0{\displaystyle {\sum}_{n=1}^{\infty }{i}^n}\frac{\left(2 n+1\right)}{n\left( n+1\right)}\left( i{b}_n{\boldsymbol{N}}_{\mathrm{oln}}^{(3)}+{a}_n{\boldsymbol{M}}_{\mathrm{eln}}^{(3)}\right) $$

(8)

Electromagnetic field that transmitted in region 1 can be represented as

$$ {E}_I(r)={E}_0{\displaystyle {\sum}_{n=1}^{\infty }{i}^n}\frac{\left(2 n+1\right)}{n\left( n+1\right)}\left({c}_n{\boldsymbol{M}}_{\mathrm{oln}}^{(1)}+{d}_n{\boldsymbol{M}}_{\mathrm{oln}}^{(2)}- i{e}_n{\boldsymbol{N}}_{\mathrm{eln}}^{(1)}- i{f}_n{\boldsymbol{N}}_{\mathrm{eln}}^{(2)}\right) $$

(9)

$$ {H}_I(r)=\frac{-{\mathrm{k}}_1}{{\upomega \upmu}_1}{E}_0{\displaystyle {\sum}_{n=1}^{\infty }{i}^n}\frac{\left(2 n+1\right)}{n\left( n+1\right)}\left({e}_n{\boldsymbol{M}}_{\mathrm{eln}}^{(1)}+{f}_n{\boldsymbol{M}}_{\mathrm{eln}}^{(2)}- i{c}_n{\boldsymbol{N}}_{\mathrm{oln}}^{(1)}- i{d}_n{\boldsymbol{N}}_{\mathrm{oln}}^{(2)}\right) $$

(10)

Here k_{1} represents the wavenumber in region 1 having μ_{1} its permeability. Electromagnetic Field present in region 2 is

$$ {E}_{II}(r)={E}_0{\displaystyle {\sum}_{n=1}^{\infty }{i}^n\frac{\left(2 n+1\right)}{n\left( n+1\right)}\left[{g}_n{\boldsymbol{M}}_{\mathrm{oln}}^{(1)}- i{h}_n{\boldsymbol{N}}_{\mathrm{eln}}^{(1)}\right]} $$

(11)

$$ {H}_{II}(r)=\frac{-{\mathrm{k}}_2}{{\upomega \upmu}_2}{E}_0{\displaystyle {\sum}_{n=1}^{\infty }{i}^n}\frac{\left(2 n+1\right)}{n\left( n+1\right)}\left[{l}_n{\boldsymbol{M}}_{\mathrm{eln}}^{(1)}+ i{l}_n{\boldsymbol{N}}_{\mathrm{oln}}^{(1)}\right] $$

(12)

The analytically solved boundary conditions at both interface are listed below

$$ \left.\begin{array}{l}{E}_{\theta}^{inc}\kern0.5em +{E}_{\theta}^{scat}\kern0.5em =\kern0.5em {E}_{\theta}^I\kern13em \rho \kern0.5em =\kern0.5em b\\ {}{H}_{\theta}^{inc}\kern0.5em +\kern0.5em {H}_{\theta}^{scat}\kern0.5em =\kern0.5em {H}_{\theta}^I\kern12.5em \rho \kern0.5em =\kern0.5em b\\ {}\\ {}{E}_{\theta}^I\kern0.5em =\kern0.5em {E}_{\theta}^{I I}\kern18em \rho =\kern0.5em a\\ {}{H}_{\theta}^I\kern0.5em =\kern0.5em {H}_{\theta}^{I I}\kern18em \rho =\kern0.5em a\end{array}\right\} $$

(13)

By using field Eqs. 5, 6, 7, 8, 9, 10, 11 and 12 in the Eq. 13 set of eight equations are obtained in terms of scattering coefficients.

$$ {h}_n\left({r}_0\right){b}_n+{j}_n\left({r}_1\right){c}_n+{n}_n\left({r}_1\right){d}_n={j}_n\left({r}_0\right) $$

(14)

$$ {r}_1{\left[{r}_0{h}_n\left({r}_0\right)\right]}^{\prime }{a}_n+{r}_0{\left[{r}_1{j}_n\left({r}_1\right)\right]}^{\prime }{e}_n+{r}_0{\left[{r}_1{n}_n\left({r}_1\right)\right]}^{\prime }{f}_n={r}_1{\left[{r}_0\ {j}_n\left({r}_0\right)\right]}^{\prime } $$

(15)

$$ {\upeta}_0^{-1}{h}_n\left({r}_0\right){a}_n+{\upeta}_1^{-1}{j}_n\left({r}_1\right){e}_n+{\upeta}_1^{-1}{n}_n\left({r}_1\right){f}_n={\upeta}_0^{-1}{j}_n\left({r}_0\right) $$

(16)

$$ {\upeta}_0^{-1}{r}_1{\left[{r}_0{h}_n\left({r}_0\right)\right]}^{\prime }{b}_n-{\upeta}_1^{-1}{r}_0{\left[{r}_1{j}_n\left({r}_1\right)\right]}^{\prime }{c}_n-{\upeta}_1^{-1}{r}_0{\left[{r}_1{n}_n\left({r}_1\right)\right]}^{\prime }{d}_n={\upeta}_0^{-1}{r}_1{\left[{r}_0\ {j}_n\left({r}_0\right)\right]}^{\prime } $$

(17)

$$ {j}_n\left({r}_2\right){c}_n+{n}_n\left({r}_2\right){d}_n-{j}_n\left({r}_3\right){g}_n=0 $$

(18)

$$ {r}_3{\left[{r}_2{j}_n\left({r}_2\right)\right]}^{\prime }{e}_n+{r}_3{\left[{r}_2{n}_n\left({r}_2\right)\right]}^{\prime }{f}_n-{r}_2{\left[{r}_3{j}_n\left({r}_3\right)\right]}^{\prime }{h}_n=0 $$

(19)

$$ {\upeta}_1^{-1}{j}_n\left({r}_2\right){e}_n+{\upeta}_1^{-1}{n}_n\left({r}_2\right){f}_n-{\upeta}_2^{-1}{j}_n\left({r}_3\right){h}_n=0 $$

(20)

$$ {\upeta}_1^{-1}{r}_3{\left[{r}_2{j}_n\left({r}_2\right)\right]}^{\prime }{c}_n+{\upeta}_1^{-1}{r}_3{\left[{r}_2{n}_n\left({r}_2\right)\right]}^{\prime }{d}_n-{\upeta}_2^{-1}{r}_2{\left[{r}_3{j}_n\left({r}_3\right)\right]}^{\prime }{g}_n=0 $$

(21)

Where the impedance is defined as \( {\eta}_n=\sqrt{\frac{\mu_n}{\epsilon_n}} \). These equations are solved for scattering coefficients *a*
_{
n
} and *b*
_{
n
} which are then used to solve the scattering efficiency, Forward-scattering efficiency, Back-scattering and Extinction efficiency as given below where *k*
_{0}
*b* = *r*
_{0} , *k*
_{1}
*b* = *r*
_{1}, *k*
_{1}
*a* = *r*
_{2} , *k*
_{2}
*a* = *r*
_{3}

The energy flow (Poynting vector) was obtained by implementing the scattered field solution and can be written as [23]

$$ {S}_i=\frac{1}{2} R e\left({E}_i\times {H}_i^{\ast}\right) $$

$$ {S}_s=\frac{1}{2} R e\left({E}_s\times {H}_s^{\ast}\right) $$

$$ {S}_{ext}=\frac{1}{2} R e\left({E}_i{H}_s^{\ast }-{E}_s{H}_i^{\ast}\right) $$

Where *S*
_{
i
} and *S*
_{
s
} are the corresponding Poynting vector associated with incident and scattered field respectively, while *S*
_{
ext
} represents the Poynting vector induced due to interaction between incident and scattered electromagnetic waves. By following standard Mie theory, if we integrate the above equation over a large sphere, various scattering efficiencies can be obtained [24]. This yields the extinction efficiency and scattering efficiencies (forward scattered and backscattered) [23, 24] as presented in Eqs. 22, 23 and 24.

$$ {Q}_{ext}=\frac{2}{{r_0}^2}{\displaystyle \sum_{n=1}^{\infty }}\left(2 n+1\right)\mathrm{\Re}\left({a}_n+{b}_n\right) $$

(22)

$$ {Q}_{forward}=\frac{1}{{r_0}^2}{\left|{\displaystyle \sum_{n=1}^{\infty }}\left(2 n+1\right){\left({a}_n+{b}_n\right)}^2\right|}^2 $$

(23)

$$ {Q}_{b ackward}=\frac{1}{{r_0}^2}{\left|{\displaystyle \sum_{n=1}^{\infty }}{\left(-1\right)}^n\left(2 n+1\right){\left({b}_n-{a}_n\right)}^2\right|}^2 $$

(24)

By using values of scattering coefficients *a*
_{
n
} and *b*
_{
n
} in above equation nihility condition is applied i.e., *ϵ*
_{2} = 0 and *μ*
_{2} = 0, and results are obtained.