Retrieval of Effective Parameters of Subwavelength Periodic Photonic Structures

Abstract

We revisit the standard Nicolson–Ross–Weir method of effective permittivity and permeability restoration of photonic structures for the case of subwavelength metal-dielectric multilayers. We show that the direct application of the standard method yields a false zero-epsilon point and an associated spurious permeability resonance. We show how this artifact can be worked around by the use of the cycle shift operator to the periodic multilayer in question.

1. Introduction

Determining the effective macroscopic parameters of a slab containing a photonic structure is often both highly relevant both from a practical and from a fundamental point of view, providing a link between the actual microscopic composition of the structure and its macroscopic optical response. The most common ways to accomplish this task are various types of ellipsometry [1,2] or reflectance and transmittance measurements [3].

However, there are two major issues with such parameter restoration techniques. One is the fundamental applicability limit of the effective medium approximation approach. It is normally mitigated by considering photonic structures with feature sizes much smaller than the wavelength (deeply subwavelength structures), although an exception exist in special cases, such as hyperbolic metamaterials [4]. The other issue is related to the artifacts of the restoration procedure itself. For example, it is known that the constitutive parameters of a slab are hard to obtain nearby the total transmission frequency or when we approach the magnetic or electric resonances of a structure [5]. These difficulties can be overcome with a modified extraction procedure, based upon a generalized homogenization technique [6].

In this work, we address another difficulty arising when extraction is to be carried out at the point of total reflection. It is shown that application of the widely used standard Nicolson–Ross–Weir procedure [7] results in spurious features near this point, such as false zero-epsilon points and permeability poles. In the optics of subwavelength structures, total reflection is often associated with the true ε-near-zero regime [8]. Therefore, being able to make a correction for artifacts in such a regime becomes of particular importance to determine the correct operational wavelengths of epsilon-near-zero behavior and the optical mechanism of subwavelength photonic structures.

The rest of the paper is organized as follows. In Section 2, we demonstrate the artifacts associated with the application of the standard restoration procedure to subwavelength metal-dielectric periodic photonic multilayers. In Section 3, we trace these artifacts to the total reflection from a photonic metamaterial slab at the point where artifacts are observed. In Section 4, we show how to amend the artifact behavior and ensure correct parameter restoration by applying the cycle shift operator to the multilayer in question. Finally, in Section 5, we summarize the results.

2. Spurious Features in the Nicolson–Ross–Weir Restoration Procedure

We consider periodic layered structures formed by alternating layers of metal and dielectric. It was shown previously that the nonlocal permittivity tensor of such plasmonic multilayers generally possesses off-diagonal components [9,10], which only vanish when light propagates normal to layer interfaces. Furthermore, in this case, no more than one eigenmode corresponds to each frequency. Hence, we will consider the normal incidence of light throughout the paper and will aim at the restoration of the longitudinal component of the permittivity tensor from the reflection r and transmission t coefficients.

We apply the standard Nicolson–Ross–Weir extraction method [7] to a finite multilayer slab of thickness D and calculate its effective permittivity ε and permeability μ as follows. First, the impedance Z and the refraction index n are given by:

$$Z=\pm \sqrt{\frac{{(1+r)}^2-t^2}{{(1-r)}^2-t^2}},n=\lambda \frac{\pm \arccos \left(\frac{1}{2t}(1-r^2+t^2)\right)+2\pi m}{2\pi D}$$

(1)

where for the given wavelength λ, one should choose the signs and the free parameter m in a proper manner (see [11,12]). Then, since $Z=\sqrt{\mu /\epsilon }$, $n=\sqrt{\epsilon \mu }$, the effective permittivity and permeability can be determined as ε = n/Z, μ = Zn.

It can be seen that there is a singular point in Equation (1) where $r=\left|r\right|e^{i{\varphi }_r}=1$. Therefore:

$$Z=\pm \infty ,n=\pm \frac{\pi }{k_{0}D}\left(\frac{1}{2}+2m\right)$$

(2)

If we suppose that R = 1 and T = 0, then Equation (1) results in Z = ±∞. Since $Z=\pm \sqrt{\mu /\epsilon }$, it follows that infinite Z implies either that μ must approach infinity or that ε must approach zero. On the other hand, the refractive index $n=\sqrt{\epsilon \mu }$ remains finite as per Equation (1); therefore, we see both μ → ∞ and ε → 0 at the same time.

Therefore, at first glance, there should always be an epsilon-near-zero, mu-near-pole response for the case of total reflection. Figure 1 shows an example of such a situation, where we observe a spurious permeability resonance. We also see a poor match between the correct effective parameters ε_eff, μ_eff and extracted ε.

Because of this apparent extraction artifact, which can be easily misinterpreted as optical magnetism in plasmonic multilayers, special care should be taken in applying the simple extraction procedure described above. In what follows, we address the subject in detail and show how the artifacts can be worked around using a cycle shift geometrical transformation on the multilayer structure.

3. Detailed Analysis of the Total-Reflection Point in Multilayers

We can characterize an arbitrary multilayer by the 2 × 2 transfer matrix M = m_jk that connects the tangential components of propagating electric and magnetic fields. Let us place one unit cell of multilayer having thickness L = l_m + l_d (l_d,m are the thicknesses of the dielectric and metal layers) between input and output host media having impedances of ${\xi }_{i,o}={\epsilon }_{i,o}^{-1/2}$. Knowing M and assuming normal incidence, the reflection (H₋) and incident (H₊) magnetic fields in the case of normal incidence [13]:

$$r=-\frac{H_{-}}{H_{+}}=-\frac{(m_{11}+m_{12}{\xi }_o){\xi }_i-(m_{21}+m_{22}{\xi }_o)}{(m_{11}+m_{12}{\xi }_o){\xi }_i+(m_{21}+m_{22}{\xi }_o)}$$

(3)

Eigenmodes $e^{i{k}_{x}L}$ of M provide us with the Bloch wave numbers $k_x={k^{\prime }}_x+i{k^{″}}_x$ of the structure. Based on the knowledge of M’s eigenmodes, we can calculate the transfer matrix for a slab composed of N unit cells (with the total thickness D = NL) simply by raising M to the N-th power. Taking into account that M is a unimodular matrix, we have [14]:

$$M^N=\left(\begin{array}{cc}{m}_{11}{U}_{N-1}-U_{N-2}& m_{12}{U}_{N-1}\\ m_{21}{U}_{N-1}& m_{22}{U}_{N-1}-U_{N-2}\end{array}\right)$$

(4)

where

$$U_N=\frac{\sin [(N+1)k_{x}L]}{\sin (k_{x}L)}$$

(5)

are Chebyshev’s polynomials. Now, it is possible to write down the reflection coefficient for the slab as follows:

$$r_N=-\frac{H_{-}-\frac{U_{N-2}}{U_{N-1}}({\xi }_i-{\xi }_o)}{H_{+}-\frac{U_{N-2}}{U_{N-1}}({\xi }_i+{\xi }_o)}$$

(6)

Now, we are ready to write down the reflection coefficient for a semi-infinite multilayer. It can be noticed that for complex k_x:

$$\frac{U_{N-2}}{U_{N-1}}=\frac{\sin [(N-1)k_{x}L]}{\sin (N{k}_{x}L)}=\frac{e^{iN{k}_{x}L}{e}^{-i{k}_{x}L}-e^{-iN{k}_{x}L}{e}^{i{k}_{x}L}}{e^{iN{k}_{x}L}-e^{-iN{k}_{x}L}}$$

(7)

If we put N → ∞, then $e^{iN{k}_{x}L}=e^{iN{k^{\prime }}_{x}L}{e}^{-N{k^{″}}_{x}L}=0$ and $U_{N-2}/U_{N-1}=e^{i{k}_{x}L}$. Finally:

$$r_{\infty }=-\frac{H_{-}-e^{i{k}_{x}L}({\xi }_i-{\xi }_o)}{H_{+}-e^{i{k}_{x}L}({\xi }_i+{\xi }_o)}$$

(8)

In the case of a semi-infinite structure, we are not interested in the output interface, and it can be assumed that ξ_i = ξ_o = ξ_h. Then,

$$r_{\infty }=-\frac{H_{-}}{H_{+}-2{\xi }_h{e}^{i{k}_{x}L}}$$

(9)

We see that the total reflection occurs when:

$$e^{i{k}_{x}L}=m_{11}+m_{12}{\xi }_h$$

(10)

If the host medium is free space, then:

$$e^{i{k}_{x}L}=m_{11}+m_{12}$$

(11)

i.e., m₁₁ + m₁₂ being the eigenvalue of M means that r_∞ = 1.

Here, we are concerned with multilayers formed by periodically alternating layers of two types, one made of a dielectric and the other made of metal. For this case, M is of the following form:

$$M=\left(\begin{array}{cc}{\kappa }_d{\kappa }_m-\sqrt{\frac{{\epsilon }_d}{{\epsilon }_m}}{\sigma }_d{\sigma }_m& -i\sqrt{{\epsilon }_d}{\sigma }_d{\kappa }_m-i\sqrt{{\epsilon }_m}{\kappa }_d{\sigma }_m\\ -\frac{i}{\sqrt{{\epsilon }_d}}{\sigma }_d{\kappa }_m-\frac{i}{\sqrt{{\epsilon }_m}}{\kappa }_d{\sigma }_m& {\kappa }_d{\kappa }_m-\sqrt{\frac{{\epsilon }_m}{{\epsilon }_d}}{\sigma }_d{\sigma }_m\end{array}\right)$$

(12)

where κ_d,m = cos(β_d,ml_d,m), σ_m = sin(β_d,ml_d,m), and ${\beta }_{d,m}=\sqrt{{\epsilon }_{d,m}}(\omega /c)$ is the wave number in the respective layers. As is the usual meaning of metal that ε_m < 0, β_m is purely imaginary:

$${\beta }_m=\sqrt{{\epsilon }_m}\frac{\omega }{c}=i{{\beta }^{″}}_m\in iℝ$$

(13)

resulting in:

$$\begin{array}{r}{\kappa }_m=\cos (i{{\beta }^{″}}_m{l}_m)=\cosh ({{\beta }^{″}}_m{l}_m)\in ℝ\\ {\sigma }_m=\sin (i{{\beta }^{″}}_m{l}_m)=i\sinh ({{\beta }^{″}}_m{l}_m)\in iℝ\end{array}$$

(14)

Based on Equations (13) and (14), we see that m₁₁ is always real, while m₁₂ is imaginary for a wave propagating across the layers. Separating the real and imaginary parts in Equation (11), we immediately obtain that. As a result,

$$e^{-{k^{″}}_{x}L}\cos ({k^{\prime }}_{x}L)={\kappa }_d{\kappa }_m-\sqrt{\frac{{\epsilon }_d}{{\epsilon }_m}}{\sigma }_d{\sigma }_m$$

(15)

defines the frequency where we observe total reflection.

We note that since |r|² + |t|² = 1 in the case of total reflection, there is no transmission through the structure, regardless of whether the structure is infinite or finite. Consequently, we will obtain false material parameters for a finite structure at this point, as can be seen from Figure 1.

4. Obtaining Correct ε and μ Using the Cycle Shift Operator

However, there is a way to correct permittivity ε_eff, and permeability μ_eff can be obtained from reflection and transmission coefficients. We recall that multilayered metal-dielectric structures have a strong inherent spatial dispersion and are typically described by a nonlocal dielectric permittivity tensor $\overline{\overline{\epsilon }}\left(\omega ,\mathrm{k}\right)$ [9]. However, as we only consider the normal incidence of light, we can move from the spatial dispersion framework to a ε(ω) − μ(ω) description of the system [15,16]. In such a paradigm, we can write the following expressions for the effective permittivity and permeability [17], valid for the case of normal incidence:

$$\begin{array}{l}{\epsilon }_{\text{eff}}(\omega )={\epsilon }_{\parallel }(\omega ,\mathrm{k}=0)\\ {\mu }_{\text{eff}}(\omega )={\left[1-\frac{{\omega }^2}{2c^2}{{\epsilon }^{″}}_{\parallel }(\omega ,\mathrm{k}=0)\right]}^{-1}\end{array}$$

(16)

where ε_║ (ω, 0) is the longitudinal component of $\overline{\overline{\epsilon }}(\omega ,0)$, and Lagrange’s notation is used for the derivative. Since we are obtaining ε_eff and μ_eff from the nonlocal effective model, they are much more accurate than the values yielded in the quasi-static limit [10].

We see that the two artifacts in the parameter retrieval seen in Figure 1 are connected. It is the mismatch in the retrieved ε that causes the faulty point where ε = 0 appears in the spectrum, which is where the pole in the effective μ appears. Calculations show that it can be traced to the mismatch in the retrieved impedance, Z, with respect to the Bloch impedance in the structure [18].

The method to avoid the appearance of the artifact features is based on a cycle shift operator applied to the multilayer structure. Taking a finite multilayer composed of N periods, this operator acts like a frame of length NL, moving across an infinite multilayer and cutting out the region that it covers, as depicted in Figure 2a.

As can be seen in Figure 2b, different values of frame shift cause a significant difference in the phase of the reflection coefficient for a semi-infinite structure, corresponding to changing of the retrieved impedance to match the Bloch impedance. This, in turn, makes it possible to pick the value of the phase that would result in correct values of the extracted parameters.

For our example with l_d = 30 nm and l_m = 60 nm and for the structure containing five periods, we start from the dielectric layer and apply the cycle shift operator, shifting the frame with a step of 1 nm. Calculating r and t at each step and extracting material parameters, we find that the best correspondence of the extracted ε and μ with the accurate values ε_eff and μ_eff is achieved when the frame is shifted by 15 nm or 60 nm, making the unit cell symmetric. Figure 3 demonstrates the extraction procedure for a 15-nm frame shift, showing a much better matching with the theoretically calculated values, as well as the disappearance of the artifact features in comparison with the extraction without the frame shift operator application (Figure 1).

Physically, the “magic” value of the shift corresponds to positioning the point where r = 1 (implying that ϕ_r = ±2πm, m ∈ ℤ) at the edge of the stop band shown in Figure 2c. One can also explain why the cycle shift operator restores the validity of the extraction procedure by noting that the impedance mismatch (the difference between the wave impedance, which enters the extraction procedure, and the Bloch impedance, which is an attribute of the infinite periodic multilayer) can result from a poorly chosen termination of the multilayer. Namely, poor termination causes high impedance mismatch between the outer space and the multilayer and, accordingly, the associated strong reflection, leading to artifacts. Applying the cycle shift operator effectively cycles through all possible terminations, and the best correspondence in parameter extraction happens when a good impedance matching between the multilayer and the outer space is achieved. Our results agree with previous accounts [19,20], where it was shown that a symmetric unit cell facilitates a good matching between the effective medium and extracted parameters.

5. Conclusions

We have demonstrated that the singularity in impedance originating from r = 1 and the zero value of its phase leads to artifacts in the retrieval of constitutive parameters of subwavelength photonic metal-dielectric multilayers, namely spurious magnetic resonance and a large mismatch of effective permittivity. We have shown that these artifacts can be worked around by applying the cycle shift operator to the multilayer. The obtained results are useful for accurate experimental determination of effective macroscopic parameters of photonic multilayers and metamaterials, especially those that work in the epsilon-near-zero regime and those that are designed to exhibit artificial optical magnetism. The proposed procedure is applicable to lossless and lossy systems within the limits assessed by the standard Nicolson–Ross–Weir technique. These results can also be used to better understand why some metamaterial geometries lend themselves to homogenization and extraction-based parameter retrieval much more readily than some others [21].