1. Introduction

Metamaterials (MMs) are engineered materials with subwavelength unit cells periodically patterned to exhibit extraordinary electromagnetic properties that do not exist in natural materials, such as negative refractive index [1–3]. Metamaterial absorbers can achieve near unity absorption by matching the effective wave impedance to that of free space and are typically designed in a three-layer configuration, consisting of a metamaterial layer, an insulating spacer layer, and a metallic ground plane [4,5]. Different absorption characteristics, such as multiple absorption bands, nonlinear response, polarization insensitivity, high quality factor, and incident angle insensitivity can be engineered from microwave to infrared by careful design of the unit cell [6–13]. Tailored perfect absorption at specific frequencies may enable applications ranging from terahertz (THz) imaging and middle infrared imaging to energy harvesting and radiative cooling [14–19]. Moreover, as the absorption peak frequency and amplitude are closely related to the permittivity of the materials surrounding the metamaterial layer, metamaterial absorbers may be precisely configured for sensing applications [20,21].

Various models have been developed to explain resonant near-unity absorption, including effective medium impedance matching [22,23]. In this approach, metamaterial absorbers are treated as homogenous materials with intrinsic complex impedance carefully designed to match free space to minimize surface reflections. The impedance is retrieved using full-wave electromagnetic simulations to obtain the S-parameters. As such, this serves as a rapid means to design and optimize metamaterials absorbers, albeit with some loss of physical insight typical of numerical approaches. Transmission line theory is another well-established approach [24,25], in which the MMs and the spacer material are modeled as equivalent lump impedances in parallel. Although the theory can describe the absorption, the mathematical complexity usually precludes further derivation to reveal deeper relationships. Coupled mode theory, while elegant, also suffers from a similar problem related to mathematical complexity [26,27]. In a more traditional approach, interference theory can be applied to understand the net reflection that results from multiple reflections between the metamaterial layer and the ground plane [28], with relatively simple mathematics offering an opportunity to explore the deeper relationships into the nature of the resonant absorption.

We employ interference theory to obtain an analytical expression for the spacer thickness needed to maximize the absorption at a given frequency. This expression gives the spacer layer thickness in terms of the reflection coefficient of the metamaterial layer in the absence of a ground plane and the complex refractive index of the spacer layer. We find that in the limit of zero spacer thickness (neglecting coupling between the metamaterial layer and the ground plane), the metamaterial absorber maxima would occur at the resonant frequency of the bare (i.e. no ground plane) metamaterial layer. With increasing spacer layer thickness, the metamaterial absorber resonances redshift. Furthermore, with increasing spacer layer thickness, phase delays result in the appearance of additional absorption resonances. To experimentally verify our analysis, we fabricated a series of metamaterial absorbers with SU8 as the spacer material with different thicknesses, and characterized the electromagnetic response using THz time-domain spectroscopy. The experimental results show excellent agreement with the analytical results. Finally, we note that our analytical approach provides insight into the effect of the spacer layer complex permittivity (ε_r + iε_i) revealing, for example, the expected result that the frequency shift of the metamaterial absorber is primarily tied to changes of ε_r.

2. Analyzation of the thickness dependence based on interference theory

We have investigated a metamaterial absorber consisting of a single layer of split ring resonators (SRRs) [geometry shown in inset of Fig. 1(b)], a dielectric spacer layer, and a ground plane. If the near field coupling between the ground plane and the MMs is negligible, the net reflection from interference theory is [28]:

 

Fig. 1 The simulated reflection and transmission coefficients S₁₁, S₂₁, S₂₂, S₁₂ in the absence of a ground plane. (a) Amplitude spectra and (b) Phase spectra. The inset shows the geometry of the SRR unit cell. The periodicity is 90 μm.

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We now consider the perfect absorber geometry, which includes a ground plane. The transmission (because of the ground plane) is negligible. As such, the frequency dependent absorption is A = 1-abs(r)², where A is the absorption amplitude and r is from Eq. (1). The absorption spectra for different spacer thicknesses are plotted in Fig. 2(a). The bare LC and dipole resonant frequencies are indicated by the vertical dashed lines. The absorption at these frequencies is nearly zero due to the strong reflection and weak transmission at the metamaterial layer.

 

Fig. 2 (a) Color plot of the absorption as a function of frequency and spacer thickness. The bare LC and dipole resonant frequencies are marked by the vertical dashed lines. The horizontal dashed line highlights the spacer thickness of 20 μm. The multiples of 2π phase delay accumulated in the spacer at the bare LC resonant frequency are highlighted with stars. The black arrowed dashed line indicates the trend of the absorption peak frequency redshift with increase in spacer thickness. (b) The analytical relation between the absorption peak frequency and the spacer thickness with different values of the integer m, also showing the transition of the contributing resonant modes as the frequency increases with shaded color.

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Typically, metamaterial absorbers are designed with thin (i.e. subwavelength) spacers. As an example, for a 20 μm spacer thickness [highlighted with the horizontal dashed lines in Fig. 2(a)], two absorption peaks are evident and with peak frequencies that are redshifted from the bare values. As the spacer thickness increases there is a clear redshift of the absorption peak. Importantly, as the spacer thickness further increases, additional absorption bands appear at frequencies that are above the bare resonant frequencies. These also redshift with spacer thickness and ultimately cross the bare resonance frequency. This phenomenon results from the phase delay (in the spacer layer) approaching integer multiples of 2π. In the following, an analytically understanding of this response is presented.

The spacer thickness d₀ to achieve the maximum absorption at a specified frequency can be derived from interference theory and is given as:

For dielectrics, n_i is usually much smaller than n_r. In this limit, Eq. (2) simplifies to:

3. Experiments and results

To verify the analysis presented above, metamaterial absorbers were fabricated [see unit cell structure and lateral dimensions in the inset of Fig. 1(b)] utilizing SU8 as the spacer material with different thicknesses (10 μm, 16 μm, 28 μm, 70 μm, 90 μm, and 105 μm). E-beam evaporation was employed to deposit Ti/Au/Ti with thickness of 20/200/20 nm as the ground plane on a silicon substrate. The first layer of titanium was applied as an adhesion layer for the subsequent gold deposition, while the second layer of titanium was utilized to promote the adhesion between the SU8 and the ground plane. Next, SU8 with different thicknesses was spin coated on the ground plane. SU8 3010 was applied to fabricate the samples with spacer thicknesses of 10 μm, 16 μm, and 28 μm. SU8 2035 was applied to fabricate the samples with spacer thicknesses of 70 μm, 90 μm, and 105 μm. The desired thickness was achieved through calibrated control of the spinner rotation. The hard bake temperature was set to 95 °C to prevent the peel-off of SU8. AZ5214E was applied to photolithography the pattern of the MMs on the SU8 films, followed by e-beam evaporation to deposit Ti/Au with thicknesses of 20/200 nm and lift-off process to obtain the final structure. A fabricated sample and the microscope image of the unit cells are shown in Fig. 3.

 

Fig. 3 (a) A fabricated metamaterial absorber sample. (b) Microscope image of the unit cells.

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The metamaterial absorbers were characterized with reflection based THz time domain spectroscopy (THz-TDS). THz radiation was focused onto the sample at normal incidence. The polarization direction is indicated by the white arrow in Fig. 3(b). The absorption spectra obtained numerically, theoretically, and experimentally were plotted in Fig. 4(a) showing high degree of agreement. The discrepancy of baseline of the absorption spectra in Fig. 4(a) is due to the imperfect referencing in a reflection geometry.

 

Fig. 4 The absorption spectra comparison among experiment, simulation, and interference theory. Results of metamaterial absorbers with spacer thickness of 10 μm, 16 μm, 28 μm, 70 μm, 90 μm, and 105 μm for (a) to (f). The absorption peaks are labeled with m values related to the corresponding m-curves in Fig. 5(a).

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In Fig. 5(a), the absorption peak frequencies from experiment, interference theory, and simulation are plotted together with the m-curves from Eq. (2). There is clearly excellent agreement between experiment and theory in addition to consistent results between Eq. (2), the full interference approach, and simulations. The simulated spectra of the metamaterial absorbers are presented in Fig. 5(b) showing the dependence with increasing spacer thickness. The bare LC resonance frequency is denoted with a vertical green dashed line [Figs. 5(a) and (b)]. Due to the strong reflection, the magnitude of the absorption is strongly diminished at the bare resonance frequency. With thin spacer thicknesses (10 μm, 16 μm, and 28 μm), the absorption peak frequency is lower than the bare resonant frequency and, as discussed above, redshifts as the thickness increases. When the spacer thickness is large (70 μm, 90 μm, and 105 μm), the absorption peak frequency is higher than the bare resonant frequency approaching the resonant frequency as the spacer thickness increases.

 

Fig. 5 (a) m-curves with absorption peak frequencies plotted for SU8 spacer metamaterial absorbers with different spacer thicknesses (10 μm, 16 μm, 28 μm, 70 μm, 90 μm, and 105 μm) from experiment, interference theory, and simulation. The vertical green dashed line demarks the bare LC resonant frequency. (b) The simulated absorption spectra showing the absorption band shifting with different spacer thicknesses.

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Figure 6 depicts the SRRs current (70 μm spacer thickness) at the resonance frequencies to illustrate the character of the resonant modes. It is evident that the LC (dipole) mode contributes to the absorption band at the frequencies of 0.4376 THz and 0.9578 THz (1.586 THz). We note that the mode character is not determined by the value of m, but rather by the bare resonant frequency, which means with increasing frequency the contributing resonant mode for the absorption gradually transfers from LC to dipolar, as indicated by the shaded color in Fig. 2(b).

 

Fig. 6 The current distribution of the SRRs (70 μm spacer thickness) at the absorption peak frequencies of (a) 0.4376 THz, (b) 0.9578 THz, and (c) 1.586 THz with red arrows indicating the current direction.

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Using Eq. (2) it is also possible to investigate the effect of the spacer permittivity on the peak absorption frequency. The m-curves (for m = 0) were numerically calculated for different permittivity combinations of the spacer material. Figure 7(a) plots the m-curves obtained upon modifying the real part of the permittivity (ε_r) of the spacer material (with the imaginary permittivity ε_i = 0.182). As ε_r increases, the resonant frequencies of both the LC and dipole modes redshift. This trend can be understood by the increase in the equivalent capacitance for larger ε_r. Figure 7(b) plots the zoomed in m-curves across a relatively small frequency scale obtained by changing ε_i (with ε_r = 2.8). Compared with the frequency shift caused by ε_r, the frequency shift caused by ε_i is miniscule as further supported by the high degree of overlap among the m-curves shown by the inset over a larger frequency range. Thus, the absorption peak frequency shift caused by the variation of imaginary permittivity can be neglected. Nevertheless, a change of ε_i will modify the absorption magnitude.

 

Fig. 7 The m-curves (m = 0). (a) Varying ε_r with ε_i = 0.182. (b) Varying ε_i with ε_r = 2.8. The inset of (b) depicts the m-curves in a larger frequency range.

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4. Conclusion

In summary, we mathematically derived the relationship between the absorption peak frequencies of the metamaterial absorbers with the resonant frequencies of the MMs and the spacer thickness. We proved that the absorption peak frequencies are closely related to the resonant frequencies of the MMs with a frequency shift determined by the phase delay in the spacer material. Moreover, we utilized SU8 as the spacer material, fabricated metamaterial absorbers with different spacer thicknesses, and characterized them with THz TDS. The measurement results exhibited high agreement with the analytical and simulation results. Also, we employed our findings to analyze the contribution of the spacer permittivity change to the absorption peak frequency shift, and revealed that the real part of the permittivity predominates the frequency shift, rather than the imaginary part. The conclusions derived above can be directly applied to any metamaterial absorber configured with metamaterial-spacer-ground plane structure, can be utilized to understand the response of sensors based on such devices, and can guide the design and optimization process.

Appendix

In the derivation below, a and b are the real and imaginary parts of r₂₁, β is the one-way phase delay in the spacer layer with β = β_r + iβ_i, d is the spacer thickness, and m is an integer. The net reflection can be expressed with Eq. (1). According to law of cosines, the amplitude of the total reflection can be written as:

(4)$${\left|r\right|}^2=A^2+2\cos \theta \cdot A\cdot B+B^2$$

in which θ is the angle between the first and secondary reflection, and A and B are the amplitude of the first and secondary reflection, which can be written as:

(5)$$\begin{array}{l}A=\sqrt{r_{12}\overline{r_{12}}}\\ B=\sqrt{\frac{t_{12}{t}_{21}\overline{t_{12}{t}_{21}}}{[(r_{21}+\exp (2{\beta }_{\text{i}}-\text{i}2{\beta }_{\text{r}})][(\overline{r_{21}}+\exp (2{\beta }_{\text{i}}\text{+i}2{\beta }_{\text{r}})]}}\end{array}$$

where the bar denotes taking the complex conjugate. At the perfect absorption condition, the first and secondary reflections are out of phase and can totally cancel each other. The angle between the first reflection and the second reflection is 180 degree. As such, in the vicinity of the perfect absorption, Eq. (4) can be simplified as:

(6)$${\left|r\right|}^2\approx A^2-2A\cdot B+B^2$$

The spacer thickness for the absorption peak at specific frequency can be calculated by setting the differentiation of Eq. (6) to zero as:

(7)$$(B-A)\frac{\text{d}B}{\text{d}d}=0$$

in which d is the spacer thickness. Around the perfect absorption condition, the equation is valid when (B is only equal to A at the perfect absorption point) dB/dd = 0, which can be expanded as:

(8)$$\frac{1}{2}\sqrt{\frac{[r_{21}+\exp (2{\beta }_{\text{i}}-\text{i}2{\beta }_{\text{r}})][\overline{r_{21}}+\exp (2{\beta }_{\text{i}}+\text{i}2{\beta }_{\text{r}})]}{t_{12}{t}_{21}\overline{t_{12}{t}_{21}}}}\frac{\text{d{[}{r}_{21}+\exp (2{\beta }_{\text{i}}-\text{i}2{\beta }_{\text{r}})][\overline{r_{21}}+\exp (2{\beta }_{\text{i}}+\text{i}2{\beta }_{\text{r}})]\}}{\text{d}d}=0$$

which gives:

(9)$$\frac{\text{d{[}{r}_{21}+\exp (2{\beta }_{\text{i}}-\text{i}2{\beta }_{\text{r}})][\overline{r_{21}}+\exp (2{\beta }_{\text{i}}+\text{i}2{\beta }_{\text{r}})]\}}{\text{d}d}=0$$

Equation (9) can be further expanded, and both sides may be divided by 4n_ikexp(2β_i):

(10)$$a\cdot \cos (2{\beta }_{\text{r}})-b\cdot \sin (2{\beta }_{\text{r}})-\frac{n_{\text{r}}}{n_{\text{i}}}[a\cdot \sin (2{\beta }_{\text{r}})+b\cdot \cos (2{\beta }_{\text{r}})]+\exp (2{\beta }_{\text{i}})=0$$

Both sides may be divided by cos(2β_r):

(11)$$a-b\cdot \tan (2{\beta }_{\text{r}})-\frac{n_{\text{r}}}{n_{\text{i}}}\left[a\cdot \tan (2{\beta }_{\text{r}})+b\right]+\frac{\exp (2{\beta }_{\text{i}})}{\cos (2{\beta }_{\text{r}})}=0$$

Rewriting the equation:

(12)$$a-\frac{n_{\text{r}}}{n_{\text{i}}}b+\frac{\exp (2{\beta }_{\text{i}})}{\cos (2{\beta }_{\text{r}})}=(b+\frac{n_{\text{r}}}{n_{\text{i}}}a)\cdot \tan (2{\beta }_{\text{r}})$$

On the left side of the equation, applying the assumption that β_r equals to mπ:

(13)$$a-\frac{n_{\text{r}}}{n_{\text{i}}}b+\exp (2\pi m{n}_{\text{i}}/n_{\text{r}})=(b+\frac{n_{\text{r}}}{n_{\text{i}}}a)\cdot \tan (2{\beta }_{\text{r}})$$

Considering β_r is near to mπ:

(14)$$2{\beta }_{\text{r}}=\mathrm{arc}\tan \left(\frac{a-\frac{n_{\text{r}}}{n_{\text{i}}}b+\exp (2\text{π}m{n}_{\text{i}}/n_{\text{r}})}{b+\frac{n_{\text{r}}}{n_{\text{i}}}a}\right)+2m\text{π}$$

Since β_r = n_rkd₀, the spacer thickness can be expressed as:

(15)$$d_0=\frac{\mathrm{arc}\tan \left(\frac{a-\frac{n_{\text{r}}}{n_{\text{i}}}b+\exp (2\text{π}m{n}_{\text{i}}/n_{\text{r}})}{b+\frac{n_{\text{r}}}{n_{\text{i}}}a}\right)}{2n_{\text{r}}k}+\frac{m\text{π}}{n_{\text{r}}k}$$

Comparing Fig. 2(a) and Fig. 2(b) in the main manuscript, it is evident that d₀ represents the spacer thickness to obtain an absorption maximum. We note that the expression was derived in the vicinity of perfect absorption, and may give larger error at the frequencies far from the perfect absorption frequency. Nevertheless, the derived result can reasonably describe the frequency shift with the change of the spacer thickness as shown in Fig. 5(a).

We also note that the m-curves in this manuscript only include the case in which the first and secondary reflections cancel out instead of adding up by only considering the round-trip phase delay [phase of r₂₁ add with delay in the spacer as Eq. (3)]. This is because we only consider the situations where the phase delay gives 2mπ (cancel out) as opposed to (2m+1)π (add up) [Eq. (14)]. To support this fact, we plot the absorption and phase of the absorber at three frequency points with different spacer thicknesses in Fig. 8. The figure clearly shows that at the spacer thickness with peak absorption [estimated with Eq. (2)] the phase difference between the first and secondary reflection approximately π.

 

Fig. 8 (a) The total absorption of the metamaterial absorbers versus the spacer thickness at the frequencies of 0.632 THz, 0.678 THz, and 0.724 THz with label spacer thicknesses for the absorption peaks. (b) The phase of first and secondary reflection versus the spacer thickness with labeled phase differences.

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Funding

National Science Foundation (NSF) Grant No. ECCS-1309835; Army Research Office under ARO W911NF-16-1-0361.

Acknowledgment

The authors would like to thank Boston University Photonics Center for technical support.

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