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Abstract
Fano resonances are observed in a composite metamaterial that consists of an electric split ring resonator eSRR and an I-shaped resonator ISR. By adjusting the length of the ISR the degree of asymmetry in the line shape of the composite metamaterial can be controlled and even made to be symmetric. In contrast to other methods to create Fano resonances, the individual modes of the eSRR and ISR have the same symmetry and are not evanescently coupled to each other. The transmission is simulated using the finite difference time domain method and a coupled oscillator model is used to obtain nominal values of the Fano asymmetry factor q. Composite metamaterials and individual eSRR and ISR metamaterials are fabricated, and their transmission is measured with terahertz time-domain spectroscopy.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Although metamaterials have enabled the creation of exotic left-handed material with a negative index of refraction [1], they can also be used to develop sensitive biosensors [2,3], high speed modulators [4], switches [5], spatial light modulators [6], slow light devices [7] and thermal absorbers and emitters [8]. The ability to engineer electro-magnetic properties is especially attractive for terahertz THz applications which have historically lacked technology found at other spectral regions [9]. For filters, sensors, and slow light devices a sharp change of the transmission - or absorption and index of refraction - with frequency is often desirable. Producing metamaterials with Fano resonances [10,11] which have highly asymmetric line shapes is one strategy to obtain sharp spectral features.
Fano resonances can be produced from the destructive interference between two modes with different damping rates and resonant frequencies [12]. Such modes can be created by deforming resonant structures in order to break their symmetry [13,14]. This enables coupling between modes with different symmetries. Since a mode with a higher multipole symmetry does not radiate as efficiently into free space as a mode with a lower multipole symmetry [15], Fano line shapes can be created. The higher symmetry mode is often referred to as the ’dark’ mode, since it weakly couples to free space radiation. While the lower symmetry is referred to as the ’bright’ mode since it strongly couples to free-space radiation in analogy with EIT in atomic systems [16]. For example, if the symmetry of a quadruple mode is broken it may couple to a dipole mode to produce a Fano resonance [17,18]. Another example is breaking the symmetry of a ring mode by introducing asymmetric cuts into it [19–21].
However, symmetry breaking is not necessary to produce a Fano resonance. What is necessary is for two modes with different damping rates - i.e. quality factors - and resonant frequencies to couple to each other [22]. Fano line shapes can thus be produced by other methods such as interference between modes that couple to external radiation via orthogonal polarizations [23,24], or interference between modes with the same symmetry [23,25,26]. Many of these schemes involve the evanescent coupling of modes belonging to spatially separated meta-atoms [26–28].
In this paper, we create Fano resonances by integrating two metamaterials with the same symmetry into a composite structure. This avoids the need for evanescent coupling or symmetry breaking. Specifically, an electric split ring resonator eSRR [29,30] with a narrow linewidth and an I-shaped resonator ISR [31–35] with a broad linewidth are used. The individual eSRRs and ISRs are joined together in such a manner that the ISR is a natural extension of the eSRR. This preserves the symmetry of the modes while allowing significant coupling between them. By varying the length of the ISR component, symmetric and asymmetric line shapes can be created when the eSRR and ISR modes are brought into and out of resonance with each other.
2. Simulations
The electric split ring resonator eSRR consists of an inner capacitor element $C$ connected in parallel with two loops that act as inductors as shown in Fig. 1. In contrast to the conventional split ring resonator [36], the magnetic response of the eSRR is suppressed and the eSRR can be excited by the electric field of an incident plane wave.
Fig. 1. a) Electric split ring resonator eSRR with dimensions $X$ = $Y$ = 54 $\mu$m, $C$ = 30 $\mu$m, $A$ = 12 $\mu$m, $W$ = 6 $\mu$m and $G$ = 6 $\mu$m. The $z-$ components of the b) electric and c) magnetic fields are shown for a frequency of 0.39 THz. The period is 64 $\mu$m in the $x-$ direction and 152 $\mu$m in the $y-$ direction. d) FDTD simulation of the transmission. An incident plane wave with the electric field polarized in the $y-$ direction is used to excite the eSRRs.
The finite difference time domain method FDTD [37] is used to simulate the transmission of a plane wave through a periodic array of eSRRs on a dielectric substrate with index n = 3.6, which is approximately the index of GaAs at THz frequencies [38]. Periodic boundary conditions were used for the $x$ and $y$ boundaries of the simulation while adiabatic absorbing boundary conditions [39], which mimic perfectly matched layers, were used for the $z$ boundaries. The simulated transmission spectra of the eSRR array shows a sharp dip at 0.39 THz in Fig. 1(d). At higher frequencies a broad dip is also present in the transmission spectra and corresponds to a dipolar mode [40].
By Fourier transforming the recorded fields of the simulations, the spatial profile of the eSRR mode can be determined as shown in Figs. 1(b) and 1(c). For a single eSRR the electric field in the $z-$ direction exhibits a symmetric/[antisymmetric] mirror symmetry in the $x$/[$y$] direction, while the magnetic field exhibits an antisymmetric/[symmetric] mirror symmetry in the $x$/[$y$] direction. The eSRR mode has dipole symmetry and is expected to couple to free space. The opposite directions of the magnetic field in the $z-$ direction of Fig. 1(c) indicates the current flows in clockwise and anticlockwise directions around the two loops of the eSRR.
An electric field component in the $y-$ direction, not shown in Fig. 1, is present in the middle of the gap of the eSRR which attests to the capacitor nature of the gap. However, the $y-$ component of the electric field is maximized on the metal edges of capacitor gap. This behavior is common for structures with a sub-wavelength gap [41,42] including field profiles of eSRR resonators with sufficient spatial resolution [43]. It is thus easier to perceive the overall symmetry of the mode by viewing the $z-$ component of the electric field on the metal surface.
The individual I-shaped resonator ISR consists of a line element, with an associated self-inductance, attached to capacitor elements on each end of the line element as shown in Fig. 2(a). The capacitor gap $G$ and length of the capacitor $C$ are identical to the eSRR metamaterial in Fig. 1. The ISR frequency can be altered by varying the length $L$ of the ISR. With increasing length, the resonant frequency of the ISR decreases as shown in Fig. 2(d). Although the mode profiles of the eSRR and ISR are different, they share the same dipole symmetry as shown in Figs. 2(b) and 2(c). Both eSRR and ISR modes will couple to plane waves with an electric field component in the $y-$ direction and can therefore be considered ’bright’ modes.
Fig. 2. a) I shaped resonator ISR with dimensions $C$ = 30 $\mu$m, $W$ = 6 $\mu$m and variable length $L$. The $z-$ components of the electric b) and magnetic c) fields are shown for a frequency of 0.47 THz for $L$ = 70 $\mu$m. The period is 64 $\mu$m in the $x-$ direction and 76 $\mu$m in the $y-$ direction for $L$ = 70 $\mu$m. The period in the $x-$ direction is identical to the period in Fig. 1, while the period in the $y-$ direction is half the period in Fig. 1. The ISRs are arranged so that the capacitor gap size between ISRs is identical to the gap size of the eSRRs in Fig. 1. d) FDTD simulations of the transmission for $L$ = 60, 70 and 88 $\mu$m. An incident plane wave with the electric field polarized in the $y-$ direction is used to excite the ISRs.
To achieve Fano line shapes, the eSRR and ISR meta-atoms are combined to form the composite metamaterial shown in Fig. 3. When the length of the ISR changes from 60 $\mu$m to 88 $\mu$m, the line shape of the transmission - the black lines in Fig. 3 - changes from being symmetric to asymmetric. Fano [44] considered the coupling of atomic resonances with the continuum in ionization spectra and found a line shape with the form $(\epsilon +q)^2/(\epsilon ^2+1)$ where $\epsilon$ is a dimensionless frequency and $q$ is the asymmetry factor. Line shapes with Fano features can also be produced by considering the power dissipated by the coupling of two damped oscillators [12,45,46] as discussed in the appendix. If the damping rate of one oscillator is taken to be zero, the line shape of the power dissipated in the coupled oscillator system approaches the Fano form $(\epsilon +q)^2/(\epsilon ^2+1)$ as the damping rate of the other oscillator increases.
Fig. 3. FDTD simulations of transmission of the composite metamaterial for lengths $L$ of 60 $\mu$m a) 70 $\mu$m b), and 88 $\mu$m c). The dotted red lines show the best fit to the coupled oscillator model described in the appendix. The parameters for the best fits are: $\omega _1 / 2 \pi = 0.42\textrm { THz}$, $\omega _2/ 2 \pi =0.44 \textrm { THz}$, $\gamma _1 / 2 \pi =0.59 \textrm { THz}$, $\omega _g / 2 \pi =0.29\textrm { THz}$ for $L$ = 60 $\mu$m; $\omega _1 / 2 \pi = 0.35\textrm { THz}$, $\omega _2 / 2 \pi = 0.43 \textrm { THz}$, $\gamma _1 / 2 \pi = 0.52 \textrm { THz}$, $\omega _g / 2 \pi = 0.25\textrm { THz}$ for $L$ = 70 $\mu$m; and $\omega _1 / 2 \pi = 0.30\textrm { THz}$, $\omega _2 / 2 \pi = 0.41 \textrm {THz}$, $\gamma _1 / 2 \pi = 0.51 \textrm { THz}$, $\omega _g / 2 \pi = 0.23\textrm { THz}$ for $L$ = 88 $\mu$m. The nominal asymmetry factors are $q=$ $-0.07$, $-0.18$ and $-0.37$ for $L=$ 60 $\mu$m, 70 $\mu$m and 88 $\mu$m, respectively.
We find that if the line width of the eSRR resonance is approximated as zero, the transmission from the FDTD simulations - solid black lines - can be fit to the coupled oscillator model - dotted red lines - in Fig. 3. From the least square fits we obtain values for the individual resonance frequencies $\omega _1$ and $\omega _2$, the damping rate of the broad oscillator $\gamma _1$, which represents the ISR resonance, and the coupling frequency $\omega _g$. From these parameters the nominal values of the asymmetry factor $q$ and a parameter $\Gamma$ defined in the appendix can be found. These parameters and the bare resonance frequency of the second oscillator $\omega _2$ are used to define the dimensionless frequency as $\epsilon \equiv [(\omega _2^2-\omega ^2)/\Gamma ^2] -q$ as discussed in the appendix.
The value of the asymmetry factor $q$ varies from −0.07 to −0.18 to −0.37 as the ISR length $L$ increases from 60 $\mu$m to $70$ $\mu$m to 88 $\mu$m in Fig. 3. The maximum asymmetry of the Fano line shape will occur when $|q| = 1$. The Fano line shape becomes symmetric when $|q| = 0$ or when $|q| \rightarrow \infty$. The increase of the magnitude of the asymmetry factor $|q|$ with ISR length $L$ indicates a line shape with an increasing degree of asymmetry which is evident in Fig. 3.
The fitted frequencies of the narrow oscillator $\omega _2/2 \pi$ range from 0.41-0.44 THz which is slightly greater than the frequency of the eSRR mode in Fig. 1. However, the fitted frequencies of the broad oscillator $\omega _1/ 2 \pi$ are 0.42, 0.35 and 0.30 THz for ISR lengths $L$ of 60 $\mu$m, 70 $\mu$m and 88 $\mu$m, respectively, while the frequencies of the ISR modes are 0.52, 0.47, and 0.41 THz in Fig. 2. The reason for this discrepancy is not entirely clear. But it should be stressed that the coupling frequency $\omega _g$ is a significant fraction of the oscillator frequencies $\omega _1$ and $\omega _2$. Thus the coupling terms in Eqs. (1) and (2) of the appendix which are linear in the coordinates $x_2$ and $x_1$ may need to be further expanded as power series. Additionally with such large coupling, the higher frequency dipolar mode of the eSRR [40] may need to be taken into account for a proper description, especially with regards to the higher frequencies. In Fig. 3 the fits become progressively worse for higher frequencies which is consistent with this explanation. The coupling frequency $\omega _g / 2 \pi$ systematically decreases from 0.29 THz at $L$ = 60 $\mu$m to 0.23 THz at $L$ = 88 $\mu$m, since the eSRR component takes up a smaller fraction of the composite metamaterial as $L$ increases.
3. Fabrication of metamaterials and experimental setup
Planar arrays of metallic resonators were fabricated on semi-insulating GaAs wafers. The metamaterials were fabricated by using contact photolithography, thermal evaporation and a two-layer lift-off process [47]. A schematic of a single metamaterial resonator on the GaAs wafers is shown in Fig. 4(a). The size of the metamaterial arrays was approximately 2 by 2 mm. The samples were placed in a THz time-domain set-up with four off-axis parabolic mirrors as shown in Fig. 4(b). Terahertz pulses were generated by femtosecond laser pulses that illuminated an interdigitated photo-conductive antenna [48]. The parabolic mirror that focused the THz beam on the sample had a f-number of 2.
Fig. 4. a) The layer structure of the samples consisted of 170 nm of Au and 5 nm Cr deposited on a 600 $\mu$m semi-insulating GaAs wafer. The first GaAs wafer was attached to a second GaAs wafer with 200 nm layer of photoresist. b) Schematic of the terahertz time-domain setup used for the transmission measurements.
The metamaterial samples were attached to an $x$-$y$-$z$ translation stage - not shown in Fig. 4(b) - that was controlled with micrometers. Aluminum masks with a central 1.8 mm diameter hole were placed on each sample to ensure that all the THz radiation passed through the metamaterial. GaAs wafers cleaved to the same dimensions as the samples were used as references with the aforementioned Aluminum masks with central circular holes. Both the samples and references were simultaneously attached to the $x$-$y$-$z$ translation stage in order for the signal and reference spectra to be taken under identical conditions. After the THz pulses passed through the sample or reference, they were detected via electro-optic sampling [48]. All data was taken under purged conditions to eliminate water lines from the broad-band spectra. The spectral resolution of THz time-domain scans was limited by Fabry-Pérot echoes in the GaAs wafers. To increase the spectral resolution the samples and references were attached to an additional GaAs wafer with a thin layer of photoresist as shown in Fig. 4(a).
4. Transmission measurements of individual and composite metamaterials
The measured transmission spectra of the individual eSRRs and ISRs are shown in Fig. 5(a) and 5(b). The measured frequency of the eSRR is 0.43 THz which is in rough agreement with the FDTD simulations of Fig. 1. The quality factor of the eSRR resonance is 8.0 which is less than the simulated value of 15.5. The degradation of the quality factor is most likely due to a form of inhomogeneous broadening, as the fabricated metamaterials vary on longer length scales than the microscope pictures shown in Fig. 5. These variations are most likely due to changes in the distances between the photomask and wafer during contact photolithography. The resonant frequency of the ISR modes are 0.56 THz, 0.52 THz, and 0.45 THz for ISR lengths $L$ of 60 $\mu$m, 70 $\mu$m, 88 $\mu$m. The measured ISR frequencies are again slightly larger than the frequencies of the simulations in Fig. 2. While the quality factors of the ISR modes are 2.1, 2.2, and 1.7 for lengths $L$ of 60 $\mu$m, 70 $\mu$m, 88 $\mu$m respectively which are slightly larger than the simulated values of 1.3, 1.1, and 1.0.
Fig. 5. a) Transmission measurement of an eSRR with identical dimensions as Fig. 1 b) Transmission measurements of ISRs with identical dimension as Fig. 2 for $L$ = 60 $\mu$m black, 70 $\mu$m red, and 88 $\mu$m blue. c) Transmission measurement of the composite metamaterials for lengths $L$ of c) 60 $\mu$m d) 70 $\mu$m, and e) 88 $\mu$m.
The transmission measurements of the composite metamaterial in Fig. 5 show that the line shape can be tuned from symmetric Fig. 5(c) to antisymmetric Fig. 5(e) by varying the length of the ISR. The spectral response of the THz time-domain measurements is such that the signal to noise ratio is too small for meaningful measurements < 0.3 THz. This prevents the lower frequency minimum in the transmission of the composite metamaterial from being completely observed. When $L$ is 88 $\mu$m Fig. 5(e) the amplitude of the transmission abruptly changes from 0.66 at 0.434 THz to 0.33 at 0.467 THz on the left side of the minimum in the transmission near 0.5 THz. The transmission drops by 50$\%$ in only 33 GHz - the resolution of the measurement. By contrast on right side of the minimum near 0.5 THz the transmission changes from 0.30 at 0.501 THz to 0.62 at 0.635 THz. The same change in transmission that takes place in less than 33 GHz on the left side of the minimum takes place in approximately 150 GHz on the right side which attests to the asymmetry in the line shape at 88 $\mu$m.
5. Measurements of complementary metamaterials
We further investigate the formation of Fano resonances from a complementary structure, where the metal regions are replaced by the non-metal regions and the non-metal regions are replaced by the metal regions, as shown in Fig. 6. According to the Babinet principle, the complementary structure inverts the transmission spectra, so that the peaks in the transmission spectra correspond to the resonant modes instead of the dips in the transmission spectra [49,50]. The Babinet principle also transforms the magnetic and electric fields of the original and complementary structures into each other. As a consequence, the complementary structures are probed with THz radiation polarized in the horizontal $x-$ direction while the THz radiation is polarized in the vertical $y-$ direction for the normal structures.
Fig. 6. a) Transmission measurements of the complementary structure black and original structure red for a composite metamaterial with reduced dimension $X$ = 28 $\mu$m, $Y$ = 27 $\mu$m, $C$ = 11 $\mu$m, $W$ = 4 $\mu$m, and $G$ = 4 $\mu$m with $L$ = 24 $\mu$m. Transmission measurements of complementary metamaterial structures for b) $L$ = 29 $\mu$m, and c) $L$ = 32 $\mu$m. d) Transmission measurement for $L$ = 32 $\mu$m with an additional GaAs substrate which permits higher resolution on account of the greater period of Fabry-Pérot echoes.
The dimensions of the Babinet metamaterials in Fig. 6 are reduced which increases their frequencies. In Fig. 6(a) the transmission of the complementary black and the normal red metamaterials are plotted on the same graph. The peaks and dips of the transmission curves of the complementary and normal structures are seen to overlap one and other. This attests to the inverted nature of the complimentary transmission. The line shapes of both the complementary and normal structures in Fig. 6(a) are symmetric when $L$=24 $\mu$m. However, when the length $L$ is increased to 29 $\mu$m and 32 $\mu$m - as shown in Figs. 6(b) and 6(c) - the line shape near 1.4 THz becomes asymmetrical. The frequency resolution of the THz scan is increased by attaching an additional 600 $\mu$m thick GaAs wafer to the original substrate for a complementary metamaterial with $L$ = 32 $\mu$m as shown in Fig. 6(d). The transmission of the metamaterial increases from 0.32 at 1.302 THz to 0.51 at 1.335 THz. Thus, the transmission changes approximately 60$\%$ on the left-side of the local maximum at 1.4 THz in only 33 GHz - which is the frequency resolution of the measurement. In contrast on the right-side of local maximum the same change in transmission takes approximately 150 GHz.
6. Conclusion
In conclusion, we show that Fano resonances can be produced by integrating an electric split ring resonator eSRR and an I shaped resonator ISR to form a composite metamaterial. By adjusting the length of the ISR element in the composite structure symmetric and asymmetric line shapes can be produced. Fabricated composite metamaterials exhibited line shapes that had sharp transmission changes on one side of local transmission extrema while having gradual transmission changes on the other side. ISR resonators fabricated on polyimide have been shown to be frequency tunable when stress is applied to produce an elongation of the metamaterial structures [32]. An integrated eSRR and ISR metamaterial on a polyimide surface could thus potentially be used to switch between symmetric and asymmetric line shapes which would be of interest for applications such as dielectric sensors [51], thin film sensors [52] and analyte detection [2,3,53].
Appendix
Fano resonances can be modeled using the differential equations that describe two coupled oscillators [12]
Here $\gamma _i$ is the damping rate of the ith oscillator, $\omega _i$ is the bare resonant frequency of the ith oscillator, $\omega _g$ is the coupling frequency and $f_1(t)$ is the driving term. The line shape function is proportional to the dissipated power $P$. If the position and driving terms are of the form $x_i(t)=x_i(\omega )e^{+i\omega t}+\textrm {c.c.}$ and $f_1(t)=f_1(\omega )e^{+i\omega t}+\textrm {c.c.}$ - where $\textrm {c.c.}$ is the complex conjugate - the time averaged dissipated power is $P=2m_1\gamma _1 \omega ^2 |{x_1(\omega )}|^2 + 2m_2\gamma _2 \omega ^2 |{x_2(\omega )}|^2$.Assuming the damping rate of the first oscillator is much greater than that of the second $\gamma _1>>\gamma _2$ we can make the approximation $\gamma _2 \approx 0$ which considerably simplifies finding the dissipated power as a function of frequency. $x_1(t)$, $x_2(t)$, and $f_1(t)$ are substituted into Eqs. (1) and (2) to find the amplitudes $x_1(\omega )$ and $x_2(\omega )$ by solving the following matrix equation
Funding
Bundesministerium für Bildung und Forschung (16KIS0867); Deutsche Forschungsgemeinschaft (TRR160).
Disclosures
The authors declare no conflicts of interest.
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