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We established a novel method to evaluate effective optical constants by terahertz (THz) time domain spectroscopy and suggested a strict definition of optical constants and an expression for electromagnetic energy loss following the second law of thermodynamics. We deduced the effective optical constants of phosphor bronze wire grids in the THz region experimentally and theoretically. The results depend strongly on the polarization of the THz waves. When the electric field is parallel to the wires, we observed Drude-like electric permittivities with a plasma frequency reduced by a factor of 10-3, whereas when the field is perpendicular, the sample behaved as a simple dielectric film. We also observed unexpected magnetic permeabilities, which originate from the non-resonant real magnetic response of finite size-conductors.
©2008 Optical Society of America
1. Introduction
Numerous artificially structured materials, which are called metamaterials, have been proposed within the last decade [1,2]. They are promising for their application as novel optical devices because of their design capability and uncommon optical properties, such as negative refraction [3]. For the design and supervision of metamaterials, the standard conceptual framework of characterizing the optical properties is needed. One candidate is the effective optical constant concept [4]. A composite metamaterial may behave as a homogeneous medium at wavelengths much longer than one of its characteristic scales, such as its period, and its optical properties may be represented by the effective optical constants: electric permittivity ε and magnetic permeability μ. There are several ways [5–7] to evaluate the effective optical constants experimentally. If the metamaterial sample is thin, which is normally the case for test samples, the preferable method is to utilize the normal complex transmission and reflection coefficients by considering multiple reflections [5,6]. However, there are controversial issues in both the homogenization approximation [8–14] and the experimental evaluation method. The issues with the former include the violation of the second law of thermodynamics. Some papers report a negative imaginary part of the electric permittivity or magnetic permeability, which seemingly contradicts the second law of thermodynamics [8–12]. The latter’s issues stem from the fact that the derived optical constants are mathematically indefinite [15,16].
In this report, we propose a novel method to evaluate the effective optical constants by THz time-domain spectroscopy (THz-TDS) following the second law of thermodynamics. Mathematical ambiguities are eliminated in the determination of the effective optical constants. Metamaterials designed in the terahertz frequency region [17–22] have significant merits compared with those designed in the microwave or optical frequency region. They are easy to fabricate precisely and have a size in the order of several centimeters, which leads to ease of handling in free space. Metamaterials are therefore useful as flexible transmission and switching devices in the THz region. Many dielectrics have non-negligible energy losses in the THz frequency region because of phonon absorption, which is not suitable for THz devices. Moreover, THz-TDS leads to a precise determination of optical constants. Since THz-TDS provides the temporal waveform of the pulse, we could derive full information of the amplitude and phase of the THz electric field from the Fourier transform of the waveform. We performed THz-TDS in both transmission and reflection geometries and obtained the complex transmission and reflection coefficients experimentally. These coefficients enable us to derive the optical constants analytically. For comparison, we also computed the transmission and reflection coefficients by the cylindrical-wave expansion method [23], and deduced the effective optical constants from the computed coefficients.
2. Theory
The extraction method of the effective optical constants utilizing the normal transmission and reflection coefficients has been reported by several independent groups and seems to be well-established [5,6,13,14]. However, there is still remaining a confusion about the elimination of the ambiguities that appear in the extracted optical constants. There has been reported that three independent ambiguities are to be fixed: the signs of the impedance and refractive index and the multivalued refractive index coming from the arbitrary choice of the phases different from each other by an integral multiple of 2π. Up to this time, three conditions have been considered to be necessary for the elimination of these ambiguities [6,13,24]. In this report, we prove that only two physical demands are needed to determine the effective optical constants completely. We employ the requirements Re z>0 and the Kramers–Kronig relation, where z is the impedance. Note that the thermodynamical demand about the refractive index n, Imn>0, is not used during the entire extraction process. Thus, we can verify the resultant effective optical constants using this demand. This examination is essentially important when we apply the extraction method to the composite metamaterials, in which we have to examine the validity of the effective optical constants concepts. This is also helpful for the correction of the experimental errors. Especially in reflection type THz-TDS, the phase data of the reflected THz waves may include some errors because of the sample-misplacement problem. In this case, the examination enables us to correct the phase errors.
In order to confirm the preceding description, we should start from the strict definition of the refractive index and the impedance, because a conventional definition such as n 2=k 2 c 2/ω 2 is confusing about their signs. Definitions of n and z that are meaningful even in negative index materials are
where ŝ is the unit vector in the direction of the energy flux, z 0 denotes the vacuum impedance (μ 0/ε 0)1/2, and e and h express the complex electromagnetic field [25].
In the following part, we consider the normal incidence of the plane electromagnetic waves onto a homogeneous slab [26] and assume the simple linear response D=ε 0 ε E and B=μ 0 μ H. First, we derive mathematically Rez>0 from the time averaged energy flux S̅ by comparing both sides of the following equation:
where S is the real Poynting vector. This also ensures that the reflection coefficients do not exceed 1. Second, we can deduce Imn>0 from the physical fact that the electromagnetic energy should not become greater in passive media. Energy absorption Q is
where U is the electromagnetic energy density [25]. By taking the time average, the last term on the right-hand side goes to zero. We can transform the time-averaged energy absorption using Eq. (1) for the plane electromagnetic waves:
where e and h are complex amplitudes of the electromagnetic field vectors, and the asterisk represents the complex conjugate [13]. Indeed, the inequality Q̅>0 requires the condition Imn>0, because of Re z>0. Therefore we have the two physical demands that Imn>0 and Re z>0. Furthermore, we can rewrite Eq. (4) in terms of ε and μ:
The inequality is then reduced to the conventional one, Imε>0 [25], when we consider that for non-magnetic and passive materials, μ=1. However, if we consider the response of magnetic materials to the plane wave, Imε>0 is not necessary.
For the normal incidence, Fresnel coefficients at the boundary surface of the two media are represented by the impedances using Eq. (1).
We can then represent the complex transmission (T) and reflection coefficients (R) of the homogeneous slab as
where d is the electromagnetic effective thickness[14,27–29]. In general, the effective thickness should be chosen carefully. We will discuss our case in the Results section. As both equations contain multiple internal reflection terms in the denominators, we could solve these equations with respect to the impedance and the refractive index:
where T′=T exp(iωd/c),R′=-R. These expressions are mathematically the same as those obtained in the transfer matrix method for the microwave region [15,16], but Eqs. (7) and (8) are deduced from the strict definition of optical constants (Eq. (1)) and make us easier to see the two ambiguities. Since the right-hand sides of Eqs. (7) and (8) consist of experimental values, we can derive the impedance and refractive index experimentally. Furthermore, the electric permittivity and magnetic permeability can be represented by the refractive index and impedance, so that we can evaluate ε and μ experimentally:
Note that the solutions of Eqs. (7) and (8) are indefinite with regard to (a) the sign of the impedance and (b) the multi-valued complex logarithm that affects the real part of the refractive index. The latter corresponds to the multi-value phase of the electromagnetic waves. First, we can determine the sign of the impedance from the condition Rez>0. Therefore, we have an infinite number of solution sets, (z,n)=(z 0,n 0+cN/fd), where f=ω/2π, and N denotes an arbitrary integer. Second, we choose the value of the complex logarithm from the Kramers–Kronig relation of n [30]. We postulate the analyticity of the refractive indices in the upper complex half-plane. This supposition is indeed justified when we assume the linear electric and magnetic responses. This determination is generally useful even if the imaginary part of the permittivity or the permeability is negative.
As described above, the optical constants are uniquely determined without using the condition Imn>0. Therefore, we can verify and correct the experimental data applying this themodynamical demand.
3. Experimental setup
The schematic diagrams of the experimental setups are summarized in Fig. 1. In the transmission geometry (Fig. 1(a)), a series of femtosecond optical pulses from a commercial fiber laser (IMRA FEMTOLITE; pulse duration 90 fs, repetition rate 50 MHz, center wavelength 780 nm, and pulse energy 0.4 nJ) is split into two parts. One is focused onto a low-temperature-grown GaAs-based photoconductive antenna for generating coherent broadband THz pulses. The other is focused onto a ZnTe crystal and used as the sampling pulse. THz pulses are detected by electrooptic sampling. In the reflection measurement, we use a Si plate as a half mirror for detection of the normal reflection, as shown in Fig. 1(b). We use an amplified Ti:Sapphire laser (pulse duration 120 fs, repetition rate 1 kHz, center wavelength 800 nm, and pulse energy 300 µJ) as the light source in the reflection measurement. In order to obtain the reference signal, we translate the sample in a direction parallel to the sample surface and measure the signal reflected from the metal frame of the sample.
Fig. 1. Schematic diagrams of (a) the transmission-type THz-TDS and (b) the reflection-type THz-TDS.
We use four phosphor bronze wire grid structures as samples. The typical structure is shown in Fig. 2, and their dimensions are summarized in Table 1. The geometrical factors for the transmission, α, shown in Table 1, are the fractions of the surface area occupied by the slits. νa is the frequency at which the vacuum wavelength is the same as the period of the wires.
Table 1. Dimensions of the samples. α denotes the geometrical factor for the transmission area.
4. Results
Figure 3 shows the four power transmission coefficients of THz pulses whose electric field is parallel to the wires. All the samples exhibit extraordinary-transmission peaks exceeding factor α [31]. Note that the shapes of the graphs are similar to each other and shift to a higher frequency. This shift may correspond to the scale up of the sample: The ratios of the peak frequencies νpeak among the four samples are almost equal to those of νa.
In Fig. 4(a), in which the electric field is parallel to the wires (parallel configuration), measured complex transmission (black curves) and reflection coefficients (purple curves) for sample A are summarized. The summation of the power transmittance and the power reflectivity (the top of Fig. 4(a)) is almost equal to 1 in low-frequency region, but beyond the vertical dashed line in Fig. 4, this value declines. This corresponds to the beginning of diffraction. In fact, the frequencies of the vertical dashed lines in Fig. 4 are equal to νa and first-order diffraction occurs at these frequencies. The extraordinary transmission (the middle of Fig. 4(a)) occurs at a frequency a little lower than νa. In a much lower-frequency region, the transmissions decrease to zero, and the reflections increase to 1. This behavior is consistent with the fact that metal wire grids are often used as the polarizer in a region with a wavelength much longer than the period of the wires. The amplitude and phase show the extrema or inflection points of the complex transmission and reflection coefficients at the same frequency, 1.5 THz. This frequency is equal to 2νa, and this phenomenon is referred to as Wood’s anomaly. We also measure the complex transmission and reflection coefficients of sample A in the perpendicular configuration (electric field is perpendicular to the wires). The result is depicted in Fig. 4(b). Diffraction and Wood’s anomaly occur at similar positions to those in the parallel configuration. In the region with a frequency lower than the vertical dashed line, there is less dispersion: The transmission coefficient is almost equal to 1, and the reflection coefficient is 0, which is also consistent with the character of the polarizer. For sample A, we also calculate the transmission and reflection coefficients by the cylindrical-wave expansion method with the approximation that the metal wires are perfect conductors (dotted curves in Fig. 4(a) and (b)). The computed results are consistent with the experimental results. Note that in the reflection measurement the potential sample-misplacement is unavoidable, causing serious errors in the phase shift of the reflected THz electromagnetic waves. In our case, the error of the position is suppressed to less than several dozen µm and is caused by slight bending of the sample. This small error does not affect the qualitative description. The phase part of the computed reflection coefficient is within the error bounds of the experimental data. Further, we can correct the phase-shift data using the physical demand Imn>0. We select the phase-shift data within the error bounds so that the resultant effective optical constants satisfy the inequality over as broad a frequency region as possible. This selection guarantees the effectiveness of the extraction method. We can say empirically that this correction method is much effective if the optical response has a large dispersion as shown in Fig. 4(a).
Fig. 4. Complex transmission and reflection coefficients of sample A in the (a) parallel and (b) perpendicular configuration; the power transmittances and reflectances (middle), their addition (top), and the phase shifts (bottom). The doted curves are the computed results. The vertical dashed orange lines indicate νa.
We determine the effective optical constants of sample A using Eqs. (7)–(9), where we regard the thickness of the wires as the effective thickness. In general the definition of the boundary surfaces in the composite materials is difficult [6,14,27–29]. However, if the materials have the simple structure and have enough symmetry, the definition is relatively uncomplicated. Our wire structures have the inversion symmetry, and therefore we must choose the symmetric boundary surfaces. Otherwise, the reflection coefficients for the two incident directions become different. Moreover, it is reasonable to assume the envelope surfaces as the boundary surfaces. Thus we chose the wire thickness as the effective thickness. We will examine the validity of our effective thickness in the last part of this section.
We deduce the effective optical constant in the parallel configuration (Fig. 5). The real part of the impedance is almost equal to 1 at νpeak. This suggests that the extraordinary transmissions are caused by an impedance match. When the impedance of the material is equal to that of the vacuum (1), the reflectivity of this material goes to 0 so that almost all electromagnetic energy transfers into it. The electric permittivity ε is equal to 0 near νpeak. ε exhibits Drude-like behavior with plasma frequency of 0.61 THz and extremely low loss. This value of the plasma frequency is much smaller than that of ordinal bulk metals by a factor of 10-3. The effective optical constants of sample A in the perpendicular configuration are plotted in Fig. 6. All the optical constants show a little dispersion, and the values are almost equal to 1. The sample behaves as a dielectric film with no loss.
We also extracted the effective optical constants from the computed transmission and reflection coefficients (dotted curves). The computed results are the almost same as those derived from the experiments.
The extracted magnetic permeabilities have no dispersions and imaginary parts of zero in both configurations. We found that if we employ the effective thickness other than the wire thickness, the extracted permeability shows an anomalous dispersion. Thus, we conclude that our choice of the effective thickness is reasonable. However, the real parts of the magnetic permeabilities μ are unexpected: The phosphor bronze is not a magnetic material, but the values of μ=0.73, 0.8 in the parallel and perpendicular configuration are smaller than 1.
Fig. 5. Effective optical constants derived from the experiments (solid curves) and computations (dotted curves) conducted on sample A in the parallel configuration. The vertical dashed orange lines indicate νa.
Fig. 6. Effective optical constants derived from the experiments (solid curves) and computations (dotted curves) conducted on sample A in the perpendicular configuration. The vertical dashed orange lines indicate νa.
5. Discussion
The reduced plasma frequency and low loss in the parallel configuration could be described by considering the effect of mutual inductance. We use Pendry’s model [4] as a basis and modify it for the one-dimensional alignment. This leads the following expressions.
where c is the speed of light in vacuum, and σ is the bulk conductivity of the wires. The theoretical plasma frequency in sample A is 0.64 THz and is consistent with the experimental value. We also deduce the effective optical constants of sample B in the parallel configuration by THz-TDS and using Eqs. (7)–(9). We can see qualitatively same behavior as that of sample A: Drude-like electric permittivity. The plasma frequency is 1.08 THz, and the loss is extremely low. This behavior is also consistent with the theoretical result (the theoretical plasma frequency is 1.08 THz).
In the perpendicular case, the effect of the capacitance between the wires is important. The deduced effective electric permittivity is
where C is the capacitance between the wires. C is very small in the present settings; therefore this model accounts for the no dispersion of the extracted permittivity.
Note that the experimentally derived magnetic permeability μ is not equal to 1 in the parallel and perpendicular configuration, which seems to be unreasonable, because the constituent phosphor bronze is a non-magnetic material. This behavior cannot be explained within the framework of the homogeneous electric current density in one wire. We must take into consideration the loop current in one wire. Non-magnetic conductors of finite size have non-zero magnetic polarizability because of the loop current [25]. A perfect or good conducting circular wire has non-dispersive magnetic polarizability per unit length under the magnetic field perpendicular to the wire axis of
where r is the radius. In the case of the magnetic field parallel to the wire axis, the magnetic polarizability is half that in the perpendicular case. Therefore, within the homogenization approximation, we derive μ=0.60 and 0.80 in the parallel and perpendicular configuration. This magnetic polariizability is derived in [25] under the assumption that there is no electromagnetic interaction between the wires, and therefore this interaction effect may explain the small deviation between the theoretical and experimental values.
It is noteworthy that we can examine the present models using the experimentally evaluated electromagnetic energy absorption Q̅ that is calculated from Eq. (5) using the extracted optical constants (Fig. (7)). Below νa, Q̅ is almost equal to 0 in the parallel and perpendicular configuration. This small energy absorption behavior can be described using the present models. In both configurations, the magnetic energy loss |ε| Imμ is 0, because the magnetic permeability is real. In the parallel case, Q̅∝γωp 2/ω 2 is almost equal to 0 in the frequency region near and over the plasma frequency. In contrast, in the perpendicular case, the total energy loss is 0 because ε is also real.
6. Conclusion
We have demonstrated the evaluation of the effective optical constants of metal wire grids by THz-TDS and have proposed a strict definition for the optical constants following the second law of thermodynamics. Experimentally obtained results are consistent with the computed results. In the parallel configuration, the samples behave as uncommon metals with plasma frequency reduced by 10-3 and very low losses. The observed extraordinary transmission results from the impedance-matching effect. In the perpendicular configuration, we can regard the sample as a simple dielectric film with very low losses. In the parallel and perpendicular configuration, the magnetic permeabilities are lower than 1,which can be explained by the inhomogeneity of the current flows in one wire. This non-resonant magnetic response may be helpful for the realization of negative-index metamaterials. Further studies into metamaterials in the terahertz frequency region are promising in the application of various optical devices, and the THz-TDS technique is suitable for metamaterial exploration.
Acknowledgment
This work was supported by a Grant-in-Aid for JSPS Fellows (19-3900), Creative Scientific Research (18GS0208) and the 21st century COE “Center for Diversity and Universality in Physics” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.
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