Terahertz polarization spectroscopy in the near-field zone of a sub-wavelength-scale metal slit

Daehoon Han; Kanghee Lee; Hanlae Jo; Yunheung Song; Minhyuk Kim; Jaewook Ahn

doi:10.1364/OE.24.021276

1. Introduction

Recent advance in science and technology involved with terahertz (1 THz = 10¹² Hz) frequency wave has made a broad impact on a variety of research fields, including physics, chemistry, material science, and electric engineering [1, 2]. Many applications of THz waves have been also developed in areas including material characterization, stand-off detection, noninvasive diagnostics, and biomedical sensing. The biomedical sensing applications are particularly promising because of the unique spectral nature of THz waves in bio-organic materials [3]. However, acquiring spectral information of biological materials requires micrometer-size spatial resolution which is not simple to achieve due to the large wavelength of THz waves (λ = 300 µm for 1 THz). To overcome the spatial resolution, limited by the Abbe diffraction of freely propagating large-wavelength waves, several methods have been considered: for example, near-field emission and/or detection [4–6], or sub-wavelength-size material platforms [7–10]. Many of these methods often use some form of metallic structures with sub-wavelength dimensions to confine, focus, guide, or bend the THz waves in the vicinity of the material, but they can strongly alter the wave properties of the interacting THz-wave itself in terms of polarization, spectral phase, and amplitude.

The wave diffraction through a sub-wavelength-size metal hole, for example, is completely different from the case of a large hole, because of the interplay between the cavity field and edge currents of the aperture [11–14]. At the limit of an extreme sub-wavelength-sized aperture transmission, the diffracted electric field results from effective magnetic dipole radiation [15,16]. Other examples of sub-wavelength optical phenomena manifesting the vectorial nature of electromagnetic wave around metal structures include extraordinary light transmission [17,18], strong electric field enhancement [19,20], diffraction phase shift [21], and anomalous Young’s double slit experiment [22], all of which are found in the diffraction from a slit, or a system of slits, with sub-wavelength dimensions.

In this paper, we report THz polarization spectroscopy of an organic material kept in a sub-wavelength-size metal slit. Using α-lactose monohydrate, which has strong an absorption line at 0.53 THz [23, 24], we investigate the spectral amplitude change of transmitted THz wave through this material within the slit as a function of the slit width. Experimental results reveal that the spectral response of the material is strongly coupled with the polarization state of the THz wave, and that the material does not interact at all with parallel-polarized (with respect to the slit direction) THz waves in the limit of an extreme sub-wavelength-sized slit. This phenomena may be explained in the basis of Bethe’s diffraction theory; the electric field in the given polarization does not penetrate the slit and the electric field measured in the far field is newly generated from the magnetic dipole radiation (which has no direct response to the material) in the near-field zone of the slit. In the remaining sections, we step-by-step describe the measurements, experimental results, and brief discussion of this polarization dependence in terms of the Bethe’s diffraction theory.

2. Experimental description

Experimental investigation was performed with a conventional THz time-domain spectroscopy (THz-TDS) setup [25], as shown in Fig. 1(a). THz pulses were produced from a commercial photo-conductive antenna (BATOP optoelectronics) pumped by femtosecond near-infrared pulses from a Ti:sapphire mode-locked laser oscillator, and measured via laser-gated electro-optical sampling with a 2-mm-thick (110) ZnTe crystal [26]. The temporal amplitude profile of the THz pulse was recorded, from which the THz spectrum was computed. The fabricated metal slit was placed at the focus of the propagating THz waves in a one-dimensional 4-f geometry THz beam delivery system comprised of two Teflon lenses with a focal length of f = 100 mm. The THz beam was focused onto the slit structure with a uniform intensity region over 2×2 mm².

Fig. 1 (a) Schematic experimental setup, where a sub-wavelength-scale slit was placed at the focus of THz wave. The inset shows the polarization angle, defined with respect to the slit orientation, which is either θ = 0 or θ = π/2, when the slit direction is along the slit length of L. (b) The geometry of the fabricated sample, in the side and front views: the sample has a pair of wedge-shaped slits, one for reference signal and the other for the material, and a large rectangular hole for calibration of the material thickness. Each slit has a width of d, varied from 5 to 30 µm, and a length of L = 20 µm. The slits are fabricated with a 500-nm-thick copper film deposited on a silicon substrate. (c) The compound of α-lactose and water was coated on a rectangular hole with an area of 7 mm × 20 mm.

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Figure 1(b) shows the schematic structure of the fabricated wedge-shaped slits in a 500 nm copper film deposited on a 525 µm thick silicon wafer of high resistivity (20,000 Ω cm). Each slit has a length of 20 mm, a wedge top of 5 µm, and a wedge bottom of 30 µm. To put α-lactose monohydrate (Sigma Aldrich) into the slit, we prepared a compound of α-lactose dissolved in deionized water. The candle wax, a hydrophobic material, was used to confine the compound into an area of 18 mm × 21 mm. The deionized water was allowed to evaporate over a period of 12 hours to completely eliminate the water. In this method, we prepared a reference slit and a sample slit as described in Fig. 1(c). The rectangular hole with 7 mm × 20 mm in Fig. 1(c) was used to estimate the thickness (d_s) of lactose on the slit. The estimated thickness d_s is about 200±6 µm, obtained from d_s = c∆t/(n_s − n_air), where c is the speed of light, ∆t is the delayed time between the maximums of the reference and sample signal in the time domain, n_s is the refractive index of lactose, and n_air is the refractive index of air [27]. We carried out measurements in two different polarizations: orthogonal (θ = π/2, E ⊥ L, or ⊥ case) and parallel (θ = 0, E ‖ L, or ‖ case) directions of the slits with respect to the THz polarization (E). Each measurement was conducted by translating the sample in XY directions with mechanical actuators of a 25 mm travel length. The THz-setup was purged with dry air to remove water vapor in the THz frequency range [28].

3. Theoretical background

Recent research [19–22] suggest that the source of the diffracted electric field through a sub-wavelength sized slit is related to the slit direction defined by the orientation (L) of length L. This section describes this behavior briefly and the details can be found in the Appendix.

The magnetic field (H-field) perpendicular or parallel to the slit direction is a constant over the slit because of the boundary condition [15]. Let H₀ and E₀ be the initial H-field and E-field for z < 0 on the left-hand side of the perfect conductor screen at z = 0 if there is no hole. By the boundary condition, H₀ and E₀ have, respectively, only tangential component and normal component with respect to the plane of incidence denoted as H₀ = H_0t and E₀ = E_0n. If there is a small hole with a radius of a, (a ≪ λ), in the screen, Bethe added the scattered field (E₁, H₁) on the left-hand side of the screen and the diffracted field (E₂, H₂) on the right-hand side of the screen in order to eliminate the discontinuity in the hole that occurs from the zero-order approximation [15,16]. By Bethe’s first-order approximation, the electromagnetic field on the right-hand side of the screen in the hole is a half of the initial field defined as [15]

H_{2 t} = \frac{1}{2} H_{0}, E_{2 n} = \frac{1}{2} E_{0},

where the normal component of the E-field in the hole is neglected when the incident field is a transverse field [15]. Since the radius of the hole is small enough compared to λ, H₀ and E₀ are considered as constants over the hole [15]. The key point is that Eq. (1) is given regardless of the shape and size of the hole, which leads to the result that the constant H-field is given regardless of the slit direction [21, 22]. However, the E-field inside the slit exhibits a strong polarization dependence. When a linearly polarized electromagnetic wave with the wavelength of λ propagates along z-direction from z < 0 to a metallic slit with a width of d and a length of L located at z = 0, each case of the slit direction perpendicular to H-field (H ⊥ L: the ‖-case) and E-field (E ⊥ L: the ⊥-case) is physically interpreted as the E-field reduction case [22,31] and the E-field enhancement case [9,12], respectively. The diffracted E-field at the far-field zone for the ‖-case can be understood by the radiation originated from the magnetic dipole moment of the sub-wavelength slit [15, 29]. On the other hand, the E-field diffracted by the slit for the ⊥-case can be thought of as a combination of the radiation from the electric dipole moment and the ordinary diffraction explained by Kirchhoff’s diffraction theory [22, 29]. The resultant diffracted E-fields for the ‖- and the ⊥-cases are given, respectively, by [15, 22, 31] (The detailed description can be found in the Appendix or Ref. [31].)

E_{‖} (z, t) = \frac{π^{2}}{8} {(\frac{d}{λ})}^{2} Z_{0} L \hat{n} \times H_{0} \frac{\exp [i (k z - ω t)]}{z},

E_{⊥} (z, t) = [β - i (\frac{d}{λ})] E_{0} L \frac{\exp [i (k z - ω t)]}{z},

where Z₀ is the impedance of free space,

\hat{n}

is a unit vector in the direction of the field point z and β is a proportional coefficient.

Therefore, when the electrically resonant material is inside the slit, we find, using Eqs. (2) and (3), that the absorption of electrically resonant material in the slit should be expected to result in complete vanishing of the otherwise resonant absorption since H₀ does not interact with the electrically resonant material.

4. Results and discussion

To verify our expectation, summarized in Eqs. (2) and (3), we used the resonant material in the slit and measured the relative absorbance α_s, defined by

α_{s} (ω) = - \ln | \frac{{\tilde{E}}_{s} (ω)}{{\tilde{E}}_{ref} (ω)} |,

where

{\tilde{E}}_{s}

and

{\tilde{E}}_{ref}

are the transmitted THz electric fields with and without α-lactose, respectively. In the frequency range from 0.1 THz to 2.0 THz, the relative absorbances α_s was measured for the ‖ and ⊥-cases, respectively, and the results are shown as a function of the slit width (d) and the measured frequency (f) in Fig. 2(a) and Fig. 2(b), respectively. We find from Fig. 2 that the absorbance at 0.53 THz in the ‖-case disappears as the slit width decreases, whereas there is no significant change in the absorbance in the ⊥-case. The absorbances near 0.53 THz for the cases at d=25.3±1.3 µm and d=6.6±1.3 µm depicted as dotted white lines in Fig. 2 are plotted in Fig. 3(a) and Fig. 3(b), respectively.

Fig. 2 The extracted absorbance α_s for (a) the ‖ and (b) the ⊥-cases as a function of the measured frequency and the relative slit width ranging from 5 µm to 30 µm. The absorbances near 0.53 THz for the both cases at d=25.3±1.3 µm and d=6.6 ±1.3 µm are respectively depicted as dotted white lines in Fig. 3.

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Fig. 3 The relative absopbance $α_{s} (ω) = - \ln | {\tilde{E}}_{s} (ω) / {\tilde{E}}_{ref} (ω) |$ is plotted around the frequency at 0.53 THz for (a) the slit width of d=25.3±1.3 µm and (b) of d=6.6±1.3 µm. All the measured data are plotted as closed circles for the ‖-case and open rectangles for the ⊥-case, and the curves are numerical fit to Eq. (5). The typical measurement uncertainty for the ‖ case is shown with red error lines, and the uncertainty for the ⊥ case is smaller than the size of the retangular symbols.

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The α-lactose monohydrate has three distinct resonances below 2 THz of which profiles are in accordance with the Lorentzian line shapes given by [23, 24]

α_{s} (ω) = Im \sum_{n} \frac{S_{n}}{ω_{0, n}^{2} - ω^{2} - i γ_{n} ω},

where S_n, ω_0,n = 2πf_0,n and γ_n are the oscillator strength, the center frequency and the line-width for the n-th resonance modes. To analyze the measured data, we fit the each α_s(ω) obtained by Eqs. (4) and (5), which leads to the result, summarized in Fig. 3. The fitting parameters of the oscillators are obtained from Ref. [24] to be f_0,1=0.53 THz and γ₁=0.025 THz. All the measured data are plotted with closed circles for the ‖-case and with open rectangles for the ⊥-case. The curves are obtained with Eq. (5) and represented with red solid lines for the ‖ case and with black dashed lines for the ⊥ case. We then clearly find, by comparing the absorbances of the two cases shown in Figs. 3(a) and 3(b), that the absorbance of α-lactose disappears at the smallest slit width. Note that the oscillator strength is defined by S = 2σ_max∆f/f² [32], where ∆f is a full width at half maximum (FWHM) and

σ_{max} = f Im [\tilde{ϵ} (f)] / 2

is the optical conductivity at 0.53 THz. The factor of

Im [\tilde{ϵ} (f)]

can be expressed in terms of Eq. (4) as

Im [\tilde{ϵ} (f)] \propto α_{s} / 2 π f

, under the assumption that

Im [\tilde{ϵ} (f)]

is given proportional to the imaginary part of the complex refractive index (

\tilde{n}

) when the real part of

\tilde{n}

is nearly constant. The modified oscillator strength (S) can then be obtained, giving S ≡ [α_s]_max∆f/2πf.

Figure 4 shows the normalized difference of the relative absorbance ∆α_s as a function of the slit width in the both cases. We can see from Fig. 4 that ∆α_s for the both cases are nearly the same above d=17.5±1.3 µm (blue dotted line); however, ∆α_s for the both cases becomes smaller below d=17.5±1.3 µm (red solid line). When the slit width decreases to the smallest slit width among the fabricated slit wedge, ∆α_s for the ‖-case becomes almost zero, which is much smaller than ∆α for the ⊥-case with a difference. Note that, although the E-field inside the slit for the ⊥-case is enhanced [9], α_s implying a ratio between a reference signal and a signal through the sample should be unchanged. ∆α_s for the ⊥-case, however, gradually decreases below d=17.5±1.3 µm, which behavior we speculate is because of the vector nature of the E-field in the near-field. From the experimental results, we therefore may conclude that that the electrically resonant material inside the slit acts as a non-resonant material when the E-field is parallel to the slit orientation.

Fig. 4 Comparison of normalized difference of the relative absorbance, ∆α_s = ([α_s]_max − [α_s]_base), between the ‖ and ⊥ cases.

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5. Conclusion

We have conducted polarization spectroscopy of α-lactose monohydrate confined in a sub-wavelength-scale wedge-shaped metal slit in THz frequency region. Experimentally we have shown that the resonance of this material disappears in the extreme sub-wavelength-scale limit transmission of parallelly-polarized THz waves. This proof-of-principle demonstration confirms our expectation based on Bethe’s diffraction theory that the absorption behavior in the far-field measurement should vanish in the presence of an electrically resonant material restricted into the sub-wavelength slit.

Appendix: Derivation of Eqs. (2) and (3)

Parallel polarization case: When the incident H-field, H₀, is perpendicular to the slit orientation direction, the diffracted E-field is the radiation from the magnetic dipole moment [15], so in the far-field zone it is given as

E (r) = - \frac{c k^{2}}{4 π} \hat{n} \times M \frac{\exp (i k r)}{r}, H (r) = \frac{k^{2}}{4 π μ_{0}} \hat{n} \times (M \times \hat{n}) \frac{\exp (i k r)}{r},

where M, P and

\hat{n}

are the magnetic dipole moment, the electric dipole moment, and the unit field vector (

r = \hat{n} r

), respectively. If there are no electric current and no electric charge, the Maxwell’s equations in free space can be written in terms of the magnetic surface charge density (η) and the magnetic surface curent density (K) [15,33] as

\nabla \cdot H = \frac{η}{μ_{0}}, \nabla \times H = ϵ_{0} \frac{\partial E}{\partial t}, \nabla \cdot E = 0, \nabla \times E = - K - μ_{0} \frac{\partial H}{\partial t} .

A constant H-field in the slit can be produced by an ellipsoidal magnetic dipole distribution having the same direction of the magnetic field [15]. The magnetic charge distribution from an ellipsoid with a height of h along the z′-axis will be identical to a surface charge distribution when h is sufficiently small. The cross section of the ellipsoid in the x′y′ plane becomes approximately a rectangular slit with a width d on the y′-axis and a height L on the x′-axis as shown in Fig. 1(a) at θ = 0. Since the ordinate (z-axis) of the ellipsoid for d ≪ L is proportional to

\sqrt{d^{2} / 4 - y^{' 2}}

[22], we obtain the surface magnetic charge density

η (x^{'}, y^{'}) = - C H_{0} y^{'} / \sqrt{d^{2} / 4 - y^{' 2}},

where C is a proportional coefficient and

H_{0} = {\hat{y}}^{'} H_{0}

, under the assumption that the initial H-field is perpendicular to the slit direction [15].

The coefficient C is obtained as follows. First, we need to calculate the infinitesimal H-field δH induced from the magnetic volume charge density ρ at the infinitesimal displacement δy′, to get ∇·δH = ρ(r′)δ(r − r′)/µ₀. When we apply the Gauss’s law to this equation and assume that the slit is wrapped by the cylindrical Gaussian surface (S) with a radius of y − y′ along the x-axis, we obtain the volume integral over S, which is

\int_{S} δ H \cdot \hat{n} d a = μ_{0}^{- 1} \int_{- L / 2}^{L / 2} d x^{'} \int_{- δ y / 2}^{δ y / 2} d y^{'} η (x^{'}, y^{'}),

where the volume charge density is defined as ρ(r′) = η(x′, y′) δ(z′). It is noted that η is a constant in δy′ and the infinitesimal area (

\hat{n} d a

) over the Gaussian surface is

\hat{n} (y - y^{'}) d ϕ d x

in which ϕ is the azimuthal angle of the Gaussian surface. Using this relation, the integral is evaluated as δH(y − y′)2πL = Lδy′η/µ₀. The H-field in the Gaussian surface can be retrieved by integrating δH over the y′ from −d/2 to d/2, giving [22]

H = - \frac{C}{2 π μ_{0}} \int_{- \frac{d}{2}}^{\frac{d}{2}} d y^{'} \frac{H_{0} y^{'}}{(y - y^{'}) \sqrt{d^{2} / 4 - y^{' 2}}} .

At the optical axis (y = 0), the integration of Eq. (10) gives the surface magnetic charge density as

η (x', y') = - μ_{0} H_{0} y^{'} \sqrt{d^{2} / 4 - y^{' 2}}

. So the magnetic dipole moment M can be obtained from η as

M = \int_{S} d x^{'} d y^{'} η (x^{'}, y^{'}) y^{'} = - \frac{π μ_{0}}{8} d^{2} L H_{0},

where the magnetic dipole is in the direction of y′-axis because η is a function of y′ with no x′-dependence. We find that the dipole moment M is anti-parallel to the initial H-field, H₀, as predicted by Bethe [15]. Therefore the diffracted E-field with harmonic time dependence exp (−iωt) by a small slit can be expressed along the optical axis as

E_{‖} (z, t) = \frac{π^{2}}{8} {(\frac{d}{λ})}^{2} Z_{0} L \hat{n} \times H_{0} \frac{\exp [i (k z - ω t)]}{z} .

Perpendicular polarization: When the incident E-field E₀ is perpendicular to the slit direction, the E-field diffracted by the slit can be thought of as a combination of the radiation from the electric dipole moment and the ordinary diffraction explained by Kirchhoff’s diffraction theory [22, 29]. From Eq. (6), the E-field from the radiation by the electric dipole moment P is equal to the H-field for the magnetic dipole M with the substitution M → P and µ₀ → ϵ₀ given by

E (r) = \frac{k^{2}}{4 π ϵ_{0}} \hat{n} \times (P \times \hat{n}) \frac{\exp (i k r)}{r} .

When K_e and η_e are the electric current density and charge density, respectively, the continuity equation is defined as ∇·K_e = iωη_e [29]. By using Gauss’s law, the relation between K_e and P is easily obtained as

\int_{S} K_{e} d^{2} r^{'} = - i ω P

, where S and r′ indicate the Gaussian surface and the source point. Equation (8) is then expressed in terms of the electric surface current K_e as

E (r) = i \frac{k Z_{0}}{4 π} \frac{\exp (i k r)}{r} \hat{n} \times (\int_{S} K_{e} d^{2} r^{'} \times \hat{n}),

which implies that the E-field is induced by the current [21,22]. Within the spherical Gaussian surface at the edge of the slit, K and the initial E-field E₀ have the relation of

K_{e} / | K_{e} | = - i E_{0} / | E_{0} |

in accordance with the Gauss’s law [22]. When

\hat{n}

is normal to the conducting screen (i.e. the optical axis), the vector terms of Eq. (9) can be simplified by the vector identity as

\int_{S} K_{e} d^{2} r^{'}

, which leads to the E-field with harmonic time dependence, exp (−iωt), by a small slit, giving [21]

E (r, t) = \frac{k Z_{0}}{4 π} \frac{\exp [i (k r - ω t)]}{r} \int_{S} d^{2} r^{'} | \frac{K_{e} (r^{'})}{E_{0} (r^{'})} | E_{0} (r^{'}) .

By Ohm’s law, the magnitude of K is proportional to the induced E-field, E_i, since we assume that the screen is a perfect conductor. It is also known that the E-field perpendicular to the slit direction is strongly enhanced in the slit when the slit width d is small enough compared with the wavelength λ of the incident E-field, which leads to the behavior between the incident E-field E₀ and the induced-E field E_i resulted as |E_i (0)/E₀(0)| ∝ λ/d, where the origin is denoted by 0 [9,12]. Therefore the E-field radiated from the electric dipole is given by

E (z, t) = β L E_{0} (0) \exp [i (k z - ω t)] / z,

where β is a proportional coefficient [22].

To complete the diffracted E-field by a slit at the far-field zone for the ⊥-case, the diffracted E-field based on the Kirchhoff diffraction theory should be also considered as [22]

E (r, t) = - \frac{i k}{2 π} \frac{e^{i (k r - ω t)}}{r} \int_{S} d^{2} r^{'} E_{0} (x^{'}, y^{'}, z^{'} = 0) \exp (- i k r \cdot r^{'} / r) .

Along the optical axis, the diffracted E-field can then be evaluated as

E (z, t) = - i (\frac{d}{λ}) L E_{0} (0) \exp [i (k z - ω t)] / z .

Therefore, the total diffracted E-field in the optical axis for the ⊥-case can be written by combining Eqs. (16) and (18) as

E_{⊥} (z, t) = [β - i (\frac{d}{λ})] E_{0} L \frac{\exp [i (k z - ω t)]}{z} .

where E₀ = E₀(0) is considered as a constant.

Acknowledgments

This research was supported by Samsung Science and Technology Foundation [SSTF-BA1301-12].

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Terahertz polarization spectroscopy in the near-field zone of a sub-wavelength-scale metal slit

Abstract

1. Introduction

2. Experimental description

3. Theoretical background

4. Results and discussion

5. Conclusion

Appendix: Derivation of Eqs. (2) and (3)

Acknowledgments

References and links

Cited By

Figures (4)

Equations (19)

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