Enhanced optical absorption and electric field resonance in diabolo metal bar optical antennas

Zeyu Pan; Junpeng Guo

doi:10.1364/OE.21.032491

1. Introduction

Optical antennas [1–4] are potentially useful for many applications such as molecular bio-sensing [5, 6], surface enhanced Raman spectroscopy (SERS) [7, 8], nonlinear optics [9], photochemistry [10], enhanced photo-detection [11, 12], and nanolithography [13, 14]. Optical antennas with various configurations such as bowties [15–28], metal nanorods [29–33] and nanoparticles [34–38] have been investigated in the past few years. Recently, rectangular metal bar optical antennas are also investigated [39–47]. Previous efforts on metal bar optical antennas focused on the effects of rectangular metal bar dimensions, array period, and substrate materials. In this paper, we investigate the fundamental mode resonance behavior of narrow waist metal bar optical antennas on a transparent substrate. Narrow waist metal bar optical antennas have similar geometric configuration as bowtie optical antennas [15–28]. A bowtie optical antenna consists of two triangular shape metal patches separated with a small gap. Electromagnetic field enhancement is inside the gap between two metal patches. A narrow waist metal bar optical antenna also consists of two triangular shape metal patches that are physically connected together without a gap. The strong electromagnetic field enhancement is near the two ends of the metal bars similar as the rectangular metal bar optical antennas. The reduced waist width of the metal bars provides an additional freedom to control the resonance behaviors of the optical antennas. Because the geometric configuration of narrow waist metal patch or metal bar optical antennas exhibits the configuration of diabolos, they are also called “diabolo antennas” [48, 49]. Previous works on diabolo antennas focused on the magnetic field enhancement caused by the reduction of the waist of diabolo antennas. In this work, we focus on the electric field enhancement of diabolo metal bar optical antennas and the effect of the reduced waist width on absorption and scattering cross-sections. In this paper, we first present finite-difference time-domain (FDTD) simulations for absorption and scattering cross-sections of narrow waist diabolo metal bar antennas of different waist widths. Then, we use a dipole oscillator model including the self-inductance force to fit the FDTD simulation results. Finally, we show that diabolo metal bar antennas have strongly enhanced electric field near two ends of the metal antenna bars and strongly enhanced electric field surrounding the metal antennas.

2. Surface plasmon resonance of diabolo metal bar optical antennas

Figure 1 shows a diabolo shape narrow waist gold metal bar optical antenna on a sapphire substrate. The top view of the diabolo antenna is shown in the insert of Fig. 1. The physical length of the metal bar l is 1.1 μm in the x direction. The width of the bar is 260 nm at the two ends of the antenna bar in the y direction. The thickness of the metal film h is 65 nm, uniformly in the z direction. The waist width in the middle of the antenna bar is w in the y direction. When the antenna assumes a rectangular shape, the waist width is equal to the width at the ends, i.e. w = 260 nm. The excitation optical wave is a plane wave normally incident onto the metal antenna from the top in the air with the polarization in the x direction and propagates in the z direction.

Fig. 1 A three-dimensional (3D) perspective of a narrow waist gold metal bar optical antenna on a sapphire substrate. The insert above shows the top view of the diabolo optical antenna.

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Using a finite difference time domain (FDTD) software code developed by the Lumerical Solution, Inc., we investigate the resonance properties of diabolo metal bar optical antennas of different waist widths. FDTD simulations obtain the full vectorial solutions of the Maxwell’s equations to find the optical response of the antennas with a plane wave excitation. Perfectly matched layer (PML) boundary conditions are used in the all boundaries of the simulation region. The simulation region has a volume of 2.9 μm by 2.9 μm in the x and y directions and 3.5 μm in the z direction. The electric permittivity of the gold film and the optical constant of the sapphire substrate are obtained from Ref [50]. A total-field scattered-field source (TFSF) is used in all simulations. The TFSF source injects a short pulse with a temporal Gaussian envelope. Two analysis groups, each of which consists of a box of power monitors, lay in the scattering-field and total-field region. The scattering-field group is used to collect the net optical power scattered from the metal bar antenna, while the total-field group is used to get the absorbed power. The scattering and absorption cross-sections are obtained by dividing the scattered power and the absorbed power by the intensity of the incident optical wave.

Scattering cross-section and absorption cross-section are calculated versus the free space wavelength for different metal bar antennas. The results are plotted in Fig. 2. It can be seen that there are two resonance modes in the metal bar optical antennas. One is the fundamental resonance mode that occurs at a longer wavelength. Another is the high order resonance mode that occurs at a shorter wavelength. The high order mode resonance is much weaker than the fundamental mode resonance. In this work, we focus only on the fundamental mode resonance. Figure 2(a) shows scattering cross-section versus the wavelength for metal bar waist widths of w = 260 nm, 210 nm, 160 nm, 110 nm, 63 nm, and 30 nm. The peak scattering wavelengths are 4.03 μm, 4.20 μm, 4.38 μm, 4.54 μm, 4.71 μm, and 4.93 μm, respectively. It also can be seen that as the waist width decreases, resonance wavelength shifts to longer wavelength, and the scattering cross-section first increases, then decreases. The scattering cross-section reaches the maximum at the waist width of 63 nm. Figure 2(b) shows the absorption cross-section versus the wavelength for diabolo gold bar antennas with different waist widths w = 260 nm, 210 nm, 160 nm, 110 nm, 63 nm, and 30 nm. The peak absorption wavelengths are 4.12 μm, 4.42 μm, 4.48 μm, 4.64 μm, 4.77 μm, and 4.99 μm, respectively. The peak absorption cross-section wavelengths are slightly longer than the peak scattering cross-section wavelengths. As the waist width decreases, both the scattering and absorption cross section peak wavelengths shift to longer wavelengths and the absorption cross-section increases significantly.

Fig. 2 (a) Scattering cross-section and (b) absorption cross-section of narrow waist gold bar diabolo antennas versus the free space wavelength.

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To understand the resonance behavior of the narrow waist metal bar optical antennas, we use a dipole oscillator model to fit the numerical simulation results [51–55]. In the dipole oscillator model, the electron location is at x(ω, t). Each electron has the electric charge of q_e = 1.60 × 10⁻¹⁹ C and the electron mass m_e = 9.11 × 10⁻³¹ kg. In the dipole oscillator model, the spring constant of the dipole is κ. The oscillator is driven by a harmonic electric field with angular frequency ω. The internal damping coefficient of the dipole oscillator is Γ_a. The radiation coefficient of the oscillator is Γ_s [51]. The self-induced inductance force due to the electric charge oscillation is characterized as L_l with the unit of henry per meter (H/m). The self- inductance force is proportional to the change rate of the electrical current as

F = - q_{e} L_{l} \frac{d I}{d t} = - q_{e}^{2} L_{l} \frac{d^{2} x (ω, t)}{d t^{2}} .

The dipole oscillator model can be written as

m_{e} \frac{d^{2} x (ω, t)}{d t^{2}} + Γ_{a} \frac{d x (ω, t)}{d t} + κ x (ω, t) = q_{e} E_{0} e^{j ω t} + Γ_{s} \frac{d^{3} x (ω, t)}{d t^{3}} - q_{e}^{2} L_{l} \frac{d^{2} x (ω, t)}{d t^{2}} .

By moving the inductance force term to the left side of the equation and defining the effective electron mass

m_{e}^{*} = m_{e} + q_{e}^{2} L_{l}

, we have

m_{e}^{*} \frac{d^{2} x (ω, t)}{d t^{2}} + Γ_{a} \frac{d x (ω, t)}{d t} + κ x = q_{e} E_{0} e^{j ω t} + Γ_{s} \frac{d^{3} x (ω, t)}{d t^{3}} .

Assuming the electron motion x(ω, t) = x₀(ω) e^jωt, the solution of Eq. (3) can be obtained as

x (ω, t) = x_{0} (ω) e^{j ω t} = \frac{(q / m_{e}^{*}) E_{0}}{ω_{0}^{2} - ω^{2} + j \frac{ω}{m_{e}^{*}} (Γ_{a} + ω^{2} Γ_{s})} e^{j ω t} .

In Eq. (4), x₀(ω) is the complex amplitude of the dipole oscillation and

ω_{0} = \sqrt{κ / m_{e}^{*}}

. Time-averaged power dissipated by the oscillator can be written as P(ω) = F*(ω)[jω x₀(ω)], where F*(ω) is the complex conjugate of the force applied to the electrons. Therefore, the total scattered and absorbed power can be written as

P_{s c a} (ω) = N^{2} ω^{4} Γ_{s} {| x_{0} (ω) |}^{2}

P_{a b s} (ω) = N^{2} ω^{2} Γ_{a} {| x_{0} (ω) |}^{2}

where N is the number of electrons participated in the oscillation. The scattering cross-section C_sca (ω) and the absorption cross-section C_abs (ω) can be obtained as

C_{s c a} (ω) = P_{s c a} (ω) / I_{0}

C_{a b s} (ω) = P_{a b s} (ω) / I_{0},

where

I_{0} = \frac{1}{2} \sqrt{\frac{μ_{_{0}}}{ε_{_{0}}}} {| E_{0} |}^{2}

is the intensity of the incident optical wave.

We use the dipole oscillator model to fit the numerically calculated scattering and absorption cross-sections in Fig. 2. From Eqs. (5)-(8), we can see that ratio of the radiation coefficient Γ_s to the internal damping coefficient Γ_a is

\frac{Γ_{s}}{Γ_{a}} = \frac{1}{ω_{}^{2}} \frac{C_{s c a t} (ω)}{C_{a b s} (ω)} .

Because Γ_s and Γ_a are frequency independent, we have

\frac{Γ_{s}}{Γ_{a}} = \frac{1}{ω_{o}^{2}} \frac{C_{s c a t} (ω_{o})}{C_{a b s} (ω_{o})} .

Equation (10) provides a constraint for Γ_s and Γ_a in the dipole oscillator model.

Figure 3 shows the dipole oscillator model fitting for the scattering cross-section and the absorption cross-section of diabolo optical antennas with waist width of w = 30 nm, 63 nm, 110 nm, 160 nm, 210 nm, and 260 nm respectively. The solid line curves are FDTD simulation results. The dashed lines are fitted curves with the dipole oscillator model. The blue and red colors indicate the scattering cross-section and absorption cross-section, respectively. It can be seen that the dipole oscillator model fits well the FDTD simulation results.

Fig. 3 Dipole oscillator model fittings for scattering and absorption cross-sections of diabolo metal bars with different waist widths of (a) 30 nm, (b) 63 nm, (c) 110 nm, (d) 160 nm, (e) 210 nm, and (f) 260 nm.

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In Table 1, we summarize the physical parameters used in the oscillator model for fitting the FDTD simulation results. The physical parameters are: effective electron mass m_e*, intrinsic oscillation frequency ω_o, internal damping coefficient Γ_a, radiation coefficient Γ_s, and number of electrons N participated in the oscillation. In the table, we list the ratio of the effective electron mass to the electron rest mass m_e*/m_e. It can be seen that as the waist width is reduced, the effective mass of electrons increases due to the increase of the self-inductance. It has been known that as the width of a metal strip is reduced, the self-inductance increases [56–59]. Here we show that the increase of self-inductance can be characterized by the increase of the effective electron mass in the dipole oscillator. It also can be seen that as the waist width of the diabolo antenna decreases, the internal damping coefficient Γ_a increases. This is due to the increase of the resistance and the energy absorption in the narrow waist metal antennas. From Fig. 3, it can be seen that as the waist width is reduced from 260 nm to 30 nm, the full width at half maximum (FWHM) line-width of the resonance curve decreases, although the internal damping coefficient Γ_a and the absorption cross-section increase. This is because the optical absorption cross section is much smaller than the scattering cross section and the effective electron mass m_e* increases quickly as the waist of diabolo metal bar is reduced. From Eq. (4), the line-width is approximately $(Γ_{a} + ω_{o}^{2} Γ_{s}) / m_{e}^{*}$ . The increase of the electron mass reduces the line-width more effectively than the line-width widening caused by the increase of the internal damping.

Table 1. Physical parameters in the dipole oscillator model for the diabolo optical antennas

View Table

From the fitting parameters shown above, it can be seen that the number of electrons participated in oscillation N decreases due to the reduction of the waist width and the volume. After dividing the number of electrons by the metal antenna volume, it is found that number of electrons per unit volume N/V increases. N/V equals to 2.90 x 10²⁹, 3.13 x 10²⁹, 3.42 x 10²⁹, 3.75 x 10²⁹, 4.10 x 10²⁹, and 4.28 x 10²⁹m^-3 for waist width w=260 nm, 210 nm, 160 nm, 110 nm, 63 nm, and 30 nm, respectively. This indicates more electrons participate in the oscillation in term of the density. It is known that due to the skin effect, not all electrons in metal antennas interact equally with the excitation electromagnetic field. Also, because the metal antenna is placed on a dielectric substrate, the coupled bound electrons in dielectric substrate also contribute to the oscillation. Additionally, we calculate the spring constant κ = m_e*ω_ο² with the oscillation frequency ω_o and the effective electron mass m_e*. The spring constant κ is 0.287, 0.289, 0.286, 0.279, 0.265, and 0.246 kg/s² for the waist width of w = 260, 210, 160, 110, 63, and 30 nm, respectively. The spring constant decreases slightly as the waist width decreases, which indicates the attraction force between the positive and negative charges in the dipole oscillator becomes weaker as the waist width is reduced.

For biochemical sensor applications, localized electric field enhancement is most desirable. To investigate the electric field enhancement, a near field point monitor is used and located 20 nm distance from one end of the metal bar along the longitudinal axis of the antenna. The distance of the near field point monitor from the sapphire substrate surface is 32.5 nm, the half of the gold film thickness. The incident optical wave is from the above of diabolo metal bars in the air as described in Fig. 1. We calculated the near electric field intensity at this point location as a function of the wavelength for different waist width antennas. Figure 4 shows the electric field intensity at the point monitor versus the wavelength for metal bars with the waist width of w = 260 nm, 210 nm, 160 nm, 110 nm, 63 nm, and 30 nm. The resonance wavelengths are 4.377 μm, 4.423μm, 4.477μm, 4.567 μm, 4.685 μm, and 4.849 μm, respectively. It can also be seen that there are two resonance modes in the narrow waist metal bars. One is the fundamental resonance mode that occurs at a longer wavelength. Another is the high order resonance mode that occurs at a shorter wavelength. The high order mode resonance is much weaker than the fundamental mode resonance. Here we focus only on the fundamental mode resonance. From Fig. 4, it can be seen that as the waist width is reduced, the resonance wavelength shifts to longer wavelengths and the near field at the resonance is enhanced due to the reduction of the waist width. As the waist width decreases, the full width at half maximum (FWHM) of the near field resonance curve also decreases.

Fig. 4 (a) Electric field intensity at a point monitor near one end of narrow waist diabolo metal bar antennas versus the free space wavelength. (b) Dipole oscillator model fitted near electric field resonance curves.

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We calculated the electric field intensity distributions of narrow waist metal bar diabolo optical antennas with waist widths of 260 nm, 210 nm, 160 nm, 110 nm, 63 nm, and 30 nm at their near field resonance wavelengths of 4.377 μm, 4.423μm, 4.477μm, 4.567 μm, 4.685 μm, and 4.849 μm respectively. The two-dimensional electric field intensity distributions at the middle plane of the gold metal bar antennas (32.5 nm above the sapphire substrate surface and 32.5 nm below the gold film top surface) are plotted in the logarithmic scale in Fig. 5. It can be seen that the metal bar diabolo antennas have stronger near electric field enhancement than the rectangular metal bar antennas. This property is very desirable for biochemical sensor applications that demand strong near field enhancement and large surface interaction areas.

Fig. 5 The electric field intensity distribution at the resonance wavelengths of narrow waist metal bar antennas with waist widths: (a) w = 30 nm, (b) w = 63 nm, (c) w = 110 nm, (d) w = 160 nm, (e) w = 210 nm, and (f) w = 260 nm. Their resonance wavelengths are (a) 4.849 μm, (b) 4.685 μm, (c) 4.567 μm, (d) 4.477 μm, (e) 4.423 μm, and (f) 4.377 μm, respectively.

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For the biochemical sensor applications, targeted molecules are bonded on the metal bar surfaces. Here we define an integration parameter U as the integration of the electric field intensity over a contour of 10 nm outside of the diabolo metal bar on the middle plane of the antenna bar (32.5 nm above the substrate) at the near field resonance wavelength, i.e.

U = \oint | E |^{2} d l .

The integration contour is shown as the white dashed line in the insert of Fig. 6(a). The resonance enhanced electric field intensity integration versus the waist width is plotted and shown in Fig. 6(a). It can be seen that as the waist width decreases, the near field intensity integration increases, which indicates more electric field energy around the diabolo antennas than the rectangular metal bar antenna. To diminish the effect of surrounding physical contour length increase due to the narrowing of the metal bar waist, we define an averaged near electric field intensity as

〈 {| E |}^{2} 〉 = \frac{\oint | E |^{2} d l}{\oint d l} .

Averaged near electric field intensities at the resonance wavelengths are also calculated for diabolo antennas of different waist widths. The result is shown in Fig. 6(b). It can be seen that as the waist width decreases, the averaged near electric field intensity increases. This property is very desirable for biochemical sensor applications.

Fig. 6 (a) The line integration of resonance enhanced near electric field intensity surrounding diabolo metal bar antennas. The insert shows the integration contour 10 nm outside of the diabolo antenna. (b) The averaged near electric field intensity at resonance versus the waist width.

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3. Summary

In this paper, we investigated the resonance properties of diabolo shape metal bar optical antennas by using FDTD simulations and a dipole oscillator model. The scattering and absorption cross-sections and the near field resonance of diabolo metal bar optical antennas were calculated with FDTD numerical simulations and then fitted with a dipole oscillator model including the self-inductance force. It is found that by reducing the waist width of the diabolo metal bars, resonance wavelengths found in the scattering cross-section, absorption cross-section, and the near electric field all shift to longer wavelengths. As the waist width decreases, absorption cross-section increases dramatically and near field enhancement at the resonance increases significantly. In the dipole oscillator model, the self-inductance force is first included to describe the resonance behaviors of the metal bar optical antennas. It is found that as the waist width of the diabolo metal bar is reduced, the self-inductance force increases, which is indicated as the increase of the effective electron mass in the dipole oscillator model.

Although diabolo metal bar optical antennas investigated have resonance wavelengths in the infrared region, findings in this work can be applied to other wavelength regions by scaling the dimensions of the metal antennas. Diabolo metal bar optical antennas support stronger near field resonance than rectangular metal bar antennas. Additionally, diabolo metal bar optical antennas have larger optical absorption cross-sections than rectangular metal bar antennas. The new optical resonance properties of diabolo optical antennas reported in this paper are potentially useful for optical sensing and energy harvesting applications.

Acknowledgments

This work is partially supported by the National Science Foundation (NSF) through the award NSF-0814103 and the National Aeronautics and Space Administration (NASA) through the grant NNX12AI09A. Zeyu Pan acknowledges the support from the Alabama Graduate Research Scholarship Program.

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Waist width (nm)		ω_ο (rad/s)	Γ_a (kg/s)	Γ_s (kg∙s)	N
260	1.412	4.72 x 10¹⁴	1.41 x 10⁻¹⁷	8.66 x10⁻⁴⁶	5.39 x 10⁹
210	1.523	4.57 x 10¹⁴	1.53 x 10⁻¹⁷	8.44 x 10⁻⁴⁶	5.26 x 10⁹
160	1.595	4.44 x 10¹⁴	1.68 x 10⁻¹⁷	7.99 x 10⁻⁴⁶	5.13 x 10⁹
110	1.670	4.28 x 10¹⁴	1.87 x 10⁻¹⁷	7.30 x 10⁻⁴⁶	4.97 x 10⁹
63	1.722	4.11 x 10¹⁴	2.16 x 10⁻¹⁷	6.38 x 10⁻⁴⁶	4.74 x 10⁹
30	1.763	3.91 x 10¹⁴	2.61 x 10⁻¹⁷	5.34 x 10⁻⁴⁶	4.44 x 10⁹

Enhanced optical absorption and electric field resonance in diabolo metal bar optical antennas

Abstract

1. Introduction

2. Surface plasmon resonance of diabolo metal bar optical antennas

3. Summary

Acknowledgments

References and links

Cited By

Figures (6)

Tables (1)

Equations (12)

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