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We simulate the early dynamics of enhanced light transmission through a subwavelength metallic slit and find that the amplitude of the transmitted light can be modulated. To understand this novel phenomenon and underlying physics, we develop a new analytical model. The field of each light period is considered as an individual unit. Each field is partially transmitted through the slit as the first subunit. The portion reflected from the exit interface travels a round trip in the slit and then partially exits again as the second subunit. There may be a gap in time between these two subunits. This process repeats so as to produce a subunit train, which is verified by the simulation of an incident sinusoidal pulse of one light period. When the wave units are continuous, the superposition of the trains produces the observed light. While the round-trip time is an integer multiple of the light period, the modulation period is the same. Besides academic importance, this study may be applicable to photonics with short laser pulses.
© 2015 Optical Society of America
1. Introduction
The extraordinary optical transmission (EOT) through a subwavelength hole array in a metallic film [1] is a fundamental scientific topic in wave mechanics and inspires researchers in the area of plasmonics to explore the possibility of applications on optical devices, such as ultrafast active plasmonics [2], particle trapping [3], bio-sensing [4], plasmonic solar cells [5], optical antennas [6], color filters [7], and light shape manipulation [8–11], etc. A single subwavelength slit is the simplest configuration to understand the fundamental mechanism and its related optical properties. For simplicity, the metal can be considered as a perfect electric conductor (PEC) for the investigation of interesting transmission phenomena. Although PEC does not support surface plasmons (SPs) [12, 13], the wave-SP-like interaction can still be studied with structured PEC because of the so-called” spoof SPs” [14].
In the last few decades, extensive theoretical works have been carried out for understanding the steady light transmission through this kind of configuration. For instance, Neerhoff and Mur [15] introduced the Green’s theorem for the calculation of light transmission through a slit with the width larger than the incident wavelength; Betzig et al. [16] confirmed its accuracy when the slit is of subwavelength scale. Harrington and Auckland [17] proposed an equivalent transmission line model for this problem. Both theories predict the transmitted light intensity varying periodically with the film thickness. In fact, the slit can be considered as an open cavity to allow a Fabry-Pérot-like resonance resultant from the reflected waves traveling back and forth in the slit [18]. The resonance has also been theoretically investigated by a mode expansion method [19, 20], as well as the phase shift of the reflection inside the slit [21].
Mechler et al. [22] analyzed the transmission of Gaussian pulses with only a few light periods and pointed out that there are three regimes of resonant excitation according to their temporal pulse widths. In fact, the time-dependent analysis was obtained by the inverse Fourier transform from the solution of Neerhoff and Mur at each frequency, i.e., solving the phasor of the time-harmonic electric and magnetic fields from Maxwell’s equations [23].
In this article, periodical variation of the transmitted energy as a function of thickness and transmission spectra due to Fabry-Pérot-like resonance for a subwavelength slit are demonstrated from monochromatic sources or a sinusoidal pulse of one light period in a finite-difference time-domain (FDTD) simulation system. With a monochromatic light, we find that the transmitted electric field through the configuration can be monotonically increased at the early stage. In fact, the increment is periodically modulated before the amplitude is saturated. The saturation time increases with the film thickness and requires at least four wave periods of the light wave, which is comparable to the time scale of a femtosecond (fs) laser pulse. We propose a new analytical model in which the wave field generated in each light period is considered as an individual unit. Each wave unit can produce a train of subunits due to partial reflections within the slit at its entrance and exit interfaces as well as the propagation in between. The repeating process is verified by the sinusoidal pulse illumination. The superposition of the wave units at the observation location yields the wave fields that agree well with the simulation results. This model illuminates the underlying new physics.
In section 2, we show the Fabry-Pérot-like resonance for the configuration and the simulation results for the increment and the modulation of the electric fields. Section 3 gives the novel analytical model. We obtain the parameters needed by the model from the FDTD simulation in section 4. In section 5, the modeling results are compared with those of the simulation. Section 6 is the discussions on the numerical issues and the implication of our model for ultrashort applications. Section 7 is the summary.
2. The Fabry-Pérot-like resonance and modulated light transmission at early stage
Consider a subwavelength slit perforated in a PEC film, as shown in Fig. 1. The slit width 2s is 40 nm and the film thickness h will vary. A p-polarized wave source of wavelength λ0 = 560 nm is located at a distance of one wavelength away (d = λ0) to illuminate the film. The light period is T = λ0/c0 = 1.87 fs, where c0 is the light speed in vacuum.
Fig. 1 Schematic of the simulated structure. A p-polarized wave with its wavelength of λ0 = 560 nm is generated at a distance of one λ0 away from the top of the PEC. The slit width 2s is 40 nm while the film thickness h will vary. In the model, the space is divided into three areas of a, b, and c.
In the FDTD simulation, the cell size of both directions is 5 nm, and the time-step is 0.005T. Perfectly matched layers (PMLs) are used as the boundary condition. The PEC is simulated by the Drude model with a sufficiently high plasma frequency ωp (= 1.0 × 1030 Hz) and the damping coefficient set to be zero.
The time dependence of the source is given as sin(ω0t) at y = h + d, where ω0 is the angular frequency. Hence, the electric field of the incident plane wave is in the x-direction and taken as the imaginary part of the complex electric field, i.e., , where y′ = y−(h+d), k0 is the wave number in vacuum, and E0 is the amplitude normalized to unity. The magnetic field in the z-direction is , where η0 is the impedance of vacuum, which is also normalized to unity.
When the incident wave propagates to the film, it will be transmitted through the slit and diffracted as a cylindrical wave. At an observation location (x = 0,y) far from the slit in region c, we can define the normalized transmitted energy flux as
where ex(0,y,t) and hz(0,y,t) are the simulated electric and magnetic fields, respectively, and tf = 40T is the FDTD simulation time to ensure the time-harmonic oscillation of the fields; in the definition, the transmitted energy flux is the time-averaged Ponyting vector multiplying while the incident energy flux is . Figure 2 shows the obtained P(ω0) at y = − λ0, where h varies from 50 nm to 800 nm in a step of 5 nm. The energy flux varies periodically because of Fabry-Pérot-like resonance [19, 20]. At optimized transmission, the corresponding film thicknesses h’s are 205 nm, 470 nm, and 730 nm, and the obtained energy fluxes are 4.28, 4.26, and 4.28, respectively.To examine the transmission spectrum for the three optimized film thicknesses, we use a sinusoidal wave field of one light period as the incident pulse:
where rect is the rectangular function. The normalized transmitted energy flux can be defined as where the Fourier transform is used for the simulated and incident fields. The obtained energy flux P(ω) as a function of wavelength λ is shown in Fig. 2(b), 2(c), and 2(d) for h = 205 nm, 470 nm, and 730 nm, respectively. The wavelengths yielded at corresponding peak transmission are 569 nm, 564 nm, and 560 nm, respectively. At λ = λ0, the normalized energy fluxes are 4.28, 4.27, and 4.28, respectively, which are in good agreement with those obtained in Fig. 2(a). To verify the spectra, we also use monochromatic sources with its wavelength λ from 400 nm to 700 nm in a step of 10 nm to obtain the normalized energy flux. The results are shown in Fig. 2; they coincide with those from the incident pulse.Fig. 2 Normalized energy flux as a function of the film thickness h (a), and as a function of wavelength when h = 205 nm (b), 470 nm (c), and 730 nm (d), respectively, where the blue solid curve is from the sinusoidal pulse of one light period, and the red dots are from the monochromatic sources.
We further show in Fig. 3 the history of the transmitted electric fields observed at the location (0, −λ0) for the three optimized film thicknesses from the monochromatic source of wavelength λ0. We find that when h = 205 nm, the amplitude is monotonically increased to a constant within a few periods; when the film is thicker, the field is periodically modulated, and it takes a longer time to saturate. For h = 470 nm and 730 nm, the modulation periods are 2T and 3T, respectively, while the amplitudes at saturation are similar. We believe that the monotonic increment and periodical modulation in the transmitted fields result from the Fabry-Pérot-like resonance.
Fig. 3 The history of the electric fields at (0, λ0) from the simulation (solid blue curve) and the modeling result of M = 20 and N = 10−(dashed red curve), where h = 205 nm (a), 470 nm (b), and 730 nm (c), respectively.
3. Analytical model based on Fabry-Pérot-like resonance
Despite the fact that it is a continuous wave source, we discretize the wave field generated in each light period as an individual unit. The electric field in the x-direction of the m’th wave unit at the source location (x,h + d) is given as
where m is a positive integer, and tm = (2m − 1)T/2 is the middle time of the m’th unit.Each wave unit propagates to the film surface in the time of d/c0 after generated. It is partially transmitted through the slit entrance interface. The unit peak is delayed and amplified due to the accumulation of the surface charges to draw adjacent energy into the slit as the so-called funnel effect [18, 24]. When the transmission coefficient is τab, the time-delay and the amplification ratio are [arg(τab)/(2π)]T and |τab|, respectively. The transmitted unit propagates to the slit exit in the time of (h/λs)T, where λs = 0.94λ0 is the wavelength in the slit measured from our FDTD simulation (the wavelength shift will be discussed later). Then, it will be partially transmitted out through the exit interface as the first subunit, n = 1, where n is the index of subunit. The charge accumulation at the exit surface delays the peak by [arg(τbc)/(2π)]T and amplifies the subunit by a ratio of |τbc|, where τbc is the transmission coefficient at the exit. After exiting the slit, the subunit diffracts as a cylindrical wave in the area c and reaches the observation location (0,y) in the time of |y|/c0. Thus, the middle time for the n = 1 subunit to propagate from the source location to the observation location is tm1 = tm + d/c0 + ts + |y|/c0, and its amplitude is Em1(y) = α(y)E0|τabτbc|, where ts = [arg(τab)/(2π)]T +(h/λs)T + [arg(τbc)/(2π)]T and α(y) is the decay coefficient of the field due to diffraction.
Besides exiting from the slit, the unit is partially reflected at the exit interface and travels back. Then, it is reflected at the entrance interface and propagates toward the exit again. The corresponding time-delay and amplitude decay due to reflection are [arg(ρabc)/(2π)]T and |ρabc|, respectively, where ρabc is the coefficient of the two reflections. The time of the round trip is estimated as trt = (2h/λs)T + [arg(ρabc)/(2π)]T. Then, the unit will be partially transmitted through the exit interface as the second subunit (i.e., n = 2) and partially reflected again at the exit interface. The middle time for the n = 2 subunit to arrive at the observation location is thus tm2 = tm + d/c0 + ts + trt + |y|/c0, and the amplitude is Em2(y) = α(y)E0|τabρabcτbc|. This process repeats when the wave unit delivers more subunits, which is similar to what is considered in the Fabry-Pérot-like resonance [18, 19].
The middle time of the n’th subunit to arrive at the observation location is
while the amplitude isWe assume that the field of the subunit maintains the ideal sinusoidal shape of one light period. Thus, the electric field of the n’th subunit of the m’th wave unit is given as
According to the superposition principle, the field of subunits produced by each wave unit can be summed. Therefore, the observed electric field is
where M (N) is the maximum number of the wave unit (subunit) included.The decay coefficient α(y) can be obtained from the energy conservation after the wave amplitude is saturated. The time-averaged Poynting vector in the y-direction at the location (0,y) is
where E(y) is the amplitude of the observed field in Eq. (9), and Z(y) is the complex impedance. The energy flux propagating through the slit should be equal to that through the semi-circumference π|y| at y ≪ 0. Thus, we haveTherefore, the decay coefficient by the definition is
Note that Eq. (12) is valid only for y ≪0.
The interference between exiting wave units in our model is similar to the multiple scattering formalism [25]. However, our model is an initial value problem in time domain while the multiple scattering is a boundary value problem in spatial domain.
4. Model parameters obtained from FDTD simulation
A slit in a film of thickness h = 10 μm is simulated to obtain the parameters needed by the model. Figures 4(a) shows the history of the simulated electric field at the entrance center of the slit. Because d = λ0, the phase zero of the first unit arrives at t = T. However, the observation is made at the arriving time of the field peak (i.e., the phase π/2). Since the phase π/2 and the peak is 0.25T after the phase zero, the peak arrives at 1.25T. With the film presented, we observe that the arriving time of the peak is 1.31T, as listed in Table 1. Hence, the phase shift is arg(τab) = (1.31T − 1.25T)×(2π/T) = 0.12π. The transmission time-delay remains constant for all wave units.
Fig. 4 The history of the simulated electric field at the centers of the entrance (a) and the exit (b) from the simulation of h = 10 μm.
Table 1. The time of first five peaks arriving at the centers of the entrance and the exit as well as their corresponding electric field amplitudes obtained from the simulation of h = 10 μm.
The first amplitude is larger than unity, as shown in Fig. 4. Then, the following amplitudes are slightly decreased and remain constant. The amplitude at steady is used to calculate the amplification ratio. As listed in Table 1, we obtain |τab| = 1.32.
Figure 4(b) shows the electric field at the exit. The first peak arrives at t = 20.31T, as listed in Table 1. Hence, the phase shift at the exit interface is arg(τbc) = [20.31T −d/c0 (h/λs)T−0.25T] × (2π/T) − arg(τab) ≃0.01π.
Compared with that at the entrance, the electric field is further increased at the exit. The amplitude at steady is used to calculate the amplification ratio, i.e., |τbc| = 2.13/τab = 1.61.
The simulation shows that the magnetic field peak arrives 0.17T before the peak of the electric field, and the steady amplitude is 1.10 (not shown). The impedance at the exit is thus estimated to be Z(0) = (2.13/1.10)exp[i(−0.17) ×2π] = 1.94exp(−i0.34π). The impedance at the observation location is Z(−λ0) = 1.01exp(−i0.06π).
The reflected wave cannot be easily separated from the incident wave because of the interference. For the case of h = 205 nm, the transmitted electric field amplitude at the early stage is monotonically increased due to the constructive interference condition trt = T and tmn = tm′n′ if m + n = m′ + n′, where m′ and n′ are positive integers. Since
we obtain arg(ρabc) ≃ 0.44π. The peak at y = 0 yielded from Eq. (9) for M ≫ 1 can be simplified as where α(0) = 1 by the definition. The peak value obtained from the simulation is 4.12 for M = 10 (not shown) so that |ρabc| ≃ 0.48.5. Modeling results and comparison with simulation
The results yielded from Eq. (9) for the m = 1 wave unit and N = 10 are shown in Fig. 5, where h is selected for optimizing the transmission. All their round-trip times are close to a multiple of T. For the h = 205 nm case, ts ≃ 0.46T, t11 = t1 + d/c0 + ts + |y|/c0 ≃ 2.96T, t1n = t11 + (n − 1)trt, where y = −λ0 and trt = T. In this case, the end of then = 1 subunit coincides with the beginning of the n = 2 subunit which just finishes a round trip; but, the amplitude of the later is decreased by the factor of |ρabc|. At the end of the n = 2 subunit, the n = 3 subunit begins, and its amplitude is further decreased by one more factor of |ρabc|. This repeats till it fades out, as shown in Fig. 5(a).
Fig. 5 The history of the electric field at (0, −λ0) of the wave unit m = 1 obtained from the model, where N = 10, h = 205 nm (trt = T) (a), 470 nm (trt ≃ 2T) (b), and 730 nm (trt ≃ 3T) (c), respectively. The grey curves show the history of the electric field from the sinusoidal pulse of one light period, while the colored curves are from a continuous sinusoidal wave source.
For the h = 470nm case, ts ≃ 0.96T, t11 ≃ 3.46T, and trt ≃ 2T because h is almost 0.5λs thicker than 205 nm. As shown in Fig. 5(b), after the n = 1 subunit ends, there is a gap of about one T because the n = 2 subunit is still in its round trip. Then, it arrives with the middle time t12 ≃ 5.46T. The n = 3 subunit arrives with the middle time t13 ≃ 7.46T. This also repeats till the field fades out. For the h = 730 nm case, ts, t11, and trt are further increased approximately to 1.46T, 3.96T, and 3T, respectively, while the gap width is about 2T, as shown in Fig. 5(c). In these two later cases, the fading time is longer than the first case of h = 205 nm. The history of the simulated electric fields from the sinusoidal pulse of one light period passing through these films are shown in Fig. 5. The simulation results (e.g., the repeating subunits) generally agree with those from the modeling.
For a continuous source, the (m + 1)’th wave unit is one T later than the m’th unit, and so are their subunits. The observed electric field is from the superposition of all units. For all these cases, the n = 1 subunit of the m = 1 unit is the first and only one that arrives at the observation location, as shown in Fig. 3. For the h = 205 nm case shown in Fig. 3(a), t21 = t12 because trt = T. The second arriving period of the observed field is from the in-phase interference between the m = 2, n = 1 subunit and the m = 1, n = 2 subunit. The amplitude is thus increased by |ρabc|. Then, the m = 3, n = 1 subunit arrives at the next period, i.e., t31 = t22 = t13. Since the amplitudes of the subunit of the m = 2 and m = 1 units have decayed by |ρabc| and |ρabc|2, respectively, the observed field amplitude has been increased by |ρabc| + |ρabc|2. The in-phase interference continues till the amplitude saturates.
For the h = 470 nm (i.e., trt ≃ 2T) case shown in Fig. 3(b), the m = 2, n = 1 subunit with the same amplitude as the m = 1, n = 1 subunit fills the first temporal gap after the later’s end while there is no overlapping with other subunits. This makes the first modulation period of 2T. At the third period of the observed field, the m = 1, n = 2 subunit, the first gap of the m = 2 unit, and the m = 3, n = 1 subunit overlap; that is, t31 = t12. While, at the forth period, the second gap of the m = 1 unit, the m = 2, n = 2 subunit, the first gap of the m = 3 unit, and the m = 4, n = 1 subunit overlap; that is, t41 = t22. Both resultant amplitudes of the observed field are increased by |ρabc| and has the second modulation period of 2T. The amplitude of the third 2T modulation is further increased by |ρabc|2 because t51 = t32 = t13 and t61 = t42 = t23. The increment occurs at every 2T till the amplitude saturates. As for the h = 730 nm case whose trt ≃ 3T, as shown in Fig. 3(c), the modulation period is 3T due to the similar reason. In fact, we can further obtain the relation that tmn = tm′n′ if m+ pn = m′ + pn′, where p is the number of wave period in one round trip and also the resultant modulation period, e.g., p = 1 for h = 205 nm, p = 2 for h = 470 nm, and so on. The modeling results agree well with the corresponding FDTD simulation results, as shown in Fig. 3.
6. Discussion
The wavelength in the slit λs shifted from λ0 is a numerical issue in FDTD simulation. In our case, λs = 0.94λ0, 0.97λ0, and 0.98λ0 when the cell sizes are 5 nm, 2.5 nm, and 1.25 nm, respectively. The difference (λ0 − λs) is much greater than the cell size resolution. Although it could vanish if the cell size approached to zero, the wavelength shift is inevitable since the cell size is always finite. We are currently developing a new numerical scheme; but, this is beyond the scope of this article. Nevertheless, this wavelength shift does not affect the physical issues of concern here because the resultant shifts of all relevant quantities (e.g., v, h, , or ) in the model are linearly dependent on λs so that their effects are canceled.
We have shown that the modeling results from the m = 1 wave unit generally agree with those from the simulation of the single sinusoidal pulse in Fig. 5. Therefore, this study could be applied to the photonics of ultrashort pulse lasers. Suppose that the temporal full-width at half-maximum of its intensity is defined as Δt. When Δt ≥ trt, the subunits partially overlap so that the temporal width of the observed field is effectively increased to be larger than that of the incident pulse [22]. When Δt is short enough (e.g., Δt ≪ trt), the whole pulse could be seen as a wave unit, and the subunits can barely interfere with each other. Therefore, the observed field will be a train of pulses with decaying amplitude, as implied in Fig. 5.
In structured PEC, spoof SPs could lead to wave dispersion [14]. For a real metal, SPs can be excited, and the plasmonic effect including both structural parameters and frequency-dependent dielectric functions can disperse the wave [26]. However, since similar Fabry-Pérot-like resonance exists in the case of real metal [18], we believe that the key physics revealed by our modeling should still be valid.
7. Summary
In summary, we have verified Fabry-Pérot-like resonance for the single slit configuration by demonstrating the periodical variation of the transmitted energy as a function of thickness and transmission spectra with the monochromatic or pulse sources. A new analytical model was proposed to study the modulated light transmission through a subwavelength slit at early stage, as obtained by our FDTD simulation. Each wave unit in the model produces a train of subunits with decaying amplitudes. The temporal gap between the subunits depends on the round-trip time of the wave traveling in the slit, which are verified by the simulation of the incident sinusoidal pulse. When the wave units are continuous, the observed field amplitude of the transmitted light is monotonically increased till saturation while the round-trip time is equal to one light period as determined by the film thickness. When the round-trip time is a multiple of one light period, the temporal gap is filled by the following wave unit(s) to form the modulation. The modulation period is equal to the round-trip time while the gap time is one light period less. The modeling results agree well with those obtained from the FDTD simulation. The resultant shifts of all relevant quantities in the model due to the wavelength shift does not affect the physical issues of concern here. Possible applications for those with pulsed laser and the plasmonic effect are discussed.
Acknowledgments
The authors are grateful for the support of Ministry of Science and Technology, Taiwan, R.O.C. and the computing resources at the National Center for High-Performance Computing.
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