Ultrashort pulse synthesis for energy concentration control in nanostructures

Ahmed K. Aziz; Lianhong Zhang; Yang Hao; Khalid Z. Rajab

doi:10.1364/OE.26.025188

1. Introduction

Antennas [1] are used extensively at radio and microwave frequencies to couple energy from a source to free space and vice versa. During transmission, the antenna modulates the spatial spread of the emitted energy by reducing the far field angular divergence of the energy. Due to reciprocity, the antenna can also collect energy efficiently from the region defined by this angular divergence. This process relies on the ability of the antenna to manipulate energy in ‘hotspots’ of deeply subwavelength dimensions. The size of these energy hotspots is on the nanoscale which results in high electric field enhancement in the vicinity of the antenna.

As the pulse length of an exciting signal gets shorter, Fourier theory dictates that spectral spread increases. This implies that the broader the spectral response of the antenna, the shorter the pulse widths it can manipulate in time leading to higher energy concentration. The dipolar nature of most nanoantennas reported in the literature leads to an inherently narrowband response. The ability to concentrate energy into such small volumes holds great promise for optical emitters [2], photovoltaics [3], optical nanocircuits [4], sensing [5–7] and spectroscopy [7–9]. Temporal and spatial control over these energy hotspots will open the way for next generation nanoscale optoelectronics, ultrafast computation and data storage [10,11]. With the requirement for precise control of ultrafast optical excitations on the nanometer scale, traditional technologies such as metal nanowires and nano-waveguides suffer from bandwidths being too narrow, losses being too high and significant delay [11,12]. Furthermore, capacitive cross-talk between elements is large and is a significant problem at optical frequencies [13].

With ultrafast temporal resolutions on the order of attoseconds, nanoplasmonics have potential in on-chip optical interconnects where nanolocalization of energy and ultrafast temporal dynamics are simultaneously important [14,15]. Nanoplasmonics involves collective electron dynamics which are a result of surface plasmons (SPs) excited by the interaction of electromagnetic waves with metallic nanostructures. The plasmonic nanosystem responds to the roughly uniform exciting signal with hotspots of energy concentration where the factor of enhancement depends on the quality of localized SP resonances. While the localization length of optical SPs is dependent on the size of the nanostructure that hosts them, their relaxation rates are in the range of $10 - 100 fs$ which facilitates coherent control of the hotspots with femtosecond laser light [15].

Excitation waveform synthesis is an elegant technique that can be used for spatial and temporal control of energy concentration mechanisms in nanoantennas. Stockman et al. utilized phase modulation of a femtosecond excitation pulse on a ‘V’ shaped nanostructure for temporal control of a spatial field distribution [16]. Aeschlimann et al. demonstrated spatial manipulation of fields at the nanoscale through adaptive polarization shaping of ultrashort laser pulses [17]. Sendur et al. used polarization modulation on a radially arranged array of nano-dipoles for broadband coherent control of field at the center of the nanostructure [18]. Existing approaches in waveform synthesis for energy concentration control include adaptive methods using the genetic algorithm [17], Green’s function method [19] and time reversal method [20]. Adaptive methods using genetic algorithms are cumbersome and the waveform arrived at is hard to interpret and implement. The Green’s function method is highly theoretical and not practical for systems that are difficult to define analytically. The time reversal method requires a reverberating chamber to house the antenna system, which results in obvious implementation difficulties at the nanoscale. Our approach towards waveform synthesis is based on prior knowledge of the resonant frequencies of the nanostructure and the understanding of the energy concentration mechanism. Compared to the existing methods, it has the advantages of being intuitive to interpret, and easy to implement.

The paper is structured as follows: in section 2 waveform synthesis is implemented to create a polarization-shaped ultrashort pulse, and used to illuminate a circular array of nano-dipoles. Temporal energy concentration control is evaluated for the structure. In section 3 phase-shaping is employed to synthesize an excitation waveform for a linear array of nano-dipoles and spatial energy concentration control is evaluated. In section 4, the microwave frequency inspired log-periodic toothed antenna is compared to the circular array of nano-dipoles for a variety of synthesized excitation waveforms in order to establish its superiority.

2. Temporal energy concentration control

A radial array of nano-dipoles is illuminated with a polarization-shaped ultrashort pulse to illustrate temporal control of energy localization. The array is shown in Fig. 1 and consists of four radially arranged nano-dipoles with a common center.

Fig. 1 Radial array of nano-dipoles with a shared center.

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The four dipoles have different lengths with $L 1 = 120 nm$ , $L 2 = 160 nm$ , $L 3 = 200 nm$ and $L 4 = 240 nm$ . Each dipole arm has a thickness of $10 nm$ , a width of $15 nm$ and there is a gap of $30 nm$ between each pair of collinear arms. The dipoles are composed of gold, with a dispersive permittivity which can be described by a Drude model with $ω_{p} = 7.79 eV$ and $ω_{τ} = 48.8 meV$ [21]. It is worth noting here that the dimensions of the nanoantennas dealt with in this paper are on the order of a few hundreds of nanometers, which does not necessitate a full quantum mechanical treatment. Nevertheless, the Drude model describing the dispersion of the nanostructure material incorporates certain quantum effects within a classical electrodynamic model [22]. The aim of waveform synthesis for the radial array is to excite each dipole in turn as the polarization of the wave rotates over the plane of the array. The spectral content of the pulse should then also vary to effectively excite individual dipoles at periodic intervals. These resonant frequencies cannot be calculated using free space wavelengths of the illuminating wave since the dispersive behavior of metals at optical frequencies as well as the coupling between nano-dipoles redshifts these frequencies. The resonant SP frequencies for each dipole in the coupled system are evaluated by recording the response of the array to a femtosecond pulse aligned with each dipole in turn. A waveform that satisfies these requirements is synthesized and expressed mathematically as

φ_{0} (t) = \sum_{n = 1}^{4} {\cos (2 π f_{n} t) \exp [- \frac{3}{2} {(\frac{t - n γ}{β})}^{2}]} .

where the frequencies

f_{1} = 135.3 T Hz

,

f_{2} = 162.0 THz

,

f_{3} = 192.4 THz

and

f_{4} = 227.1 THz

correspond to the resonant SP frequencies of the respective dipoles when embedded in the array. The location of the Gaussian pulse in time is controlled by the time delay

γ = 0.2 ps

and

β = 64.9 fs

defines the spread of the Gaussian pulse envelope.

Four Gaussian sine pulses at the resonant frequencies of the system are first generated and displaced in time to form a pulse train as shown in Fig. 2(a). The required circularly polarized waveform can be synthesized by the superposition of two orthogonal linearly polarized waveforms in phase quadrature. The linearly polarized waveform depicted in Fig. 2(a) is modulated with a slowly oscillating sine function polarized in the $\hat{x}$ direction, and a cosine function polarized in the $\hat{y}$ direction. This is represented mathematically in Eqs. (2), (3) and (4)

φ_{x} (t) = φ_{0} (t) \cos (2 π \frac{t - γ}{α}),

φ_{y} (t) = - φ_{0} (t) \sin (2 π \frac{t - γ}{α}),

φ (t) = \hat{x} φ_{x} (t) + \hat{y} φ_{y} (t) .

where

α = 1.6 ps

represents the period of the sine and cosine signals. The superposition of the two equations results in a circularly polarized waveform that propagates along the

- \hat{z}

direction, as depicted in Fig. 2(b). In this way the first pulse at the lowest frequency is aligned with the largest nano-dipole at the peak amplitude of the pulse. The rotation of the polarization causes the remaining pulse peaks to align with their corresponding nano-dipoles until the final pulse at the highest frequency aligns with the shortest nano-dipole at its peak amplitude.

Fig. 2 Waveform synthesis for radial nano-dipole array, (a) Train of Gaussian sine pulses at SP resonance frequencies corresponding to Eq. (1), (b) Polarization-shaped pulse train.

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A time domain simulation is used to measure the electromagnetic response of the nano-dipole array. The simulation is performed using the commercial full-wave software CST Microwave Studio [23]. The temporal response at the center of the radial array is shown in Fig. 3. Four peaks are evident, occurring at times slightly after $t = 200, 400, 600, 800 fs$ which correspond to peak amplitudes in the excitation waveform. The excitation waveform can be redesigned to control the temporal spacing between the response peaks, while the bandwidth of operation is a function of the different nano-dipole lengths used to construct the array. The number of response peaks depends on the number of nano-dipoles in the array. By varying these parameters, we can exercise complete temporal control over the electromagnetic response of the nanostructure.

Fig. 3 Temporal electric field amplitude response at the center of the gap of the radial nano-dipole array.

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3. Spatiotemporal energy concentration control

In this section a linear array of nano-dipoles is illuminated with a phase modulated femtosecond pulse to illustrate spatial control of energy localization. Time reversal of the excitation is shown to add a limited temporal control dimension to the setup. The nano-dipoles from the radial array analyzed in the last section are rearranged into a linear array as shown in Fig. 4. The separation between collinear arms and thickness of the dipoles are $30 nm$ and $10 nm$ respectively. The dipoles are uniformly separated with a periodicity of $250 nm$ .

Fig. 4 Four element nano-dipole array with uniform spacing. Response is measured at the points labeled P1 through P4.

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A linear chirp is a signal with an instantaneous phase term. This means that the frequency increases (positive chirp) or decreases (negative chirp) linearly with time. A plane wave excitation is designed by encapsulating a linear chirp in a Gaussian pulse envelope. The wave is polarized in the $\hat{y}$ direction, propagates in the $- \hat{z}$ direction and is defined as follows

φ_{0} (t) = \hat{y} \cos [2 π f_{0} (t + ρ t^{2})] \exp [- \frac{3}{2} {(\frac{t - τ}{ξ})}^{2}] .

where

f_{0} = 40 THz

is the start frequency,

ρ = \pm 10 THz

is the rate of frequency change,

ξ = 0.259 ps

is the spread of the Gaussian pulse width and

τ = 0.333 ps

is the time delay of the Gaussian pulse envelope. The electromagnetic response at the gap center of each dipole excited by the positive chirp signal is shown in Fig. 5(a) as measured at the points P1 through P4. The peak of the response moves sequentially from P4 to P1 as the excitation frequency increases. This is because despite the detuning of each dipole’s resonance frequency due to mutual coupling in the array, the larger dipoles resonate at lower frequencies than the smaller ones as would be expected at microwave frequencies. This accounts for the sequence of hotspots created as the nanostructure is illuminated. The use of a negative chirp signal reverses the sequence of hotspots since the smallest dipole is excited first. Figure 5(b) shows the response of the nanostructure to the negative chirp signal. The temporal resolution is about

30 fs

and spatial resolution is between

λ / 8

and

λ / 6

.

Fig. 5 Temporal electric field amplitude response as a result of chirp signal excitation, (a) Positive chirp excitation response is sequential, (b) Negative chirp excitation response is the sequence in reverse.

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To explore the spatial resolution capabilities of the nanostructure, more finely spaced arrays were analyzed. It was seen that an identical array with an element spacing of $50 nm$ was the minimum spacing where a clear distinction between resonant peaks was seen. Smaller periodicity caused the peaks to overlap and identifying the sequential response became difficult. A temporal resolution of $24 fs$ , and a spatial resolution between $λ / 40$ and $λ / 24$ were achieved with a periodicity of $50 nm$ . Given that the temporal resolution is of the order of typical semiconductor switching speed, such a technique can be applied to ultrafast optoelectronic switching and spectroscopy.

The techniques explored in this section combine the ability to choose where a hotspot of electric field is created and when it is created. The number of elements in the array and the spectral content of the chirped waveform can be tailored to elicit a desired response.

4. Log-periodic nanoantenna for broadband field enhancement

The nano-dipoles investigated in the previous sections are inherently narrowband structures and the rich spectral content of their response stems from the multiband behavior of an array of narrowband nanostructures. To find a truly broadband design, one good starting point is the theory of frequency independent antennas from microwave antenna design theory. It follows from antenna scale modelling, that if the shape of an antenna is completely specified by angles, its performance would have to be independent of frequency [1].

Duhamel and Isbell showed that a linear periodic structure when mapped with an exponential conformal transformation would have a periodic response similar to that of the original linear structure if fed at the vertex [24,25]. The log-periodic toothed antenna is a radial structure obtained by an exponential conformal transformation of a corrugated waveguide structure. The consequent periodicity of the frequency response of the radial antenna on a logarithmic frequency scale is what gives the log-periodic toothed antenna its broadband frequency response. While its linear form, the Yagi-Uda antenna array is ubiquitous at radio frequencies, the radial log-periodic toothed antenna has been used with success as a broadband antenna at radio and microwave frequencies.

In this section, the electromagnetic response of the log-periodic toothed nano-antenna is compared to that of the radial array of nano-dipoles. The aim is to demonstrate the superiority of the broadband response of the former to a variety of synthesized waveforms inspired by the synthesis methods developed in the previous sections. The final nanostructure is formed by placing two log-periodic toothed antennas orthogonally to one another in one plane with a common center. It closely parallels the frequency independent concept of Ref [24]. although not strictly because angles cannot solely define its entire shape. The geometry of the log-periodic toothed antenna is defined by the angles $S_{1} = 45 °$ and $S_{2} = 22.5 °$ and the parameters $R_{n} = 240 nm$ , $r_{n} = 186 nm$ , $k = 0.775$ and $n = 11$ where $k = R_{n + 1} / R_{n} = r_{n + 1} / r_{n}$ and is illustrated in Fig. 6.

Fig. 6 Two log-periodic toothed nanoantennas placed orthogonally with 11 elements in each arm.

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The radial nano-dipole array previously investigated is used for comparison. Both nanostructures have a conductor thickness of $10 nm$ and the circular gap in the center of each structure has a diameter of $30 nm$ . The radial dimension of the log-periodic toothed antenna is chosen so that it is equal to the longest nano-dipole in the radial array. This ensures that both antennas have the same spatial footprint, which is a concern when the antennas are fabricated in array form for sensing and spectroscopy.

Three excitation waveforms are used to compare the response of the nanostructures, each in the form of a femtosecond pulse. A Gaussian sine pulse is first used, without the use of any waveform synthesis techniques, to interrogate the broadband response of the structure. Next the benefit of phase shaping is investigated by using a positive chirp waveform to illuminate the nanostructure. Finally, a combination of phase and polarization shaping is used to attain the best possible response from the radial array and compared to the response of the log-periodic nanoantenna.

A femtosecond Gaussian pulse that spans the spectrum of interest for both nanostructures is first used as the excitation. Figure 7(a) compares the spectral response of both nanostructures and illustrates that the log-periodic nanoantenna generally has a stronger response than the radial array across the spectrum of interest.

Fig. 7 Electromagnetic response for radial array and log-periodic nanoantenna at the gap center for Gaussian pulse excitation, (a) Spectral response to Gaussian pulse, (b) Time domain response to Gaussian pulse

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Figure 7(b) shows the time domain response at the gap center of both structures to the Gaussian pulse. With a peak electric field amplitude of $5 V/m$ for the nano-dipole array and $12 V/m$ for the log-periodic nanoantenna, the latter is marginally superior with a higher field concentration peak than the radial array.

Next a phase-shaped exciting femtosecond pulse identical to that described in Eq. (5) is used to illuminate both nanostructures. Figure 8(a) shows the time domain response of the nanostructures to the chirped waveform.

Fig. 8 Electromagnetic response for radial array and log-periodic nanoantenna at the gap center for synthesized waveforms, (a) Response to positively chirped excitation, (b) Response to simultaneously chirped and polarization-shaped excitation.

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The radial nano-dipole array has a peak electric field of $30 V/m$ which is an improvement on the response to the Gaussian pulse. The log-periodic nanoantenna has a peak electric field of $65 V/m$ which is significantly higher and demonstrates the benefit of using the chirp waveform over the Gaussian waveform. The log-periodic nanoantenna benefits from its frequency independent nature. While SP modes coupled to the nano-dipole array are dictated by the resonant frequencies of individual dipoles, the log-periodic nanoantenna resembles a truncated continuum of dipoles with an associated continuum of SP resonance frequencies. More photons are momentum matched to free electrons in the structure and coupled to surface modes which concentrate energy within the nanostructure.

Illumination with a Gaussian pulse results in radiation with a rich normally distributed spectral content incident on the nanostructure simultaneously. The energy and hence the momentum of incident photons is concentrated at a peak value but falls off on either side of the center frequency. The pulse has the ability to couple multiple SP modes to the nanostructure but since they couple simultaneously, interference between the modes leads to a lower overall field. On the other hand, phase modulation results in an excitation where the energy of the photons is varied smoothly over the course of the pulse. It is evident that the latter technique results in more constructive interference between different SP modes and results in a higher peak electric field.

It was shown in section 2 that the radial nano-dipole array responds more strongly to a circular polarization than a linear polarization. In order to evaluate the benefit of both phase and polarization shaping on the nanostructures, they are illuminated by a circularly polarized version of the chirp waveform used previously and described by Eq. (5), and the time domain response is shown in Fig. 8(b).

The resulting peak electric field of $35 V/m$ is the best response elicited from the radial array using the waveform synthesis methods of phase and polarization shaping. The mechanism of this improvement is the variation of the orientations of photon momentum as they strike the surface. This allows each dipole to be optimally excited as the polarization of the wave rotates over the plane of the nanostructure.

The log-periodic nanoantenna responds to the excitation waveform with a higher field concentration, as shown in Fig. 8(b). The electric field strength in the gap center presents a series of peaks, the highest of which has an amplitude of $58 V/m$ . While the phase shaping of the excitation waveform results in a continuum of photon momentums, the polarization shaping adds a continuum of orientations to these photon momentums which elicits the improved response. The result is a similar enhancement to that seen with phase-shaping only, but a longer lifetime for the hotspot and multiple secondary temporal peaks which are typically as strong as the fundamental peak on the radial array previously analyzed.

To understand the mechanism of this behavior, multiple temporal peaks are labeled in Fig. 9 as point of interest for analyzing the surface electric field on the nanostructure. Figures 10(a) through (j) show the electric field intensity distribution on the surface of the antenna receiving a polarization-shaped positive chirp signal at various instances in time that are labeled on Fig. 9. It is understood that the log-periodic toothed antenna with two elements in dipolar configuration differs from a linear dipole antenna since its fundamental mode of operation is induced by an excitation waveform that is polarized perpendicularly to the axis that joins both elements [26]. The tooth shaped segments of each element then behave like an array of dipoles, and two opposite neighboring teeth of each nanoantenna arm are preferentially excited at each resonant frequency.

Fig. 9 Time domain response of log-periodic nanoantenna to a simultaneously chirped and polarization-shaped excitation with peaks in response labeled ‘a’ through ‘j’.

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Fig. 10 Evolution of electric field intensity distribution on the excited face of the log-periodic nanoantenna at different instances in time corresponding to labeled peaks in Fig. 9. The color bar indicates the base ten logarithm of the electric field intensity in Volts/meter.

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The currents excited in each arm of the nanoantenna at these resonances induce a local dipole moment between the antenna arms creating a hotspot at the center. The extended lifetime of this hotspot can be seen in Fig. 10 as it retains the highest field concentration for each instance in time, while the resonant teeth of each arm vary with time. It can also be seen from the temporal evolution of the field over the surface of the antenna that the active region of the antenna moves towards the center of the antenna as time progresses. This is expected for a positively chirped excitation since the higher frequency dipoles are formed by the smaller inner teeth and are excited at later times.

5. Conclusion

Spatial and temporal control of energy concentration has been demonstrated using nanostructure design and waveform synthesis. Phase and polarization shaping have been used individually and simultaneously for waveform design and their benefits established. The radial nano-dipole array has been shown to respond very well to a polarization shaped pulse train while the linear array responds best to a phase shaped waveform. In both these designs, there is a lot of room to redesign and tailor the spatiotemporal response for a particular application. Changing the number of resonant elements, the dimensions of these elements and the temporal and spectral content of the illuminating waveform allow us to tailor the femtosecond pulse manipulation capabilities of these nanostructures. The tools available allow for complete control over where energy concentrates on the nanostructure, when it peaks and what the spectral content of the hotspot is. The use of polarization-shaping alone is seen to improve electric field strength at hotspots by a factor of six while the simultaneous use of polarization-shaping and phase-shaping is seen to enhance the energy concentration in the hotspot by a similar factor and additionally extend its lifetime.

Finally, the log-periodic toothed antenna is depicted as combining the design principles and advantages of the linear and radial nano-dipole arrays. Its temporal energy concentration control mechanism is shown to outperform the radial array across the spectrum. It shows a greater than twofold improvement in response, in comparison to the radial array, when only phase shaping is employed. It shows an almost threefold improvement when phase and polarization shaping are used simultaneously. The improvement in the response has been verified with time domain simulations and explained using concepts from antenna theory and Fourier theory. Given the applications of broadband nanostructures to spectroscopy and bio-sensing, the designs presented are hoped to be a step forward towards wideband and multiband plasmonic devices.

Funding

United States Air Force Office of Scientific Research (FA8655-12-1-2091).

Acknowledgements

The authors would like to acknowledge the support of the Commonwealth Scholarship Commission UK.

References

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Ultrashort pulse synthesis for energy concentration control in nanostructures

Abstract

1. Introduction

2. Temporal energy concentration control

3. Spatiotemporal energy concentration control

4. Log-periodic nanoantenna for broadband field enhancement

5. Conclusion

Funding

Acknowledgements

References

Cited By

Figures (10)

Equations (5)

Optics Express