## Analytic coherent control of plasmon propagation in nanostructures

Optics Express, Vol. 17, Issue 16, pp. 14235-14259 (2009)

http://dx.doi.org/10.1364/OE.17.014235

Acrobat PDF (1158 KB)

### Abstract

We present general analytic solutions for optical coherent control of electromagnetic energy propagation in plasmonic nanostructures. Propagating modes are excited with tightly focused ultrashort laser pulses that are shaped in amplitude, phase, and polarization (ellipticity and orientation angle). We decouple the interplay between two main mechanisms which are essential for the control of local near-fields. First, the amplitudes and the phase difference of two laser pulse polarization components are used to guide linear flux to a desired spatial position. Second, temporal compression of the near-field at the target location is achieved using the remaining free laser pulse parameter to flatten the local spectral phase. The resulting enhancement of nonlinear signals from this intuitive analytic two-step process is compared to and confirmed by the results of an iterative adaptive learning loop in which an evolutionary algorithm performs a global optimization. Thus, we gain detailed insight into why a certain complex laser pulse shape leads to a particular control target. This analytic approach may also be useful in a number of other coherent control scenarios.

© 2009 Optical Society of America

## 1. Introduction

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*et al*. demonstrated theoretically that coherent control of nanosystems is possible with chirped laser pulses [19

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4. T. Brixner, F. García de Abajo, J. Schneider, and W. Pfeiffer, “Nanoscopic Ultrafast Space-Time-Resolved Spec-troscopy,” Phys. Rev. Lett. **95**(9), 093,901-4 (2005). [CrossRef] [PubMed]

19. M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Coherent Control of Femtosecond Energy Localization in Nanosystems,” Phys. Rev. Lett. **88**(6), 067,402 (2002). [CrossRef]

*et al*. showed theoretically that with suitable elliptically polarized light one can control the propagation of electromagnetic energy in a T-junction consisting of Ag nanoparticles [15

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*et al*. [25

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39. L. Polachek, D. Oron, and Y. Silberberg, “Full control of the spectral polarization of ultrashort pulses,” Opt. Lett. **31**(5), 631–633 (2006). [CrossRef] [PubMed]

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20. T. Brixner, F. García de Abajo, J. Schneider, C. Spindler, and W. Pfeiffer, “Ultrafast adaptive optical near-field control,” Phys. Rev. B **73**(12), 125,437–11 (2006). [CrossRef]

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43. T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, “Femtosecond pulse shaping by an evolutionary algorithm with feedback,” Appl. Phys. B **65**(6), 779–782 (1997). [CrossRef]

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## 2. Methods

### 2.1. Basic idea

20. T. Brixner, F. García de Abajo, J. Schneider, C. Spindler, and W. Pfeiffer, “Ultrafast adaptive optical near-field control,” Phys. Rev. B **73**(12), 125,437–11 (2006). [CrossRef]

42. M. Ninck, A. Galler, T. Feurer, and T. Brixner, “Programmable common-path vector field synthesizer for femtosecond pulses,” Opt. Lett. **32**(23), 3379–3381 (2007). [CrossRef] [PubMed]

6. S. Maier, P. Kik, H. Atwater, S. Meltzer, E. Harel, B. Koel, and A. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nature Mater. **2**(4), 229–232 (2003). [CrossRef]

15. M. Sukharev and T. Seideman, “Phase and polarization control as a route to plasmonic nanodevices,” Nano Lett. **6**(4), 715–719 (2006). [CrossRef] [PubMed]

24. M. Sukharev and T. Seideman, “Coherent control of light propagation via nanoparticle arrays,” J. Phys. B **40**(11), S283–S298 (2007). [CrossRef]

*y*axis. The

*y*arm is coupled to the long chain in the middle between the tenth and the eleventh sphere. The spheres have a diameter of 50 nm and are separated by a 10 nm gap, which corresponds to a 60 nm unit cell. The nanostructure is excited at the beginning of the long chain with a tightly-focused shaped laser pulse (focal Gaussian beam diameter of about 200 nm intensity FWHM) propagating along the

*z*axis perpendicular to the chain with the beam center located at the center of the first sphere. Two excitation polarizations are chosen: polarization 1 along the

*x*axis and polarization 2 along the

*y*axis. After coupling to the nanostructure, the pulse energy is guided by plasmons away from the focus to remote spatial positions on the two branches. The inset of Fig. 1 shows the total scattering cross sections of this nanostructure excited by plane waves with polarization components 1 and 2 (black solid and red dashed curves, respectively). Clearly, two resonances are observed: the resonance of the long chain along the

*x*direction at

*ω*~ 4.3 rad/fs and the resonance of the short chain along the y direction at

*ω*~ 5.4 rad/fs which are mostly excited by polarizations 1 and 2, respectively.

*et al*. theoretically showed the control of propagation direction after a junction by scanning the ellipticity of the excitation light using a two-dimensional parameter space [15

**6**(4), 715–719 (2006). [CrossRef] [PubMed]

24. M. Sukharev and T. Seideman, “Coherent control of light propagation via nanoparticle arrays,” J. Phys. B **40**(11), S283–S298 (2007). [CrossRef]

42. M. Ninck, A. Galler, T. Feurer, and T. Brixner, “Programmable common-path vector field synthesizer for femtosecond pulses,” Opt. Lett. **32**(23), 3379–3381 (2007). [CrossRef] [PubMed]

### 2.2. Field calculation

46. F. García de Abajo, “Interaction of Radiation and Fast Electrons with Clusters of Dielectrics: A Multiple Scattering Approach,” Phys. Rev. Lett. **82**(13), 2776 (1999). [CrossRef]

*A*

_{α}^{(i)}(

**r**,

*ω*)∣ with

*α*=

*x,y,z*describe the extent to which the two far-field polarization components

*i*= 1,2 couple to the optical near-field, whereas the phases

*θ*

_{α}^{(i)}(

**r**,

*ω*) = arg{

*A*

_{α}^{(i)}(

**r**,

*ω*)} determine their vectorial superposition and dispersion properties. These quantities are characteristic of the nanostructure and depend on the focussing conditions. However, the response is independent of the applied pulse shape that will be considered below.

*ω*

_{0}= 4.6 rad/fs (409 nm) with a FWHM of 0.35 rad/fs corresponding to ~ 10 fs pulse duration. The field distribution in a tight focus is represented as a superposition of plane waves. For realistic simulations we use a Gaussian focus which can be achieved by a high numerical aperture. The focal spot size (~200 nm) is close to the diffraction limit, obtained by a coherent superposition of 1245 partial waves.

**A**

^{(i)}(

**r**,

*ω*), as can be verified by inverse Fourier transformation to the time domain. In order to exemplify coherent propagation control, we will concentrate on two spatial positions,

**r**

_{1}and

**r**

_{2}, as marked in Fig. 1, that are reached after plasmon propagation along the

*x*arm or

*y*arm, of the structure, respectively. The goal will then be to control linear and nonlinear signals at these two positions, especially contrast and pulse compression, even though the illumination region is spatially separated. In effect, this corresponds to control over direction (“spatial focusing”, Section 3) and time (“temporal focusing”, Section 4).

*E*

_{1}

^{in}(

*ω*), oriented along the

*x*axis, and

*E*

_{2}

^{in}(

*ω*), oriented along the

*y*axis, consisting of spectral amplitudes

*φ*(

_{i}*ω*) which can all be varied independently using the most recent pulse shaper technology [42

**32**(23), 3379–3381 (2007). [CrossRef] [PubMed]

**E**(

**r**,

*ω*) is obtained by calculating the near-field for each far-field polarization separately and taking the linear superposition [4

**95**(9), 093,901-4 (2005). [CrossRef] [PubMed]

**E**(

**r**,

*t*) can be obtained by inverse Fourier transforming

**E**(

**r**,

*ω*) for each vector component separately.

*ω*) and the spectral amplitudes

*φ*

_{1}(

*ω*) provides a handle to manipulate the temporal evolution of the local fields.

### 2.3. Definition of signals

**E**(

**r**,

*ω*) at any position

**r**induced by a vector-field-shaped laser pulse. We then use this quantity to define different signals in analogy with far-field optics: we define local spectral intensity as

*FT*indicates Fourier transformation and the parameters

*b*describe which local polarization components are included in the signals. Setting

_{α}*b*=1 and

_{x}*b*=

_{y}*b*= 0, for example, describes field-matter interactions with transition dipoles oriented along the

_{z}*x*axis. In the following calculations, we always use

*b*=

_{x}*b*=

_{y}*b*= 1, corresponding to an isotropic distribution of dipole moments, unless mentioned otherwise. We define local linear flux

_{z}*ω*

_{0}we can neglect frequencies where the intensity is sufficiently small. Hence, we only have to integrate over an appropriate interval

*ω*=

_{min}*ω*

_{0}- Δ

*ω*to

*ω*=

_{max}*ω*

_{0}+ Δ

*ω*), where Δ

*ω*is a suitable width.

*δω*, the frequency integral in Eq. (7) can be replaced by a sum over all frequencies of the local spectrum defined in Eq. (6):

*δω*/2

*π*for simplicity as we employ the same grid for all comparisons. By inserting the definition of the optical near-field of Eq. (3) into Eq. (6), we obtain the local spectrum as a function of external laser intensities

*I*(

_{i}*ω*) and phases

*φ*(

_{i}*ω*):

*ω*) is defined in Eq. (5) and Re is the real part.

*A*

_{mix}(

**r**,

*ω*) is the complex scalar product with amplitude ∣

*A*

_{mix}(

**r**,

*ω*)∣ and phase

*θ*

_{mix}(

**r**,

*ω*) describing the mixing of the two near-field modes

**A**

^{(1)}(

**r**,

*ω*) and

**A**

^{(2)}(

**r**,

*ω*), which can be calculated independently of the external field once the MESME calculation is done.

### 2.4. Adaptive Optimization

*et al*. [43

43. T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, “Femtosecond pulse shaping by an evolutionary algorithm with feedback,” Appl. Phys. B **65**(6), 779–782 (1997). [CrossRef]

*φ*

_{1}(

*ω*),

*φ*

_{2}(

*ω*),

*F*

_{lin}(

**r**), and for nonlinear flux

*F*

_{nl}(

**r**), as input for the fitness function, and contrast control is achieved by regarding flux differences between different positions as explained below. For a better identification of the lines in the figures, where the results of adaptive optimizations are compared to the analytic solutions, only every second data point of the adaptively optimized phases or amplitudes is plotted.

## 3. Spatial focusing of propagating near-fields

### 3.1. Optimization of linear flux at one position

#### 3.1.1. Polarization shaping by phase-only modulation

*F*

_{lin}(

**r**) at one specified location. In the examples below, this location will be chosen at either

**r**

_{1}or

**r**

_{2}as marked in Fig. 1. Equation (9) provides insight into the near-field control mechanisms. To optimize linear flux, the two laser phases

*φ*

_{1}(

*ω*) and

*φ*

_{2}(

*ω*) can be adjusted independently from the amplitudes

*A*

_{mix}(

**r**,

*ω*)∣, is a measure of how much the near-field modes project onto each other and determines the controllability at this point. For example, if the two near-field modes do not project onto each other, i.e. if the modes are perpendicular, they do not interfere and it is not possible to control the local linear flux with the laser pulse phases because

*A*

_{mix}(

**r**,

*ω*) = 0. Maximum controllability is obtained for parallel near-field modes, i.e. having a maximum projection.

*θ*

_{mix}(

**r**,

*ω*), determines how the phase difference between the two external laser polarization components, Φ(

*ω*), should be chosen in order to make the interference term of Eq. (9) positive or negative. Constraints for the constructive [Φ

_{max}(

*ω*)] and destructive [Φ

_{min}(

*ω*)] interference are

*ω*) only is due to the interference of the two near-field modes as the single control mechanism responsible for the linear signal. In other words, the local field is determined by the polarization state of the incident light and

*A*

_{mix}(

**r**,

*ω*) is a measure of the controllability that can be achieved by polarization shaping (i.e. adjusting the phases of the two far-field polarization components). Setting, for example,

*b*= 1 and

_{x}*b*=

_{y}*b*= 0 one can get a good understanding of this effect, since the phase difference

_{z}*θ*

_{mix}(

**r**,

*ω*) =

*θ*

_{x}^{(1)}(

**r**,

*ω*) -

*θ*

_{x}^{(2)}(

**r**,

*ω*) of the two near-field modes induced by the nanostructure is then just compensated exactly, leading to optimal constructive or destructive interference for given amplitudes. Including more than one component, e.g.

*b*=

_{x}*b*=

_{y}*b*= 1, the sum in Eq. (10) performs a weighting of the phases of each component by their amplitudes. If the near-field modes are not parallel, some part of the field will still remain even for destructive interference.

_{z}**r**

_{1}[Fig. 2(a)] and

**r**

_{2}[Fig. 2(b)] as solid blue lines (maximum flux) and dashed red lines (minimum flux). The plots can be understood as follows: for example, as shown in Fig. 2(a), linearly polarized light at

*ω*~ 4.65 rad/fs oriented along the (1,1,0) direction (Φ = 0) minimizes the local flux at

**r**

_{1}, whereas linearly polarized light oriented along the (-1,1,0) direction (Φ =

*π*) maximizes the local flux at

**r**

_{1}. In contrast right (Φ= -

*π*/2) and left (Φ =

*π*/2) circularly polarized light at

*ω*~ 4.25 rad/fs generates the maximum and minimum local flux at

**r**

_{1}, respectively. The control at other frequencies is achieved similarly using elliptically polarized light.

*F*

_{lin}(

**r**) using an evolutionary algorithm, and the resulting optimal spectral phases are plotted as blue circles (maximization) and red squares (minimization) in Fig. 2. Analytic and adaptive results are in excellent agreement in the region of relevant laser spectral intensity (black dotted line). The predicted difference of

*π*between the phase differences [see Eqs. (12) and (13)] can be seen as an offset between the red and blue curves, and the shape of the curves reflects the spectral response properties of the nanostructure as contained in the scalar product of Eq. (10). Parseval’s theorem guarantees that the control can be done separately for each frequency component, as shown above using analytical methods. The adaptive optimization is however performed here for the entire pulse simultaneously, and therefore the agreement with the analytic model is a non-trivial cross validation of the results.

*I*(

_{G}*ω*) (see Section 3.2.2):

**r**

_{1}(blue lines) and

**r**

_{2}(red lines) are shown in Fig. 3 for the optimal incident laser phase differences from Fig. 2. The maximum and minimum local linear flux phase differences are used to generate the maximum and minimum local response intensities, respectively. As can be seen in Fig. 3, the control of the near-fields is achieved over the whole spectral range, i.e. the blue dashed line is higher than the blue solid line, and the red dash-dotted line is higher than the red dotted line. In addition, it can be seen that the local response intensity at

**r**

_{2}(red) exceeds the local response intensity at

**r**

_{1}(blue) over a large part of the spectral range, which is due to better coupling of the two excited modes to the

*y*arm. However, the minimized response at position

**r**

_{2}is smaller than the responses at position

**r**

_{1}in the region of

*ω*~ 4.7 rad/fs, which will be relevant for the discussion of amplitude shaping in Section 3.2.2.

#### 3.1.2. Polarization shaping with additional amplitude modulation

*I*

_{1}(

*ω*) and

*I*

_{2}(

*ω*) first without choosing the optimal laser pulse phases

*φ*

_{1}(

*ω*) and

*φ*

_{2}(

*ω*). In that case, the solutions for linear flux control at one spatial position are trivial as can be inferred from Eq. (9). Given that amplitude shaping can only decrease the intensity of light at a particular frequency, the optimal solution for maximum linear flux is full pulse-shaper transmission, i.e. making use of the full available intensity over the whole laser pulse spectrum. Likewise, the solution for minimum local flux is given for zero transmission, i.e. for both

*I*,(

_{i}*ω*) = 0.

**r**. According to Eqs. (9) and (10), the destructive interference from Section 3.1.1 can be made perfect if

**A**

^{(1)}(

**r**,

*ω*) =

*β*(

*ω*)

**A**

^{(2)}(

**r**,

*ω*)i.e. if the local responses excited by the two laser pulse polarizations are parallel to each other, with any ratio

*β*(

*ω*) ∈ ℂ For such a case, the external laser intensities should be chosen such that their ratio fulfills

*I*

_{1}(

*ω*)/

*I*

_{2}(

*ω*) = ∣

*β*(

*ω*)∣

^{2}. In that case, selection of the phase difference Φ(

*ω*) according to Eq. (13) resulting in Φ(

*ω*) = - arg{

*β*(

*ω*)} -

*π*leads to the desired zero flux due to perfect destructive interference. If

**A**

^{(1)}(

**r**,

*ω*) and

**A**

^{2}(

**r**,

*ω*) are not parallel, the same procedure can be used to cancel out just one component

*E*(

_{α}**r**,

*ω*).

### 3.2. Controlling the direction of propagation

*x*arm or the

*y*arm of the nanostructure (Fig. 1). In contrast to Section 3.1 the optimization goal is now determined simultaneously by the local response at two different locations, whereas in Section 3.1 both locations were treated independently. A suitable observable that characterizes this goal is the difference of linear local flux at the two spatial points

**r**

_{1}and

**r**

_{2},

**r**

_{1}(flin maximum) or

**r**

_{2}(

*f*

_{lin}minimum). These extrema will be found by calculating first the correct phases (Section 3.2.1) and then the optimal amplitudes (Section 3.2.2).

#### 3.2.1. Polarization shaping by phase-only modulation

*f*

_{lin}is a linear sum over the individual frequency components, each frequency can be considered separately. Thus, the extrema of

*f*

_{lin}are found for

*g*

_{lin}(

*ω*) = 0. Assuming

*I*

_{1}(

*ω*) ≠ 0 and

*I*

_{2}(

*ω*) ≠ 0 [if one or both of the intensities are zero in any frequency interval, the phases

*φ*

_{1}(

*ω*) and

*φ*

_{2}(

*ω*) are irrelevant for linear control targets and can be chosen arbitrarily], the optimal spectral phase difference is then

*k*= 0,1,2 is chosen such that Φ(

*ω*) ∈ [-

*π*,

*π*]. This results in two solutions that can be assigned to the global maximum or minimum by evaluation of Eq. (16) or by investigating the second derivative. For the special case of a vanishing denominator in Eq. (20), the solutions are Φ(

*ω*) = π/2 and Φ(

*ω*) = -

*π*/2, which correspond to left and right circular polarization respectively.

*ω*) of the external polarization components and the maximum and minimum solutions differ by

*π*. The analytically determined optimal phase difference does not depend on the pulse intensities

*I*

_{1}(

*ω*) and

*I*

_{2}(

*ω*). However, as we show in Section 3.2.2, shaping the amplitudes additionally results in improved contrast.

*F*

_{lin}(

**r**

_{1})-

*F*

_{lin}(

**r**

_{2}) forthepoints

**r**

_{1}and

**r**

_{2}of the chosen nanostructure are shown in Fig. 4 (lines) and are again compared to the results of an adaptive optimization (symbols). For directional control along the

*x*arm toward

**r**

_{1}(blue) as well as along the

*y*arm toward

**r**

_{2}(red), both approaches agree well, and the phase difference of

*π*between maximization and minimization of the difference signal is also confirmed.

**r**

_{2}only [Fig. 2(b)]. In this particular example, the point

**r**

_{2}has more influence on the optimal phase because the absolute value of the optimal local response intensity

*R*(

**r**,

*ω*) is larger at

**r**

_{2}than at

**r**

_{1}over most of the spectral region (cf Fig. 3). Therefore, the maximization (minimization) of the linear flux difference [Eq. (15)] results to a large extent from the minimization (maximization) of

*F*

_{lin}(

**r**

_{2}) for the chosen nanostructure. If one chose to control flux contrast between such positions where the individual fluxes were of more similar magnitude then the optimal phase would deviate more strongly from the optimizations of both of the separate fluxes. However, with the analytic approach one has the guarantee to nevertheless find the global optimum.

**r**

_{1}and minimizing at

**r**

_{2}separately is (0.599 - 0.410 = 0.189), while maximizing the difference directly leads to 0.088. The difference between minimization at

**r**

_{1}and maximization at

**r**

_{2}is (0.401 -0.984= -0.583), and direct contrast control yields -0.483. The optimal solution for the difference signal provides a good compromise between control at the individual points

**r**

_{1}and

**r**

_{2}for this nanostructure.

*x*arm or the

*y*arm of the structure. In the following section, additional amplitude shaping will even improve the control performance. It is important to point out that the results of the present section obtained for the optimal phases are valid without dependence on the particular intensities

*I*

_{1}(

*ω*) and

*I*

_{2}(

*ω*) of the two external polarization components. Thus the optimal amplitudes can be found in a separate step.

#### 3.2.2. Polarization shaping with additional amplitude modulation

*i*, by weighting amplitude coefficients

*γi*(

*ω*) varying from 0 to 1:

*ω*:

*A*

_{mix}(

**r**,

*ω*)∣,

*θ*

_{mix}(

**r**,

*ω*),

*C*(

_{i}*ω*), and Φ(

*ω*) are known from Eqs. (10), (17), and (20), respectively, and the weighting amplitude coefficients for both polarization components

*γ*

_{1}(

*ω*) and

*γ*

_{2}(

*ω*) are unknown.

*γ*

_{1}(

*ω*) ≤ 1 and 0 ≤

*γ*

_{2}(

*ω*) ≤ 1 yields the solutions

*C*

_{1}(

*ω*),

*C*

_{2}(

*ω*), and

*C*

_{mix}(

*ω*), and is found by substitution into Eq. (22). By locating the desired minimum or maximum in this fashion separately for each frequency, the optimal amplitude shape for each laser polarization component can be obtained. These solutions provide the optimal amplitudes which in turn depend on the chosen phases

*φ*

_{1}(

*ω*) and

*φ*

_{2}(

*ω*) through the parameter

*C*

_{mix}(

*ω*).

*F*

_{lin}(

**r**

_{1})-

*F*

_{lin}(

**r**

_{2}) while using the optimal phases obtained in Section 3.2.1. Again, an evolutionary algorithm was employed for comparison, in which the phase difference as well as the amplitude weighting coefficients were optimized. In Fig. 5, analytic (lines) and adaptive results (symbols) for both polarizations are compared. Similar to Sections 3.1.1 and 3.2.1 for the case of phase shaping, the analytic and adaptive results for amplitude shaping agree well. The deviation of the amplitude coefficients from the adaptive optimizations appearing in the regions of low laser pulse intensities do not have any physical significance.

*F*

_{lin}(

**r**

_{1}) -

*F*

_{lin}(

**r**

_{2}) = 0.201. This value should be compared to the phase-only shaping result of 0.088 (cf. Table 1). Thus, the maximum of the linear flux difference is increased significantly. However, in the case of minimization [Fig. 5(b)], the optimal amplitudes are at their maxima over most of the spectrum (both weighting coefficients equal 1). This is because amplitude shaping cannot further decrease the linear flux difference of -0.483 (cf. Table 1). For interpretation of these results, the local response intensity

*R*(

**r**,

*ω*) shown in Fig. 3 for the two points

**r**

_{1}and

**r**

_{2}is used. The small spectral region around

*ω*~ 4.7 rad/fs where the two responses at position

**r**

_{1}exceed the minimized response at position

**r**

_{2}is also imprinted in the weighting coefficient

*γ*

_{1}(

*ω*) in Fig. 5(a) (green). The incident polarization component 1 is reduced in that spectral part where the local response intensity

*R*(

**r**

_{2},

*ω*) (Fig. 3) exceeds

*R*(

**r**

_{1},

*ω*). In the small part around

*ω*= 4.7 rad/fs, where

*R*(

**r**

_{1},

*ω*) dominates, both weighting coefficients equal 1 to ensure maximum contribution from the desired components.

*γ*

_{1}(

*ω*) predicted by the analytic theory.

### 3.3. Controlling the local spectral intensity

**r**

_{1}and of the other part to

**r**

_{2}. For illustration, we have chosen to guide the “red” half of the spectrum (3.9 – 4.6 rad/fs) to the point

**r**

_{1}, whereas the “blue” half (4.6 – 5.3 rad/fs) was guided to

**r**

_{2}. The analytic results for the required optimal phases and amplitudes of the external control field are hence obtained in complete analogy to Section 3.2, only employing the results from maximization [Fig. 4 (blue) and Fig. 5(a)] in the “red” half of the spectrum and the results from minimization [Fig. 4 (red) and Fig. 5(b)] in the “blue” half. The resulting local spectrum for this control target is plotted in Fig. 6(a). The sharp peaks observed in the center of the spectrum are due to the steep slope of the amplitude coefficient

*γ*

_{1}(

*ω*) [Fig. 5(a)] in the applied laser pulse shape, which results from the spectral part in the local response where

*R*(

**r**

_{1},

*ω*) exceeds the minimized local response

*R*(

**r**

_{2},

*ω*) shown in Fig. 3. In addition, we show the local spectrum normalized to the sum of the local spectra at positions

**r**

_{1}and

**r**

_{2}in Fig. 6(b). Note that the switching efficiency varies significantly with frequency. For some spectral components the switching efficiency is negligible (e.g., 4.5 rad/fs) whereas it is almost 100% in other regions (e.g., 4.8 rad/fs). This reflects the fact that each frequency component interferes with itself and thus the local switching efficiency is controlled by the local spectral response for each wavelength independently [20

20. T. Brixner, F. García de Abajo, J. Schneider, C. Spindler, and W. Pfeiffer, “Ultrafast adaptive optical near-field control,” Phys. Rev. B **73**(12), 125,437–11 (2006). [CrossRef]

**r**

_{1}(dashed blue) than at

**r**

_{2}(solid red), and for the higher frequencies it is higher at

**r**

_{2}than at

**r**

_{1}.

## 4. Temporal focusing

### 4.1. Nonlinear flux

*ω*) =

*φ*

_{1}(

*ω*) -

*φ*

_{2}(

*ω*) between the two excitation laser polarization components, and the prescription of Section 3.1.2 or Eq. (24) to find both intensities

*I*

_{1}(

*ω*) and

*I*

_{2}(

*ω*). Since the linear signals depend only on the phase difference Φ(

*ω*), Eq. (5), which was already introduced at the beginning of this manuscript (Section 2.2), can be used to vary the phase

*φ*

_{1}(

*ω*) under the constraint for linear flux control, i.e. given all quantities in the curly brackets. The remaining control parameter

*φ*

_{1}(

*ω*) can then be used to control the time evolution at a certain position, in particular to compress the near-field temporally [18]. Thus, for example, we can optimize the nonlinear flux defined in Eq. (11) at that position where the linear flux was guided to.

#### 4.1.1. One field component

*b*= 1 and

_{α}*b*=

_{β}*b*= 0 with {

_{γ}*α*,

*β*,

*γ*} ∈ {

*x*,

*y*,

*z*} in Eq. (11 ). We choose

*b*= 1 and

_{x}*b*=

_{y}*b*= 0 and set the control target to optimize the

_{z}*x*component of the near-field. The issue of automated laser pulse compression in the case of conventional far-field optics has been addressed more than 10 years ago by Silberberg’s group [44

44. D. Yelin, D. Meshulach, and Y. Silberberg, “Adaptive femtosecond pulse compression,” Opt. Lett. **22**(23), 1793–1795 (1997). [CrossRef]

43. T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, “Femtosecond pulse shaping by an evolutionary algorithm with feedback,” Appl. Phys. B **65**(6), 779–782 (1997). [CrossRef]

*b*= 1) the required phase

_{α}*φ*

^{α}_{1}(

*ω*) is obtained in a straightforward manner via Eq. (5) by requiring the phase of the local field to be uniformly zero, i.e. arg{

*E*(

_{α}**r**,

*ω*)} ≡ 0. Thus, we get

**r**and component

*α*.

*b*=1 and

_{x}*b*=

_{y}*b*= 0) as a basis and have used the parameters for maximal

_{z}*f*

_{lin}=

*F*

_{lin}(

**r**

_{1}) -

*F*

_{lin}(

**r**

_{2}) as constraints in Eq. (25). In addition to switching the plasmon propagation along the

*x*arm, we request temporal compression at the target point

**r**

_{1}. The resulting analytic phases and amplitudes are shown in Figs. 7(a) and 7(c) (lines), respectively, and are again compared to those found by the evolutionary algorithm (symbols). In this example, however, the fitness function for the adaptive optimization was chosen directly as the difference of the nonlinear flux at the two points

**r**

_{1}and

**r**

_{2}:

*F*

_{lin}(

**r**

_{1}) -

*F*

_{lin}(

**r**

_{2}) from Section 3.2 (with

*b*= 1 and

_{x}*b*=

_{y}*b*= 0) as a basis and compressing the pulse temporally at position

_{z}**r**

_{2}. The optimal phases and amplitudes are compared with adaptive optimizations in Figs. 7(b) and 7(d), respectively. Here, some variations between the two approaches can be seen. However, the amplitude coefficients [Fig. 7(d)] agree reasonably well and some of the relevant features in the phases [Fig. 7(b)] are also reproduced, such as the separation between the red and the blue curves in the region above

*ω*~ 4.8 rad/fs. Below

*ω*~ 4.6 rad/fs, the intensities in the incident fields is reduced [Fig. 7(d)], which partly explains the deviation of the phases between the two approaches. Comparing the non-linear flux difference obtained with the analytic optimization (-1.208·10

^{-9}) to the results obtained with an adaptive optimization (-0.377·10

^{-9}) shows again the validity of our analytic approach, i.e. in this case the analytic approach performs better.

**r**

_{1}and

**r**

_{2}are spatially separated from the excitation spot in our example, plasmon propagation as a function of time is relevant. The propagation time corresponds to a linear spectral phase with a positive slope such that the dominant negative slopes of Fig. 7 lead to an arrival time of

*t*= 0. By adding any linear phase this timing can be modified. Only the nonlinear part of the phase is responsible for compression.

#### 4.1.2. Three field components

*φ*

_{1}(

*ω*) as one (scalar) degree of freedom it is not possible to compensate for all three phases simultaneously. We therefore consider now different approximate solutions that find a suitable compromise between compensation of the different components. However, we have to divulge that due to the geometry of the chosen nanostructure, the longitudinal components, i.e. the

*x*component in the

*x*arm and the

*y*component in the

*y*arm, are dominant and are compressed in roughly the same way using both approaches.

### 4.2. Interpretation of optimized fields in the time domain

*φ*

_{1}(

*ω*) for compression of the largest local near-field component [cf. (Eq. 27)], and the temporal near-fields intensities reached with an unshaped pulse. The switching of the local near-field intensity is visible by comparing the optimization of nonlinear flux guiding to

**r**

_{1}(red lines) with guiding to

**r**

_{2}(blue lines). The temporal near-fields at

**r**

_{1}and

**r**

_{2}are shown with solid and dashed-dotted lines, respectively. Clearly, the red solid line is higher than the red dashed line and the dashed blue line is higher than the solid blue line, thus confirming the successful linear flux control.

**r**

_{1}and

**r**

_{2}can be seen better in Figs. 8(b) and 8(c), respectively. Here, we compare the normalized and time-shifted temporal intensity under the constraints of guiding the linear flux, but choosing

*φ*

_{1}(

*ω*) ≡ 0 (green lines) with the normalized temporal intensity when additionally choosing the optimal

*φ*

_{1}(

*ω*) for pulse compression (red and blue lines in Fig. 8(b) and 8(c), respectively). The compression is small in Fig. 8(b). However, for guiding to

**r**

_{2}[Fig. 8(c)] the dispersion is larger and the compression is more pronounced, i.e. the optimally compressed near-field (dash-dotted blue) is shorter than the near-field for pulses with

*φ*

_{1}(

*ω*) ≡ 0 (dash-dotted green).

*x*component of the propagating near-fields excited by optimally shaped ultrashort laser pulses. Two snapshots from these movies are shown in Fig. 9, where in addition to the projections of the two laser polarization components we also show the full quasi-3D profile [36

36. T. Brixner and G. Gerber, “Femtosecond polarization pulse shaping,” Opt. Lett. **26**(8), 557–559 (2001). [CrossRef]

37. T. Brixner, G. Krampert, P. Niklaus, and G. Gerber, “Generation and characterization of polarization-shaped femtosecond laser pulses,” Appl. Phys. B **74**(0), s133–s144 (2002). [CrossRef]

**r**

_{1}with the excitation phases and amplitudes of Figs. 7(a) and 7(c), respectively, and Fig. 9(b) (Media 2) shows guidance to

**r**

_{2}with the optimal field shapes from Figs. 7(b) and 7(d). Both of the snapshots are taken at

*t*= 0, since the phase

*φ*

_{1}(

*ω*) was chosen as in Eq. (25), which includes the linear spectral phase of the propagated near-field. Therefore, the excitation pulse appears at times

*t*< 0 and the propagating mode arrives at the target location at

*t*= 0. It can be clearly seen that after propagation from the excitation position, the field mode “switches ” into the desired arm after the junction and is guided to and compressed at either

**r**

_{1}[Fig. 9(a)] or

**r**

_{2}[Fig. 9(b)].

### 4.3. Analytic space-time control

**95**(9), 093,901-4 (2005). [CrossRef] [PubMed]

**95**(9), 093,901-4 (2005). [CrossRef] [PubMed]

**73**(12), 125,437–11 (2006). [CrossRef]

**r**. However, the near-field response at other locations is different, and therefore does not vanish. This can be used to achieve space-time control by splitting the spectrum into two parts and determining optimal pulse shapes for each of the parts independently. The temporal shape of the near-field can be optimized using the remaining free laser pulse shaping parameter

*φ*

_{1}(

*ω*):

*α*=

*x*,

*y*,

*z*.

*x*component of the near-field at

**r**

_{1}, while the “blue” half is used to cancel out the

*y*component at

**r**

_{2}. This gives the possibility to shape the largest local component of the signal appearing in one arm of the nanostructure independently from the signal in the other arm.

*t*= 0, but here a time delay of

*τ*= 90 fs was introduced by requesting additional linear spectral phases with different slopes in the two spectral regions. The spectral amplitudes for the largest near-field components, i.e. the

*x*component at

**r**

_{1}and the

*y*component at

**r**

_{2}, are plotted in Fig. 10(a). However, in Fig. 10(b) the temporal intensities, which include all three components, are shown for the same polarization-shaped far-field pulse, and it can be seen that the desired spatial-temporal sequence can be reached with extremely good contrast. The pump and probe pulses peaking at

*τ*= -45 fs and

*τ*= +45 fs are limited exclusively to positions

**r**

_{2}(dashed) and

**r**

_{1}(solid), respectively.

## 5. Discussion and conclusions

**r**

_{1}and point

**r**

_{2}can be obtained if the mixed scalar products

*A*

_{mix}(

**r**

_{1},

*ω*) and

*A*

_{mix}(

**r**

_{2},

*ω*) as defined in Eq. (10) have a phase difference of

*θ*

_{mix}(

**r**

_{1},

*ω*) -

*θ*

_{mix}(

**r**

_{2},

*ω*) =

*π*, such that the signal at

**r**

_{1}is maximized and at

**r**

_{2}minimized for the same phase difference Φ =

*φ*

_{1}-

*φ*

_{2}between the two external polarization components. Considering this relation, we have also analyzed the responses of the symmetric T-structure investigated by Sukharev and Seideman [24

**40**(11), S283–S298 (2007). [CrossRef]

**95**(9), 093,901-4 (2005). [CrossRef] [PubMed]

**A**

^{(i)}(

**r**,

*ω*) as a function of spatial position and frequency for two perpendicular incident light polarizations. It is not even necessary to calculate the response in the frequency domain, but it can also be calculated using time-domain methods, such as Finite-Difference Time Domain (FDTD), and then Fourier-transformed [18]. Similar to the theoretically obtained optical response, it should also be possible to measure it [48

48. M. Sandtke, R. J. P. Engelen, H. Schoenmaker, I. Attema, H. Dekker, I. Cerjak, J. P. Korterik, B. Segerink, and L. Kuipers, “Novel instrument for surface plasmon polariton tracking in space and time,” Rev. Sci. Instrum. **79**(1), 013,704 (2008).
[CrossRef]

30. M. Shapiro and P. Brumer, “Laser control of product quantum state populations in unimolecular reactions,” J. Chem. Phys. **84**(7), 4103–4104 (1986). [CrossRef]

29. D. J. Tannor and S. A. Rice, “Control of selectivity of chemical reaction via control of wave packet evolution,” J. Chem. Phys. **83**(10), 5013–5018 (1985). [CrossRef]

31. R. S. Judson and H. Rabitz, “Teaching lasers to control molecules,” Phys. Rev. Lett. **68**(10), 1500 (1992).
[CrossRef] [PubMed]

## Acknowledgements

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**OCIS Codes**

(320.5540) Ultrafast optics : Pulse shaping

(250.5403) Optoelectronics : Plasmonics

(310.6628) Thin films : Subwavelength structures, nanostructures

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: May 12, 2009

Manuscript Accepted: June 29, 2009

Published: July 31, 2009

**Citation**

Philip Tuchscherer, Christian Rewitz, Dmitri V. Voronine, F. J. García de Abajo, Walter Pfeiffer, and Tobias Brixner, "Analytic coherent control of plasmon propagation in nanostructures," Opt. Express **17**, 14235-14259 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-14235

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### References

- P. Vasa, C. Ropers, R. Pomraenke, and C. Lienau, "Ultra-fast nano-optics," Laser & Photon. Rev. (2009). [CrossRef]
- L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006).
- S. A. Maier, Plasmonics: Fundamentals and Applications, 1st ed. (Springer, Berlin, 2007).
- T. Brixner, F. Garc´ıa de Abajo, J. Schneider, and W. Pfeiffer, "Nanoscopic Ultrafast Space-Time-Resolved Spectroscopy," Phys. Rev. Lett. 95(9), 093,901-4 (2005). [CrossRef] [PubMed]
- S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A Route to Nanoscale Optical Devices," Adv. Mater. 13(19), 1501-1505 (2001). [CrossRef]
- S. Maier, P. Kik, H. Atwater, S. Meltzer, E. Harel, B. Koel, and A. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nature Mater. 2(4), 229-232 (2003). [CrossRef]
- P. Andrew and W. L. Barnes, "Energy Transfer Across a Metal Film Mediated by Surface Plasmon Polaritons," Science 306(5698), 1002-1005 (2004). [CrossRef] [PubMed]
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