1. Introduction

The reflection of light incident upon a metal film is among the most basic phenomena in optics. While it is common to describe reflection (and transmission) in terms of the Fresnel equations based on boundary conditions [1,2], reflection can also be described by secondary radiation from current induced in the metal. To explain, suppose that initially there is only an incident wave and no film. Then, the film is introduced such that the incident wave is normal to it. Provided the film is sufficiently thick, one observes no wave on the transmitted side and finds the superposition of the incident and reflected waves on the illuminated side, i.e., a standing wave. The electric field of the incident wave drives a thin layer of surface current across the film, which oscillates with the driving field, and thus, radiates a secondary wave. The absence of a wave on the film’s transmitted side is then explained by this secondary radiation being $180^{\circ }$ out of phase with the incident wave. On the reflected side, the secondary wave remains $180^{\circ }$ out of phase but counter-propagates with the incident wave.

Notice that this way of describing reflection involves the recognition that an electromagnetic wave cannot be “blocked” in the same way that a water wave, e.g., is blocked in a mechanical sense by a barrier. Because electromagnetic waves require no medium to support their propagation, the only way to remove a wave from a region where it would otherwise be is for the wave to destructively interfere with another wave; this is the gist of the Ewald-Oseen extinction theorem [3–6]. The same logic can be used to explain why partial transmission occurs if the film is thin compared to the skin depth. There, the induced current is not sufficiently strong to radiate a secondary (extinction) wave with the needed magnitude to fully cancel the incident wave, leaving a portion of the incident wave intact on the film’s transmitted side.

From the point-of-view of the observed phenomena, there is no difference between either way of understanding reflection and transmission. Taking the secondary-radiation perspective, however, raises the possibility of temporarily altering the metal’s optical properties by interfering with the induced current via the application of an additional wave. For example, if it were possible to use pulsed light to briefly alter the density of conduction electrons in the metal, then when the film is illuminated by the incident wave the current it induces would be altered from what it otherwise would be. The result would be a change in the film’s reflectance and transmittance, caused by the action of the additional pulsed light. To be plausible, this process must involve pulses shorter than the characteristic time at which conduction electrons can diffuse, i.e., the relaxation time $\tau _{\text {e}}$. Choosing gold as an example, $\tau _{\text {e}}\sim 15$ fs [7–9], and thus, this concept could only be tested by experiments involving few-femtosecond pulses.

The hypothesis is that a modification of the free-electron density in a thin gold film can be created by $\sim$10 fs laser pulses resulting in changes to the film’s reflectance and transmittance. For brevity, the concept will be called Electromagnetically Induced Modification (EIM). The process envisioned is temporary and reversible, being active only while the pulses are present. It is also entirely classical in nature, in contrast, e.g., to electromagnetically induced transparency, occurs at room temperature, and would apply for any wavelength at which the metal exhibits high conductivity and no inter-band transitions or multi-photon processes. A simplistic model for the EIM effect will be proposed and a proof-of-principle experiment conducted to search for evidence of its occurrence. The changes in transmittance of thin films that are observed, and the dependence of these changes on the film-thickness and laser pulse-length, are then systematically studied.

Of course, the application of ultrafast laser pulses to study and manipulate the properties of matter is hardly new [10–32]. In some studies, effects that appear similar to EIM are observed. For example, it is found in [18–30,33–36] that a time-dependent change in the reflectance and transmittance is observed due to ultrafast pulses. However, these effects result from thermalization of the electron distributions via processes on the timescale of one hundred femtoseconds to a few picoseconds, and thus, are well-separated from the timescales involved here. Effects involving few-cycle laser pulses, meaning on the order of 10 fs or less, are of particular interest and studied by many [10–15,18–30,33–36]. One relevant application is the processing of signals on a femtosecond timescale [10–15] where inter-band transitions are involved. For such excitation to occur, the pulse spectrum and intensity are critical parameters. This is because either the photon energy needs to be greater than the transition threshold for excitation or the pulse intensity should be high enough for multi-photon processes to occur; neither condition will apply here. While our work cannot definitively prove the validity of EIM, the nature of the experiment conducted presents difficulty for attempts to explain the results with well-known processes such as those mentioned above. Thus, the article will conclude with an assessment of potential alternatives and suggest future methods to further clarify a conclusion.

2. Model for EIM

Consider a uniform thin film of gold with thickness $\ell$ on a transparent substrate that is illuminated along the $z$-axis by a beam of pulses with pulse length $\tau$. From a classical perspective, the electric field $\mathbf {E}$ of a pulse will exert a Coulomb force on conduction electrons, driving a current density according to Ohm’s law. Focus now on a small volume $V$ residing in the skin-depth region of the metal that is much smaller (along $z$) than the center wavelength $\lambda _{\text {o}}$ of the pulse spectrum. If $\tau >\tau _{e}$, electrons will follow $-\mathbf {E}$ to flow out of $V$ and the corresponding decrease of charge density within $V$ will be nearly instantaneously compensated by a flow of electrons from elsewhere in the metal. In other words, one can envision of two currents at play. The first is the flow of electrons due to the pulse’s electric field. This current is within the skin depth $\delta$ of the metal. Deeper than $\delta$, that field is negligible, and so, electrons freely diffuse through the metal lattice to compensate the Coulomb-force imbalance due to the altered density within $\delta$. In this case, no exotic effects (like EIM) are expected.

However, if $\tau <\tau _{e}$, it is conceivable that electrons are driven out of $V$ over a period ($\tau$) shorter than the time required ($\tau _{e}$) for electrons external to $V$ to diffuse and compensate the charge-density imbalance. There would be a modification to the distribution of conduction electrons in $V$. If so, this could temporarily affect the conductivity $\sigma$ and $\delta$, and thus, alter the film’s optical properties albeit only over the brief period $\tau$. The concept of EIM could then be supported if properties related to changes in $\delta$, such as a change in the film’s transmittance and reflectance, are observed. To test the concept would require laser pulses shorter than $\tau _{e}\sim 15$ fs. Also, one must employ techniques that are able to resolve small changes to the reflectance and transmittance occurring on timescales of $\tau$ and that would not affect the EIM process itself.

The approach taken to test this hypothesis involves illuminating the film with three beams of pulses, each with $\tau <\tau _{e}$, in a configuration equivalent to transient reflection gratings [37] (TRGs). Two of the beams create a standing-wave pattern across the film and are intended to establish the regions of temporarily modified electron density. These conditioning beams illuminate the same spot on the film at equal, but opposed, angles of incidence $\theta$ as shown in Fig. 1(a)-(b). Note that in this configuration, the total field distribution forms linear bands of high-to-low intensity. The third and less intense beam with pulse length $\tau$, called the signal beam, simultaneously illuminates the same spot on the film along the normal direction and is used to infer the presence, or absence, of any modification. Note that since the EIM effect is expected to occur at timescales of $\tau _{e}$, the pulse length of the signal beam must also be approximately $\tau _{e}$, or shorter, in order to resolve the effect.

 

Fig. 1. Electromagnetically Induced Modification (EIM). Two conditioning beams of $\tau \sim 10$ fs pulses converge at an angle $\theta$ to a common spot on a thin film of gold of thickness $\ell$ deposited on a fused silica window. The pulse field, $\mathbf {E}^{\text {con}}$, is either s- or p-polarized with respect to the film, with the latter, $\mathbf {E}_{\text {p}}^{\text {con}}$, being shown in (b). For each pulse pair, a standing wave pattern forms where the beams overlap. In (c) is a plot of the tangential component of the net electric field $\mathbf {E}_{\parallel }^{\text {con}}$ across the film in this standing wave region, i.e., the black outlined region in (b) in the overlap region inn the $x$-$y$ plane. Sketch (d) depicts the undisturbed electron distribution as an array of evenly spaced electrons (circles) in the same region, which is then redistributed via $\mathbf {E}_{\parallel }^{\text {con}}$ to form temporary regions of partly enhanced or partly reduced charge, $\Delta q$, in (e).

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We pause to note an apparent similarity of the conditioning and signal beams with that of pump-probe spectroscopy. Pump-probe studies generally involve the initial excitation of a quantum state by a pump beam, or beams, followed by delayed illumination of the film by a less-intense probe beam. The probe then monitors changes in the film’s optical properties due to the pump. The key difference is the implied quantum mechanical character of pump-probe studies. In EIM, the hypothesized “excitation” is the temporary creation of nonuniform free-electron density. Because the laser spectrum used in the experiment (below) has neither the energy nor intensity to cause inter-band excitation, the process is viewed as classical in character. Thus, the terminology of pump-probe spectroscopy is avoided to highlight this difference although the beam configuration used is essentially the same.

Consider Fig. 1(c), which shows the instantaneous tangential component of the standing-wave field, $\mathbf {E}_{\parallel }^{\text {con}}$ and how it alternates in direction along the $x$-axis. As time advances, a given fringe will oscillate between positive and negative in direction, but its location on the film remains stationary due to the standing-wave nature of the field. Suppose that a portion of conduction electrons are redistributed to form alternating regions of partly enhanced and partly depleted charge-density. Figure 1 conceptually illustrates this by representing the undisturbed electron-density as an array of equally spaced circles in Fig. 1(d) that are then moved by the field to form the arrangement in Fig. 1(e). As the field oscillates, regions of enhanced and depleted charge density alternate, but again, the overall pattern remains stationary.

Now, consider the top fringe in Fig. 1(c), i.e., the red band where $\mathbf {E}_{\parallel }^{\text {con}}$ is directed downward. This field will redistribute conduction electrons to create a region of partly enhanced negative charge, shown by the blue band at the top of Fig. 1(e), accompanied by a region of partly reduced negative charge shown by the red band immediately below. A change in the charge density, whether it be an enhancement or reduction, would affect the metal’s conductivity, and thus, should alter $\delta$. Consequently, it may be possible to detect that the process is occurring by using an additional beam of short pulses and observe whether that additional beam has reflection or transmission behavior differing from what is expected. The additional beam is the signal beam and consists of less-intense pulses that illuminate the same spot on the film and do so simultaneous with the conditioning-beam pulses. To prevent interference, the polarization of the conditioning and signal beams are crossed [38].

Using a classical approach, namely the Drude model, an order of magnitude estimate for the transmittance change of the signal beam can be made. Here, conduction is described as the motion of a gas of non-interacting electrons within a positively charged ionic lattice [7,8,39]. Under the influence of the conditioning beams’ field $\mathbf {E}^{\text {con}}(\mathbf {r},t)$, assume that the electrons are driven ballistically to form the altered charge-density pattern $\Delta q$ in Fig. 1(e). Then focus on a single fringe, the one shown in Fig. 2(a) with volume $V$, and consider an instant in time $t$ when the p-polarized field $\mathbf {E}^{\text {con}}_{\text {p}}(\mathbf {r},t)$ drives electrons into the volume as shown. The volume is given by the fringe spacing $\Delta x$ and the area $A_{\perp }$ extending into the metal in the $y$-$z$ plane. Note that there are two areas $A_{\perp }$ on the top and bottom of $V$. Approximating the field as uniform across the top area, i.e., $\mathbf {E}^{\text {con}}(\mathbf {r},t)\sim E^{\text {con}}(t)\hat {\mathbf {x}}$, the current density of electrons $\mathbf {J}_{\text {e}}$ crossing $A_{\perp }$ will be $\mathbf {J}_{\text {e}}(t)=-\sigma E^{\text {con}}_{\text {p}}(t)\hat {\mathbf {x}}$, where $\sigma$ is the conductivity. Because $A_{\perp }$ is perpendicular to the current, a rate equation for the charge $q$ entering $V$ is obtained from $\text {d}q/\text {d}t=-J_{\text {e}} A_{\perp }$ as

The factor of two in Eq. (1) is due to current flowing into $V$ from the top and bottom areas $A_{\perp }$, recall Fig. 2(a). As the field oscillates, it will alternate from driving (electron) current into and out from $V$ after each half-cycle of the oscillation. Take $2N_{\text {c}}$ to be the approximate number of half-cycles in the pulse $\tau$, and focus on a single cycle when the field is driving current into $V$. For that cycle [40], take $E^{\text {con}}_{\text {p}}(t)\sim E_{\text {o}}\sin \left (\omega _{\text {o}} t\right )$ and integrate Eq. (1) over the half cycle to get

Equation (3) describes the total amount of charge displaced by the laser pulse to enhance the charge density relative to the undisturbed density. Recalling Fig. 2(a), this enhancement occurs for half of the fringes formed by $\mathbf {E}^{\text {con}}$ across the film. For the other half, the inverse occurs, where the relative charge is reduced by the same fraction, $\eta$. Using the DC conductivity [41] as a crude estimate $\sigma _{\text {o}}=4.5\times 10^{7}\,\Omega ^{-1}\,\text {m}^{-1}$, a free-charge density [8] of $n_{\text {e}}=5.9\times 10^{28}\,\text {m}^{-3}$, $\omega _{\text {o}}=2.4\times 10^{15}\,\text {s}^{-1}$, and $E_{\text {o}}\sim 4.0\times 10^{8}\, \text {N}/\text {C}$, one finds that $\eta \sim$1.1% for one cycle. Thus, the effect, should it occur, is small. Yet, as one will see below, the experiment finds a small change in the film’s transmittance that appears reasonable in light of this estimate.

 

Fig. 2. Electron redistribution in EIM and its effect on transmission. The sketch in (a) shows a cross section of the metal film being exposed to the incident conditioning beams as in Fig. 1(a)-(b) travelling along $\hat {\mathbf {n}}^{\text {con}}$ and $\hat {\mathbf {n}}^{\text {con'}}$. The field pattern in the metal, which is tangential to the film [2], i.e., $\mathbf {E}_{\parallel }^{\text {con}}$ has the form of a stationary standing-wave, or fringe, pattern with fringe spacing $\Delta x$. Detail of a small volume $V$ within the skin depth $\delta$ is shown in (a) by the dashed box. The current density of flowing electrons $\mathbf {J}_{\text {e}}$ passing through the area $A_\perp$ of $V$ is shown and is directed into $V$. Sketch (b) depicts the simultaneous exposure of the film to the signal beam field, $\mathbf {E}^{\text {sig}}$. Electrons redistributed by $\mathbf {E}_{\parallel }^{\text {con}}$ present regions of reduced and enhanced transmission to the signal light through perturbations of the skin depth $\delta$.

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Meanwhile, the signal beam illuminates the same area of the film and its polarization is rotated $90^{\circ }$ with respect to the conditioning beams so that interference is avoided. The signal-beam field $\mathbf {E}^{\text {sig}}$, which is significantly weaker than $\mathbf {E}^{\text {con}}$, will drive a current of its own in the metal. However, each band of partly enhanced, or reduced, charge density in Fig. 1(e) disrupts the secondary radiation that would otherwise be produced by the signal beam. In other words, the modified charge density $\eta$ amounts to small changes to the skin depth encountered by the signal beam. The reason is because $\delta$ depends on the conductivity $\sigma$ as $\delta =\sqrt {2/\mu \sigma \omega }$ where $\mu$ is the metal’s permeability [2]. Thus, this simplistic model suggests that $\delta$ is slightly smaller in regions with enhanced electron density and slightly larger in the regions with reduced electron density.

In Fig. 3, two measurements of the transmitted signal-beam intensity by a photodiode are depicted. When the conditioning beams are blocked in Fig. 3(a) only the signal beam illuminates the film, partly transmits through it, and is then received by the photodiode. The power registered by the photodiode in this case is $P^{\text {tra}}_{0}=I^{\text {tra}}_{0}\Delta A$ where $I^{\text {tra}}_{0}$ is the transmitted signal intensity and $\Delta A$ is the area of the photodiode. Then, the measurement is repeated after the beam blocks are removed and the conditioning beams illuminate the film simultaneous with the signal beam as in Fig. 3(b). Now the power received by the photodiode is $P^{\text {tra}}_{1}=I^{\text {tra}}_{1}\Delta A$, where $I^{\text {tra}}_{1}$ is again the transmitted signal intensity. Notice that because the conditioning beams are incident at an angle to the film, they are prevented from reaching the collection lens in Fig. 3.

 

Fig. 3. Schematic of the two-step procedure used to search for a change of the transmittance of the film due to EIM. In (a), the conditioning beams are blocked from reaching the film and the photodiode measures the transmittance of the signal beam, $T_{0}$. In (b), the same measurement is performed except with the conditioning beams illuminating the film simultaneous with the signal beam. The photodiode again responds only to the signal beam, but may have a different response, $T_{1}$, if EIM occurs.

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To connect the photodiode measurements to the EIM model, let $I_{\text {s}}$ represent the intensity of the signal beam incident on the film. Then, in the absence of conditioning beams, the transmitted signal-beam intensity $I^{\text {tra}}_{0}$ is given by [2]

Note that $\Delta \delta$ represents the magnitude of the change in skin depth whereas the sign of the change is accounted for in the denominators in Eq. (5). Although conductivity is an intrinsic property of any material in bulk form, it is not at the nanoscale, and can depend on the film thickness $\ell$ [42]. For very thin films, the effect of surface roughness and electron surface-scattering may significantly modify the conductivity. Further detail on this point is given in the Supplement 1 and shown not to be a concern in this work.

Using two calibrated photodiodes as described in Sec. 3 below, the reflected and transmitted signal beam from the glass substrate without a film can be recorded. The sum of the two measurements is the power of the signal beam $P_{0}$ assuming no loss in the glass. Then, when the film is present, the response of the photodiode in Fig. 3 can be regarded as proportional to $I^{\text {tra}}_{0}$ or $I^{\text {tra}}_{\text {1}}$ when divided by the beam-spot area $A_{\text {s}}$ of the signal beam across the film. Finally, dividing $I^{\text {tra}}_{0}$ and $I^{\text {tra}}_{\text {1}}$ by the incident (signal) intensity $I_{\text {s}}$ gives spectral transmittance values $T_{0,\lambda }$ and $T_{1,\lambda }$. The subscript $\lambda$ indicates that these quantities depend on wavelength.

The wavelength dependence of $T_{0,\lambda }$ and $T_{1,\lambda }$ originate from the wavelength dependence of $I_{\text {s}}$, $\delta$, and $\Delta \delta$ in Eqs. (4)–(6). A spectrometer can be used to measure $I_{\text {s}}(\lambda )$ and then summed over the spectrum give the total, i.e., $I_{\text {s}}=(1/N_{\text {s}})\sum I_{\text {s}}(\lambda _{i})\Delta \lambda$, where $N_{\text {s}}$ is the number of spectral points measured by the spectrometer in the range $\lambda \in \left [633\,\text {nm},1050\,\text {nm}\right ]$. Then, the transmittance values can be determined in a similar manner as $T_{0}=(1/N_{\text {s}})\sum _{\lambda }T_{0,\lambda }\Delta \lambda$ and $T_{1}=(1/N_{\text {s}})\sum _{\lambda }T_{1,\lambda }\Delta \lambda$ where $N_{\text {s}}$ divides out given the definition of $I_{\text {s}}$. Lastly, the difference in transmittance with and without the conditioning beams illuminating the film $\Delta T=T_{1}-T_{0}$ can then be found. Using the laser spectrum measured in Sec. 3 and $\sigma _{\text {o}}(\ell )$, this model for $\Delta T$ is plotted in Fig. 4 where further details are given in the Supplement 1. The model predicts a decrease in the transmittance of signal light due to the influence of the conditioning beams. One may find this result surprising since the small decrease of electrons in one region (fringe) is compensated by a small increase of electrons in another region as depicted in Fig. 2(b). However, recall the exponential dependence on $\delta$ in Eq. (5), which shows that the change in transmission for increased electron density in one region and a decrease in another will not cancel.

 

Fig. 4. Prediction of the EIM effect as a function of the film thickness. Here, the change in transmittance of the film $\Delta T$ is plotted by combining Eqs. (4)–(6), using the $\sigma _{\text {o}}(\ell )$ discussed in the Supplement 1, and summing over the same spectral components of the laser pulses used in the measurements of Sec. 3.

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Before proceeding, note that a key assumption of this simplistic model is that the EIM process is linear. Indeed Eqs. (2) and 3 exhibit a linear dependence on the field. As explained by Boyd et al. [43], free electrons have no contribution to a nonlinear optical response within the electric-dipole approximation for a flat, featureless metal surface. Moreover, inter-band transitions from the valence band to conduction band are not excited due to the laser energy spectrum, which is discussed below.

3. Experiment and results

The experiment used to test the EIM hypothesis is shown in Fig. 5. An 80 MHz repetition rate Ti:Sapphire laser (Laser Quantum, Venteon One) produces $\tau \sim$ 10 fs pulses with pulse energy $\mathcal {E}_{\text {p}}\sim 3$ nJ and center wavelength $\lambda _{\text {o}}=790$ nm. To characterize the laser pulses [40,44], an interferometric autocorrelator (Thorlabs, FSAC) is used. The autocorrelator is pre-compensated with a set of chirped mirrors and a CaF$_2$ wedge pair to add negative group delay dispersion (GDD). By assembling the pre-compensation optics and autocorrelator on a dedicated breadboard, the laser pulse length at different locations in the experiment can be measured. An example of the pulse autocorrelation (AC) trace measured close to the metal film is shown in Fig. 6. The peak pulse-intensity is $I_{\text {o}}\sim 10^{10}$ $\text {W}/\text {cm}^{2}$, and thus, ionization of electrons from the film and associated ablation is avoided [45–48].

 

Fig. 5. Optical layout used to test the EIM concept. As described in the text, a GDD pre-compensated beam of $\tau \sim 10$ fs pulses from a Ti:Sapphire laser is split into the signal beam and two conditioning beams. The beams are brought to a focus on the gold film, where photodiodes PD2 and PD3 measure the film’s reflectance and transmittance, respectively. The angle of incidence of the signal beam on the surface of the film is a small nonzero value to enable the reflectance measurement. The inset (a) shows the linear polarization states of the conditioning beams, where both beams are either s- or p-polarized. Inset (b) shows an example of an observed fringe pattern due to the conditioning-beams standing wave as imaged through the removable microscope objective (MO) by a CCD sensor.

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Fig. 6. Laser pulse characterization. An interferometric autocorrelator (IAC) is placed in the beam path near the location of the metal film in Fig. 5. The main plot shows the observed IAC trace in blue and the pulse autocorrelation (AC) trace in red. The AC trace defines the FWHM laser-pulse duration of $\tau \sim$ 10 fs. Shown inset is the measured spectrum of the pulses, covering the range $\lambda \in \left [633\,\text {nm},1050\,\text {nm}\right ]$.

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Following the laser, a dispersion-compensation mirror pair (Laser Quantum, DCM7) along with a CaF$_2$ wedge pair is used to account for the GDD over the full beam-path [49]. This is critical because the short pulses become longer due to the GDD added by optical elements or simply by air [40,44]. For example, at $\lambda _{\text {o}}\sim$790 nm, the GDD of air is $\sim 65~\text {fs}^2$, meaning that a 10 fs laser pulse becomes $\sim 20$ fs after travelling three meters in air. Each optical element in the beam path also adds a small amount of GDD.

To keep the amount of added GDD to a minimum, metallic femtosecond optimized mirrors (FM) are used whenever possible to redirect the beam. A 20:80 Reflection:Transmission (R:T) low GDD beamsplitter, BS1, reflects 20% of the beam to be used as the signal. Then, a 50:50 (R:T) beamsplitter, BS2, divides the transmitted beam from BS1 into the two conditioning beams. A delay stage (DS1) is used in the path of one of the conditioning beams and another (DS2) is used in the signal beam, which allows the pulses of all three beams to be overlapped in time at the film. The polarization of the beams is controlled by low GDD half waveplates WP1-WP3, which allow the measurements to be done with cross-polarized conditioning and signal beams, thus preventing interference [38]. The polarization states are labeled s or p as defined in Fig. 5(a).

The conditioning beams are focused onto the metal by two $15^{\circ }$ off-axis parabolic mirrors (OPM1) with a focal length of 381 mm and a $90^{\circ }$ off-axis parabolic mirror (OPM2) focuses the signal beam onto the film. The films are electron-beam deposited gold on uncoated fused silica glass windows. A variety of films with different thickness $\ell$ are made ranging from $\ell =20$ Å to $\ell =400$ Å. The typical error in $\ell$ reported by the manufacturer (Ferrotec Corp.) is $< \pm 10$ Å (for $\ell =60$ Å).

To ensure that the focused signal and conditioning beams overlap at the same spot, and that the pulses arrive at the same time, the beam spots on the film are imaged through a long working-distance microscope objective (MO) by a CCD camera. It is critical to find this overlap as the intent is to study only the local effects due to the conditioning beams. The overlap is achieved by small adjustments to the delay stages and confirmed by the observation of interference fringes. In the language of pump-probe spectroscopy, the delay between the pump and probe pulses is less than the pulse length itself, $\tau \sim 10$ fs, and thus, the pulses are essentially simultaneous. Figure 5(b) shows an example of the fringes observed, with a fringe spacing of $\Delta x\sim 3$ $\mathrm{\mu}$m. This procedure is repeated each time the film is changed or moved and the MO is removed prior to transmittance or reflectance measurement.

Three photodiodes with a rise time of approximately 1 ns are used for data collection. The first shown in Fig. 5, PD1, monitors the pulse-to-pulse laser power fluctuations. The second photodiode, PD2, measures the reflected signal-light power from the film via a small pick-off mirror labeled DM (Thorlabs, PFD10-03-P01). The transmitted signal light is measured by PD3 and is the same photodiode discussed in Fig. 3. A power meter (Gentec XLP12-3S-H2-D0) with a noise equivalent power of 0.5 $\mathrm{\mu}$W is used to calibrate the PD2 and PD3 photodiodes.

For each film, the data collection process includes three steps. First, the reflected and transmitted power are recorded with the conditioning and signal beams illuminating the metal. Dividing these by the power of the signal beam, $P_{\text {s}}$, gives the reflectance and transmittance, $R_{1}$ and $T_{1}$, respectively. Next, the conditioning beams are blocked and the data is collected again with only the signal beam reaching the film, providing $R_{0}$ and $T_{0}$. Lastly, the data is collected with the conditioning beams reaching the film but with the signal-beam blocked, which is done to verify that there is no scattering or stray light from the conditioning beams that reach the photodiodes. These measurements are “single shot,” meaning that the photodiode signals are recorded for each pulse over a period that includes approximately $10^{5}$ pulses, such that after averaging, effects due to fluctuations in the laser power are suppressed. The standard deviation is then calculated, and the error bars are found by propagating the standard deviation values [50].

Figure 7 presents the change in transmittance and reflectance, $\Delta T$ and $\Delta R$, measured for films of varying thickness $\ell$. As seen in Fig. 7(a), $\Delta T$ depends sensitively on $\ell$ with the greatest magnitude effect occurring at $\ell =50$ Å. Note that the skin depth in this wavelength range is $\delta \sim 30$ nm [8], which is much larger than the film-thicknesses where an effect is seen. Interestingly, $\Delta R\sim 0$ within the error bars in Fig. 7(b) for all thicknesses, yet nonzero $\Delta R$ is expected for nonzero $\Delta T$. Nonzero $\Delta R$ is not found within the experimental error bars, indicating that if a change in reflectance occurs, the current experiment does not have sufficient sensitivity to measure it.

 

Fig. 7. Experimental test of the EIM effect. In (a) is shown the measured change in transmittance $\Delta T$ as a function of film thickness $\ell$, and in (b) the change in reflectance $\Delta R$. Both polarization combinations are presented by the solid red and blue curves. The dashed black curve in (a) re-plots the model prediction of Fig. 4 to compare the functional form of the theoretical and experimental results. The finding that $\Delta T\ne 0$ demonstrates that the conditioning beams’ action the film does indeed affect the transmission of the signal beam, supporting the concept of EIM. The effect is small, however, being approximately $\Delta T\sim -2.7\%$ for $\ell =50$ Å, and showing no measurable change in reflectance within the experimental error.

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In the case of no film, no change to either $\Delta T$ or $\Delta R$ is expected as there are no free electrons to be redistributed by the conditioning beams. To verify that the observed changes are due only to the metal, $\Delta T$ and $\Delta R$ are measured for a glass window with no film and both $\Delta T$ and $\Delta R$ are zero within the experimental error. For films thicker the skin depth, changing the skin depth by a few percent (via EIM) would not affect the transmittance. This is because much of the light is absorbed or reflected regardless of whether the skin depth is changed slightly due to the action of the conditioning beams. Moreover, by comparing the measurements to the black dashed curve in Fig. 7(a), which is the model prediction from Figs. 4, one sees agreement with respect to the functional dependence (curve shape) on $\ell$. The disagreement in magnitude between the curves is partially due to the fact that there is $<\pm 10$ Å error bar on the thickness of the films. Of course, the agreement is only qualitative as the model is highly simplistic. Recalling Sec. 2, one can see that the model does not predict the difference seen in the EIM effect between the p-polarized and s-polarized conditioning beam cases. It is not presently clear why such difference is observed in the context of the model in Sec. 2. The EIM effect should disappear as the pulse length becomes longer than the relaxation time, $\tau _{\text {e}}$. By adding GDD to the beam before it is split into the signal and conditioning beams, $\tau$ is increased and the measurements for $\Delta T$ are repeated. The result is shown in Fig. 8(a) for a the $\ell =50$ Å film. As expected, $\Delta T$ approaches zero rapidly with increasing $\tau$ and has vanished once $\tau \gtrsim 2\tau _{\text {e}}$.

 

Fig. 8. Conditioning pulse length $\tau$ and pulse intensity $I_{\text {o}}$ dependence of the EIM tests. In (a) is shown the change $\Delta T$ measured as the laser pulse duration $\tau$ is increased for the $\ell =50$ Å film in Fig. 7. The conditioning beams are p-polarized and signal beam is s-polarized. The model for EIM predicts $\Delta T \to 0$ as $\tau$ exceeds the metal’s relaxation time $\tau _{\text {e}}\sim 15$ fs; behavior that is seen here. Note that the intensity in the top axis goes from higher to lower from left to right. This is the peak pulse-intensity $I_{\text {o}}$ corresponding to each $\tau$. The laser fluence is constant. In (b) is shown the intensity dependence of $\Delta T$. Here, $\tau$ is kept approximately constant at $\sim$10 fs and $I_{\text {o}}$ is varied. Again, $I_{\text {o}}$ ranges from higher to lower from left to right on the bottom axis. Note that although the intensity dependent study is performed with pulses that are less intense by about an order of magnitude compared to the pulse duration study, a nonzero $\Delta T$ is observed indicating that the EIM effect is more sensitive to $\tau$ than $I_{\text {o}}$.

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4. Discussion

It is necessary to consider the possibility that processes other than EIM could explain the observations. First, consider transient thermal effects in a pump-probe context. For pump-pulse energy that is less than the inter-band transition, conduction electrons are instantaneously heated through electron-electron interactions (free-carrier absorption) creating a non-equilibrium temperature distribution [51]. Then, thermalization occurs as the electrons lose energy via electron-phonon interactions to equilibrate with the lattice. It is now well-known that this thermalization process changes the gold’s optical properties and is delayed from the pump by $\tau _{\text {th}}$, turning-on over a period of $\tau _{\text {th}}\sim 500$ fs [30,43] and disappearing entirely in several picoseconds [18–30,33–36]. Block et al. [52] have even spatially resolved these changes. Thus, it appears that the changes in transmittance found in Fig. 7(a) cannot be explained by delayed thermal effects given that the experiment probes the transmittance at the same time as the pump, i.e., the conditioning and signal pulses are simultaneous and present only for $\tau \sim 10\,\text {fs}\ll \tau _{\text {th}}$. Thermal effects persisting from pulse-to-pulse are also unlikely as the period of time between consecutive pulses is $12.5\,\text {ns}\gg \tau _{\text {th}}$. Now, recall Fig. 8(a) where the dependence on $\tau$ is presented. These measurements are done with constant fluence, meaning that making $\tau$ twice as long decreases the intensity by a factor of two. One may then wonder how changes in transmittance depend on the intensity of the pulses for fixed $\tau$? Fig. 8(b) investigates this question for $\tau \sim 10$ fs, again for the $\ell =50$ Å film. The conditioning pulse intensity $I_{\text {o}}$ is varied by successive attenuation with neutral density filters where the dispersion of the filters is balanced by negative GDD to ensure that $\tau$ is approximately constant for each measurement. The results show that $\Delta T$ is nearly constant even as $I_{\text {o}}$ is decreased by a factor of two, see the left three data points (remember that $\Delta T$ is the relative change of intensity of the signal beam). For yet lower intensity, $\Delta T=0$ within the error bars. Also, note that the intensity dependent study [Fig. 8(b)] is performed with pulses that are less intense by about an order of magnitude compared to the pulse duration study [Fig. 8(a)], and yet a nonzero $\Delta T$ is observed. If the transmittance changes were due to thermal effects, however, one would expect $\Delta T$ to continuously decrease with the pulse intensity. In other words, the process behind $\Delta T$ is much more sensitive to the pulse duration, Fig. 8(a), than it is to the pulse intensity, Fig. 8(b). Adding a variable delay between the conditioning and signal pulses, as is done in pump-probe measurements, could provide further insight.

Next, consider the possibility that inter-band processes, meaning excitation of electrons from the valence d-band to the conduction band, cause absorption and account for the decrease in transmittance. The inter-band transition threshold is generally cited as $\sim$2.45 eV [19,30,53,54], i.e., $\sim 506$ nm where wavelengths greater than this do not excite inter-band transitions [19,30]. The laser spectrum in Fig. 6 extends from 633-1050 nm (1.18-1.96 eV), and thus, no inter-band transitions are expected. However, Guerrisi et al. [54] show another transition at 1.94 eV, or 639 nm, which is often neglected in the literature and others [53] cite a transition at $~1.8$ eV or 689 nm. These wavelengths reside just within the spectrum, where the intensity is comparatively low. Nevertheless they imply a degree of inter-band excitation could occur. If so, one would not expect the behavior seen in Fig. 8(a) since increasing $\tau$ (via GDD addition) does not affect the pulse’s spectral content. Multi-photon inter-band transitions also appear unlikely due to the absence of intensity dependence for the nonzero data points in Fig. 8(b).

Surface plasmon (SP) effects are another possible mechanism that could account for the observations. For SPs to be excited, both momentum and energy conservation conditions must be satisfied [55–69]. The excitation is required to occur from light incident on the film from a medium with $n>1$, where $n$ is the refractive index. For this condition to be satisfied, there would have to be a layer of oxide or another dielectric on the film, which is unlikely for gold. Plasmonic effects are also highly sensitive to the film thickness [55]. For gold the optimum thickness is $\ell \sim 500$ Å for a strong plasmonic response [55], which is substantially greater than the $\ell =50$ Å feature seen in Fig. 7(a). Sub-wavelength holes in the film or nanometer-scale surface roughness can also excite SPs. If present, holes would enhance the transmittance [55,57,58,66,67] and SPs due to surface roughness would reduce the reflectance [55], both of which are not observed Fig. 7. Conversely, Guo et al. [70] show that a suitably patterned array of nanobar antennas on silicon-dioxide coated silver can affect reflection in unique ways. That effect differs, however, from what is seen here as no oxide is present on the film nor are nanostructures. To further rule out plasmonic effects, the experiment is performed with two crossed-polarization combinations (signal/conditioning: s/p and p/s). Because SPs are excited on a smooth metal film only with p-polarized light [55], the observation of nonzero $\Delta T$ in Fig. 7(a) for both polarization combinations render SP effects an unlikely explanation.

The difference between the measurements and those of transient reflecting gratings [37] (TRGs) should be explained. Indeed, the linear interference pattern formed by the conditioning beams, shown in Fig. 5(b), is equivalent to a TRG. The pattern causes a periodic modulation of temperature in the metal that persists after the pulses vanish. Such gratings can then be observed by the associated modulation of the metal’s reflectivity via a probe (signal) pulse. Heat diffusion causes the temperature modulations to spread until the grating eventually erases itself. As such, the process constitutes a non-local effect in both space and time. Using the extended two-temperature model, Sivan et al. [27] show heat diffusion can be faster than the electron-phonon transfer rate. In all cases considered, they find that the thermal effects in TRGs occur over timescales ranging from hundreds of femtoseconds to approximately one picosecond. These timescales are at least an order of magnitude longer than the signal pulse in Sec. 3, and thus, heat diffusion is an unlikely candidate to explain the measurements.

Lastly, there is an intriguing aspect of Fig. 7(a) regarding the simplistic model of Sec. 2. Equation (6) employs the DC conductivity $\sigma _{\text {o}}$ whereas one would expect to use the real part of the frequency dependent (complex) conductivity $\sigma (\omega )$ as it describes optical attenuation, (the imaginary part describes a phase shift [71]). Olmon et al. [8] shows that for the energy range of the laser used here, $\text {Re}\{\sigma (\omega _{\text {o}})\}/\sigma _{\text {o}}\sim 10^{-2}$, meaning that the relative enhancement (or reduction) of electron charge, $\eta$, should be $10^{-2}$ smaller. One interpretation for this inconsistency could be that, if the EIM process is indeed real, the current flow involved has little frequency dependence. While it is difficult to see how this could be true, we point out that an understanding for the intra-band behavior of conduction electrons in such extreme conditions ($\tau \sim 10$ fs, $E_{\text {o}}\sim 4\times 10^{8}\,\text {N}/\text {C}$, $\ell \sim 50$ Å) appears to be nascent in the literature. Thus, further study could clarify the significance of these observations.

5. Conclusions

Based on classical arguments, this work proposes and tests an apparently new ultrafast optical effect for thin metal films. The hypothesis is that a standing wave formed by laser pulses shorter than the relaxation time can temporarily redistribute electrons in a non-equilibrium manner. Interim variations in conduction-electron density would then affect the metal’s optical properties and be observable. The process is envisioned to occur classically via Coulomb forces and not through quantum excitation. If so, such changes are expected only when the pulses act on the metal and not at a delayed time, in contrast to well-known transient thermal effects and inter-band transitions. An experiment is conducted, finding that the transmittance of gold films is reduced by as much as 2.7% during exposure to pulses approximately 10 fs in duration. As the film thickness or pulse length are increased, the changes in transmittance diminish, disappearing entirely for film thickness greater than 300 Angstrom or pulse length longer than 20 fs. Alternative explanations involving established ultrafast processes are examined in the context of the experiment. Although the alternatives can either be eliminated or shown to have difficulty explaining the observations, it is not possible to definitively conclude that a new effect occurs and further investigation is justified. This work could spur a variety of applications useful in photonics assuming the EIM effect is eventually validated and can be made stronger by, e.g., increasing the laser peak-intensity. The reversibility of the effect and the ultrafast timescale at which it occurs makes it a candidate for the development of high-speed optical gating devices and suggests the possibility of creating optically formed and temporary metamaterials.