1. Introduction

When a laser beam with frequency ω is incident on a material without a center of symmetry, the second order nonlinear polarization of the material has ω+ω=2ω and ω-ω=0 components. The two terms are responsible for second-harmonic generation (SHG) and optical rectification, respectively [1]. The former is utilized to generate shorter wavelength while the latter as a photodetector. Conventionally the symmetry in this context occurs on the atomic scale and therefore these effects are observed only in a certain class of material or at surface. Recently Ishihara et al. reported SHG from photonic crystal slabs (PCS) consisting of centrosymmetric materials [2]. They found that SHG can be generated if the laser is incident obliquely or if the photonic unit cell is asymmetric and the effect was associated with the spatial derivative term Q_ijklE_j∇_kE_l rather than χ ⁽²⁾ _ijkE_jE_k in the second order nonlinear polarization, where E_j is the j-th Cartesian component of the optical field and Q_jikl, χ_ijk are fourth-rank and third-rank tensors, respectively. Thus in this case, asymmetry on the scale of light wavelength is important for generating SHG.

The DC polarization due to the spatial derivative term results in photon drag effect, and is often distinguished from optical rectification. In 1970s Gibson et al. [3] and Danishenskii et al. [4] independently observed photo-induced voltage by irradiating Germanium crystals with pulse lasers. The phenomenon was named “photon drag” because it was explained in terms of drift of carriers pushed by photons. Since then extensive studies of the photon drag effect were carried out in semiconductors [5] and metals [6]. Electromotive force and current due to photon drag in simple metals were discussed in terms of the momentum conservation and the hydrodynamic theory by Goff and Schaich [7]. Surface plasmon enhanced photon drag was reported for Au thin film [8].

The purpose of this paper is to present experimental observation of photon drag voltage in metallic photonic crystal slabs with asymmetric unit cells and elucidate it with a simple momentum conservation argument. We modify the approach in [7] for the periodic systems having diffraction. In case of normal incidence, the photo-induced voltage is understood as optical rectification, because asymmetry in the photonic unit cell is responsible for the effect. The photovoltaic effect in metallic PCS with symmetric unit cells was briefly reported previously [9]. Further details on this will be published elsewhere.

2. Samples and experimental set up

A 40 nm Au film was evaporated on a quartz substrate with a nominally 3nm thick intermediate layer of Cr for better adhesion. Resist layer coated on the Au film was patterned by electron beam and the Au film was etched to form a periodic structure with symmetric (a) and asymmetric (b) unit cells as seen in the AFM images of Fig. 1. Both the samples have a period of 1220 nm. In the unit cell of Sample (a), symmetry planes normal to the periodic axis x can be found, while in the unit cell of Sample (b), the inversion symmetry is broken along x-axis. The experimental set up is shown in Fig. 2. Light from optical parametric oscillator (OPO) pumped by tripled YAG laser is sent to the sample with p-polarization, for which stronger diffraction and therefore a stronger signal is expected.

The patterned area is 0.6×0.6mm² and is electrically connected to a coaxial cable through two lead patterns on the substrate. The laser pulse width and repetition rate were 5 ns and 10 Hz, respectively. We measured photo-induced voltage across the sample with an oscilloscope (Tektronix TDS3012B) through a fast-speed amplifier with a gain of 25. All the voltages referred to in this paper are those that actually occur at the sample, and not the voltages measured after the amplification. The signal intensity was averaged for 64 pulses in order to eliminate the pulse-to-pulse fluctuation of OPO and normalized to the unit laser intensity for each wavelength. Because the signal intensity is confirmed experimentally to be proportional to the laser intensity, this normalization is justified

 

Fig. 1. AFM images of periodic structures fabricated on Au layers and their profiles for (a) symmetric and (b) asymmetric structures, respectively. (c) Schematic diagram of asymmetry in sample profile and diffractions. Coordinate and voltage sign are also defined.

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Fig. 2. Experimental set up. Nanosecond pulsed laser light is sent to sample with incident angle θ. The figure is drawn for θ _.>0.

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3. Angle dependence of photovoltage

When the laser pulse is incident on metallic photonic crystal slabs, a voltage pulse is observed on the oscilloscope except for those wavelength-dependent particular angles where the signal vanishes. The voltage pulse width (typically 7ns) of the signal is slightly larger than the laser pulse (5ns), which can be ascribed to the parasitic impedance of the electric connection. Figure 3 (a) and (b) show the peak voltage for incident wavelength of 1200 nm as a function of the incident angle θ for symmetric and asymmetric photonic crystals, respectively. In case of the symmetric structure, the voltage is anti-symmetric as a function of the incident angle θ. This can be readily understood from the symmetry of the system. As a result, the voltage is zero at the normal incidence. Besides, signal enhancement appears at θ=±2 and ±25 deg. These angles are the diffraction channel opening angles for the incident light wavelength (1200nm). For the same angles, photovoltage features are seen as well for the asymmetric structure with the same period, while the signal is asymmetric as a function of θ. In particular, the photovoltage at normal incidence for asymmetric structure is clearly non-zero. The fact that the magnitude of the voltage observed at normal incidence in the asymmetric sample is comparable to the maximum voltage signal seen in the symmetric sample suggests that the two have the same origin: In the former case the symmetry is broken in the structure by the asymmetric unit cell, while in the latter by the finite incident angle.

 

Fig. 3. Photovoltage at the laser wavelength 1200 nm, as a function of incident angle θ for symmetric (a) and asymmetric (b) structures. At 2 and 25 deg, spiky structures induced by diffraction channel opening of diffraction appear for both (a) and (b)

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4. Wavelength dependence of photovoltage at normal incidence

Figure 4(a) shows peak photovoltage at normal incidence as a function of the incident wavelength. For the symmetric structure, Sample (a), there is no significant signal (black line), while for the asymmetric structure, Sample (b), there is a finite signal (red curve) depending on the wavelength. In most of measured wavelength range, the photovoltage was positive for the geometrical arrangement shown in Fig. 1(c) with spiky structures appearing at 890 nm and 1200 nm. The signal was negative in the range of 750–870 nm. A maximum voltage of 3 mV was measured at 1200 nm for a laser intensity of 1 MW/cm².

In order to account for the behavior of the photo-induced voltage, we focus our attention on the x-component of light momentum given to the sample. This momentum pushes free electrons in the samples, and is balanced with the electromotive force to keep a non-accelerating current. The main scheme of this discussion was shown in reference [7]. But we have to take diffraction terms into account in our case.

At the periodic structure, incident light is transmitted, reflected, diffracted and absorbed. Momentum conservation in the x-direction is given by

where k is the wave vector of incident light at the air, θ and θ′ the incident and refraction angles, R reflection coefficient, k′ the wave vector in the substrate, T transmission coefficient, g index of diffraction order, D_g diffraction coefficient, G_g the reciprocal lattice vector at the g-th order. Momentum in x-direction given to the sample is ħK. Here, we assume that all the excess momentum goes to free electrons in the sample. Total lateral momentum flux given to the sample is expressed as follows:

where I [W/cm²] is the peak light intensity, λ the wavelength of incident light, c the light velocity in vacuum, and Λ the pitch of the sample. The index g runs over the open channels. At normal incidence, equation reduces to

In this case, there is a pair of diffraction into air if λ<Λ. For symmetric structure, the positive and negative diffractions have identical intensity, D ₁=D _-1. Therefore, the photovoltage vanishes according to eq. (3). For the asymmetric structure, on the other hand, D ₁ and D _-1 are no longer the same, and the sample receives a finite lateral momentum. Similar arguments apply for higher order diffraction and that into the substrate. Thus the wavelength dependence of the photovoltage should be related to the diffraction spectrum. Fig. 4(b) shows diffraction spectra calculated by the scattering matrix method [10,11]. Solid and broken red lines are for diffraction of g=1 and g=-1 into air, while solid and broken black lines are for diffraction of g=+1 and g=-1 into the substrate, respectively. Gray lines are for the second order diffractions for the substrate. Because angular frequency ω and reciprocal lattice vector must satisfy a condition $\mid g\mid G<\frac{n\omega }{c}$ to have diffraction of the light, there are thresholds in the diffraction spectrum. For the asymmetric structures set as in Fig. 1(c), diffractions to the +x direction are found to be stronger at wavelength range between 900 and 1220 nm. Thus free electrons are pushed to the -x direction as recoil of uneven diffraction momenta, which contributes to positive photovoltage for electrons with negative charge. On the other hand, photovoltage becomes negative for wavelength shorter than 900 nm due to dominant contribution of the -2nd order diffraction to the substrate. At the opening point of a diffraction channel where light is diffracted to the sample surface, the lateral momentum of light given to the free electrons in Au becomes maximum. Therefore the strongest photovoltage is expected at 1200 nm. The calculated spectrum is shown as a blue line in Fig. 4(a). In this calculation, all the diffractions are taken into account. The wavelength dependence of signal (jump of photovoltage at the channel opening, gradual decrease above it and a spike feature at 900 nm) is fairly well described. Appearance of spiky structures in photovoltage signal at the channel opening point of diffraction was not reproduced by calculation, however.

Because signal at normal incidence was observed only for the asymmetric structures, it can be referred to as a photo-rectification effect due to the asymmetry in the unit cell of the photonic crystal slab. But we need to keep in mind that in this case the spatial derivative term Q_ijkl E_j∇_kE_l in the second order nonlinear polarization is responsible for the photo-induced voltage rather than χ_ijkE_jE_k term, which is the origin of optical rectification in the conventional sense. In any case, the optical rectification signal is proportional to I, as is different from SHG which is proportional to I ².

Now let us estimate the voltage quantitatively. For the rectangular photonic crystal sample with length L and width W, the lateral force on the sample is PWL, which balances against the electric field E acting on nWLd free electrons. Thus the equation of balance is

where n is free electron density of Au, d the effective thickness of the sample, and e elementary charge. The photo-induced electromotive force is calculated by integrating the field along the sample.

One problem in estimating this equation is the uncertainty in the thickness d. One needs to know what thickness is appropriate for the metallic photonic crystal slabs. Here we comment that d turned out to be 0.5 nm in order to reproduce the experimental observation in the above simple model, where electromagnetic field distribution within the unit cells is not taken into account. If we assume d=40 nm which is the thickness of the deposited Au film, the theoretically expected signal would be 80 times smaller. In order to relate the effective thickness with the sample structure, we believe that it is necessary to calculate the local DC electric field from the precise field distribution of the surface plasmon, which is a subject for future investigation.

 

Fig. 4. (a) Photovoltage at normal incidence as a function of laser wavelength for symmetric (black line) and asymmetric (red line) structures. Photovoltage estimated from diffraction spectra is also shown (blue line). (b) Calculated diffraction spectra of lower indices.

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5. Conclusion

We fabricated metallic photonic crystal slabs with symmetric and asymmetric photonic unit cells and observed finite photo-rectification voltage at normal incidence for the asymmetric sample. The wavelength and angle dependence of photo-induced voltage is understood in terms of the lateral momentum given to the free electrons in the sample and therefore the phenomenon can be interpreted as a generalization of the photon drag effect to periodic systems. In case of normal incidence, symmetry breaking due to asymmetric unit cells provides photo-induced voltage, which can be referred to as optical rectification. Unlike the ordinary optical rectification in materials without inversion center, this effect is accounted for by the spatial derivative term in the second order optical polarization.