We measure second-harmonic generation from arrays of sub-wavelength apertures in transmission using fundamental input at 800 nm. Lattice arrangements include disordered, Penrose (quasi-periodic or aperiodic), and square (periodic). Strong angular dependence of SHG is observed, with maxima located at angular positions that roughly correspond to incidence angles of extraordinary optical transmission (EOT) for the fundamental. In addition, even at incidence normal to the sample, strong secondary maxima are observed at off-normal scattering angles for the arrangements with higher degree of order. Breaking the inversion symmetry of the aperture allows second harmonic peaks at normal incidence and detection. These measurements help to resolve the role that symmetry plays in second-harmonic generation from arrays of apertures.
©2007 Optical Society of America
Extraordinary optical transmission (EOT) through metal films modulated with a 2-D array of sub-wavelength apertures  was reported in 1998. There now exists experimental evidence [2, 3] that upon the condition of enhanced light transmission, intensity buildup within the apertures occurs. This phenomenon suggests that enhancement of nonlinear processes within the apertures can be obtained. Resonant enhancement of second-harmonic generation (SHG) by a factor of 104 has been demonstrated for a single sub-wavelength aperture surrounded by periodic annular corrugation . SHG has also been studied in arrays of sub-wavelength apertures of round , triangular , coaxial , double-hole [7, 8], and rectangular  shapes, using disordered [10, 5] and periodic [5, 6, 7, 9] arrangements. The effect of the symmetry of the aperture has been shown , in that, at normal incidence, apertures with inversion symmetry produce much weaker SH than non-centrosymmetric apertures, while at off-normal incidence, SH can be produced with centrosymmetric apertures [5, 8]. Nevertheless, SHG has been studied at normal incidence where inversion symmetry holds [4, 9].
Recently, EOT has been observed in quasi-periodic arrangements of sub-wavelength apertures [11, 12, 13]. While not fundamentally different than EOT in periodic arrangements , the insight gained from the generalization of the phenomenon from periodic to aperiodic  is expected to yield new applications. We focus here on the effects of aperture symmetry and lattice arrangement on SHG in transmission through sub-wavelength metallic apertures and show that quite different behavior is exhibited in the scattering of the SH signal. In addition, these results help to resolve the role of symmetry in the recent publications on SH in sub-wavelength aperture arrays.
2. Experimental methods
Arrangements of sub-wavelength apertures were produced in 100 nm thick gold films on quartz, using both focused ion beam (FIB) milling and electron beam (e-beam) lithography. Numerous samples were produced via FIB with round apertures of varying sizes and lattice arrangements, but the only sample used in this study is a square lattice of round apertures (see Fig. 5). All other samples were produced with e-beam lithography. Briefly, a 5 nm chromium adhesion layer was sputter deposited onto the quartz substrate, followed by 100 nm of gold and 20 nm of chromium. ZEP520A e-beam resist of about 300 nm thickness was spin coated. Following exposure, the upper chromium layer was dry etched with chlorine, and the e-beam resist removed. The chromium layer served as a hard mask for argon ion milling of the gold. A wet etch removed the upper chromium layer (and likely resulted in some undercut in the underlying chromium adhesion layer). Lattice arrangements in this paper include disordered, square, and Penrose.
The experimental setup is a modification of the one used previously , as shown in Fig. 1. A Ti:Sapphire laser is used at 800 nm wavelength and roughly 30 fs pulse duration. The modified setup allows the rotation of the sample with respect to the incident fundamental beam as well as rotation of the detector around the sample so that the SHG radiation pattern can be measured in transmission. The detector is a blue-sensitized PMT and two spectral filters are used to block the transmitted fundamental at 800 nm and minimize the influence of two-photon luminescence . The angular acceptance of light collection is roughly 1 degree. Lock-in detection is performed by modulating the 86 MHz pulse train at 2 kHz, and average incident power is 75 mW (measured after the chopper).
3.1. Disordered lattice
An SEM image of the disordered lattice of round apertures is shown in Fig. 2, along with a scan of the transmission of the fundamental beam and SH output (along the direction of the transmitted fundamental, i.e. γ=θ) versus angle of incidence. As expected due to the inversion symmetry of the aperture, the SH signal is minimum at normal incidence. This is a similar result to that obtained previously .
A double-angle scan for the disordered arrangement of round apertures is shown in Fig. 3. Since the lattice is disordered, the only SH signal observed is along the direction of the fundamental beam. The far-field radiation pattern indicates minimal contribution from two-photon luminescence (which would have a broad angular distribution), and is the result of sources of SH radiation localized at the apertures. For off-normal incidence, the transmitted SH signal increases rapidly due to symmetry breaking at the aperture [15, 5]. The SH wavevector satisfies the following momentum-matching condition - k 2ω t = 2k ω t , where t represents transverse components, so that γ= θ.
Symmetry breaking due to off-normal incidence can be understood by considering the distribution of intensity within the aperture as a function of fundamental incidence angle θ for an x'-polarized input field. Calculations were performed using FDTD Solutions (Lumerical Solutions, Inc), the results of which are shown in Fig. 4 for θ = 0°, 30° and 60°. The first row of images shows the x,y and z components of the intensity pattern at normal incidence, at a position 5 nm below the aperture entrance. As expected, there is a reflection symmetry along the y axis. By examining the magnitude of the three orthogonal components, Ex dominates along the x axis, while Ey dominates along the diagonals, suggesting that the local second-harmonic signal arises from the surface χ (2) ⊥⊥⊥ component between gold and air. As shown in cross-section, there are locations where Ez dominates, complicating any detailed microscopic analysis. The aperture symmetry dictates that the effective second-order susceptibility of the aperture χ (2,eff) ttt = 0, where t is a transverse component, such that the local SH sources exactly cancel. As shown in the second row of images, the field distribution throughout the depth of the aperture is non-uniform, with local maxima at the entrance and exit  (note that Ey is of negligible magnitude in this xz cross section). This asymmetry along the z direction  allows nonzero effective aperture susceptibilities indexed by zzz,ztt, and tzt = ttz, in principle enabling some SH emission at normal incidence/exit, as exhibited in Fig. 3.
For off-normal incidence angles, asymmetry in the lateral intensity distribution develops, breaking the symmetry of the aperture, such that χ (2,eff) ttt ≠ 0, where χ (2,eff) xxx is assumed dominant. Note that similar intensity patterns were obtained by asymmetrically placing a point source near a sub-wavelength aperture . At small incidence angles, the observed increasing SH is initially driven by symmetry breaking, but at larger angles, it is clear that peak intensity increases as well.
3.2. Square lattice
An SEM image of the square lattice of round apertures is shown in Fig. 5. EOT peaks are observed in the transmission of the fundamental beam, but SH output is only observed at angles significantly off-axis. Again, this result is similar to previous measurements  and is due to symmetry breaking by off-axis illumination.
A double-angle scan for the square lattice is shown in Fig. 6. Again, the SH signal is minimum at normal incidence and detection; however, well-defined SH peaks are detected off-axis, indicating that symmetry-breaking also occurs in detection [19, 17]. This effect cannot be observed with the disordered lattice as the only SH signal produced is along the direction of the fundamental (i.e. there is no coherent addition of periodic SH sources to produce off-axis beams). Another interesting result is that the peak SH signal is about 25-times greater than for the disordered arrangement.
The far-field radiation pattern is the result of sources of SH radiation which are periodically arranged. From this observation, the SH wavevector satisfies the following momentum-matching condition:
where K is a reciprocal lattice vector. It is assumed that, at the SH wavelength, there is no coupling between the apertures due to surface waves, as the SPP propagation distance in gold is less than the aperture spacing . If the SH SPP were not negligible, additional phase matching considerations would come into play . For the square lattice, and assuming that the optical wavevectors have only x̂ and ẑ components, this equation can be written
where Λ is the lattice constant and λ is the fundamental wavelength. As shown in Fig. 6, this equation closely describes the trajectories of the SH peaks.
This simple diffraction model does not, however, explain the absence of the zeroth-order beam; in other words, it does not take into account the symmetry of the aperture. Referring back to Fig. 4 at normal incidence, it is reasonable to approximate the locations of the SH from each aperture as two point sources on opposing side of the aperture (and aligned along the direction of fundamental polarization). Inversion symmetry demands that these two point sources are out of phase. Mathematically, the array of SH source terms can be expressed by
in real space, where d is the aperture diameter. By taking the 2-D Fourier transform (with fx and fy the transverse spatial frequencies), the SH can be expressed in angular space by
where sin γ=λfx/2.
Because of the sin() factor, the SH diffraction pattern has no zeroth order (i.e. where fx = 0). Further, opposing diffraction orders must be out of phase with respect to each other, meaning that if detected symmetrically, they would interfere destructively and cancel. This expression also predicts that for higher diffraction orders, the intensity follows sin2 (πfxd), which has been shown for SH from arrays of symmetric particles .
In order to obtain a SH signal at normal incidence/detection, the inversion symmetry of the aperture must be broken so as to introduce asymmetry in the local field distribution  (a further requirement is that the unit cell must also be non-centrosymmetric [22, 23]). Square arrays of non-centrosymmetric apertures were also fabricated, as shown in Fig. 7, which satisfy these requirements. Also in this figure is a plot of fundamental transmission and SH versus incident angle, which shows that strong SH is obtained at normal incidence where a peak in fundamental transmission (EOT) occurs.
The asymmetry of this aperture shape is exhibited in the intensity distribution upon normal incidence, as shown in Fig. 8, in the xy plane 5 nm below the entrance. Here, the intensity distribution is more complex than for the round aperture, with Ex and Ey components reaching roughly the same magnitudes, even though these magnitudes are less than that obtained for Ex with the round aperture.
The double-angle SH scan for the asymmetric aperture is shown in Fig. 9, where the strong zeroth-order peak is observed at normal incidence. This scan also clearly shows the role of the aperture shape in distributing energy to the diffraction orders.
3.3. Penrose lattice
An SEM image of the Penrose lattice with round apertures is shown in Fig. 10. The Penrose tiles are constructed similarly to that used in , with five-fold rotational symmetry. The reciprocal space, with 10-fold rotational symmetry, is also plotted, with peaks indexed by the transverse spatial frequencies fx and fy. Since the reciprocal space is densely packed as compared to that for a square lattice (i.e. there are no primitive vectors), not all peaks are plotted, only those of sufficient magnitude that can be distinctly identified from our measurements.
The same SH measurement has been performed on the Penrose lattice, as shown in Fig. 11. One noticeable feature of this plot is that the SH diffraction orders are more numerous, but significantly weaker, than for the square lattice. This is a direct result of the reciprocal space for the Penrose lattice. Reciprocal vectors along the fx axis are used to plot the trajectories in Fig. 11, which are in good agreement with the measurement.
A Penrose lattice was also fabricated with the same asymmetric aperture shape as used for the square lattice. The measured fundamental transmission and SH for this lattice is shown in Fig. 12, with the full angular scan shown in Fig. 13. A clear zeroth-order SH peak is seen at normal incidence, while many of the diffracted peaks are more clearly defined than with the round aperture.
A recent theoretical paper on EOT in quasi-periodic aperture arrays  has shown that the intensity distribution of incident light is not uniformly distributed at the apertures as it is for a periodic lattice. Because SHG scales as the square of intensity, it is a highly sensitive indicator of this distribution. From the good agreement of the measured SH diffraction pattern to what would be predicted from the reciprocal space (which assumes that the SH intensity from each aperture is equal), it is clear that the hot spots must maintain the same underlying arrangement as the lattice.
4. Discussion and conclusions
We have studied three mechanisms of symmetry-breaking in SHG from arrays of sub-wavelength apertures: off-axis illumination, off-axis detection, and non-centrosymmetric aperture shape. We have also studied the effects of the lattice arrangement, from disordered to periodic.
A simple model of SH emission from centrosymmetric apertures shows why SHG is weak under the conditions of normal incidence and detection, but can be measured under normal incidence and off-axis detection . This model also predicts that opposing diffraction orders have opposite signs, and would cancel under symmetric detection, such as collection of SH emission through a high NA objective. Clearly, however, as shown through our asymmetric measurement technique, these orders wouldn’t completely cancel due to other sources of asymmetry.
As we have shown, non-centrosymmetric apertures result in symmetry breaking in the intensity distribution of the fundamental driving field, such that strong SH can be produced under otherwise symmetric conditions. Symmetry breaking is also obtained for a centrosymmetric aperture under off-axis illumination.
Very different behavior in the second harmonic emission pattern is observed as a function of lattice arrangement. The disordered arrangement cannot produce off-axis SH peaks because of the lack of any short or long range periodicity that would allow for coherent addition of local sources in directions off that of the fundamental. The Penrose arrangement is quasi-periodic, allowing this coherent addition to occur, but with weaker diffraction efficiency than obtained for a periodic lattice. Finally, the SH emission pattern closely follows what would be predicted for a Penrose arrangement of SH sources.
The authors thank A. Nahata and A. Agrawal for many useful discussions. This research was sponsored in part by grant ECS 0622225 from the National Science Foundation, and by the Army Research Office. G. P. Zhang was supported by “The Hundred Talent Plan” of the Chinese Academy of Sciences.
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