Double Resonant SHG from a V-Groove

Figure 5.27: Sketch of the V-groove geometry, designed by Andreas Hille.

Another structure which I investigated during my time as a doctoral student was a so-called V-groove. It was de-signed by Andreas Hille and some as-pects of SHG from this structure have already been discussed in a publica-tion [M2]. This discussion however still lacked the thorough mode analysis which I am about to present. The V-groove is a two-dimensional structure, 100 nm by 20 nm, with a V-shaped cut-out on one side (Fig.5.27). The idea behind the design is to have a low frequency resonance pertaining to the full length of the system (100 nm) and higher frequency resonances pertaining to the two shorter pieces to the left and to the right of the groove. If the dimensions are chosen appropriately, one can find a resonance atω0 so that there is also a resonance at2ω0. In this sense, the structure is geometrically tuned to be double-resonant. Part of the mesh which was employed for the numerical simulations is shown in Fig. 5.28.

0 25 50

− 10 0 10

x [nm]

y [nm]

Figure 5.28:Part of the mesh used for the V-groove calculations. For better visibility, only one half of the total field region is displayed. The maximal edge length within the scatterer is 0.65 nm. The scattered field region (not shown) extends to±350nm in x direction and ±250nm in y direction, beyond that, there are 50 nm of PMLs in every direction (not shown).

As mentioned in Sec. 3.4, the V-groove has certain similarities to a dimer but features lower symmetry because it cannot be mirrored upside down. Formally, the V-groove corresponds to the symmetry point-groupC₂ (Sec 3.4). Due to the lower symmetry, there are only two different classes of modes. I recall

0.2 0.4 0.6 0.8 1

Figure 5.29:Logarithmic plot of the linear spectra of the V-groove. I)incidence along the short axis (from the top in Fig. 5.27),II)incidence along the long axis (from the left in Fig. 5.27). The resonances indicated by vertical lines will be of importance in the discussion of SHG.

the findings from Sec. 3.4 for the excitation of modes in a V-groove structure: For incidence from the top (Fig. 5.27), i.e. with theE-field polarized inx-direction and thek-vector pointing downward, only one class of modes (class I) is excited. If the incidence is rotated by 90 degrees, both classes, class I and II, are excited. For the second harmonic signals the situation is as follows: For incidence from above (along the short axis) the nonlinearly excited modes are of class II, i.e. precisely those modes that were not excited linearly. For angle from the left, along the long axis, where there are modes of class I and II on a linear level, the SHG signal also features modes of both classes I and II.

The linear spectra are displayed in Fig. 5.29. Close inspection of the spectra reveals that for at every frequency at which there is a resonance in Fig. 5.29 I), there is also a resonance in Fig. 5.29II), but the inverse does not hold. This backs the group theory which suggested that both classes of modes are excited for incidence along the long axis, while only one class is excited for incidence along the short axis. All peaks in the spectra in Fig. 5.29Imust pertain to class I. Three of the those peaks in question are labeled byω1,ω2andω4.

On the other hand, the peaks in Fig. 5.29II)could in principle be either from class I or from class II. The peaks labeled byω^∗₁,ω₂^∗andω₄^∗are found at the same frequencies asω1,ω2andω4. Therefore, they are either the same modes of class I which were excited by the other angle of incidence, or they

are modes from class II which are found at the same frequency. Only for the peak labeled byω₃^∗ it is obvious that it must belong to a class II mode, since there is no peak to be found at the same frequency for the other angle of incidence. Below, an analysis of the SHG spectra makes it possible to determine whether the starred peaks pertain to class I or class II.

0.4 0.5 0.6 0.7 0.8 0.9

Figure 5.30: SHG from a double resonant V-groove. a) In-cidence along the short axis. Strong double resonant behav-ior is found as the linear signal coincides with ω¹. b) Inci-dence along the long axis. Double-resonant enhancement is not found as theω₁^∗ resonance on the linear level for this an-gle of incidence is weak. Note that the linear signal is only rastered across theω1peak, from0.25ωpto0.35ωpto capture double resonant behavior. This is not a full frequency scan.

The frequencies ω2 = ω₂^∗, ω3 = ω₃^∗ and ω4 = ω₄^∗ are at approximately twice the frequency ofω₁. Thus, send-ing in a light pulse near ω1, generat-ing a second harmonic signal at those peaks, will be a double-resonant exci-tation.

For incidence from the top the linear signal atω₁is very strong, which yields a strong double-resonance, while for the other angle of incidence the sig-nal (ω₁^∗) is extremely small and there-fore the effect of double-resonance is weak, if not negligible. This is shown in Fig. 5.30. For incidence from the top there is strong SHG enhancement, yielding a signal which is approxi-mately five orders of magnitude larger than the signal for the other angle of in-cidence. For the single cylinder, a res-onance at the incoming frequency has been shown to give rise to strong SHG fields (Fig. 5.6) – over an order of mag-nitude stronger than if the resonance is at the SHG frequency. Here, a combi-nation of both leads to significant en-hancement. The SHG efficiency for this type of excitation is actually com-parable to the estimate for dielectrics which was given in Eq. (2.52). If the incoming field was increased to 10¹¹V /mthe SHG signal and the lin-ear signal would be of the same order of magnitude. Having taken note of the strength of the SHG signal, the mode analysis of the SHG signals still has to be done. Two very pronounced peaks are found in 5.30a)at the frequencies ω2 =ω₂^∗andω₃^∗. Above, I already pointed out thatω₃^∗is with certainty a mode of class II, which is dark for this angle of incidence. Therefore a SHG signal is expected. The fact that there is also a resonance at

ω₂ =ω^∗₂means that theω₂^∗peak comes from a class II mode which is degenerate in frequency with the class I modeω2. The fact that there is no SHG signal atω₄^∗on the other hand means thatω^∗₄is the same class I mode asω4. The SHG signal for the other direction of incidence, Fig. 5.30b)exhibits peaks at all three frequenciesω₂ =ω₂^∗,ω₃ =ω₃^∗ andω₄ = ω^∗₄. This confirms once again the group-theoretical prediction that modes from both symmetry classes are excited linearly and nonlinearly for this angle of incidence, i.e. the modeω₃^∗which is clearly of class II is excited linearly and nonlinearly and so is the mode ω₄ which was identified as a class I mode. Obviously the modeω₂ =ω^∗₂ also delivers a signal since it is a mixed signal of a class I and a class II signal at the same frequency.