The physical mechanism of the large polarization rotation effect in direct transmission of the all-dielectric artificially chiral nanogratings is explored by experiment and numerical analysis. It is shown that the different coupling of right- and left-circularly polarized components of the normally incident light to the leaky guided modes or Fabry-Pérot resonance modes lead to the enhanced circular dichroism, resulting in the giant polarization rotation effect. The mode profile and local field calculations demonstrate intuitive images of the different coupling performance at resonances.
©2009 Optical Society of America
Artificially-engineered periodic chiral nanostructures [1–11] have attracted considerable attention in recent years because they can produce polarization rotation effects (or optical activity) much larger than those in natural chiral media in the optical regime. Most of the previous studies are on metallic two-dimensional (2D) chiral structures [1–5] because the optical activity can be enhanced by surface-plasmon resonance. Tens-of-degree polarization rotation in non-zeroth diffraction orders of such devices has been observed and studied ; whereas the observed maximum polarization rotation in the zeroth order, though more favorable in applications, does not exceed few degrees [2–4]. To reduce the optical loss in metallic structures, dielectric 2D chiral structures have also been investigated [6,7]; however, still there is no evident optical activity found in the zeroth order. Meanwhile, dielectric three-dimensional chiral photonic crystals [9–11] have successfully demonstrated large optical activity and circular dichroism in direct transmission, even though the fabrication process is more complicated and time-consuming.
Recently we successfully designed and fabricated  some all-dielectric 2D artificially chi-ral nanogratings (ACNGs) exhibiting giant polarization rotation effect (up to 26.5 °) in direct transmission . This shows that large optical activity is also achievable in direct transmission of 2D chiral structures and can be effectively enhanced by optical resonances other than the surface-plasmon resonance. However, the specific enhancement mechanism is still unclear. In this work, we thoroughly investigate the light-matter interaction in the dielectric ACNGs and thereby interpret the physical origin of the large polarization rotation effect.
2. Resonance property of the dielectric ACNGs
The fabricated ACNG sample (see 1Fig. 1 in  for the geometry and the notations of structural parameters) consists of a silica substrate, a homogeneous TiO2 waveguide layer, and a TiO2 gammadion-shaped grating layer. The structural parameters are d = 600 nm, w = 120 nm, l = 70 nm, t = 410 nm, and h = 820 nm. Due to structure imperfection, w and l are just approximate values averaged along the normal direction of the sample.
Such a ACNG structure is a typical waveguide grating that can support leaky guided modes [14–16]. Due to grating modulation, the normally incident light can be coupled into and out of the waveguide through the higher spatial harmonics by phase matching with the guided modes. At the lower interface of the waveguide layer, the trapped and re-diffracted light interferes destructively with the directly transmitted light, leading to the abrupt redistribution of diffracted energy and manifesting as the transmission dips on a smooth Fabry-Pérot (FP) background. This is known as the guided-mode resonance effect [14–16], which can be seen clearly in the spectra of the ACNG sample as shown in Fig. 1.
We used the rigorous Fourier modal method  to implement the numerical simulation, which can well reproduce the main features of the measured spectra (see Fig. 1). The discrepancies, such as the shift of resonances, are caused by many practical factors such as structural imperfection, variation of material properties, and even thermal effect. For example, our deposited TiO2 film is not ideally lossless, but exhibits a small absorption due to the scattering of light from structure defects and rough interfaces. We have measured the refractive index of the TiO2 film nt = n+iκ by ellipsometer and found that κ is on the order of 10-3 (since the signal-to-noise ratio of our measurement of this small quantity is not good enough to produce a dispersive curve, we adopted κ = 0.005 in our simulation) and n is fitted to Sellmeier’s formula
where λ is expressed in nm. By using the measured nt data, the calculated spectra are closer to the measured ones than the lossless case (not shown in Fig. 1). So the numerical simulation is sufficiently reliable to be used for the following analysis.
Figure 1 shows that the optical activity can be enhanced by both the guided-mode resonance and FP resonance because the polarization rotation peaks are all in accordance with the transmittance dips. The enhancement factor, if defined as the ratio of polarization rotation angles at resonance and off-resonance wavelengths, could be as high as 102. However, it is difficult to distinguish the two kinds of resonances intuitively, especially in the shorter wavelength range where the resonances are so close to each other and some resonance dips are too weak to see. To distinguish the resonances, as the first approximation, we have previously employed a conventional (achiral) planar waveguide model to calculate its TE/TM modes and allocate the mode for each resonance by oblique-incidence measurement . Nevertheless, with that simplified model, the role of structural chirality was not taken into account and therefore the physical mechanism of the enhanced polarization rotation effect could not be properly explained.
Considering the structural chirality, in this work, we start with a planar chirowaveguide model  with the TiO2 layers replaced by an effective homogenous chiral layer with thickness h + t, an effective chirality parameter γ, and an effective refractive index
where f ≈ 0.85 is the filling factor of TiO2 in the volume containing both the waveguide layer and the gammadion layer and nc = 1 is the refractive index of cover medium (air). The chirality parameter γ is defined in the Drude-Born-Federov constitutive equation for chiral media 
In the chirowaveguide the eigenmodes are no longer TE and TM, but left- (LEP) and right-elliptically polarized (REP) . To calculate the dispersion curves of the modes in the ACNG, we need to find a proper chirality parameter γ for the effective chirowaveguide, which seems rather difficult if we want to directly retrieve this parameter from the optical response of the ACNG. Since the purpose of using the chirowaveguide model is to qualitatively demonstrate the light coupling behavior in the structure, we take an easier method by increasing the γ value from zero up until the calculated resonance positions are in good correspondence with those in Fig. 1. In this way, we have found γ = 0.5 nm to be a reasonable value, and the corresponding dispersion diagram is shown in Fig. 2. Note that neff = β/k 0 is the effective index of a waveguide mode with propagation constant β and wavenumber in vacuum k 0 = 2π/λ, but not the effective refractive index ng of the chiral layer. It is seen that, owing to the circular birefringence in the chiral media, the free-space refractive indices of REP and LEP waves (i.e., the light lines n + and n -) are different in the chiral slab, which leads to the splitting of degeneracy of the REP and LEP modes. Recalling the propagation condition of a mode max[n c k 0,nsk 0] < β ± < n ± k 0, we can understand that the dispersion curves asymptotically approach the light lines n ± and the modes are bounded only when the curves are above the light line ns.
Now taking into account the periodicity of the ACNG, we know that the bounded mode may become leaky and couple to the incident light when it matches the phase condition (under normal illumination)
through the (m,n)th spatial harmonic of the grating (where m and n are integers 0,±1,±2,…). With Eq. (4) we can plot the phase-matching curves n eff = √νλ/ d for different spatial harmonics with order indices ν = m 2+n 2. Therefore, in Fig. 2, the intersections of the mode dispersion curves and the phase-matching curves, indicated by CνM, are the places where the incident light can be coupled to the RM or LM modes through the spatial harmonics with ν= 1,2,or 4.
The above analysis, though still based on an effective-medium model, helps us to phenomenologically realize the mode structure in the ACNG. Then we can try to find the corresponding resonance dip in Fig. 1 for each particular mode coupling process CνM. This is easy to do in the longer wavelength range as the doublet structure of the resonance dips (such as C 11 in Fig. 1) is quite evident. However, in the shorter wavelength range, the guided-mode resonance dips are closer to each other (and some of them are too weak to see) and mixed with the FP resonances, which makes them very difficult to distinguish. To help find the correct resonance places, we rigorously calculated (by using the Fourier modal method) the electric field amplitude variation along the normal direction of the ACNG sample at each resonance wavelength (some weak resonances such as C 21 can be located by referring to the lossless spectra), some of which are shown in Fig. 3. In this way, we can allocate different mode coupling processes CνM to the narrow resonances in Fig. 1 according to the characteristics of the mode profiles (i.e., the number of minima) ; the FP resonances are easy to distinguish because their field amplitude variation does not have such a regular characteristic as the guided-mode resonance.
Compared with the guided-mode resonance, the FP resonances manifest as smoother and broader dips in the spectra (such as those at 852 nm, 792 nm, and 642 nm in Fig. 1), which can be characterized by basic FP theory . For simplicity, if we still consider the effective waveguide model, the FP resonance spacing Δλ and the full-width at half-maximum (FWHM) of each resonance δλ can be estimated by Δλ ~λ 2[2ng(h+t)] and δλ ~Δλ(1 - R)/(πR 1/2), with R the averaged surface reflection coefficient. Therefore, in the wavelength range λ ∊ [550,850] nm, we can get an estimation of Δλ ∊[60,140] nm and δλ ∊ [15,35] nm, whose magnitudes are consistent with the FP characteristics of the experimental and calculated spectra in Fig. 1.
3. Mechanism of the resonance-enhanced polarization rotation effect
It is seen that both the guided-mode resonance and FP resonance can enhance the polarization rotation effect. To reveal the underlying mechanism, we should recall the circular birefringence in the ACNG. We know that linearly polarized light can be decomposed into a pair of right-(RCP) and left-circularly polarized (LCP) components. For the gammadion-shaped ACNG with four-fold-rotational symmetry, a normally incident RCP (LCP) wave is still RCP (LCP) after direct transmission ; the polarization rotation effect originates from the different transmittance (i.e., circular dichroism, CD) and phase shifts of the RCP and LCP waves. If we define the complex transmission coefficients of the RCP (+) and LCP (-) components as t ± = a ±exp(iϕ ±) with a ± and ϕ ± the amplitudes and phases of t ±, then the polarization rotation angle θ and ellipticity angle χ are given by
To examine the CD experimentally, we have measured the transmittance T ± of RCP and LCP waves through the ACNG, as shown in Fig. 4(a). Then keeping in mind Eq. (5) and a ± = (ncT ±/ns)1/2, we can derive the ellipticity χ cd and thereby the polarization rotation angle θ KK (by Kramers-Kronig transformations, see ). In Fig. 4(b) we compare the angles with χ meas and θ meas that were directly measured by polarization modulation technique , and find the good correspondence. The small discrepancy between χ cd and χ meas is due to the experimental errors in uncorrelated measurements, and that between θ kk and θ meas is contributed in addition by finite integral range of λ when doing Kramers-Kronig transformations. This experiment verifies that CD, which is enhanced by the resonances, is the origin of the large polarization rotation effect in our ACNG sample. Our numerical simulation of the CD and the induced ellipticity [shown in Fig. 4(c)] also coincides with the experiment well.
If we have a closer look at the CD spectra in Figs. 4(a) and (c), we may find an interesting fact: at longer wavelengths where there is only one transmitted order, the resonances are very strong, whereas the CD is very weak leading to poorly-enhanced polarization rotation; however, at shorter wavelengths where there are more than one transmitted order, the CD is much larger leading to well-enhanced polarization rotation, even though the resonances of the RCP and LCP waves are weaker. That is why we always observed larger polarization rotation at shorter wavelengths (the maximum rotation 26.5° is at λ = 638 nm). This phenomenon is attributed to the coupling performance of RCP and LCP waves in the waveguide grating. Figure 4 shows that both the RCP and LCP incident waves can be coupled to the same supported guided mode (regardless of whether it is REP or LEP mode); however, the coupling performance of them may be quite similar at longer wavelengths while significantly different at shorter wavelengths, manifesting as similar or different resonance dips, respectively.
To have an intuitive insight into the coupling performance of the RCP and LCP waves, we calculated the distribution of E∥, the electric field component in the grating plane, at the lower interface of the waveguide layer. We chose this typical plane because the interference of trapped light and directly transmitted light happens in this plane at resonances [14–16] and it is also the last interface before light leaves the waveguide grating after the light-matter interaction. According to our calculation, the local field distributions at two resonances have the same characteristics if they are in the same (either one-transmitted-order or multi-transmitted-order) diffraction regime. Therefore, in Fig. 5 we demonstrate the local field distributions of E∥ at two typical resonances: C12 in the one-transmitted-order regime and C 24 in the multi-transmitted-order regime. It is seen that, at resonance C 12, the local fields for RCP and LCP illuminations look quite similar and have the surprising mirror symmetry, implying no priority for the coupling of either the RCP or LCP wave to the guided mode and that the structural chirality almost has no impact on the local field; whereas, at resonance C 24, the local fields are excited and twisted rather differently for RCP and LCP illuminations, manifesting the substantial influence of the structural chirality on the mode coupling performance of the RCP and LCP waves. This is consistent with our anticipation based on Fig. 4.
It would be beneficial if the coupling efficiency of the RCP and LCP waves could be evaluated more quantitatively. However, unlike the conventional planar waveguide or fiber with bounded modes, it seems rather difficult to define a coupling efficiency for the waveguide grating because the guided modes are leaky. Nevertheless, the strength of the local field may provide a quantitative description of the coupling because it reflects, from an aspect, the power of transferring the incident field into the local field in the waveguide grating. For example, if we define the local field strength S as the average of the normalized amplitude of E∥, then at resonance C 12 we can calculate (from the data of Fig. 5)S RCP = 2.38 and S LCP = 2.27, while at resonance C 24 we have S RCP = 1.64 and S LCP = 0.56. Since the strength is normalized with the incident field amplitude, S can be used as an absolute value to compare the coupling efficiencies at different resonances directly: the larger value S takes, the sharper and stronger resonance dip would appear; and the bigger difference between S RCP and S LCP at the same resonance, the larger CD would be produced. All of these can be seen clearly in the spectra of Fig. 4.
The enhancement of CD by FP resonance can be understood if we recall the different refractive indices of RCP and LCP waves in the ACNG due to circular birefringence, which naturally affect the multiple interference process and thereby causes the different resonance performance of the RCP and LCP waves. Moreover, unlike the guided-mode resonance, the FP resonance wavelengths under RCP and LCP illuminations may slightly split due to the index difference (as seen in Fig. 4), which also contributes to the enhanced CD and polarization rotation effect.
In conclusion, we have investigated the enhancement mechanism of the giant polarization rotation effect in direct transmission of the all-dielectric artificially chiral nanogratings by numerical analysis and experiment. The polarization rotation effect originates from circular dichroism that is enhanced by either guided-mode resonance or Fabry-Pérot resonance. The dispersion diagram and field profiles of the guided modes in the grating structure as well as the local field calculation demonstrate the different coupling performance of RCP and LCP components of the incident light to the guided modes at resonances. In the multi-transmitted-order regime the coupled field is affected more drastically by the structural chirality than in the one-transmitted-order regime, leading to larger polarization rotation effect at shorter wavelengths.
The study of resonance-enhanced polarization rotation effect in dielectric ACNGs may provide new insight to the full understanding of the physical mechanism of enhanced optical activity in artificially chiral nanostructures. The new dielectric ACNGs may also bring inspiration to application designs of novel compact polarization-sensitive devices in the integrated optical systems.
We acknowledge the support by the Academy of Finland (Contracts 209806, 129155, 115781, and 118951), the Network of Excellence in Micro-optics (NEMO, www.micro-optics.org), the Japan Society for the Promotion of Science, and the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.
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