Leaky and waveguide modes in biperiodic holograms

This study details a theoretical analysis of leaky and waveguide modes in biperiodic all-dielectric holograms. By tuning diffraction orders and subsequently confining local density of optical states at two distinct resonance wavelengths, we present a new class of highly sensitive refractive index biosensing platforms that are capable of resolving 35.5 to 41.3 nm/RIU of spectral shift for two separate biological analytes.


Leaky mode in waveguides
Distinct from microwave wavelengths, the optical regime supports leaky waves only by multilayer stacks and periodic structures. This can be understood from the Helmholtz wave equation for which leaky waves, surface waves and lateral waves, are all possible solutions. In the dispersion relation, if k is the propagation constant, lateral waves will contribute to the branch cuts, while surface waves are related to real poles, and leaky waves are related to the complex poles in the complex plane of k = k′ + ik′′ ( i = √ −1 is the unit imaginary number, i) 9 . These complex pole singularities will only emerge in the solution of the dispersion relation for multilayer and periodic structures.
Moreover, leaky waves, like surface waves, will have a discrete spectrum because they are related to pole singularities and not branch cuts 10,11 . The power flow (Poynting vector) for leaky waves will generally be at an acute angle relative to structure surface. The modes are called either forward or backward leaky waves depending on whether the in-plane component of the power flow is in the same or opposite orientation as the power flow inside the structure, respectively. Multilayer structures can only support forward leaky waves e.g. in a planar waveguide 12 , while periodic structures or gratings support both forward and backward leaky waves, which for certain applications make them desirable 11 .

Periodic leaky wave design
Hessel and Oliner were the first to discuss leaky waves in periodic structures and gratings 13 . Although they focused on metallic gratings, the theory applies equally well for all-dielectric structures 9,14 . Because of the periodicity of the grating structure, the scattered energy from the surface of the structure to the medium covering the grating, will leave the surface by an angle θ m , and thus one can write 10 www.nature.com/scientificreports/ where the left side of the equation is the projection of the wave vector in the propagation plane. Also, k is the wave propagation factor in the cover which is equal to 2πn/λ, with λ being the wavelength of light in free space and n the refractive index of the cover medium. β 0 is the propagation factor of the wave moving along the grating, Λ is the period of the grating, and β is the propagation factor along the grating for the m-th space harmonic. For the energy to leak out of the grating, θ m has to be a real angle and thus for the values of m that Eq. (2) is valid, there exists a leaky wave. Forward leaky waves are related to positive values of m, while the negative values will lead to the backward leaky waves. As experimentally realized in 15 , the layer structure of the designed leaky waveguides is based on a relief grating made of Si 3 N 4 assumed on top of a glass substrate and is in contact with an external aqueous medium (illustrated in Fig. 1a). The thickness, t, of the waveguide-grating is 200 nm and is covered by a thin layer of a biological analyte with a coating thickness of d bio . In our simulations, it is assumed that an aqueous medium with a refractive index of 1.333 covers the structure surface. For the structure and material used here, the above calculations suggest the periodic hologram will support leaky waves in the wavelength, λ, range of 500-650 nm for the periods of the grating, Λ between 300 and 450 nm.
The reflectance spectra in the Transverse Magnetic (TM) mode for the gratings with different periods were calculated using the finite-difference time-domain method (Lumerical Inc.) and are shown in Fig. 1b. The results show that the first resonances of the reflectivity spectra occur in the predicted wavelength range (Fig. 1b). In addition to these predictable leaky modes, we observe other resonances, corresponding to distinct waveguide modes, that occur at longer wavelengths in the reflectivity spectrum for each structure.
To gain insight into these higher wavelength resonances, we calculated the normalized electric field profile at the wavelengths of the two resonances for one such structure (Fig. 1c,d). The results show that the second resonance contributes to a waveguide mode in Si 3 N 4 grating structure. However, as Fig. 1b suggests both resonances can be controlled by the period of the grating, while the grating amplitude would only affect the coupling efficiency and not the resonance wavelengths. Thus, one can envision further tuning of such structures by optimizing the amplitude of the grating and hence the coupling efficiency.

Biperiodic all-dielectric holograms
Alternative to leaky waveguide-gratings with lateral dimensions 16 , binary sequences 17 and quasi-guided modes 18 , another diffractive architecture that supports dual resonant holograms is the multispectral plasmonic supercell, described previously by us in 19 . Surface development along with field confinement for this bidiffractive structure is shown in Fig. 2b,c. The surface profile of the relief grating is a function of x where Λ 1 and Λ 2 are two different periods for each sinusoidal function, while a 1 and a 2 are the amplitude of each sinusoidal function. For simplicity, we consider equal amplitudes for both functions. Each sinusoidal function will have a peak resonance in its reflectivity spectrum. However, by using the combination of two functions, a periodic function is formed that is expected to have two peak resonances. The period of f = (x) can be found by calculating the Least Common Multiple (LCM) of Λ 1 and Λ 2 . To avoid the excitation of leaky modes in these structures, the periods for the sinusoidal function to form f = (x) were chosen to be Λ 1 = 450 nm and Λ 2 = 500 nm.
Simulation results for each of these sinusoidal functions and their combination are shown in Fig. 2a. The results show that the periodic structure, which has a period of 4500 nm, has two distinctive resonance peaks in the reflectivity curve. The electric field distribution profile for each resonance peak of the periodic structure shows that, although a part of the field power is concentrated in the sensing-layer, the modes related to both of these resonances are waveguide modes in the Si 3 N 4 diffraction grating layer (Fig. 2b,c). Like for the leaky structure introduced before, the amplitude and periods of the bimodal grating can similarly be optimized to obtain the best coupling efficiency.

Analysis and sensitivity calculations
The principles in designing the structures described above can be used in biosensing platforms to detect subtle changes in the refractive index of analyte in the biolayer. Adding the biological analyte on top of the waveguide structures will effectively form a new layer. The reflectivity spectrum and hence, the described waveguide modes will be affected by the refractive index of analyte and the thickness of the biolayer. Thus, recording the reflectivity spectra on defined grating structures can in principle be used to determine the refractive index of biological analyte if the layer thickness is controlled.
Calculations showing how a change of thickness of the biolayer will affect the respective resonances, for both the single-period (i.e. leaky) and biperiodic (i.e. waveguide) structures, is shown in Fig. 2d,e. The data shows that increasing the thickness of the biolayer will red-shift resonance wavelengths of the reflectivity spectrum. Interestingly, the change in resonance wavelength with respect to a change in thickness is not the same for the two different geometries. Specifically, the single period grating is more sensitive to a thickness change than the biperiodic grating as shown in Fig. 2d,e. The first resonance is notably more sensitive to changes in d bio for the www.nature.com/scientificreports/ biperiodic structure (Fig. 2e), in which both modes are waveguide modes. This suggests that for cases where d bio are less well known (as may be the case for less stringent practical applications), a longer wavelength resonance is superior for estimating the refractive index. It also follows that due to this distinct thickness-scaling of the two resonances, a single spectral measurement encompassing both resonances can allow one to extract d bio as well as its refractive index.
To illustrate the efficiency of these structures for biosensing application, we calculate the surface sensitivity i.e. dn/dt to changes in the Refractive Index Unit (RIU) for d bio = 50 nm (see Table 1). The values in this table have been calculated by assuming a spectral resolution of 0.01 nm, which is considered reasonable for a modern off-the shelf compact single grating spectrometer. As shown for both single-period and biperiodic grating, the sensitivity will be different for the short and long wavelength resonances. Furthermore, long wavelengths are associated with a lower sensitivity for the case of the single-period holograms, while biperiodic holograms provide higher sensitivity. The overall sensitivity of fitting both modes is also seen to be significantly better for the biperiodic holograms. However, the situation appears to be reversed for the sensitivity to the thickness, with both the leaky and waveguide mode of the periodic hologram showing a significantly better sensitivity (independently and when taken together) to that of the dual-resonant hologram. As such one can envision that the biperiodic hologram may find potential applications where d bio needs to be determined and cannot be inferred a priori or measured using other techniques, particularly for high-throughput analysis 20,21 .

Conclusions
Through simulations and calculations based on diffraction grating and Helmholtz wave equations, we have studied two different types of holograms that each support two resonant modes: leaky and waveguide modes in a periodic structure, and two distinct waveguide modes in a biperiodic structure. The performance of these diffractive resonant structures is assessed specifically for potential biosensing applications in aqueous medium. A single-period grating can represent a leaky mode excited at a lower resonant wavelength than its waveguide mode. Though, over a sufficiently broad wavelength window it will be possible to make use of both leaky and waveguide modes of such grating structure. Due to the interferometric nature of the resonances (i.e. multiple resonances in a given wavelength range), the excited modes are effectively narrower and exhibit a higher finesse, and in turn allow for a higher precision. Localizing multiple peaks simultaneously in itself results in a higher precision of localization than a single peak. Our models showed a refractive index sensitivity of 0.033 and 0.043 RIU for leaky and waveguide modes in the single-period grating. The sensitivity to d bio for this structure was found to be 0.102 and 0.079 for the two modes. The sensitivity is then compared to a superimposed bidiffractive grating structure that incorporates two independent resonances with two encoded periods. In result, a refractive index sensitivity of 0.028 and 0.024 RIU for the first and the second waveguide modes. The sensitivity to d bio for this bimodal structure was estimated to be 0.12 and 0.139 for both modes. These findings reveal that the leaky-mode waveguide structures, while having a lower refractive index sensitivity, have more than an order of magnitude higher thickness sensitivity than the dual-resonant structure supporting only waveguide modes. This considers the fitting modal sensitivity of both the short and long wavelength resonances for each case. We expect this latter property will have beneficial applications for biosensing, especially where d bio cannot be accurately controlled or measured.