A Theoretical Description of Node-Aligned Resonant Waveguide Gratings

: Waveguide gratings are used for applications such as guided-mode resonance ﬁlters and ﬁber-to-chip couplers. A waveguide grating typically consists of a stack of a single-mode slab waveguide and a grating. The ﬁlling factor of the grating with respect to the mode intensity proﬁle can be altered via changing the waveguide’s refractive index. As a result, the propagation length of the mode is slightly sensitive to refractive index changes. Here, we theoretically investigate whether this sensitivity can be increased by using alternative waveguide grating geometries. Using rigorous coupled-wave analysis (RCWA), the ﬁlling factors of the modes of waveguide gratings supporting more than one mode are simulated. It is observed that both long propagation lengths and large sensitivities with respect to refractive index changes can be achieved by using the intensity nodes of higher-order modes.


Introduction
Passive and low-loss planar optical waveguides can transport light over large areas [1][2][3][4]. When they are combined with optical elements such as diffraction gratings (termed waveguide gratings), they can be used for applications such as optical filters [5][6][7][8][9][10] and sensors [11][12][13][14][15][16][17] via exploiting guided mode resonances. Commonly, only the spectral positions of resonance are sensitive to refractive index changes, while the corresponding propagation length L prop remains almost constant. This circumstance indicates the necessity of spectrometers for such devices based on waveguide gratings. With a large sensitivity of L prop , small refractive index changes could be directly translated into a spatial variation in the outcoupled guided light. This can be detected by an array of simple broadband photodetectors.
Beyond passive refractive index sensors, a large sensitivity would allow for electrical control of L prop , which opens up new possibilities such as active beam deflectors or modulators. To meet the requirement of long propagation lengths, it is desirable to use fast and loss-free effects such as the electro-optic Pockels or Kerr effect. However, those effects enable small refractive index tuning in the order of ∆n ≈ 10 −4 . . . 10 −3 [18][19][20] only, which requires large sensitivities of L prop . Thus, it is of no surprise that reports about electrooptic detuning of waveguide gratings can only rarely be found in the literature to date and rather show the control of the spectral positions of resonances than the control of the propagation length [21,22].
In fact, there is a way to overcome these limits. It has been shown that intensity nodes of TE modes (s-polarized modes with a transversal electric field node) can be used to maximize the propagation length [23][24][25] by placing a lossy, diffractive, or scattering structure at the node position. This way, spectrally narrow resonances can also be obtained [26]. Conceptually, it has been estimated that such node modes should provide high sensitivities, Optics 2022, 3 61 as a slight shift of the node position largely affects the propagation length [27]. However, this concept has not been discussed in the scientific literature yet.
Here, we explicitly and exemplarily show that the node of the TE 1 mode in planar waveguide gratings allows obtaining long propagation lengths of more than 10 5 λ and large sensitivities in the order of L prop (n)/L prop (n + ∆n) ≈ 1.5 × 10 6 for ∆n = 1 × 10 −4 .

Definition of Geometry and Symmetry Parameters
A visualization of the sensitivity of the propagation length with respect to refractive index changes is shown in Figure 1a,b. Light propagates through a waveguide grating mode in the positive x-direction. Due to the interaction with the grating, light is emitted into free space with a propagation length L prop under a mean angle θ 0 and with an angular divergence of ∆θ. Without a change in the refractive index, the propagation length is large, and ∆θ is small. When a refractive index change ∆n is introduced, the propagation length is much shorter, and ∆θ is larger. Detailed equations on these relations are provided later in this text.
Optics 2022, 3, FOR PEER REVIEW 2 structure at the node position. This way, spectrally narrow resonances can also be obtained [26]. Conceptually, it has been estimated that such node modes should provide high sensitivities, as a slight shift of the node position largely affects the propagation length [27]. However, this concept has not been discussed in the scientific literature yet.
Here, we explicitly and exemplarily show that the node of the mode in planar waveguide gratings allows obtaining long propagation lengths of more than 10 and large sensitivities in the order of ( )/ ( + ∆ ) ≈ 1.5 × 10 for ∆ = 1 × 10 .

Definition of Geometry and Symmetry Parameters
A visualization of the sensitivity of the propagation length with respect to refractive index changes is shown in Figure 1a,b. Light propagates through a waveguide grating mode in the positive x-direction. Due to the interaction with the grating, light is emitted into free space with a propagation length under a mean angle and with an angular divergence of ∆ . Without a change in the refractive index, the propagation length is large, and ∆ is small. When a refractive index change ∆ is introduced, the propagation length is much shorter, and ∆ is larger. Detailed equations on these relations are provided later in this text. To present a strategy for how a large sensitivity of the propagation length can be achieved, we used the geometry of a waveguide grating, which is defined by the parameters in Figure 1c. It consists of an infinitely extended rectangular grating of period Λ, refractive indices and , and a duty cycle . Two dielectric layers of thicknesses and , as well as refractive indices and , surround the grating. For all To present a strategy for how a large sensitivity of the propagation length can be achieved, we used the geometry of a waveguide grating, which is defined by the parameters in Figure 1c. It consists of an infinitely extended rectangular grating of period Λ, refractive indices n g1 and n g2 , and a duty cycle D. Two dielectric layers of thicknesses t d1 and t d2 , as well as refractive indices n d1 and n d2 , surround the grating. For all simulations in this research, we considered s-polarized plane-wave incidence (TE) in the x-z plane, with a lateral momentum k x,0 .
To provide a measure of symmetry for both the waveguide grating's refractive indices and thicknesses, we defined the symmetry parameters as where "gp" represents the grating position, and "n" represents the refractive index profile. Values of χ gp = 1.0 and χ n = 1.0 indicate a fully symmetric waveguide grating.

Geometry with n g1 = n g2
To introduce some measures of interest and explain the role of the TE k mode, as well as the role of symmetry, we defined a waveguide grating with the parameters t d = 0.632 λ, t g = 0.079 λ, n s = 1.0, n d1 = n d2 = 1.5 (χ n = 1.0), and n g1 = n g2 = 1.275, and compared the cases of χ gp = 0.0 (Figure 2a,b) and χ gp = 1.0 (Figure 2c,d). As it is known from the literature, such a waveguide grating exhibits eigenmodes (TE k ) that can be found for distinct real-valued lateral momenta k x,0,k = k 0 n e f f ,TE,k , whereby n e f f ,TE,k is the effective refractive index of an eigenmode. The index k counts the number of nodes of the electric field distribution Re(E y ) attributed to an eigenmode. Thus, the TE 0 has no nodes of Re(E y ), while the TE 1 mode exhibits exactly one node of Re(E y ). This node causes an interesting behavior of the filling factor of the grating layer where z g and z g + t g define the first and second interface of the grating layer with respect to z. While relatively large values of FF between 0.033 and 0.129 occur for all asymmetric cases as well as for the TE 0 mode at χ gp = 1.0, we observe a substantially lower value of FF = 0.002 for the TE 1 mode when χ gp = 1.0.
Optics 2022, 3, FOR PEER REVIEW 3 simulations in this research, we considered s-polarized plane-wave incidence (TE) in the x-z plane, with a lateral momentum , .
To provide a measure of symmetry for both the waveguide grating's refractive indices and thicknesses, we defined the symmetry parameters as where "gp" represents the grating position, and "n" represents the refractive index profile. Values of = 1.0 and = 1.0 indicate a fully symmetric waveguide grating.

Geometry with =
To introduce some measures of interest and explain the role of the mode, as well as the role of symmetry, we defined a waveguide grating with the parameters   The grating (t g > 0, n g1 = n g2 ) acts as a discrete lateral momentum reservoir providing a set of lateral momenta k m,k = k x,0,k + m 2π Λ as a result of Floquet's theorem [28]. The physical consequence of this set of momenta is that a mode of initial lateral mo- As a result, the initially real-valued n e f f ,TE,k becomes complex-valued. The intensity of an excited mode dampens to 1/e of its initial value by radiation over the normalized propagation length Im(n e f f ,TE,k ) due to radiation into free-space modes. Radiated light enters the free space under an angle of θ 0 = arcsin( Re(k m,k ) n s k 0 ) and angular divergence of the diffracted light of ∆θ = 2Im(n e f f ,TE,k ) n s cos(θ 0 ) . Here, we chose n g1 = 1.0, n g2 = 1.5, D = 0.5 with otherwise identical parameters as for the waveguide grating discussed in Figure 2. λ , ∆θ, and the field distributions originates from the small value of FF discussed in Figure 2: empirically, for thin gratings (t g < 0.1 t WG ), we find the relation We observed that the TE 1 mode at χ gp = 1.0 exhibits p = 6, and thus, L prop λ ∝= t −6 g . In comparison, the TE 0 mode at χ gp = 1.0 and χ gp = 0.0 as well as the TE 1 mode at χ gp = 0.0 show p = 2. These dependencies presumably occur because the radiative loss rate α of the grating scales with α ∝ t 2 g and the filling factor approximately scales with FF ∝ t 3 g and FF ∝ t 1 g , respectively. As a side note, gratings with dominant Ohmic losses (e.g., metallic gratings) show scalings of p = 3 (TE 1 mode, χ gp = 1.0) and p = 1 for all other cases. Figure 4a,b show L prop /λ and ∆θ with variation in t g /λ at a fixed value of t d1 + t d2 = 0.632 λ. Decreasing values of t g /λ lead to increasing values of L prop /λ and decreasing values of ∆θ with the explained proportionalities. These trends can be observed up to a value of t g = 0.4 λ, corresponding to t g t WG ≈ 0.39. Figure 4c,d show L prop /λ and ∆θ with variation in t WG /λ at a fixed value of t g = 0.079 λ. L prop /λ is strongly increased for the TE 1 mode at χ gp = 1.0 in comparison to all other displayed cases for all values of t WG /λ above the cutoff of the TE 1 mode.
Thus, as long as t g is small, compared with t WG , and t WG is large enough to support the TE 1 mode, its increased values of L prop /λ and decreased values of ∆θ can be obtained over a broad range of waveguide grating thicknesses in the case of χ gp = 1.0 and χ n = 1.0.
To present an impression of the meaning of these values in an optical application, we considered the TE 1 mode and χ gp = 1.0 for a wavelength of λ = 632.8 nm with a grating thickness of t g = 0.079 λ = 50 nm. For these values, we obtained L prop = 7.6 cm. In comparison, the standard scenario of a TE 0 mode and χ gp = 0.0 leads to L prop = 110 µm. To reach the same L prop as for the TE 1 mode at χ gp = 1.0, the grating thickness would have to be reduced to 0.88 nm (a factor of 1/56) or the waveguide grating thickness (for t g = 50 nm) would have to be increased to approximately t WG = 7 µm (a factor of 15). Therefore, for a given grating geometry and waveguide grating thickness, using the TE 1 mode at χ gp = 1.0 allows for a drastic increase in the propagation length in comparison to standard waveguide gratings using the TE 0 mode. This behavior of , ∆ , and the field distributions originates from the small value of discussed in Figure 2: empirically, for thin gratings ( < 0.1 ), we find the relation ∝ 1

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We observed that the mode at = 1.0 exhibits = 6, and thus, ∝= .    χ gp = 0.0 (dashed lines) and χ gp = 1.0 (solid lines) and a fixed normalized waveguide grating thickness t d1 + t d2 = 0.632 λ; (c,d) corresponding plots with variation in the normalized waveguide grating thickness t WG /λ and a fixed normalized grating thickness t g = 0.079 λ. Figure 5a shows the dispersion relation of the TE 1 mode at χ gp = 1.0 and the TE 0 mode at χ gp = 0.0, with fixed ratios of all other geometry parameters (identical values to the ones in Figures 3 and 4). Remarkably, large values of L prop /λ between 10 4 and 10 6 are observed for the TE 1 mode over a broad spectral range between λ t WG = 0.9 and λ t WG = 2.0. This behavior occurs due to the symmetry of the waveguide grating, which enforces the node of the TE 1 mode to remain at the center plane of the waveguide grating. Therefore, regardless of the wavelength, an equivalent situation as discussed in Figure 3 is apparent when λ t WG is below the cutoff of the TE 1 mode. In comparison, the TE 0 mode exhibits values of L prop /λ between 10 2 and 10 3 for all values of λ t WG .
For an asymmetric geometry (χ gp = 1.0, χ n = 1.0), L prop /λ shows a maximum of around 10 5 at a distinct value of λ t WG = 0.96 (Figure 5b) for the TE 1 mode. This maximum occurs since the position of the node of Re E y shifts through the waveguide grating with respect to the z-direction as a function of λ the ones in Figures 3 and 4). Remarkably, large values of / between 10 and 10 are observed for the mode over a broad spectral range between = 0.9 and = 2.0. This behavior occurs due to the symmetry of the waveguide grating, which enforces the node of the mode to remain at the center plane of the waveguide grating. Therefore, regardless of the wavelength, an equivalent situation as discussed in Figure 3 is apparent when is below the cutoff of the mode.

Sensitivity to Asymmetric Refractive Index Changes
In the last part of this study, we analyze the sensitivity of the waveguide grating with respect to an asymmetric change in the refractive index at = 1.0. Such an asymmetric change in the refractive index means that is varied, and remains fixed at a value of 1.5. As a side note, variations in both and at = 1.0 show almost no sensitivity for the mode for any symmetric geometry, as is always minimized. For this exemplary case, the grating thickness was chosen to be = 1.5 × 10 , whereby all other remaining geometry parameters were chosen to be the same as in   Figure 3 (only the wavelength is varied with respect to the geometry) and χ n = 1; (b) dispersion relations and normalized propagation lengths for an asymmetric geometry with χ gp = 0.91, χ n = 1.03 for both the TE 0 mode and TE 1 mode with t g t d1 +t d2 = 0.045.

Sensitivity to Asymmetric Refractive Index Changes
In the last part of this study, we analyze the sensitivity of the waveguide grating with respect to an asymmetric change in the refractive index at χ gp = 1.0. Such an asymmetric change in the refractive index means that n d1 is varied, and n d2 remains fixed at a value of 1.5. As a side note, variations in both n d1 and n d2 at χ n = 1.0 show almost no sensitivity for the TE 1 mode for any symmetric geometry, as FF is always minimized.
For this exemplary case, the grating thickness was chosen to be t g = 1.5 × 10 −3 λ, whereby all other remaining geometry parameters were chosen to be the same as in Figures 3 and 4.
The following two simulation observations are of interest in order to investigate the sensitivity of the waveguide grating: (1) For small changes of the refractive index, the figure of merit provides a measure for the sensitivity.
(2) For more practical considerations, the refractive index is commonly switched between two distinct values, with a difference of ∆n. The sensitivity can be defined by Figure 6 shows both L prop /λ and the corresponding FoM, as well as S ∆n as a function of n d1 for the TE 0 mode and the TE 1 mode. Similar to the variation in χ gp in Figures 3-5, the TE 1 mode exhibits high values of L prop /λ around 10 12 of when χ n approaches 1.0. Most strikingly, L prop /λ strongly varies with changing values of n d1 (Figure 6a). In comparison, the TE 0 mode exhibits nearly constant values of L prop /λ around 10 3 . The FoM of the TE 1 mode reaches values of up to 2 × 10 4 , whereas the maximum FoM of the TE 0 mode in the displayed range is 5.6 ( Figure 6b). The reason for this large FoM for the TE 1 lies in a strong decrease in L prop /λ consequent to symmetry breaking. Concerning S ∆n (Figure 6c), for a value of ∆n = 1 × 10 −4 (e.g., in the Pockels effect [18,19]), the values of S ∆n are close to 1 for the TE 0 mode. ures 3-5, the mode exhibits high values of / around 10 of when approaches 1.0. Most strikingly, / strongly varies with changing values of (Figure 6a). In comparison, the mode exhibits nearly constant values of / around 10 . The of the mode reaches values of up to 2 × 10 , whereas the maximum of the mode in the displayed range is 5.6 ( Figure 6b). The reason for this large for the lies in a strong decrease in / consequent to symmetry breaking.
Concerning ∆ (Figure 6c), for a value of ∆ = 1 × 10 (e.g., in the Pockels effect [18,19]), the values of ∆ are close to 1 for the mode. However, for the mode, it can be observed that ∆ exhibits a maximum at = 1.0, with a substantially higher value of around 1.5 × 10 , in comparison to the However, for the TE 1 mode, it can be observed that S ∆n exhibits a maximum at χ n = 1.0, with a substantially higher value of around 1.5 × 10 6 , in comparison to the TE 0 mode. Although fully bound modes ( L prop → ∞ ) cannot be reached, as the filling factor cannot be set to zero, the TE 1 mode allows obtaining much higher propagation lengths and sensitivities with respect to asymmetric refractive index changes than the TE 0 mode using the same geometry. In comparison, reports in the literature regarding the sensitivity of the propagation length with respect to refractive index sensitivity are around FoM ≈ 3 and S ∆n ≈ 1.005 for ∆n = 1 × 10 −4 [21,22].

Conclusions
The results presented in this study show a way to drastically increase both the propagation length and sensitivity of waveguide grating by using the TE 1 mode, as long as the grating thickness is small, compared with the waveguide grating thickness. As all results were obtained with the help of an exemplary set of waveguide grating geometries, further optimizations for specific applications such as different refractive indices and grating shapes should be considered in future studies. Nonetheless, as the increased propagation length and sensitivity result from symmetrical conditions, the concept at hand can be applied to a broad range of geometry parameters and wavelengths in general. Control over the propagation length cannot be provided only by changing the refractive index but also by breaking the geometric symmetry of the waveguide grating, e.g., by thermomechanical effects. To be sure, this property also implies the necessity of the accurate control of thickness homogeneity. From the practical point of view, symmetric and homogeneous waveguide gratings can be approached by lamination [29] and are, therefore, in principle, accessible with precise standard fabrication techniques such as roll-to-roll coating [30] in combination with lithography methods [31,32]. Thus, we anticipate this concept to be suited for largearea applications requiring control of the propagation length and divergence angle over many orders of magnitude or sensitivity to environmental changes. Obvious applications are spatially resolved refractive index sensors and light modulators.

Methods
All simulations in this research were conducted using rigorous coupled-wave analysis (RCWA) [28]. To ensure the stability of the simulation, we checked both the convergence of the simulated values as well as the conservation of energy (see the Supporting Information). Naturally, as the presented data were calculated for TE polarization, fast convergence and large stability were already obtained for a low number of Fourier orders.
Supplementary Materials: The following supporting information can be downloaded at: https: //www.mdpi.com/article/10.3390/opt3010008/s1, Figures S1: (a,b) The convergence of L prop as a function of the number of Fourier orders at the smallest (t g = 1.5·10 −3 λ) and thickest grating layer thicknesses t g = 0.4·10 −4 λ; (c) energy conservation for various grating thicknesses at 15 Fourier orders. For all values, the energy is conserved, confirming the stability of the simulation. Figure S2: (a,b) L prop /λ and ∆θ as a function of t g /λ for a waveguide grating with a blaze grating geometry and otherwise identical parameters as in Figure 3. The black and red dots indicate the TE 0 and TE 1 modes for χ gp = 1.0, respectively. Figure S3: (a,b) L prop /λ and ∆θ as functions of t g /λ for a waveguide grating with a lossy grating (n g1 = n d1 and n g2 = 0.06 + 4.24j) and otherwise identical parameters as in Figure 3. The black and red lines indicate the TE 0 and TE 1 modes for χ gp = 1.0, respectively. Figure S4: (a,b) L prop /λ and ∆θ as a function of Λ/λ for a waveguide grating with otherwise identical parameters to the geometry in Figure 3 for χ gp = 1.0.  Data Availability Statement: The data in this study are available on request from the corresponding author. Supporting data are available in the Supporting Information.