Sensitivity enhancement of guided mode resonance sensors under oblique incidence

Liang Guo; Lei Xu; Liying Liu

doi:10.1364/PRJ.530126

1. INTRODUCTION

The guided mode resonance (GMR) effect is a peculiar resonance phenomenon that occurs in waveguide grating structures [1]. When light illuminates a GMR, a sharp resonance peak generates when the evanescent diffractive wave of the grating couples to a guided mode of the waveguide. GMR has a narrow linewidth ( $< 1 nm$ ), high efficiency ( $\sim 100 %$ ), and high sensitivity to the change of the surrounding refractive index [2 –4]. Therefore, GMR devices can be used as filters [5 –10], light modulators [11], light switches [12], and polarizers [13].

GMR sensors have become an increasingly attractive research topic because of their real-time, label-free detection capability [14 –17]. Up to now, almost all researches and applications are based on normal incidence [18 –23]; little attention has been paid to GMR sensors under oblique incidence. Qian et al. discovered that under a specific angle of incidence, sensitivity of the sensor ( $S$ ) has a 2.9-fold improvement compared to that under normal incidence [24]. Unfortunately, the mechanism of enhancement is unknown; no systematic research and theoretical analysis can be found.

In this paper, we employed approximate slab waveguide theory to analytically explore the sensitivity of GMR sensors and elucidate the sensing mechanism under oblique incidence. We found that the total sensitivity of a GMR sensor consists of grating sensitivity and waveguide sensitivity; when the two parts add up, the total sensitivity can be much higher than that at normal incidence. Further results from numerical calculations and experiments agree well with the analytical results. Our study paves a way to the possibility of obtaining higher sensitivity of GMR by effectively using the addition of grating sensitivity and waveguide sensitivity.

2. THEORY FOR SENSITIVITY OF GMR SENSORS UNDER OBLIQUE INCIDENCE

Assuming that the grating layer of a GMR is sufficiently thin to function predominantly as a diffractive element, the GMR can be approximately treated as a three-layer planar waveguide. Consequently, the resonant condition and sensing properties of the GMR can be analyzed using the grating equation and the waveguide eigenvalue equation [21].

Figure 1 depicts the schematic structure of a GMR sensor. Light of wavelength $λ$ illuminates the grating at an incident angle $θ_{i}$ , and we have [21,25] $Λ (n_{c} \sin θ_{i} - n_{w} \sin θ_{d}) = j λ, j = 0, \pm 1, \dots .$ (1)

Here, $Λ$ is the grating period, $n_{w}$ is the refractive index of the waveguide, $θ_{d}$ is the diffraction angle in the waveguide region, $n_{c}$ is the refractive index of the cover medium, and $j$ is the diffraction order of the grating (only the $+ 1$ st and $- 1$ st diffraction orders are considered here). If the wave vector of the diffracted light matches that of a waveguide mode, the incident light couples to the waveguide mode, and resonance is generated. Therefore, we can obtain the following equation: $k_{0} n_{w} d_{w} \cos θ_{d} - Φ_{w, c} - Φ_{w, s} = m π, m = 0, 1, 2, \dots .$ (2) $k_{0} = 2 π / λ$ is the wavenumber of light in vacuum, $d_{w}$ is the thickness of the waveguide, and $m$ is the order of the guided mode. $Φ_{w, c}$ and $Φ_{w, s}$ represent the phase shift due to total internal reflection at the waveguide-cover medium interface and waveguide-substrate interface; they can be expressed as the following: $Φ_{w, c} = \arctan {(\frac{n_{w}^{2}}{n_{c}^{2}})}^{ρ} {(\frac{N^{2} - n_{c}^{2}}{n_{w}^{2} - N^{2}})}^{\frac{1}{2}},$ (3) $Φ_{w, s} = \arctan {(\frac{n_{w}^{2}}{n_{s}^{2}})}^{ρ} {(\frac{N^{2} - n_{s}^{2}}{n_{w}^{2} - N^{2}})}^{\frac{1}{2}} .$ (4)

Here, we denote $ρ = 0$ for TE (transverse electromagnetic wave) and $ρ = 1$ for TM (transverse magnetic wave) polarization; $n_{s}$ is the refractive index of the substrate, and $N$ is the effective refractive index of the guided mode. When resonance occurs, the $j$ th diffracted wave and the $m$ th guided mode satisfy the phase matching condition. For a slab waveguide, $N$ can be expressed as $N = n_{w} \sin θ_{d} .$ (5)

Therefore, combining Eq. (1), $N$ can be rewritten as $N = n_{c} \sin θ_{i} - j λ / Λ, j = \pm 1, \dots .$ (6)

At normal incidence (e.g., $θ_{i} = 0 °$ ), $N$ is the same for $+ 1$ st and $- 1$ st diffracted light. However, when $θ_{i} \neq 0 °$ , $+ 1$ st and $- 1$ st diffraction orders correspond to different $N$ . For a given structure and incident condition, $n_{w}, d_{w}, θ_{i}, Λ$ are fixed; resonance wavelength $λ$ is a function of $n_{c}$ . Define a function $F (λ, n_{c})$ that has a form of the left side of Eq. (2) (only the fundamental mode of the waveguide is considered, i.e., $m = 0$ ): $F (λ, n_{c}) = \frac{2 π}{λ} n_{w} d_{w} \cos θ_{d} - Φ_{w, c} - Φ_{w, s} .$ (7)

$λ$ is an implicit function of $n_{c}$ . The sensitivity of the GMR sensor can be written as $S = \frac{d λ}{d n_{c}} = - \frac{F {(λ, n_{c})}_{n_{c}}}{F {(λ, n_{c})}_{λ}} .$ (8)

Here, $F {(λ, n_{c})}_{n_{c}}$ denotes the derivative of $n_{c}$ with respect to the equation [Eq. (7)]; $F {(λ, n_{c})}_{λ}$ denotes the derivative of $λ$ with respect to the equation.

$Schematic of a typical GMR sensor and the incident light propagation path under oblique incidence. It usually consists of substrate, waveguide, grating, and cover medium. Light of wavelength λ is incident on the structure; part of the light undergoes reflection (R) and transmission (T), while the remaining light is coupled into the waveguide through the +1st and −1st diffraction orders to form guided modes.$

Figure 1.Schematic of a typical GMR sensor and the incident light propagation path under oblique incidence. It usually consists of substrate, waveguide, grating, and cover medium. Light of wavelength $λ$ is incident on the structure; part of the light undergoes reflection (R) and transmission (T), while the remaining light is coupled into the waveguide through the $+ 1$ st and $- 1$ st diffraction orders to form guided modes.

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3. PERFORMANCE OF THE GMR SENSOR UNDER DIFFERENT STRUCTURAL PARAMETERS

Figure 2 is the heatmaps of $S$ versus $Λ$ and $d_{w}$ calculated by utilizing Eqs. (1 )–(8). $n_{w}, n_{c}, n_{s}, θ_{i}$ are set to be 2.12, 1.33, 1.45, 30°, respectively. The resonance excited by the $- 1$ st ( $+ 1$ st) diffraction order under TE/TM polarization is represented as $TE - 1 st / TM - 1 st$ ( $TE +1 st / TM +1 st$ ); meanwhile the resonance at normal incidence is represented as $TE ±1 st / TM ±1 st$ .

Figure 2.Calculated sensitivity heatmap of the GMR sensor about grating period and waveguide thickness: (a) $TE ±1 st$ ; (b) $TM ±1 st$ ; (c) $TE −1 st$ ; (d) $TM - 1 st$ ; (e) $TE +1 st$ ; (f) $TM +1 st$ .

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It is evident from the figures that $S$ at oblique incidence is higher than at normal incidence. Moreover, $S (TE +1 st / TM +1 st) ≫ S (TE −1 st / TM −1 st)$ and $S (TE ±1 st / TM ±1 st), S (TM +1 st) > S (TE +1 st)$ , $S (TM ±1 st) > S (TE ±1 st)$ . Conversely, $S (TE −1 st) > S (TM −1 st)$ ; this phenomenon will be discussed in the following section. Note that we have neglected the sign of $S$ because only its absolute value is meaningful.

The dark blue areas to the left of the white dotted lines stand for the cut-off regions. The cut-off region forms because a cut-off waveguide thickness exists for an asymmetric waveguide when the effective refractive index of the waveguide mode $N$ equals the refractive index of one of the cladding layers. Moreover, for GMR, $N$ depends on grating period $Λ$ [see Eq. (6)]. For example, when $Λ = 500$ and 600 nm, the cut-off waveguide thickness of $TE ±1 st$ mode is 26.67 and 32.00 nm, respectively.

Note that $S$ increases monotonically with $Λ$ for all resonance modes. Conversely, $S$ of $\pm 1$ st mode increases with $d_{w}$ initially and then decreases, which is consistent with the previous reports [21,26 –28]. $S$ of $+ 1$ st mode behaves similarly to that of $\pm 1$ st mode; however, for $- 1$ st mode, $S$ decreases initially and then increases.

We found that $S$ also changes dramatically with $θ_{i}$ . Figure 3 illustrates the relation of $S$ with $θ_{i}$ , where $Λ$ and $d_{w}$ are set to be 720 nm and 200 nm, respectively; it can be seen that as $θ_{i}$ increases, $S (+ 1 st)$ increases, meanwhile $| S (- 1 st) |$ decreases first and then increases. Therefore, there must exist a specific incident angle at which the sensitivity is zero. The green (TE) and blue (TM) dots in the figure represent zero-sensitivity points. Here we call these angles “zero angles”; they are 7.23° (TE) and 15.98° (TM), respectively. In addition, there must exist specific angles at which the sensitivity equals that at normal incidence, and we call these angles “critical angles”; they are represented as green (TE) and blue (TM) triangles in the figure at 14.79° and 34.73°, respectively.

Figure 3.GMR sensitivity as a function of incident angle. The grating period and waveguide thickness are 720 nm and 200 nm, respectively; calculated “zero angles” are 7.23° (TE) and 15.98° (TM), and “critical angles” are 14.79° (TE) and 34.73° (TM), respectively.

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4. VALIDITY OF THE PROPOSED MODEL

In order to confirm the validity of the proposed theory, we use FDTD Solutions to numerically calculate $S$ . In the calculation, $n_{w}, n_{c}, n_{s}, θ_{i}, Λ$ are set to be 2.12, 1.33, 1.45, 30°, 720 nm, respectively. The grating height $d_{g}$ is set to be 10 nm, 30 nm, and 60 nm; the grating refractive index $n_{g}$ is the same as the waveguide (the grating can be considered as the extension of the waveguide), and the filling factor of the grating $f$ is 0.5. Figure 4 plots the relation of $S$ with $d_{w}$ (only $TM +1 st$ mode results are presented; similar results apply to the other modes). In the figure, the solid line comes from analytical calculations by the theoretical model; the scattered points are numerical calculations by FDTD Solutions. It is evident that analytical and numerical results agree well when $d_{w}$ is larger than 240 nm. As $d_{w}$ decreases, the evanescent field increases, and the electromagnetic field penetrates into the grating layer, making $d_{g}$ non-negligible. Nevertheless, agreement is better for smaller $d_{g}$ obviously.

Figure 4.GMR sensitivity of $TM +1 st$ mode as a function of waveguide thickness. Grating height is 10 nm, 30 nm, 60 nm.

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Note that our theoretical model is built on the assumption that $d_{g}$ is very small; the impact of other parameters on $S$ will show up when $d_{g}$ becomes large. Figure 5 plots the changes of $S$ when the filling factor $f, n_{g}$ , and grating profile change; $d_{g}$ is set to be 60 nm in the calculation. Figure 5(a) illustrates the variation of $S$ with $d_{w}$ for different $f$ . Analytical and numerical calculations show good agreement when $d_{w}$ is larger than 240 nm. Moreover, the agreement becomes better for smaller $f$ , because a smaller $f$ has less influence on the evanescent field of the grating. Figure 5(b) depicts the relation of $S$ with $d_{w}$ for two different $n_{g}$ . When $d_{w}$ is less than 240 nm, two calculation results agree well for smaller $n_{g}$ . On the other hand, when $d_{w}$ is larger than 240 nm, the agreement becomes better for larger $n_{g}$ ; it may be because in our theoretical model, we assume $n_{g} = n_{w}$ , and the grating can be considered as an extension of the waveguide. When $n_{g}$ decreases, more electromagnetic field is localized in the waveguide layer, leading to reduced sensitivity. Figure 5(c) plots the change of $S$ versus $d_{w}$ for rectangular and sinusoidal grating profiles. It can be seen that the grating profile exerts a relatively minor influence on $S$ ; this can be attributed to the small effective refractive index difference between the rectangular and sinusoidal gratings, such that their impact on the evanescent field difference is negligible.

$GMR sensitivity of TM+1st mode versus waveguide thickness. (a) Grating filling factor is 0.3, 0.5, and 0.7; (b) grating refractive index is 2.12 and 1.60; (c) grating profile is rectangular and sinusoidal.$

Figure 5.GMR sensitivity of $TM +1 st$ mode versus waveguide thickness. (a) Grating filling factor is 0.3, 0.5, and 0.7; (b) grating refractive index is 2.12 and 1.60; (c) grating profile is rectangular and sinusoidal.

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Upon the above observations, we can conclude that the theoretical model works well, although there are more or less discrepancies between our model and FDTD calculation. $d_{g}, f$ , and $n_{g}$ have a relatively large influence on $S$ while the influence of the grating profile is relatively small. These results give important guidelines for further experiments.

5. ANALYSIS ON MECHANISM OF SENSITIVITY ENHANCEMENT UNDER OBLIQUE INCIDENCE

Rewrite Eqs. (2), (6) to a form as Eqs. (9), (10), respectively: $λ = \frac{2 n_{w} π d_{w} \sqrt{1 - \frac{N^{2}}{n_{w}^{2}}}}{\arctan {(\frac{n_{w}^{2}}{n_{s}^{2}})}^{ρ} {(\frac{N^{2} - n_{s}^{2}}{n_{w}^{2} - N^{2}})}^{\frac{1}{2}} + \arctan {(\frac{n_{w}^{2}}{n_{c}^{2}})}^{ρ} {(\frac{N^{2} - n_{c}^{2}}{n_{w}^{2} - N^{2}})}^{\frac{1}{2}}},$ (9) $λ = Λ (n_{c} \sin θ_{i} - N) / j, j = \pm 1, \dots .$ (10)

The intersection of Eqs. (9) and (10) is the resonance wavelength of GMR, which clearly varies with $n_{c}$ . Figures 6(a)–6(c) are plots of Eqs. (9) and (10) for $TM ±1 st$ , $TM +1 st$ , $TM −1 st$ , respectively. Lines for $n_{c} = 1.33$ and $n_{c} = 1.39$ are plotted in each figure, $n_{w}, n_{c}, n_{s}, θ_{i}, d_{w}, Λ, Δ n_{c}$ are set to be 2.12, 1.33, 1.45, 30°, 200 nm, 720 nm, 0.06, respectively, in the calculation. In the figures, $C_{1}, C_{2}$ are curves of Eq. (9), $C_{3}, C_{4}$ are curves of Eq. (10), and the insets in each figure are enlarged region of the intersections. Obviously $C_{3}$ and $C_{4}$ coincide at normal incidence [Fig. 6(a)], because $λ$ is independent of $n_{c}$ when $θ_{i} = 0 °$ [see Eq. (10)]; therefore the change of $λ$ only comes from Eq. (9). In addition, it can be seen that the change of $λ$ for $TM ±1 st$ ( $Δ λ = 12.08 nm$ ) is higher than that for $TE ±1 st (Δ λ = 5.51 nm)$ .

$Calculated resonance wavelength versus the effective refractive index of the guided mode: (a) TM±1st; (b) TM+1st; (c) TM−1st. Insets are enlarged areas of the red circles.$

Figure 6.Calculated resonance wavelength versus the effective refractive index of the guided mode: (a) $TM ±1 st$ ; (b) $TM +1 st$ ; (c) $TM −1 st$ . Insets are enlarged areas of the red circles.

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When $θ_{i} \neq 0 °$ , both Eqs. (9) and (10) contribute to the change of $λ$ ; we call them waveguide sensitivity and grating sensitivity. For $+ 1$ st diffraction mode [see Fig. 6(b)], when $n_{c}$ changes from 1.33 to 1.39, $C_{1}$ shifts to $C_{2}$ , and $C_{3}$ shifts to $C_{4}$ . The entire pathway can be considered as moving from $A_{1}$ (the intersection of $C_{1}$ and $C_{3}$ ) to $A_{2}$ (the intersection of $C_{1}$ and $C_{4}$ ) and then to $A_{3}$ (the intersection of $C_{2}$ and $C_{4}$ ). The variation of $λ$ is accordingly $Δ λ_{1}$ and $Δ λ_{2}$ , with the total variation $Δ λ = Δ λ_{1} + Δ λ_{2}$ . Note that both $Δ λ_{1}$ and $Δ λ_{2}$ are positive; thus $Δ λ$ is a constructive addition of the two contributions. On the other hand, for $- 1$ st mode [see Fig. 6(c)], $Δ λ_{1}$ and $Δ λ_{2}$ have opposite signs; thus $Δ λ$ is a destructive addition of the two contributions. This also explains the presence of “zero angle” ( $Δ λ_{1} = Δ λ_{2}$ ) and “critical angle” for $- 1$ st as shown in Fig. 3.

In addition, when $d_{w}$ increases, the evanescent field becomes weaker. Grating sensitivity is the dominant contribution; hence $| S |$ can be much larger than that at normal incidence.

6. EXPERIMENTAL RESULTS AND DISCUSSION

We conducted an easily implementable experiment to verify the theoretical predictions. In our experiment, the materials of the substrate, waveguide, and grating are ${SiO}_{2}$ , ${Ta}_{2} O_{5}$ , and S1805 photoresist, respectively.

First, a 200 nm ${Ta}_{2} O_{5}$ film was deposited on a flat ${SiO}_{2}$ substrate via electron beam vapor deposition; the thickness of the waveguide was measured using a spectroscopic ellipsometer (Horiba, UVISEL). Then, an $\sim 100 nm$ thick photoresist film was spun coated on the ${Ta}_{2} O_{5}$ waveguide layer to record the grating fringes by using a 442 nm helium cadmium laser. After exposure, 2.38% tetramethylammonium hydroxide (NMD-3) solution was used as the developer to obtain one dimensional grating patterns. The prepared GMR sample is shown in Fig. 7(a). An atomic force microscope (AFM, Bruker, Dimension 5000) was employed to characterize the fabricated sample, as shown in Figs. 7(b) and 7(c); it can be seen from the figures that the depth of the grating is 89.50 nm and the measured period of the manufactured grating is 722.5 nm (take the average of six periods), which is slightly different from the designed value of 720 nm.

Figure 7.(a) Photograph of the fabricated GMR sensor; (b) 3D image of the holographic grating pattern obtained by AFM; (c) profile of the holographic grating scanned by AFM; the measured grating period is 722.5 nm, and grating depth is 89.50 nm.

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Figure 8 is the schematic experimental setup for measuring the reflection spectra. Light emitted from a super-continuum light source (YSL Photonics, SC-5, spectral range 470–2400 nm) is collimated using a fiber collimator. The collimated light beam passes through an attenuator, a diaphragm, and a polarizer before irradiating on the GMR sample. The reflected light beam is then collected by a spectrometer (Acton Research, SpectraPro-2750). The sample is immersed in aqueous glycerol solutions with concentrations varying from 0% to 40% by volume in 10% increments. The corresponding refractive indices for these solutions are 1.3330 (0%), 1.3466 (10%), 1.3602 (20%), 1.3740 (30%), 1.3880 (40%).

Figure 8.Schematic illustration of the experimental setup for measuring the reflectance spectra of the GMR sensor.

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Figure 9 presents both experimental and simulated results. Figure 9(a) is the reflection spectra under different concentrations of aqueous glycerol solutions at normal incidence, with the inset displaying corresponding simulated spectra from FDTD Solutions. Figures 9(b) and 9(c) depict the experimental reflection spectra for $TM +1 st$ at 10° and $TE −1 st$ at 40°, respectively. Figure 9(d) is the corresponding fitting curves of the peak wavelength to refractive index of the solution for $TM +1 st$ at 10°, $TE ±1 st$ at 0°, and $TE −1 st$ at 40°; the slopes of the fitting curves give the sensitivity of the GMR sensor. The fitted experimental and simulated sensitivities are 281.85 and 292.46 nm/RIU for $TM +1 st$ at 10°, 75.70 and 70.15 nm/RIU for $TE ±1 st$ at 0°, and $- 363.61$ and $- 339.11 nm / RIU$ for $TE −1 st$ at 40°, respectively. Both the resonance wavelength and sensitivity measured in the experiment agree well with simulated results.

$(a) GMR reflection spectra of TE mode at normal incidence, with the refractive index of the aqueous solutions varying from 1.333 to 1.388 with a step of 0.0136. Inset: resonance spectra simulated by FDTD Solutions. (b), (c) Experimental reflection spectra for TM+1st at 10° and TE−1st at 40°. (d) Resonance wavelength change versus refractive index. Blue dots are experimental resonance wavelength, and red dots are calculated resonance wavelength by FDTD Solutions. Lines are linear fittings representing sensitivity.$

Figure 9.(a) GMR reflection spectra of TE mode at normal incidence, with the refractive index of the aqueous solutions varying from 1.333 to 1.388 with a step of 0.0136. Inset: resonance spectra simulated by FDTD Solutions. (b), (c) Experimental reflection spectra for $TM +1 st$ at 10° and $TE −1 st$ at 40°. (d) Resonance wavelength change versus refractive index. Blue dots are experimental resonance wavelength, and red dots are calculated resonance wavelength by FDTD Solutions. Lines are linear fittings representing sensitivity.

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Figure 10 compares the experimental results with analytically calculated results of $S$ as a function of $θ_{i}$ . The resonance wavelength for $+ 1$ st mode is out of the response of our detector when $θ_{i} > 10 °$ ; therefore experimental data are mainly for $- 1$ st. It is evident that the experimental results agree well with the theoretical calculations. For $+ 1$ st, the sensitivity increases proportionally with the incident angle, whereas for $- 1$ st, sensitivity transitions from positive to negative as the incident angle increases. After crossing the “zero angle”, the sensitivity goes higher as the incident angle increases. The measured sensitivity of $TM +1 st$ is 281.85 nm/RIU at $θ_{i} = 10 °$ , which is almost equivalent to $TE −1 st$ at 35° (measured sensitivity is $- 325.60 nm/RIU$ ) and $TM −1 st$ at 40° (measured sensitivity is $- 281.82 nm/RIU$ ), and this result is 1.65-fold higher than that of normal incidence ( $TM ±1 st$ at 0°; measured sensitivity is 171.19 nm/RIU). Moreover, $S$ of $TM +1 st$ is higher than that of $TE +1 st$ . For example, the measured sensitivity of $TE + 1 st$ at 5° is 130.43 nm/RIU, while $S$ of $TM +1 st$ at 5° is 225.95 nm/RIU. On the other hand, $S$ of $TE −1 st$ is slightly higher than that of $TM −1 st$ . The measured sensitivity of $TE −1 st$ at 5° and $TM −1 st$ at 15° is $- 6.50 nm/RIU$ and $- 17.81 nm/RIU$ , respectively, indicating the two angles are very close to the “zero angle” as predicted from our theoretical analysis. The highest sensitivity measured in the experiment is $- 363.61 nm/RIU$ at $TE −1 st$ at 40°, which has a 4.8-fold enhancement in contrast to normal incidence ( $TE ±1 st$ at 0°; measured sensitivity is 75.70 nm/RIU).

Figure 10.GMR sensitivity as a function of incident angle. The solid lines are calculated curves by the proposed theory (T), and the scattered points are the measured values in the experiment (E).

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Deviation of the measured data from the calculated data is mainly because the grating height is neglected. The fabricated grating has a height of 89.50 nm and a refractive index of approximately 1.60 (photoresist), which affects the sensitivity. Less deviation is expected if a smaller height grating is used.

Higher sensitivity is expected by selecting $TM +1 st$ mode at a larger incident angle. For instance, take $n_{w}, n_{c}, n_{s}, Λ$ to be 2.12, 1.33, 1.45, 720 nm, respectively; when $θ_{i} = 60 °$ and $d_{w} = 280 nm$ , $S$ can reach 775 nm/RIU, which is more than five times larger than that at normal incidence (147 nm/RIU). Furthermore, using porous glass ( $n = 1.17$ ) as the substrate with $d_{w} = 161 nm$ can further increase sensitivity to 1320 nm/RIU at the same incident angle.

7. CONCLUSION

To summarize, we found that the sensitivity of GMR sensors under oblique incidence comes from the superposition of grating sensitivity and waveguide sensitivity. The sensitivity of $- 1$ st diffraction mode is a destructive superposition of the two contributions while the sensitivity of $+ 1$ st mode is a constructive superposition. The sensitivity at large incident angle can be five times higher than that at normal incidence. The simplified approximate slab waveguide theory gave quantitatively similar results to numerical calculations, especially when the grating layer is very thin. Our study paves a new way to the possibility of obtaining higher sensitivity of GMR.

Category: Nanophotonics and Photonic Crystals

Received: May. 24, 2024

Accepted: Sep. 2, 2024

Published Online: Nov. 1, 2024

The Author Email: Liying Liu (lyliu@fudan.edu.cn)

DOI:10.1364/PRJ.530126