1. Introduction

Waveguided grating structures (WGS) [1–6] is a kind of narrow-band optical filters, which have also potential applications in optical switch [7,8] and sensors [9,10] if using some special design to construct the grating and/or the waveguide. Spectral tunability makes this kind of device very flexible in practical applications to fit different spectral ranges. The tuning properties result from the dependence of the resonant waveguide mode on the grating period, the thickness and dielectric constant of the waveguide, and the angle of incidence, which are also dependent on the polarization of the incident light. Therefore, giving a comprehensive analysis of these tuning properties is important for the design of the WGS and for exploiting new optical devices. Furthermore, the angle-resolved tuning rate, which might be understood by evaluating the changing rate of the waveguide resonance mode as a function of the incident angle, should be considered differently for different applications. For instance, in angle-sensitive device we expect a tuning rate as high as possible so that an extremely small tilting of the sample can give rise to a strong detection signal. However, the angle-resolved tuning rate should be as low as possible in biosensors, so that any noise resulting from the vibration, the unintentional tilting, or the necessary remounting of the sample introduces very little influence on the bio-reaction signal. Thus, the signal-to-noise ratio can be maintained at a satisfying level. Theories have been developed very well [1–6] for analyzing the physics in the WGS device. In this paper, we make use of the ray picture theory [6] to give a more specific investigation of the tuning mechanisms of this kind of important photonic structures. Our experimental analysis of the angle-resolved tuning properties using the device described in section 2 shows good agreement with the simulation, which laid a good basis for our further theoretical investigations in the following sections.

2. Fabrication and microscopic characterization of the WGS device

The one-dimensional grating was fabricated using interference lithography on top of an indium tin oxide (ITO) glass substrate [11], where S1805 photoresist (PR) from Rohm and Haas Electronic Materials Ltd. was used to record the structures and a 355-nm diode-pumped solid-state laser from Advanced Optical Technology Ltd. was used as the UV light source. The ITO layer acts as the waveguide and has a thickness of about 200 nm. Fig 1(a) and (b) show the atomic force microscopic (AFM) images of the WGS device and scanning electron microscopic (SEM) studied in this paper, respectively. The inset of Fig. 1(a) gives the profile of the grating structures, showing more clearly the modulation depth of the PR grating. However, due to the size of the tip in the AFM measurement and the width of the grating grooves, the practical modulation depth of the grating structures is actually larger than 50 nm. The period of the PR grating is about 300 nm.

 

Fig. 1. Microscopic images of the grating on top of the WGS: (a) AFM height and the profile (inset), (b) SEM.

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We characterize this WGS device by the optical extinction spectrum measurement for both the TM and the TE polarization at different angles of incidence. The broadband whitelight for the measurement is a 100-W halogen light source, which is collimated and spatially filtered before being sent to the sample. A plastic polarizer mounted on a rotation stage and placed before the sample is used to control the polarization of the incident light. The sample of the WGS device is mounted on a home-built holder with the gratings extending along the vertical axis, which is installed on the rotation stage so that the WGS sample can be rotated in the horizontal plane and about a vertical axis. A spectrum analyzer from Ocean Optics (USB 4000) is used to measure the optical extinction spectrum. Similar procedures as have been illustrated in our previous work [11–12] are employed in the optical extinction measurement. For the TM polarization, the incident light is polarized perpendicular to the extending direction of the grating, whereas, the TE polarization is parallel to the grating. Fig. 2(a) and (b) show the optical extinction measurement (-log₁₀T(λ)) over a spectral range from 500 to 800 nm for the TM and TE polarizations, respectively, where T(λ) is the transmission through the structures at wavelength λ with reference to that through the substrate. For TM polarization, the resonance of the waveguide mode is degenerate at 540 nm at normal incidence (θ=0) and the longer-wavelength branch evolves to about 755 nm when the angle of incidence increases to 52 degrees. For TE polarization, the resonance of the waveguide mode takes place at 552 nm at normal incidence and the longer-wavelength branch is tuned to 756 nm with θ increased to 52 degrees, indicating a lower angle-resolve tuning rate than the TM polarization, which is defined as the shift of the resonant waveguide mode over the change of the incident angle in nanometers per degree.

 

Fig. 2. Angle-resolved tuning properties of the resonance mode of the WGS for (a) TM and (b) TE polarizations for an incident angle varying from 0° to 52° in steps of 4°.

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It can be found in Fig. 2 that the bandwidth of the resonance mode is approximately 9 nm at FWHM for both TM and TE polarization. The experimental results in Fig. 2 will be used to compare with and to confirm our theoretical analysis in the following sections.

3. Modeling and discussions

3.1 The theoretical model

The WGS device consists of a substrate, a thin waveguide layer, and a grating fabricated on the top of the waveguide, as shown in Fig. 3.

 

Fig. 3 Schematic diagram of the WGS.

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The incident light is firstly diffracted by the grating and excites the propagation mode in the waveguide, which is diffracted further by the grating while propagating in the waveguide, producing multiple beams parallel to that transmitting directly through the WGS.

The diffraction by the grating structures is governed by the following equation

where m denotes the diffraction order and ψ is the diffraction angle, and n_wg and n_sp are the refractive indices of the waveguide and the superstrate (or of the environment where the grating is located), and Λ is the period of the grating.

The resonance of the waveguide mode takes place when these beams interfere destructively, or the phase matching condition satisfied

where k_wg =2n_wgπsin(ψ)/λ is the transverse component of the propagating wave vector, and 2k_wgh denotes the phase shift due to the optical path length of one propagation cycle over the waveguide of thickness h. ϕ₁ and ϕ₂ are the two polarization-depending phase shifts introduced by the total internal reflections at the top and bottom surfaces of the waveguide. M denotes the number of the modes which can propagate in the waveguide and takes the values of 0, ±1, ±2….

According to the ray picture model [6], the resonance of the WGS is mainly dependent on six parameters: the angle of incidence θ, the refractive index of the waveguide n_wg, the thickness of the waveguide h, the period of the grating Λ, the refractive index of the substrate n_sb, and the refractive index of the medium on top of the grating nsp. The tunability of the WGS device is thus achieved by changing one or more of these parameters.

To characterize the waveguide mode, we need to define the phase offset from the resonance Δ, which is dependent on the polarization of the light and is equals 0 at the resonance. For TM polarization, where the light is polarized perpendicular to grating, Δ can be expressed as:

For thin waveguide, only the zero-order (M=0) mode exists, which is the case in our following discussions.

For TE polarization, where the light is polarized parallel to the grating,

Equations (1), (3) and (4) are the basis for our theoretical analysis of the tuning properties of the waveguide resonance mode and for the comparison with the experimental results.

3.2 Angle- resolved tuning dynamics of the waveguide resonance mode

Changing the angle of incidence (θ) is the straightforward method to tune the resonance of the waveguide mode, as can be conclude after substituting Eq.(1) into Eq.(3) for TM polarization or into Eq.(4) for TE polarization. Two branches of the resonant modes, which correspond to the +1 and -1 orders of diffraction, evolve into opposite directions with increasing θ. For simplicity, we study in the following sections only the +1 branch that evolves in the longer-wavelength direction. The squares (TE polarization) and the solid circles (TM polarization) in Fig. 4 summarize the measurements of the angle-resolved tuning properties in Fig. 2 for the device shown in Fig. 3.

The extinction peaks in the measurement in Fig.2 are taken as the resonance wavelengths (λ_R) of the waveguide mode shown in Fig. 4 for the comparison with the simulations. In the simulation results in sections 3.2, 3.3, and 3.4, we fixed the refractive index of the substrate at 1.52 (n_sb=1.52) and that of the medium on top of the device at 1.0 (n_sp=1.0). Using the same parameters of the WGS device in Fig. 3 and the theoretical model in section 3.1, we simulate the angle-resolved tuning properties, as demonstrated by the solid curves for the TM (black) and TE (red) polarizations. The spectral peak of the resonance mode increases with increasing the angle of incidence, as demonstrated by the solid red and black tuning curves for TE and TM polarizations, respectively.

 

Fig. 4. Angle-resolved tuning properties of the sample (red line: simulation, TE polarization; red solid square: measurement, TE; black line: simulation, TM; black solid square: measurement, TM), and the angle-resolved tuning rate with the red dashed line for the TE and the black dashed line for TM polarizations.

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Basically, the simulation agrees with the measurement, implying that the theoretical model can be used as a solid tool to investigate tuning dynamics of the WGS device. As has been mentioned above, the tuning rate, which is defined as the spectral shift of resonant waveguide mode over the change of the incident angle (Δλ_R/Δθ) in nanometers per degree, is important in the design of a WGS device for the applications in sensors and in filters. The dashed curves in Fig. 4 give the tuning rate as a function of incident angle for TM (black) and TE (red) polarizations using the simulation results in Fig. 4. The tuning rate has been calculated by making a numerical point-by-point derivative of simulated results illustrated by the solid curves in Fig. 4 and the same procedure is used for all of the simulation results of the tuning rate properties in the following sections. The data points in the simulated results have an interval of 1 degree as scaled by the horizontal axis. This resolution is found high enough to give a precise evaluation on the tuning rate. The angle-resolved tuning dynamics of TE polarization actually exhibits stronger nonlinearity than the TM. We can conclude from Fig. 4 that the TE polarization is more sensitive to the change of incident angles than the TM and the tuning rate is higher at smaller angles of incidence.

3.3 Dependence of the tuning dynamics on the parameters of the waveguide

The parameters of the waveguide interesting to us include the thickness and refractive index of the waveguide layer. Fig. 5 shows the angle-resolved tuning properties of a WGS device with a grating period of 300 nm and a waveguide thickness of 200 nm for different refractive indices of the waveguides for both the TE (solid) and TM (dashed) polarizations. Clearly, the resonance of the waveguide mode and the tuning range shift to the red with increasing the refractive index of the waveguide layer. The most important feature that can be resolved in Fig. 5 is that the separation between the TE- and TM-polarization resonance becomes larger with increasing the refractive index of the waveguide, which can be as large as 52 nm for n_wg=2.8 at normal incidence. However, this separation is smaller than 9 nm for n_wg=1.8. Generally, the bandwidth of the waveguide resonance mode is less than 10 nm at FWHM. Thus, the spectral response of the TE- and TM-polarized waveguide resonance mode can be separated completely with a large free spectral range for n_wg>2.2. This is very useful for exploring new optical devices for special applications. For example, in a polarization-sensitive device that needs to switch between TE and TM polarization or to respond at the same wavelength, we need to use lower-refractive-index materials to fabricate the waveguide so that the resonance of TE- and TM-polarization can overlap as much as possible. In contrast, if we need to have the polarization-sensitive device respond at two different wavelength or spectral range with high contrast, we should use high refractive index for the waveguide. Furthermore, the allowed dynamic range for angle-resolved tuning becomes smaller with increasing the refractive index of the waveguide. An additional feature is that the tuning rate of TE polarization is always higher than that of the TM polarization. All of these properties need to be considered in the design of a practical device based on WGS.

 

Fig. 5. Simulated angle-resolved tuning properties for the refractive index of the waveguide ranging from 1.0 to 3.0 with Λ=300 nm, h=200 nm, n_sp=1.0, and n_sb=1.52 (dashed: TM, solid: TE).

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Figure 6(a) and (b) demonstrate the simulated angle-resolved tuning properties (θ=0-60 degrees) for different thickness (h=200-500 nm) of the waveguide for both the TM (solid) and TE (dashed) polarizations, where the grating period is fixed at 300 nm and the refractive index of the waveguide is a constant of 2.0.

 

Fig. 6. (a) Simulated angle-resolved tuning properties for the thickness of the waveguide, ranging from 200 nm to 500 nm, with Λ=300 nm, n_wg=2.0, n_sp=1.0, n_sb=1.52. (b) Calculated resonant wavelength as a function of the thickness of the waveguide for an incident angle of 0° and 40° degrees. (c) Calculated angle-resolved tuning rate for different ranges of the waveguide thickness for TE polarization.

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Figure 6(a) indicates clearly that the resonance of the waveguide, as well as the tuning range, shifts to longer wavelengths with increasing the thickness of the waveguide. The TM polarization is more sensitive to the change of the thickness of the waveguide than the TE, and this sensitivity becomes higher at larger angle of incidence, as illustrated in Fig. 6(b), where we show the resonant wavelength for different values of the thickness of the waveguide at normal and 40-degree incidence. A comparison is made for the angle-resolved tuning rate between different thickness of the waveguide in Fig. 6(c). It can be observed that the angle-resolved tuning rate, which decreases with increasing the angle of incidence, becomes larger with increasing the thickness of the waveguide from 100 nm to 300 nm, implying that this tuning rate becomes faster with the increasing the thickness of the waveguide. However, this becomes completely different when the thickness of the waveguide is less than 100 nm, as shown in Fig. 6 (c). It is clear that the angle-resolved tuning rate becomes smaller when the thickness of the waveguide increases from 50 nm to 70 nm. This might be useful for tailoring the filtering effect of the WGS device [12].

3.4 Dependence on the grating period

The grating period (Λ) is the main parameter that determines the optical properties of the WGS device. Therefore, the optical response of the device changes dramatically with changing the grating period. Fig. 7 shows the angle-resolved tuning properties of the WGS device for the TM (dashed) and TE (solid) polarizations at different grating periods. Obviously, the tuning range of the resonant waveguide mode shifts quickly to the red with increasing the grating period from 280 to 400 nm for both the TM and the TE polarizations. For instance, at normal incidence the resonance shifts from 501.73 nm to 675.25 nm for the TM polarization when the grating period changes from 280 to 400 nm.

 

Fig. 7. Simulated angle-resolved tuning properties for a grating period ranging from 280 nm to 400 nm, with h=200 nm, n_wg=2.0, n_sp=1.0, and n_sb=1.52.

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Furthermore, the tuning rate also increases with increasing the grating period. As an example, when the angle of incidence increases from 0 to 20 degrees, the resonance of the waveguide mode shifts from 501.73 to 581.31 nm for a grating period of 280 nm for TM polarization; whereas, the resonant waveguide mode shifts from 675.25 to 782.94 nm for a grating period of 400 nm. This implies not only that the tuning range of the WGS device can be adjusted largely by changing the grating period, but also that any spectral range in the visible can be reached by resonance of the waveguide mode through a correct combination of the grating period and the angle of incidence. Additionally, smaller grating period should be employed if the WGS device is required to be less sensitive to the change of the incident angle, which is important in the design of sensor and filter devices.

3.5 Dependence on the parameters of the medium on top and on the bottom of the WGS device

The refractive indices of the superstrate (the environmental medium surrounding the WGS device) and the substrate can also be adjusted to modify the tuning dynamics of the WGS device. Fig. 8 shows the angle-resolved tuning properties of the WGS device as a function of the refractive indices of the superstrate and the substrate, respectively, where the refractive index of the waveguide is fixed at 2.0. In Fig. 8(a) the refractive index of the substrate is fixed at 1.52 and that of the superstrate is changed from 1.0 to 1.8. In Fig. 8(b), the refractive index of the superstrate is fixed at 1.0 and that of the substrate is changed from 1.4 to 1.8. The tuning dynamics of the TM and TE polarization are illustrated by the dashed and solid curves, respectively.

 

Fig. 8. (a) Simulated angle-resolved tuning properties for the refractive index of the superstrate ranging from 1.0 to 1.9 with Λ=300 nm, h=200 nm, n_wg=2.0, and n_sb=1.52. (b) Simulated angle-resolved tuning properties for the refractive index of the substrate ranging from 1.0 to 1.8 with Λ=300 nm, h=200 nm, n_wg=2.0, and n_sp=1.0.

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Figure 8 shows that the separation between the tuning curves of TM and TE polarizations becomes smaller with the increasing the refractive indices of the substrate or the superstrate. For example, in Fig. 8(b) the separation between the black dashed curve and the black solid curve is 31.4 nm at normal incidence when n_sb=1.4, however, it reduces to 9.7 nm (the separation between the dashed and solid light-blue curves) when n_sb=1.8. Controlling the separation between the TE and TM resonance modes enables flexible design of the narrowband polarization filters. The angle-resolved tuning rate also increases with the increasing the refractive indices of the super- and substrate. A simple comparison between Fig. 8(a) and (b) implies that the variation of the refractive index of the substrate induces stronger influence on the optical response than that of the superstrate.

Figure 9 gives an example of the tuning dynamics for both TM and TE polarizations, which is shown by the resonant wavelength (λ_R) as a function of the refractive indices of the superstrate (Fig. 9(a)) and substrate (Fig. 9(b)) at normal incidence, where the grating period and the refractive index of the waveguide are fixed at 300 nm and 2.0, respectively. As shown in Fig. 9, the tuning rates, which are defined as Δλ_R/Δn_sp for the superstrate and Δλ_R/Δn_sb for the substrate, are higher at larger refractive index, implying that the device is more sensitive to larger-refractive index materials, which should be considered in the design and application in sensors [11].

 

Fig. 9. Calculated resonant wavelength (λ_R) as a function of the refractive indices and the corresponding tuning rate of (a) the superstrate (Δλ_R/Δn_sp) and (b) substrate (Δλ_R/Δn_sb) at normal incidence.

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From the simulation results in Fig. 8 and Fig. 9, we can find clearly that the resonance and the tuning range of the waveguide mode shift to the red with increasing the refractive index of the sub- or superstrate, and the TM mode is more sensitive to the change of these refractive indices than the TE mode. This is favorable for the applications in biosensors, which employs the TM polarization so that the coupling between the plasmon resonance of the gold nanowires and the waveguide mode is used to sense the change of the environmental parameters for one-dimension devices.

However, in above discussions and in the simulation results in Fig. 8 and Fig. 9 we did not take into account the dispersion of the substrate and superstrate, which was found to have negligible influence on the angle-resolved tuning properties in the interesting spectral range in this paper.

4. Conclusions

We have investigated theoretically the tuning dynamics of the waveguided grating structures, and its dependence on the structural and geometric parameters, aiming to give a practical guidance for the design of WGS device in the applications of biosensors and optical switch:

(1) We should make the angle-resolved tuning rate of WGS device as low as possible, so that small tilting or the vibration of the sample will not influence the detected optical extinction signal. In this consideration, the grating period should be as smaller as possible provided that the waveguide mode is resonant in the visible spectral range and is suitable for the biomolecules to be detected (the absorption or emission spectrum of the biomolecules is better not overlapped with the detection range), which can be concluded from Fig. 7 in section 3.4. The refractive index of the waveguide has little influence on the tuning rate (Fig. 5 in section 3.3). However, adjustment of the refractive index of the waveguide layer can be used to optimize the resonant waveguide mode to the expected the spectral position in combination with the grating period. Although the thickness of the waveguide has little effect on the tuning rate, there exists an optimal value of the waveguide thickness. For instance, a waveguide thickness around 100 nm is recommended in the simulation result in Fig. 6(c) in section 3.3. If allowed, a larger angle of incidence is always recommended in the geometry of the measurement, so that the tuning rate is reduced significantly.

(2) The WGS device should be designed so that modulation of the resonant waveguide mode is as sensitive as possible to the change of the dielectric constant of the environment (refractive index of the superstrate in this work). In this consideration, a larger angle of incidence is also recommended. This can be concluded from Fig. 8(a) in section 3.5, when we fix the angle of incidence and look at the changing of the resonant waveguide mode as a function of the refractive index of the superstrate. This also shares the same requirement as described in (1).

Furthermore, the TM polarization exhibits better performance than TE for the considerations in both points (1) and (2), which favors the application in biosensor using waveguided gold nanowires.[11]

(3) Optical switching is another important application of the WGS device, where the superstrate or the waveguide layer should be made of photorefractive materials. The WGS device should be made as sensitive as possible to the change of the refractive indices of the superstrate and the waveguide layer. This requires a larger angle of incidence if the optical switch is based on a photorefractive superstrate, as can be concluded from Fig. 8(a). However, if comparing Fig. 5 and Fig. 8(a), we can find clearly that the optical response of the WGS is much more sensitive to the change of the refractive index of the waveguide layer than that of the superstrate. Therefore, in constructing an optical switch we’d better use photorefractive materials to fabricate the waveguide layer. This is more preferable than fabricating a photorefractive superstrate, which is really an important result that can be resolved from our theoretical work in this paper.