## Abstract

We present the nonpolarizing resonance properties of two types of one-dimensional (1D) guided-mode resonance (GMR) gratings consisting of the sinusoidal-profile grating substrate and the conformal dielectric thin films. The optimization with respect to the grating height and the phase of the conformal graded-index layer is important for the design of nonpolarizing type-I GMR gratings. The thin films design of the conformal step-index multilayer and the optimization with respect to the grating height are of two critical steps to obtain the nonpolarizing type-II GMR gratings. Both of the two types of nonpolarizing GMR gratings can be designed to support single-mode resonance and multimode resonance under normal incidence.

©2013 Optical Society of America

## 1. Introduction

Guided-mode resonance (GMR) of dielectric-layer waveguide gratings have been extensively studied for years [1–4] and continue to appear new aspects and applications such as biosensors, nanoplasmonics [5] and comb-like filters [6]. Recently, in order to extend its attributes in the field of narrowband filters which are insensitive to the polarization state, great effort has been made to investigate the nonpolarizing GMR structures which are usually based on either one-dimensional (1D) GMR gratings under conical incidence [7] or two-dimensional (2D) GMR gratings [4,8]. However, the complexity in the incident mountings and the difficulty of fabrication make them unfeasible in the extensive practical applications. In contrast, Alasaarela et al. [9] proposed a simple design of the single-layer rectangular profile 1D nonpolarizing GMR grating and the single-layer-coated sinusoidal profile 1D nonpolarizing GMR grating under normal incidence, respectively. Then, Saleem et al. [10] present the experimental realization of the single-layer rectangular profile and the single-layer-coated rectangular profile 1D nonpolarizing GMR gratings, respectively. As the single-layer-coated GMR structures can use the thermoplastic grating substrates which are replicable by using the nanoimprinting with the master stamp grating, this kind GMR structure will be the most desirable candidate for the mass production of low-cost nonpolarizing GMR filters. However, the resonance properties of the single-layer-coated sinusoidal or rectangular profile 1D nonpolarizing GMR gratings have yet to be improved because of the obvious discrepancy in the resonance linewidths of TE and TM polarization. In addition, much past research on nonpolarizing 1D GMR devices has focused on the thin periodic layers supporting only a single leaky mode. Multimode nonpolarizing 1D GMR gratings are seldom investigated.

In this work, we present the design of a conformal graded-index-layer sinusoidal-profile 1D GMR grating (Type-I) and a conformal step-index multilayer sinusoidal-profile 1D GMR grating (Type-II), which are both investigated to exhibit the single-mode nonpolarizing GMR and multimode nonpolarizing GMR properties under normal incidence. The numerical results indicate the potential application of the two types of GMR gratings for the narrowband nonpolarizing filters.

## 2. Model of the GMR gratings

The proposed 1D GMR grating with period of *p* and grating height of *h* is shown in Fig. 1
. The sinusoidal profile grating substrate is coated by the conformal dielectric films. The physical thickness of the thin films is represented by *T*. Graded-index layer is used for type-I GMR grating. The refractive index variation throughout the physical thickness of the layer is described by

*n*

_{ave}and Δ

*n*are the average index and the amplitude of the index variation, respectively,

*t*

_{0}is the physical thickness of one sinusoidal period in the layer,

*z*is the thickness variable and Φ is the phase at the grating substrate. Let

*m*be the number of the sinusoidal periods in the graded-index layer, the physical thickness of the film is

*T*=

*m*×

*t*

_{0}, consequently.

The type-II GMR gratings use the conformal step-index multilayer instead of graded-index layer coated on the grating substrate. Two kinds of materials with refractive index of *n*_{L} and *n*_{H} are used for the thin films. The physical thickness of each layer is *t*_{i}, where *i* represents the sequence of the thin films growing from the grating substrate. The refractive index of the first layer next to the substrate is *n*_{L}.

For the calculation, the incident media is air and the refractive index of the grating substrate is 1.516. SiO_{x}N_{y} is considered to be the most suitable material for the conformal dielectric films as its refractive index which is dependent on the deposition condition can vary form 1.52 to 1.92. The GMR grating is operated under normal incidence of a linearly polarized plane wave whose electric field vector is perpendicular (TE) or parallel (TM) to the grating lines.

## 3. Design of type-I GMR gratings

#### 3.1 Single-mode nonpolarizing type-I GMR gratings

We first demonstrate a theoretical design of a nonpolarizing type-I GMR grating possessing a single resonance peak. The initial structural parameters are *p* = 320 nm, *h* = 170 nm, *n _{ave}* = 1.713 and Δ

*n*= 0.15. Figure 2(a) shows the refractive index distribution of the graded-index layer with

*m*= 2 and

*T*= 321.7 nm. Optimization was performed for the remaining parameter Φ. Numerical calculation is based on the rigorous coupled-wave analysis method [11] using stack matrix methods to solve for the interlayer boundary conditions with self-developed codes. Figure 2(b) shows the relationship between Φ and the resonance wavelength. There are two cross points on the two curves which closely resemble the sinusoidal waveform. At the cross point Φ = 0.44π or Φ = 1.10π, the nonpolarizing GMR effect can be achieved. For example, the reflectance spectra of the GMR grating with Φ = 0.44π is calculated and shown in Fig. 3 . The resonance caused by the phase matching of the first diffraction order with the leaky waveguide mode occurs at 511.398 nm and 511.329 nm with the resonance linewidths evaluated by a full width at half maximum (FWHM) of 2.469 nm and 0.377 nm for TE and TM polarization, respectively. The two resonance wavelengths are close enough to produce the nonpolarizing effect. However, for incident wavelength slightly away from the resonance peak, the nonpolarizing effect will degenerate rapidly due to the discrepancy in the resonance linewidths.

In our situation, changing the value of Φ will leads to the slightly variation of the effective refractive index of the waveguide layer. As we know, the effective refractive index of the waveguide layer is different for TE and TM polarization [12]. According to the slab waveguide theory [1], solving the eigenvalue equations for TE and TM polarization will likely provide the same resonance wavelength as long as we properly construct the graded-index layer to match the requirement of the effective refractive index for TE and TM polarization simultaneously.

At the resonance wavelength of the nonpolarizing GMR grating, TE and TM polarizations have the same propagation constant of the leaky guided-mode. However, the light intensity distribution in the waveguide grating are different in the two cases. We employ the finite-difference time-domain (FDTD) method to perform the numerical simulation for the designed type-I GMR grating with a grid size of 0.2 nm in Fig. 4 . Periodic boundary conditions are used for the side boundaries. Perfectly matched layer are used at the top and bottom boundaries to absorb the outgoing electromagnetic wave. A time dependence of sinusoidal plane wave source is used to generate the incident field. The FDTD simulation demonstrates that the incident light of TE polarization is mostly concentrated in the central area of the conformal layer, while the light intensity of TM polarization is mainly concentrated in the convex area along the upper boundary of the conformal layer and the concave area along the lower boundary of the conformal layer forming the zigzag distribution.

The type-I GMR grating is difficult to model in FDTD because of the excessively large computational domain required to resolve the small geometrical feature and the slight variation of the refractive index. It should be note that, the grid spatial discretization in FDTD will divide the conformal graded-index layer into a large number of sufficiently thin conformal step-index layers. Due to this structural approximation in simulation, we can see weak transmitted light intensity in the substrate both for TE and TM polarization at the resonance wavelength in Fig. 4.

The deviation slightly from the theoretical design will cause the mismatch of the resonance wavelength Δλ defined by the resonance wavelength of TE polarization λ_{TE} minus the resonance wavelength of TM polarization λ_{TM}. Figure 5(a)
shows the sensitivity of the resonance wavelength versus the grating height *h*. The slopes are −0.038 and −0.107 for TE and TM resonance, respectively. Figure 5(b) shows the sensitivities of resonance wavelength and Δλ versus the grating period *p*. λ_{TE} and λ_{TM} almost coincide with each other while *p* is changed from 300 nm to 350 nm. If the grating period is set to be 335 nm, Δλ will tend to zero with resonance wavelength of 531.64 nm, which is the true sense of an nonpolarizing GMR effect. In order to check the feasibility of this GMR structure, we have calculated the relative variation needed on the grating parameters to entail Δλ = 1 nm [3]: grating height 5.9% and grating period 15.6% at least. Such tolerance requirements can be easily reached with a careful nanoimprinting calibration process.

Furthermore, according to Fig. 5(b), it will be easy to shift the nonpolarizing resonance wavelength just by changing the grating period and re-optimization. So, it is possible to design nonpolarizing GMR grating for a specified operating wavelength.

Actually, the design of single-mode nonpolarizing type-I GMR gratings is a multi-parameter problem. However, it is not necessary to run extensive optimization with all variables in order to find the best structural parameters resulting the expected effect. In order to accelerate the design process, first, the refractive index and the thickness of the conformal layer should be fixed. A temporary value in the range from 0 to 2π should been set for the parameter Φ for initial calculation. Then, the value of the grating period is set to approximately locate the resonance wavelength. After that, the optimization is performed for the grating height to make the resonance wavelengths get much closer leading to the quasi-nonpolarizing effect. The parameter Φ is optimized at last to obtain the exact nonpolarizing effect for the type-I GMR gratings.

#### 3.2 Multimode nonpolarizing type-I GMR gratings

The thickness of the graded-index layer should be increased to support multimode resonance [13]. The first way is by increasing the number *m* and keeping *t*_{0} fixed. The second way is by increasing *t*_{0} and keeping *m* fixed. Here, we chose the first way to design the type-I GMR grating possessing two resonance peaks. Initially, the parameters are set at *m* = 4, *T* = 643.3 nm, *n _{ave}* = 1.713, Δ

*n*= 0.15 and Φ = 0. After several calculations, the grating parameters are chosen to be

*p*= 340 nm,

*h*= 200 nm to locate the resonance peaks in the wavelength range form 520 nm to 580 nm and get the quasi-nonpolarizing effect. Then, the optimization is performed for the parameter Φ. Figure 6 gives out the relationship between Φ and the resonance wavelength. The first nonpolarizing resonance wavelength λ

_{1}is seen at Φ = 0.87π or Φ = 1.75π with the difference of the second resonance wavelength Δλ

_{2}at 0.088 nm and 0.025 nm, respectively. The second nonpolarizing resonance wavelength λ

_{2}is seen at Φ = 0.25π with Δλ

_{1}at 0.421 nm.

Since it is difficult to realize the exact nonpolarizing GMR effect at two resonance peaks simultaneously, we have to choose the value of Φ which will give the minimum difference (Δλ_{1}^{2} + Δλ_{2}^{2})* _{min}*. According to this criterion, Fig. 7
shows the calculated reflectance spectra of the nonpolarizing GMR grating with Φ = 1.75π. The wavelength location and FWHM of each resonance peak are listed in Table 1
. We can see the mismatch of the resonance linewidths especially of the first resonance peak.

Figure 8
shows the sensitivities of the resonance wavelength and the difference Δλ of the designed multimode type-I GMR grating versus the grating period *p*. The slopes of λ_{1} and λ_{2} are 1.35 and 1.56, respectively indicating that the spacing of the two resonance peaks is gradually enlarged as *p* is increasing. Due to the small value of Δλ_{1} and Δλ_{2}, it is possible to shift both λ_{1} and λ_{2} by changing the grating period and then get the nonpolarizing effect again by re-optimization with respect to the grating height and the parameter Φ.

Multimode type-I GMR grating possessing three resonance peaks can also be achieved by changing *m* to be 6 and *T* to be about 965 nm. However, it is really hard to get the nonpolarizing effect at the three resonance wavelengths simultaneously during the last optimization with respect to the parameter Φ. More variable parameters are needed to get the expected effect. We find that if *m* is further increased while *T* remains basically unchanged, which requires a corresponding decrease in the value of *t*_{0}, the three resonance peaks will still exist and the resonance wavelengths can be slightly shifted within different magnitudes for TE and TM polarization. This is very helpful to obtain the nonpolarizing multimode GMR effect. For example, Fig. 9
shows the calculated reflectance spectra of the optimized type-I GMR grating designed with parameters set at *p* = 320 nm, *h* = 206 nm, *m* = 9, *t* = 955.7 nm, *n _{ave}* = 1.713, Δ

*n*= 0.152 and Φ = 0.75π. Three nonpolarizing resonance peaks are almost achieved simultaneously. The wavelength and FWHM of each resonance peak are listed in Table 2 .

Based on the analysis presented above, we can see that the design of a multimode nonpolarizing type-I GMR grating is a little different from that of a single-mode nonpolarizing type-I GMR grating. First, the thickness of the graded-index layer should be increased to get the right number of resonance peaks. Second, the optimization not only with respect to the grating height and parameter Φ but also with respect to parameter *m* is needed to realize the multimode nonpolarizing resonance.

It is obvious that the precise control of the refractive index during the graded-index layer deposition on the grating is the most important for the fabrication of type-I GMR grating. Plasma enhanced chemical vapor deposition (PECVD) [14] and reactive sputtering [15] which are of increasing interest for the deposition of high-precision optical coatings can been used to fabricate the graded-index layer for the type-I GMR grating. Before the running of the deposition process, the initial flow of the reactive gases should be carefully set to produce the correct refractive index determined by the parameter Φ. Then, the refractive index of the deposited material can be simply controlled by modifying the flow of the reactive gases. The conformal coating can be realized by using reactive magnetron sputtering [16] or dual ion beam sputtering [17] which can carry out the deposition on the corrugated substrate without shadowing effect. The inherently stable deposition process of sputtering can lead to the precise thickness control by time. So, we believe that the fabrication of the type-I GMR gratings is technically feasible.

## 4. Design of type-II GMR gratings

The nonpolarizing type-I GMR gratings can be constructed easily. However, its practical application is limited to the minus filters for the ultra-narrowband laser due to the obvious discrepancy in the resonance linewidths as mentioned above. In order to match the resonance linewidths to produce the high quality GMR filters, we use conformal step-index multilayer instead of graded-index layer coated on the sinusoidal profile grating substrate to construct the type-II GMR grating.

In order to design the nonpolarizing type-II GMR grating efficiently, the refractive index and the thickness *T* of the conformal layers are fixed. Initially, all of the step-index layers have equal thickness. First, the value of the grating period is set to locate the resonance peaks in the wavelength range of interest and a temporary value should been set for the grating height, for example, we use *h* = 0.625*p* for initial calculation. Then, thin films design is carried out for the step-index multilayer for the purpose of matching the resonance linewidths as well as possible. Meanwhile, the GMR resonance is still polarization-dependent. So, after that, the grating height becomes variable to match the resonance wavelength to obtain the nonpolarizing effect. Of course, the optimization with respect to the grating height may introduce slight mismatch of the resonance linewidths simultaneously. If this happens, it will be necessary to perform the thin films design again to improve the result. This is the most effective way we have found to design the nonpolarizing type-II GMR gratings.

#### 4.1 Single-mode nonpolarizing type-II GMR gratings

The theoretical design of a nonpolarizing type-II GMR grating possessing a single resonance peak is demonstrated. The initial structural parameters are *p* = 320 nm, *h* = 200 nm, *n*_{L} = 1.650, *n*_{H} = 1.781 and *T* = 302 nm. There are two conformal layers on the grating substrate. The thickness of each layer should be carefully determined, because the resonance properties especially the linewidths are directly influenced by the thin films structure. Figure 10(a)
shows the relationship between resonance properties and the thickness of the first layer *t*_{1}. Taking into account of the resonance lineshapes, we choose the thickness parameters: *t*_{1} = 120 nm and *t*_{2} = 182 nm with a small difference of 0.4 nm in the resonance linewidths of TE and TM polarization. However, the resonance wavelengths are still separated by 3.7 nm. This GMR structure needs to be further optimized by adjusting the grating height to tune the dispersion relations of the leaky guided modes to obtain the nonpolarizing effect [9]. Figure 10(b) shows that TE and TM resonance peaks will have the same central wavelength of 511 nm with a smaller difference of 0.3 nm in the resonance linewidths when *h* is 154 nm. The calculated reflectance spectra of this type-II GMR grating are shown in Fig. 11
. The Q-factor defined in terms of the ratio of the resonance wavelength to the resonance linewidth is 121.7 and 131.0 for TE and TM polarization, respectively. Such kind of nonpolarizing GMR resonance with well-matched linewidth under normal incidence can also be realized with an existing rectangular profile 1D GMR grating with Q-factor only less than 30 [10].

We can also shift the resonance wavelength of the type-II GMR grating by changing the grating period. However, the slight mismatch of the resonance wavelengths and linewidths will happen simultaneously. As a result, not only the thin films design but also the optimization of grating height should be performed again to obtain the good nonpolarizing filtering effect for the new wavelength.

By calculating the reflectance of the type-II GMR grating designed in Fig. 11 as a function of wavelength *λ* and incident angle *θ*, we plot the dispersion relations of the leaky guided modes represented by the peak reflectance loci of TE and TM polarization in Fig. 12
where *k*_{x} = 2*π*sin*θ*/*λ* is the tangential component of incident wave vector and *K* = 2*π*/*p* is the grating momentum. There is a cross point of the dispersion curves of TE and TM polarization at *λ* = 511 nm and *k*_{x} = 0.

If the incidence angle deviates from normal, the resonance will be polarization-dependent and very narrow side lobes will emerge. According to Fig. 12, as the incidence angle shifts from normal to 0.1° off normal, the main resonant peak of TE polarization slightly redshifts 0.05 nm and the main resonant peak of TM polarization slightly blueshifts 0.02 nm leading to Δλ = 0.07 nm. By Comparing with the corresponding value of 0.2 nm shift obtained in nonpolarizing 2D GMR grating reported in Ref [8], this type-II 1D GMR grating is less sensitive to the incidence angle. The tolerance of the incidence angle will extend to 0.41° if the limitation of Δλ is 1 nm for the designed nonpolarizing type-II GMR grating.

The FDTD simulation with a grid size of 0.5 nm is also performed for the type-II GMR grating in Fig. 13 . We can find that the incident light of TE polarization is mostly concentrated in the first conformal layer next to the grating substrate, while the light intensity of TM polarization is mainly distributed along the interface of two conformal layers.

#### 4.2 Multimode nonpolarizing type-II GMR gratings

The multimode type-II GMR grating possessing two resonance peaks is constructed by the sinusoidal profile grating substrate and four conformal layers with low refractive index of *n*_{L} = 1.674 for the first and third layer, high refractive index of *n*_{H} = 1.753 for the second and fourth layer, and total thickness *T* = 664 nm. Following the same design procedure, we obtained the optimal structural parameters: *p* = 316 nm, *h* = 200 nm, *t*_{1} = 187 nm, *t*_{2} = 288, *t*_{3} = 88 nm and *t*_{4} = 101. The calculated reflectance spectra are shown in Fig. 14
and the specifications of each resonance peak are listed in Table 3
. Figure 15
shows the calculated dispersion relations of the leaky guided modes. We can find two cross points of the dispersion curves of TE and TM polarization at normal incidence.

In principle, it is possible to design multimode nonpolarizing type-II GMR gratings possessing more resonance peaks by following the same procedure which may involve the repetition of the conformal thin films design and the adjustment of the grating height step by step. Of course, we can also carry out the two steps simultaneously which may extend the design space. However, this multi-parameter optimization problem will lead to extensive and time-consuming calculation work. So, taking into account the efficiency, we always set the other structural parameters to be invariable during the optimization with respect to the grating height.

As far as we know, the multimode nonpolarizing 2D GMR gratings have yet to be designed. The proposed multimode nonpolarizing type-II GMR gratings can be used in substitution for the traditional multi-notch filters [18] which contain hundreds of layers with total thickness of more than 50μm in Raman spectroscopy and laser-based fluorescence instruments.

The type-II GMR gratings demonstrate better nonpolarizing filtering effect than the type-I GMR gratings. However, the sidebands are still not suppressed and the resonance lineshapes are also out of control. We are taking further study to improve the resonance features both for the type-I and type-II GMR gratings (for example, by adding quintic layer and anti-reflection layer to suppress the sidebands and change the lineshapes).

Note that, due to the unknown effective refractive index of the waveguide layer including the conformal thin films, the resonance wavelengths of the type-I or type-II GMR gratings are hardly determined at the very beginning of the design procedure by analytical methods. In case of multimode nonpolarizing resonance, we find that the optimization with respect to the structural parameters cannot modify the spacing of two adjacent resonance peaks significantly. As a result, it is still difficult to design a nonpolarizing GMR grating which can selectively address several fixed resonance wavelengths.

## 5. Conclusion

We have presented the design of two types of 1D sinusoidal profile nonpolarizing GMR gratings with conformal dielectric thin films. Both of them exhibit the single-mode nonpolarizing GMR and multimode nonpolarizing GMR properties under normal incidence. The optimization with respect to the grating height and the phase of the conformal graded-index layer plays an important role in the design of nonpolarizing type-I GMR gratings. The thin films design of the conformal step-index multilayer and the optimization with respect to the grating height are of two critical steps to obtain the nonpolarizing type-II GMR gratings. We believe the presented results provide great potential for the applications based on the GMR gratings with conformal thin films and could be helpful for the development of high quality narrowband filters and other novel photonic devices.

## Acknowledgments

We acknowledge the support by the National Natural Science Foundation of China (NSFC) (project 61108037), National Science and Technology Programme of China (project 2012BAI08B04), Science and Technology Department of Zhejiang Province (project 2010C03002), and Eye Hospital Innovation Guide Foundation (project YNCX201003).

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