Theoretical investigation of hierarchical sub-wavelength photonic structures fabricated using high-order waveguide-mode interference lithograph^*

Micro- and nanostructures of various shapes and sizes have tremendous prospects for applications in diverse fields, including physics,^[1–3] material science,^{[4, 5]} chemistry,^{[6, 7]} and medical science.^[8] One-dimensional sub-wavelength gratings can be used to enhance Raman scattering intensity.^[9] Patterning of TiO₂ ^[10] nanostructures, for which geometries and spacings can be precisely controlled, may play an important role in the improvement of sensors for gas detection. Different methods^[11] have been used to fabricate micro- and nanostructures for research and applications. Electron beam lithography,^{[12, 13]} atomic force microscope lithography,^[14] UV photolithography,^[15] and x-ray lithography^[16] are the most commonly used fabrication methods. These methods all present disadvantages, such as complicated techniques, high costs, and low yields.

Recently, Radwanul Hasan Siddique^[17] et al. demonstrated the fabrication of hierarchical photonic nanostructures, which utilized the laser interference pattern in the vertical direction in addition to the conventional horizontal one. They found that the structural color of the final photonic structure can be influenced directly by various factors, including the period of the vertical standing wave and the photoresist material. Although they demonstrated that the method has an advantage for photonic nanostructures with nanoscale elements in the vertical direction, the period of the photonic nanostructures was not sub-wavelength in the horizontal direction.

Here, we present a theoretical approach showing that high-order waveguide-mode interference lithography can fabricate hierarchical sub-wavelength photonic structures (HSPS) in both the vertical and horizontal directions. Furthermore, this method supports fabrication of sub-wavelength photonic structures with different periods and layers. This HSPS nanolithography method is simple, low-cost, and maskless.

Figure 1(a) shows a schematic of the lithographic system. Here, the system configuration, which is similar to one that we reported previously, supports fabrication of a sub-wavelength grating by utilizing the low-order waveguide mode interference lithography technique^{[18, 19]} and is based on an asymmetric metal-cladding dielectric waveguide structure. An index-matching oil is used to connect the glass substrate and prism. The refractive index (RI) of the index-matching oil and the glass substrate are similar to the RI of the prism. A laser beam is split into two identical beams with a beam splitter. The resulting beams then are reflected into the prism using two mirrors. When the two laser beams irradiate the sample at an angle θ_m at which high-order waveguide modes are excited, two waveguide modes are generated. In the photoresist layer, one mode propagates to the left, and another propagates to the right. The interference between the two waveguide modes creates standing waves, which are used to inscribe HSPS. The photoresist layer employs a positive photoresist. After the exposure and developing processes, HSPS are fabricated. Figure 1(b) shows the enlarged schematic view of the inscribed HSPS inscribed; the x–y plane lies along the interface of the Ag and photoresist layer, and the z axis points to the air from the photoresist.

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The eigenmode equation of the asymmetric metal-cladding dielectric waveguide structure can be expressed as^[20]

$$\begin{eqnarray}&&{\kappa }_{2}d=m\pi +\arctan \left({\rho }_{1}\displaystyle \frac{{\alpha }_{3}}{{\kappa }_{2}}\right)+\arctan \left({\rho }_{2}\displaystyle \frac{{\alpha }_{1}}{{\kappa }_{2}}\right),\end{eqnarray} \tag{ 1 }$$

where m is the mode-order index, d is the thickness of the photoresist, and κ₂, α₁, and α₃ can be expressed as

$$\begin{eqnarray}&&\left\{\begin{array}{l}{\kappa }_{2}={\left({k}_{0}^{2}{\varepsilon }_{2}-{\beta }^{2}\right)}^{1/2},\\ {\alpha }_{1}={\left({\beta }^{2}-{k}_{0}^{2}{\varepsilon }_{1}\right)}^{1/2},\\ {\alpha }_{3}={\left({\beta }^{2}-{k}_{0}^{2}{\varepsilon }_{3}\right)}^{1/2}.\end{array}\right.\end{eqnarray} \tag{ 2 }$$

Here, β is the propagation constant and ${k}_{0}=2\pi /{\lambda }_{0}$ is the wavenumber corresponding to wavelength λ₀ in a vacuum. Furthermore, ρ₁ and ρ₂ are given by

$$\begin{eqnarray}{\rho }_{1}=\left\{\begin{array}{ll}1, & {\rm{for}}\,{\rm{TE}}\,{\rm{modes}},\\ {\varepsilon }_{2}/{\varepsilon }_{3}, & {\rm{for}}\,{\rm{TM}}\,{\rm{modes}},\end{array}\right.\end{eqnarray} \tag{ 3 }$$

and

$$\begin{eqnarray}{\rho }_{2}=\left\{\begin{array}{ll}1, & {\rm{for}}\,{\rm{TE}}\,{\rm{modes}},\\ {\varepsilon }_{2}/{\varepsilon }_{1}, & {\rm{for}}\,{\rm{TM}}\,{\rm{modes}},\end{array}\right.\end{eqnarray} \tag{ 4 }$$

respectively, where ε₁ is the dielectric constant of Ag. Because the imaginary part of ε₁ is much less than its real part,^{[21, 22]} the imaginary part, for the sake of convenience, can be neglected in the analysis. Similarly, ε₂ and ε₃ are the dielectric constants of the photoresist and air, respectively. In this structure, the allowed range of the effective RI for the waveguide modes is

$$\begin{eqnarray}&&\sqrt{{\varepsilon }_{3}}\lt {\beta }_{m}/{k}_{0}\lt \sqrt{{\varepsilon }_{2}}.\end{eqnarray} \tag{ 5 }$$

Moreover, the coupling excitation of high-order waveguide modes should satisfy the wave vector-matching condition, which is defined by

$$\begin{eqnarray}&&{k}_{0}{n}_{0}\sin {\theta }_{m}={\beta }_{m},\end{eqnarray} \tag{ 6 }$$

where n₀ is the prism RI and θ_m is the resonance angle of the m-order waveguide mode. Furthermore,

$$\begin{eqnarray}&&{n}_{0}\sin {\theta }_{m}={n}_{\mathrm{eff}}=\sqrt{{\varepsilon }_{0}}\sin {\theta }_{m},\end{eqnarray} \tag{ 7 }$$

where ${n}_{\mathrm{eff}}$ is the effective RI of the waveguide modes.

When one uses the above-mentioned asymmetric metal-cladding dielectric waveguide structure, the period of HSPS inscribed in the x direction can be expressed as

$$\begin{eqnarray}&&{\rm{\Delta }}x=\displaystyle \frac{{\lambda }_{0}}{2{n}_{\mathrm{eff}}}=\displaystyle \frac{{\lambda }_{0}}{2\sqrt{{\varepsilon }_{0}}\sin {\theta }_{m}}=\displaystyle \frac{\pi }{{\beta }_{m}}.\end{eqnarray} \tag{ 8 }$$

The number layers of HSPS, which depends on the order of the exciting waveguide modes, is determined in the z direction. Theoretical analysis of the dispersion curve and the distribution of the interference field of high-order waveguide modes by the finite-element method demonstrates that the layer number of HSPS is equal to the mode-order index plus 1, that is $m+1$.

Equation (8) shows that the HSPS period is determined by λ₀ of the incident light, the prism RI, and θ_m . Combining Eq. (1) through Eq. (6), θ_m can be determined from the dispersion curve, which can be calculated by using relative parameters, namely, the waveguide mode index, the incident light wavelength, and the RI and thickness of both the metal and photoresist. In this study, we used a 442-nm laser as the exciting light source and an Ag film as the metal layer. The dielectric constant of Ag is $-5.7018+0.7514{\rm{i}}$ ^[23] at 442 nm. A 45-nm-thick Ag film was considered in the calculations.

When the prism RI is 1.6 and the photoresist RI and thickness are 1.7 and 950 nm, respectively, the θ_m values of the TE₅ and TM ${}_{5}$ waveguide modes are 45° and 50°, respectively. The periods of HSPS inscribed by the TE₅ and TM₅ waveguide modes are 196 nm and 180 nm, respectively. The calculated results of the periods for HSPS align with the results obtained from the simulation. Figures 2(a) and 2(b) show the simulated electric field $| {E}_{y}|$ of the TE₅ waveguide-mode interference and the magnetic-field component $| {H}_{y}|$ of the TM₅ waveguide-mode interference in the multilayer film, respectively.

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3.1. Photoresist thickness and RI

The proposed method can fabricate HSPS with various periods. In pursuit of this outcome, we investigated in detail the various dependences of HSPS characteristics on the photoresist thickness and RI, and on the waveguide modes. As shown in Eq. (8), the HSPS period is inversely proportional to β_m . Therefore, we first studied the dispersion curves of TE₂ modes using different RIs for the photoresist while maintaining the prism RI at 1.7.

Figure 3(a) shows the calculated dispersion curve. With the same photoresist RI, β_m is larger for thicker photoresists, yielding smaller HSPS periods. With the same photoresist thickness, the larger the photoresist RI, the larger the β_m value and the smaller the HSPS period. Figure 3(b) exhibits the relationships among the HSPS period, photoresist thickness, and RI. The HSPS period decreases as the photoresist thickness increases. Figures 4(a) and 4(b) show $| {E}_{y}|$ for the TE₂ waveguide-mode interference for photoresist thicknesses of 345 nm and 500 nm, respectively, where the prism and photoresist RIs are 1.7 and 1.8, respectively. HSPS periods are 221 nm and 159 nm when the photoresist thicknesses are 345 nm and 500 nm, respectively. When the prism RI is 1.7 and the photoresist thickness and RI are 500 nm and 1.5, respectively, the HSPS period is 213 nm. This HSPS period is larger than when the RI is 1.8. Figure 4(c) shows $| {E}_{y}|$ for TE₂ waveguide-mode interference when the photoresist RI and thickness are 1.5 and 500 nm, respectively. The results clearly demonstrate that a larger photoresist RI will yield an inscription of HSPS with a smaller period, as shown in Fig. 3(b). The TM modes also follow the same trend. Thus, HSPS with various periods can be fabricated by varying the photoresist thickness or RI.

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3.2. Waveguide modes

The value of β_m also depends on the polarization and m. Figure 5(a) shows the dispersion curves for different waveguide modes when the photoresist RI is 1.7. If the photoresist thickness and RI are constant, β_m is larger for m-order TM modes than for same-order TE modes. Therefore, the period of HSPS inscribed using m-order TM modes is smaller than the HSPS period inscribed using TE modes. For the same polarization modes, when the photoresist thickness and RI are identical, β_m is larger for low-order waveguide modes than for high-order ones. This characteristic implies that the HSPS period is smaller in the former case than in the latter. Figure 5(b) shows the relationship between the HSPS period and the photoresist thickness for different waveguide modes with prism RI 1.85. Figure 6(a) shows $| {H}_{y}|$ for TM₃-mode interference in the multilayer film for an HSPS period of 159 nm. Figures 6(b) and 6(c) present $| {E}_{y}|$ for TE₃- and TE₄-mode interference, which can inscribe HSPS with periods of 173 nm and 221 nm, respectively. The photoresist thickness was considered to be 700 nm in the simulation. The values of RI for the photoresist and prism are identical to those shown in Fig. 5(b). These results indicate the possibility of selecting appropriate waveguide modes to write HSPS with a required period and number of layers. Thus, HSPS with various periods and different numbers of layers can be fabricated.

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We theoretically demonstrated hierarchical sub-wavelength photonic structures fabricated using an asymmetric metal-cladding dielectric waveguide structure. HSPS with various periods and layers can be fabricated using this method. Furthermore, HSPS can be controlled effectively using the excited waveguides modes and photoresist parameters, in particular, the photoresist thickness, which can be easily changed when performing experiments. Our analysis of the interference field distribution of the high-order waveguide-modes indicates that a positive photoresist should be used to fabricate hierarchical sub-wavelength photonic structures for experimental uses. The HSPS fabrication method presented here is both simple and low-cost; it provides a possible experimental method for fabricating hierarchical sub-wavelength photonic structures.