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Abstract
An achromatic response is required in most optical systems for wideband and straightforward configurations. The chromatic response of the optical system depends on the optical dispersion of the elements in the system. Here we study the dispersion of subwavelength grating (SWG) known to have a form birefringence. The birefringence of SWG was numerically analyzed with Bloch wave analysis (BWA) and finite element method (FEM). The sandwiched SWG with two identical substrates was studied for practical applications. We successfully demonstrated the negative dispersion form birefringence of SWG with an optimal duty cycle. This extraordinary dispersion was also shown considering the intrinsic dispersion of materials. Dispersion- and the angular response were in a tradeoff relationship while they depended on periodicity. The optical interference between the grating and the substrates can be eliminated by controlling the duty cycle. Our analysis offers optimal SWG with achromatic birefringence and high transparency, promising in the widespread applications of polarization control devices.
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1. Introduction
A grating with a periodicity (P) smaller than the wavelength (λ) of incident light (i.e., P < λ) is called subwavelength grating (SWG) [1–3]. SWG has a forbidden band (bandgap) where electromagnetic (EM) waves cannot exist with small high-order diffraction widely used in many optical applications such as antireflection layers, spectral filter, optical absorber, and waveguide [4–7]. Meanwhile, the light propagating through the periodically layered structures could have retardation, as in birefringent materials. When the light propagates through the birefringent medium, the light splits into two orthogonally polarized components: ordinary (o)- and extraordinary (e)-rays. The effective refractive index (RI) of o- and e-rays are called ordinary (no) and extraordinary (ne) RIs, respectively. Birefringence is defined as Δn≡ne-no, and retardation can be defined as Re=Δn.d, where d is the thickness of the medium. The birefringence originated from the geometrical structure is known as form birefringence and can be described by the effective medium theory (EMT) [8–11].
The dispersion of the birefringence results in the chromatic response of optical systems. In most optical applications, the chromatic response degrades the performance and makes the system complicated. Thus, the achromatic retarder is more useful for practical applications. In order to make the achromatic retarder with constant phase retardation (Γ=Re·2π/λ) over the spectral range, the birefringence should show negative dispersion (ND) where the magnitude of birefringence (|Δn|) increases with longer λ. However, birefringent materials in nature represent a positive dispersion (PD) where the |Δn| decreases with longer λ. Synthetic organic materials were studied to fabricate ND retardation films [12–17]. For example, copolymers or polymer-blends of disk- and rod-like monomers, and the H-, T-, or X- shaped reactive mesogens [12–14,16]. The multiple-layered inorganic materials (e.g., birefringent crystals) can work as an ND retarder [18–22]. On the other hand, the crystalline retarders are fragile, requiring a multi-step manufacturing process and elaborate design [19,22]. The ND retarder can be also made using the concept of form birefringence, but it is not known well compared to other methods [12–22].
The composition of synthetic organic materials (copolymers or reactive mesogens) is the most well-known method, although such works lack theoretical analysis and background. Most of the results are based on empirical results without a description of the working mechanism. Therefore, they are hard to reproduce or develop. The achromatic retarders made of inorganic materials are durable and easy to analyze. Conventional inorganic achromatic retarders required a multi-step manufacturing process, elaborate design, or had limitations of materials. Our proposed SWG has a simple one-dimensional lattice structure and has a well-established manufacturing method. Any transparent materials (e.g., glass, SiO2, ZnO) can be used in the proposed design. The dispersion engineering of SWG is guided in detail, supported by BWA and FEM analysis.
The form birefringence can be controlled by tailoring the geometrical parameters such as periodicity, groove depths, and duty cycle [23–25]. Most of the previous studies focused on the form birefringence of the periodic structure with significantly smaller P than λ (P≪λ). In such a condition, the dispersion of birefringence is not dependent on the structure. Limited numbers of research groups have reported ND form birefringence of SWG with large P, but they have not suggested a deep analysis about the factors affecting birefringence dispersion of SWGs [10,26,27]. Therefore, it is important to understand the role of geometrical parameters on the dispersion of form birefringence. In this work, the form birefringence of SWG was studied with variable geometric parameters, such as the periodicity P, groove depth d, materials, duty cycle α, and RIs. Furthermore, the dependence of form birefringence on substrates’ RIs and incident angle were analyzed for practical applications. We employ Bloch waves analysis (BWA) [1,9] and numerical simulation based on the finite element method (FEM) to investigate the form birefringence of SWG. The FEM simulation was done by using Wave Optics Module in COMSOL Multiphysics (COMSOL) [28].
Fabrication methods for the SWG are well established, such as e-beam lithography (EBL), laser interference lithography (LIL), nanoimprinting lithography (NIL), focused ion beam (FIB), and deep-ultraviolet (DUV) lithography fabrication techniques [29–34]. For instance, the SWG was carried out by the e-beam lithography (EBL) process, with the pitch and duty cycle being about 270 nm and 30%, respectively [29,30]. For LIL, we can obtain 180 nm-pitch metallic gratings successfully fabricated on a silicon wafer [31], while a 60 nm half-pitch can be achieved for NIL [32]. In addition, FIB [33] and DUV [34] are valuable methods for SWG structures with ∼200 nm pitch.
2. Theoretical background
2.1 Theory of electromagnetic Bloch waves
The grating structure is a periodic medium and can be analyzed using BWA [1]. The transverse electric (TE) and transverse magnetic (TM) waves in the periodic mediums are given from the solution of the Bloch waves. Then, retardation or form birefringence can be obtained by analyzing the TE and TM waves. The eigenmodes of EM waves propagating through the periodic mediums are Bloch waves, given by,
where ω is the angular frequency, t is time, β and K are x and y components of the Bloch wave vector. The dispersion relation between ω, β and K for of TE and TM waves is given by [1]We emphasize that most previous studies focused on the form birefringence of the periodic structure with significantly smaller P than λ (P≪λ). The small P results in smaller high-order terms of the Bloch waves and weaker dispersion. As a result, there is less room to control the dispersion of the form birefringence [25,35]. In small periodicity condition (P≪λ), Eqs. (2) & (3) are approximated as a zeroth-order term then RIs can be written as [25],
Fig. 1. Schematic of SWG sandwiched between the substrates. a and b correspond to the width of the less and more dense medium with refractive indices n1 and n2.
According to Eqs. (6) and (7), ne and no depend on the width of the alternating layers a and b and intrinsic materials dispersion of n1 and n2. However, as the grating periodicity increases (P < λ/2), the higher-order terms in the expanding transcendental functions in Eq. (2) are no longer negligible and Eqs. (6), and (7) are no longer valid [3]. Note that only SWG was considered in BWA and the presence of substrates was ignored.
In this paper, the form birefringence of SWG with large P will be analyzed and verified by the BWA and FEM, respectively, in section 3.1. At the same time, we will show the influence of the interference on the performance of the SWG structure in section 3.2 by using FEM. The BWA on the dispersion of form birefringence, which consists of the contribution of the intrinsic material dispersion, will be addressed in section 3.3. The angular response of the SWG structure and conclusions are presented in sections 3.4 and 4, respectively.
2.2 Finite element method (FEM)
In our previous works [6,36], FEM simulations were performed to calculate the optical properties of light through a periodic unit cell structure. Our simulation approach is a full-field optical propagation analysis in a periodic structure based on the electromagnetic field’s theory, expressed as follows [37].
Here, the distance of the optical propagation between the sourced and destinated points are denoted as rsrc and rdst. E and H are electric and magnetic fields traveling in the medium, respectively. ɛr is the complex relative permittivity of the material, given by ɛr =(n-ik)2 with the real and imaginary parts n and k of the refractive index, kF and k0 are the wave vector and wavenumber, respectively.
3. Results and discussions
3.1 Fundamental analysis with Bloch waves, FEM, and Cauchy’s curve fitting
In this section, we investigated the degree of agreement between BWA and FEM in the dispersion of form birefringence given various geometrical parameters (P, α, d) and RI (n2). Here, α is the ratio between b and P (α=b/P). We assume that the geometrical parameters are P = 220 nm, α=0.6 (b = 132 nm), and d = 1 µm, while RIs of component layers are ns = 1.8, n1 = 1.0, and n2 = 2.0. Figure 2 shows the birefringence dispersion of SWG calculated by BWA and FEM, where the lines are fit by Cauchy’s equation [38]. Although the constituent materials intrinsically have PD, the calculated birefringence showed ND whose absolute value is increasing with longer λ. The form birefringence calculated by BWA and FEM had an excellent agreement with each other.
Fig. 2. Comparison of the birefringence dispersion of SWG using the theory of electromagnetic Bloch waves (black square symbol) and FEM (red circular symbol). The lines correspond to the fitted results by Cauchy’s equation.
Both results using BWA and FEM show that |Δn| is increasing with longer λ (i.e., ND). We employed Cauchy’s equations to evaluate the dispersion property of the form birefringence. Cauchy’s equation is generally used to define the dispersive relationship between the RI (or Δn) and λ, which is given by,
where A, B, C are the coefficients determined by approximating the data. This dispersion model is derived from an empirical model of transparent materials and is known to fit well with ND and PD, as shown in Fig. 2. We use the three terms form of the equation to fit sufficiently the birefringence dispersion curve. Cauchy’s equations with only the first two terms are insufficient to express form birefringence, resulting in deviation at the beginning and end of the wavelength range. In other words, the B coefficient of the second term best represents the dispersion property near the center. Thus, we focused on exploiting B to verify the dispersion property of the form birefringence in SWG.The dependence of form birefringence on the nominal wavelength (λ0) was studied [11,39]. The form birefringence at three different nominal wavelengths (λ0=450, 550, and 650 nm) were calculated with three different structures P = 0.4λ0, 0.5 λ0, and 0.6 λ0 using Bloch waves theory. Figure 3 shows the calculated form birefringence in the space of normalized wavelength (λ/λ0). The magnitude and dispersion of the form birefringence are identical regardless of nominal wavelength λ0=450 (line), 550 (solid marks), and 650 nm (empty marks), with given P. Note that form birefringence of λ0=550 nm (solid marks) and 650 nm (empty marks) in the full spectrum was omitted at some wavelengths for better visualization. Therefore, the form birefringence's spectral properties (magnitude and dispersion) are independent of λ0. This result means that there would be a universal design that can be used for any working wavelength region. λ0 was chosen as 550 nm in this study.
Fig. 3. Δn (λ) vs. normalized wavelength λ/λ0 when P is 0.4λ0 (black), 0.5λ0 (red), and 0.6λ0 (blue). The solid line data were obtained with the nominal wavelength λ0 at 450 nm, while the solid square and empty square data were obtained with the nominal wavelengths λ0 at 550 and 650 nm, respectively. This result means independence of the form birefringence (Δn) to nominal wavelengths (λ0).
Figure 4(a) shows the form birefringence at λ0 (Δn(λ0)) with various α and P. In small periodicity conditions (P/λ0<0.3), the magnitude of Δn(λ0) increases with the increase of α until α=0.5 and gradually decreases when α > 0.5. On the other hand, Δn(λ0) depends on P rather than α in extensive periodicity conditions (P/λ0 >0.3). The |Δn(λ0)| increases with the increasing of P and represents the maximum 0.36 at P/λ0 = 0.65, α=0.27. Therefore, the periodicity strongly affects the magnitude of the form birefringence. Meanwhile, P is more negligible on |Δn(λ0)| when α is far away from 0.27.
Fig. 4. (a) Contour plot of Δn(λ0) as a function of duty cycle α and P/λ0. (b) Δn vs. α for several n2 values with n1= 1.0. (c) Contour plot of the coefficient B in Cauchy’s equation vs. α and P/λ0 (n2 = 2.0). (d) Γ vs. λ/λ0 for various d (P/λ0 and α are 0.4 and 0.8, respectively).
Figure 4(b) shows a plot versus α of Δn(λ0) for several n2 values with n1 = 1.0. It is observed that the magnitudes of birefringence monotonically increase when the refractive index of n2 increases. The |Δn(λ0)| showed maximum at α=0.32 given n2 = 3.0, and α showing the maximum was moved to greater value with decreasing n2.
Figure 4(c) shows the coefficient B of the form birefringence with various α and P/λ0. The coefficient B in Cauchy’s equation [Eq. (1)1] was used to describe the dispersion of the form birefringence. Because SWG has a negative birefringence (Δn < 0), Δn increases with negative B and decreases with positive B. Thus, |Δn| of SWG increases with longer λ (ND) when B is positive. The dispersion of form birefringence is small when P is much smaller than λ0 (P/λ0 <0.2), as shown in many previous studies [40–42]. On the other hand, the ND which is represented by B becomes greater with increasing P in longer periodicity conditions (P/λ0 >0.2). It is also observed that B becomes positive, i.e., ND of Δn could be achieved in the ranges of duty cycles α ≥0.5 while PD is obtained with α <0.5. We also examined the coefficient B of the form birefringence with various α and P/λ0 when n2 increases. Further details of the results can be found in Fig. S1 of Supplement 1.
The thickness d may have errors or variances from the target d during fabrication. In BWA, d is assumed to be infinite, and the retardation of SWG is independent of d. Therefore, the magnitude of retardation is directly proportional to d. d should be maintained with a reliable range to obtain the SWG with targeted retardation. Meanwhile, the unwanted error of retardation might occur in the finite d (e.g., FEM). Figure 4(d) shows Γ for several d of 1.5, 2.0, and 2.5 µm. As a result, the shifts of Γ increased at long wavelengths and the maximum was 0.54% at a normalized wavelength of 1.45. Therefore, the effect of d is negligible even in practical conditions. For practical applications, a stack of multiple gratings may be preferred. A sandwiched SWG can be served as a unit cell for this solution. In both solutions (thick or stacking), the refraction between the SWG and the substrates is not desired. The optimization process can suppress the reflection in substrate/SWG or SWG/substrate interfaces. In detail, we will discuss this in the next section.
3.2 Influence of the interference effect on the performance of the SWG structure
In order to prevent contamination and deformation of the nanostructures, bottom and top substrates are desirable to protect SWG for practical applications. When the light passes through the boundary of two mediums, some light reflects due to a mismatch of the refractive index. The pair of partially reflective substrate spaced micron apart may cause the light interference, so-called Fabry–Pérot interference [43]. For instance, M. Goto et al. presented a thin achromatic quarter-wave film for antireflection of OLEDs based on the sandwiched SWG [26]. In their experimental results, the oscillation of Re was observed.
As the first step to analyze interference by the SWG structure, the influence of the substrates RI was studied. Transmittance through the SWG structure was calculated as a function of wavelength with various ns [ Fig. 5(a)]. Transmittance intensely oscillated when the substrates RI was 1.5. The oscillation was decreased with greater RI and minimized with ns = 1.8. Further increase of the substrates RI (ns = 2.0) resulted in the increases of oscillation, but it is still smaller than that in the case of ns = 1.5 due to the gradual mismatch refractive index between the two mediums [44]. The transmittance was also sensitive to α [Fig. 5(b)]. At α = 0.3 and 0.5, it vigorously oscillates with the large amplitudes at resonance peaks. Given α = 0.8 (b = 0.32λ0, P = 0.4λ0), the transmittance spectra are slightly oscillated in the wavelength region of interest due to the matching refractive index mechanism, which will explain in the below paragraphs.
Fig. 5. (a), (b) Transmittance of the SWG structure as a function of wavelength with various ns and α. α is 0.8 in (a) and ns is 1.8 in (b). (c) Dispersion of retardation vs. λ/λ0 calculated by the FEM simulation and BWA with various duty cycles α.
To highlight the appearance of the interference effect, we set α=0.3, 0.5, and 0.8 corresponding to a>b, a=b, and a<b conditions, and investigated its effect on the Re dispersion [Fig. 5(c)]. We carry out the calculation using both BWA and FEM simulation. BWA and FEM simulation well agreed while FEM results show some oscillation. The interference effect occurs in the simulated results due to the mismatch of the refractive indices between the mediums similar to Fig. 5(a)–5(b). The effect of the interference was not concerned in BWA [Eqs. (2) & (3)]. The form birefringence of SWG is PD when α = 0.3, while it is ND when α =0.5 and 0.8. In the result of the FEM analysis, the oscillation of the retardation is largest, weak, and almost vanished when α =0.3, 0.5, and 0.8, respectively [straight line in Fig. 5(c)]. Therefore, not only the transmittance but also the dispersion strongly depends on α.
The transmission spectrum can be interpreted with the theory of the Fabry-Pérot interferometer and Fresnel’s equations [43,45]. The minimum transmittance Tm of Fabry–Pérot interferometer is given by a function of reflectance R between two mediums as below:
where ni and nt are the refractive indices of the incident and transmitted mediums at the normal incident, respectively. When Tm reaches maximum transmittance, the interference is minimized. The reflectance between two mediums is minimized when ni ≈ nt. As a result, the transmittance spectra are near unity due to the minimization of interference by matching the substrates RI with effective RIs of SWG (ns is 1.8).The effective RIs of SWG in small periodicity condition (P≪λ) [Eqs. (6) and (7)] can be rewritten as,
Accordingly, the effective RIs are strongly dependent on α, and the interference can be minimized by optimizing α to match the effective RIs of SWG with the substrates. Thus, the reduced oscillation in Fig. 5(c) can be explained.
3.3 Subwavelength grating structure in the distribution of intrinsic materials
The effect of the intrinsic dispersion of the SWG constituent material on the form birefringence was investigated in this section. The effective RIs with and without intrinsic dispersion were calculated using BWA [ Fig. 6]. The SWG structure consisted of alternating layers of air and zinc oxide (ZnO) with lengths of a and b, respectively, which were considered in the calculation. The RI data of constituent material ZnO has been extracted from Ref. [46]. First, the form birefringence without intrinsic dispersion was calculated (n1 = 1.0 and n2 = 2.0). Figure 6(a) and 6(b) shows the dispersion of RIs and form birefringence without the intrinsic material dispersion for the various periodicity P (P/λ0 = 0.2, 0.27, and 0.4). When P increases, the dispersion and magnitude of both ne and no increase due to the increasing finite layer thickness (the thicknesses of the alternative layers a, b) [Fig. 6(a)]]. Figure 6(b) shows the substantial dispersion of form birefringence in increasing P. |Δn| decreased, and the dispersion became stronger with larger P. The change of Δn is significantly affected by the dispersion of ne rather than no [Fig. 6(a)] Because ne is more sensitive to P than no, the strong ND is given with large P. This kind of dispersion, so-called structural dispersion, comes from the geometry dependence of the Bloch wave [25].
Fig. 6. Dispersion of (a) RIs and (b) Δn vs. λ/λ0 when the intrinsic dispersion of the SWG constituent material was not considered in the calculation. (c) and (d) are the corresponding data when the constituent material's dispersion was considered.
As a second step, an intrinsic dispersion of ZnO was considered in the calculation [Fig. 6(c)–6(d)]. The RI of the ZnO is much larger than 2.0 in shorth wavelength region (λ<λ0) while it is smaller than 2.0 in the long-wavelength region (λ>λ0) [40]. In Fig. 6(c), ne and no are significantly greater than those calculated without intrinsic dispersion at the short wavelength region. Meanwhile, ne and no in the long-wavelength region is slightly smaller than the results shown in Fig. 6(a). The strong dispersion of ZnO affected dispersion of form birefringence Δn. Δn at P/λ0 = 0.2 was ND, but it became PD when intrinsic birefringence was considered [Fig. 6(d)]. The dispersion property was strongly converted from PD to ND at P/λ0 = 0.27 and became ND completely at large P (P/λ0 = 0.4). The effect of intrinsic dispersion was similar in any P, unlike the structure dispersion. Therefore, the effective RIs of SWG with large P are strongly influenced by the structural dispersion and weakly affected by the intrinsic dispersion.
3.4 Angular dependence
The angular dependence of the form birefringence needs to be considered for exploiting light efficiently. The wide field-of-view (FOV) property is desirable in most of the optical components such as the retarder films for display devices and geometric phase holograms [47–49]. However, previous works about the form birefringence mainly focus on the normal incidence. This section evaluates the angular dependence of the form birefringence with respect to the structural parameters of the SWG structure.
At small periodicity (P = 110 nm (P/λ0 = 0.2)), the Γ is almost unchanged when the incident angle (θ) varies in the range from 0 to 10° and have a good agreement in both BWA [ Fig. 7(a)] and FEM [Fig. 7(b)]. A linear decrease in the magnitude of $\Gamma $ occurs with further increasing θ. Γ at θ = 40° was 4.8% smaller than Γ at θ = 0° in the whole wavelength range in BWA. For FEM, the decrease in magnitude was also observed when θ was greater than 10°. At λ0, the decrease of Γ reaches ∼15% with an θ = 40° compared to the Γ at normal incidence.
Fig. 7. Phase retardation (Γ) vs. λ/λ0 at various incident angles. The periodicity was varied as (a), (b) P = 110 nm (P/λ0 = 0.2), (c), (d) P = 165 nm (P/λ0 = 0.3), (e), (f) P = 220 nm (P/λ0 = 0.4). BWA was used in (a), (c), (e) and FEM was used in (b), (d), (f). The parameter of n1, n2, ns, and α are 1.0, 2.0, 1.8 and 0.8, respectively.
For further increase periodicity (P = 165 nm (P/λ0 = 0.3)) [Fig. 7(c) and Fig. 7(d)], no change in the Γ was observed for θ less than 30°, while there was a slight decrease in the magnitude of the Γ at θ = 40° compared to the Γ at 0° in BWA [Fig. 7(c)]. Meanwhile, we obtain the almost unchanged Γ within 20° of the oblique incidence [Fig. 7(d)] in FEM similar with BWA. Nevertheless, the Γ gradually deviates from the original orbit when θ is over 20°. At the short-normalized wavelength (∼0.82), the spectra strongly deviate from the original orbit at 0°. On the other hand, such deviations were also observed in the long-normalized wavelength (1.45) comparing the case at θ=0°.
For large periodicity (P = 220 nm (P/λ0 = 0.4)) [Fig. 7(e) and Fig. 7(f)], a slight decrease of the Γ at θ = 0° was observed when θ is 40° at the short-normalized wavelength (∼0.82) in BWA [Fig. 7(e)]. It continues to show high consensus for the angles less than 20° compared to its states in FEM [Fig. 7(f)]. To the best of our knowledge, the extraordinary oscillations in Γ at the oblique angles can be understood by referring to the theory of Bragg angle [50]. Accordingly, the oscillations in Γ with the zeroth-order term decrease when θ is farther away from the Bragg angle. Therefore, the oscillation occurs strongly in the increasing oblique angles. Furthermore, we investigate Γ vs λ/λ0 at various incident angles when α changes. As a result, the magnitude of Γ significantly enhanced with the decreasing of α. Details of these results are shown in Fig. S2 of Supplement 1.
The relation between the oscillation of retardation and Bragg angle could be shown by transmittance. At the normal incidence, SWG is transparent in any periodicities as shown in Fig. 8. The near-unity transmittance is maintained for any oblique angles with P = 110 nm (P/λ0 = 0.2) [Fig. 8(a)]. Decreasing transmittance at oblique incidence angle in short wavelength region is due to RI of SWG larger than the substrate. The transmittance of SWG with P = 165 nm (P/λ0 = 0.3) does not oscillate in small incident angles (θ<40°), but the abnormal transmittance is shown when θ=40°. [Figure 8(b)]. In a larger periodicity of 220 nm (P/λ0 = 0.4), the transmittance of the SWG is unusual unlike that of transparent materials even in the small incident angles [Fig. 8(c)]. The band of wavelengths with abnormal transmittance expands with a larger incident angle and transmittance decreases in the band. These kinds of behaviors are observed in some Bragg reflectors. In addition, the fluctuation of the transmittance in Fig. 8 is not because of Fabry-Pérot interference [Fig. 5]. Therefore, the oscillation of the retardation seems related to the Bragg reflection [51] [Fig. 7].
Fig. 8. The transmittance of the SWG structure vs. λ/λ0 with different incident angles. (a) P = 110 nm (P/λ0 = 0.2), (b) P = 165 nm (P/λ0 = 0.3), (c) P = 220 nm (P/λ0 = 0.4). The parameter of n1, n2, ns, and α are 1.0, 2.0, 1.8, and 0.8, respectively.
4. Conclusions
We analyzed the form birefringence of SWG using BWA and FEM. Both methods showed excellent agreement and fitted well by Cauchy's equation. The dispersion of form birefringence was investigated with various geometrical parameters (P, α, and d). The simulation results showed that SWG with long periodicity has the potential to act as a dispersion control platform. In addition, the effect of interference on the performance of the SWG has been investigated in consideration of duty cycles and the substrate's refractive index. By adjusting the duty cycle or substrate's refractive index, light reflection at both boundaries of SWG and substrates and interference could eliminate.
Furthermore, we examined the contributions of the intrinsic dispersion on the dispersion of form birefringence. The negative dispersion of birefringence is still maintained firmly in the longer periodicity and the intrinsic material's dispersion gives less influence than the structural factor. Finally, we evaluated the angular response of SWG. Our results showed angular dependence of phase retardation was decreased with short periodicity. The retardation slightly changed in oblique incidence given the short periodic structure.
Meanwhile, the longer the periodicity of SWG, the more limited the operating range of the incident angles. Abnormal retardation was observed in the short wavelength region at the oblique incidence. Similarly, the near-unity transmittance is maintained for small oblique angles with short periodicity, and abnormal transmittance was observed in wide oblique angles in the longer periodicity of SWG. Our result will be helpful to develop the optical components with any desired dispersion (PD or ND) of birefringence and can be applied on many optical applications such as ellipsometry, polarimetry, and geometric phase lens.
Funding
Ministry of Trade, Industry and Energy (20011031); National Research Foundation of Korea (2019H1A2A1074764, 2019R1A2B5B01069580, 2019R1A6A1A09031717).
Disclosures
The authors declare no conflicts of interest
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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