In order to easily analyze and design the transmittance characteristics of an antireflective surface called the ‘moth-eye structure’, the validity of both scalar diffraction theory and effective medium theory is quantitatively evaluated by a comparison of diffraction efficiencies predicted from both simplified theories to exact results calculated by a rigorous electromagnetic theory. The effect of surface microstructure parameters including the normalized period and the normalized depth has been determined at normal incidence. It is found that, in general, when the normalized period is more than four wavelengths of the incident light the scalar diffraction theory is useful within the error of 5%. Besides, the effective medium theory is accurate for evaluating the diffraction efficiency within the error of less than 1% when the higher order diffraction waves other than zero order wave is not to propagate. In addition, the limitation of scalar diffraction method and effective refractive index method is dependent on not only the normalized period of surface profile but also the normalized groove depth.
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With the development of microfabrication technologies, it is well known that antireflective ‘moth eye’ surface microstructure has attracted more and more attention [1–3]. These advanced diffractive optical elements are investigated from the view points of manufacture and design not only in the performance of antireflection for wide range of spectrum and wide incident angle range [4,5] but also in high laser damage threshold [6,7]. In the aspect of design and analysis, rigorous electromagnetic vector theories such as rigorous coupled wave analysis (RCWA)  and Fourier model method (FMM) [9,10] have been commonly applied to yield accurate diffraction characteristics regardless of the feature size of surface microstructure. These methods give us the exact solution of Maxwell’s equation. However, rigorous vector methods are difficult to be used for computationally intensive and are limited in applicability to periodic structures. Fortunately, scalar diffraction theory (SDT) and effective medium theory (EMT) are frequently used in the design and analysis of diffractive optical elements when the normalized period of diffraction optical components is much greater and is much smaller than the wavelength of the incident light, respectively. Then, the both simplified methods for the two ranges of the microstructure-period-to-wavelength ratio enable us to understand easily the optical characteristics of advanced optical elements since they are intuitive and are much less computationally intensive than rigorous treatments.
In general, scalar diffraction method is often considered inadequate for predicting diffraction efficiencies where (period Λ)/(wavelength λ) <20 [11–13]. Moreover, when , the EMT is frequently considered to yield inaccurate for estimating diffraction efficiencies [14,15]. However, the accuracy of the SDT and the EMT has not been fully quantified for antireflective surface microstructure in the literatures. Therefore, it is important to determine at what normalized period and at what normalized depth SDT and EMT will emerge inaccurate results for analyzing and designing an antireflective optical component.
In this paper, the transmittance characteristics of ‘moth eye’ surface microstructure with respect to the normalized period and the normalized groove depth at normal incidence have been investigated. Through the comparison of the transmission efficiencies predicted by the scalar theory and the EMT, respectively, with those calculated by FMM, we found the minimum microstructure period that scalar diffraction method can be used within the error of 5% is about four wavelengths of incident light, and the maximum microstructure period which the effective medium method is valid within the error of 1% is that the higher order diffractive waves other than zero order begins to propagate. Besides, the impact of groove depth is also determined for the limitation of the both simplified methods.
2.1 Fourier modal method
An antireflection surface microstructure called the ‘moth eye’ structure with a normal incident angle is depicted in Fig. 1 . Λ and d represent the period and groove depth, respectively, K is the profile vector, and and are the refractive indices of the incident medium and microstructure material, respectively. Here, we choose and. And the refractive index of the surface region is equal to that of substrate medium. We consider that the light-wave propagates from the air through surface into the substrate material at normal incidence and the absorption loss of the medium can be ignored. Besides, the dispersion effect which describes the dependence of refractive index of the medium on frequency is ignored in this medium.
The antireflective surface structure can be approximated by a multilayer lamellar grating according to Fourier modal method [16,17]. For each of the lamellar grating layers, the electromagnetic field can be obtained using the Maxwell equations. Then, the reflection and transmission coefficient matrix propagation algorithm (RTCM) , a numerically more efficient variant of S-matrix algorithm than any of known form , is used to calculate the amplitude coefficient matrix of modal fields with the boundary conditions. Hence the reflectivity and transmittance of this kind of optical elements can be derived.
In addition, it is known that the accuracy of FMM is dependent on the number of spatial harmonics used to represent the periodic electromagnetic field, and the diffraction efficiencies calculated with the FMM converge to the exact solution as the number of spatial harmonics is increased. Hence a sufficient number of harmonics were retained in all the diffraction efficiency calculations to ensure accuracy.
2.2 Scalar diffraction theory
The diffraction efficiencies of the scalar method were calculated by utilizing the scalar Kirchhoff diffraction theory which neglects the vectorial, polarized nature of light but gives reasonably accurate results when the periodicity of surface profile is much larger than wavelength of incident light.
The general equation for diffraction efficiency η in the scalar approximation for a periodic structure is given by 
Therefore, the scalar transmittance efficiency of the zero order for an antireflective surface structure at normal incidence is
2.3 Effective medium theory
When an antireflective surface microstructure has a period much smaller than the light wavelength, i.e. in the quasi-static limit (the period-to-wavelength ratio), zeroth-order EMT can be utilized which is formulated by making static approximations that entail assuming that electromagnetic fields do not vary spatially within a given material because of small. Besides, it is worth mentioned that all the background of the EMT was developed from a rigorous modal formalism . However, when one is interested in profile periods that are only several times smaller than the incident wavelength or in substrate material whose permittivity is large compared to that of the incident medium’s, the use of higher-order EMT becomes necessary. It is well known that the second-order EMT was derived primarily by Rytov who made no static-field approximations and was able to expand the expressions for the effective optical properties asymptotically in terms of . As becomes larger, more higher-order terms from the series expansions of transcendental functions must be included in the analysis. In this paper, the zeroth-order and the second-order EMT are used to predict the diffraction efficiencies compared with the results calculated by FMM for an antireflective surface structure, which is to evaluate the accuracy of the EMT.
In our discussion of effective optical properties, we specify the orientation of the electric field E with respect to the profile vector K, which depend on electric field E of incident wave parallel to the profile vector K (TM polarization) or perpendicular to the profile vector K (TE polarization) at normal incidence. These two incident vectors are important since an antireflection surface behaves as a uniaxial crystal whose crystal axis lies along the profile vector.
To second order EMT in the, the effective indices of refraction (assuming that both the incident and substrate medium have permeabilities that do not deviate from the permeability of free space) are represented by [23,24]Eqs. (4) and (5), f is called the filling factor of lamellar grating, and and represent the zero-order EMT approximations to effective indices of refraction, which are given by
For an antireflective surface profile, it can be approximated a large number lamellar grating layers having different filling factor f in depth direction shown in Fig. 2 . Hence the region of microstructure profile can be considered to a stack of homogenous thin film with the effective gradient indices of refraction, and the incident and substrate medium are supposed to be homogeneous and isotropic dielectrics. With this film-stack method, the transmittances of this layered structure are then determined by using thin-film theory. Thus Eqs. (4-7) can be used to calculate the zeroth and the second order effective indices of refraction for each layer.
In figure.2, a single period of an antireflection structure with the total thickness d is divided into N layers. Therefore, the thickness of each layer is , and the filling factor for the q-th layer is25], the characteristic matrix of an assembly of N layers is
In addition, one should note that it is inappropriate to use EMT for describing the surface when higher order diffraction waves other than zeroth order begin to propagate since EMT is based on the premise that only the zeroth diffraction orders are propagating. Whether a given orders propagates or not is determined by the grating equationEq. (11) this requirement sets an upper bound on the period-to-wavelength ratio, specified by
3. Calculated results and discussion
3.1 The results of the rigorous vector diffraction
It is known, in general, that the convergence of a rigorous vector method such as RCWA works well for the TE polarization, but it exhibits poor convergence for the TM polarization and non-rectangular groove profile. Thus the convergence of FMM using RTCM as a function of Fourier order is given in Fig. 3 for ensuring all convergent results calculated in this paper. It is shown that the convergence with sufficient divided multilayer lamellar grating for the surface profile is good for both TE and TM wave with the Fourier order greater than 13. Afterwards, the convergent results in our study were given for both TE and TM polarization with enough Fourier order of 21.
Figure 4 shows plot of transmittances calculated by FMM for antireflection periodic microstructure with respect to the normalized period at normal incidence. The transmission performance with two fixed groove depths, and , is computed for TE and TM polarizations, respectively.
It is clear that the diffraction pattern do change significantly with the normalized period regardless of the groove depth and incident polarization. The transmittances of zeroth order are approximate to 100% when the period of surface structure is much smaller than the wavelength of the incident light. And the transmission efficiencies of both zeroth order and orders trend to a constant when the period increases to a few wavelengths of the incident light in Fig. 4. Therefore, it seems that the simplified methods of both EMT and SDT can be used for analyzing the performance of surface microstructure for the two ranges of the period to wavelength ratio. But, when the period of microstructure approaches to the wavelength of illuminating light the characteristics of transmittance is complex, and a rigorous vector theory must be applied for calculating the optical characteristics. The quantitatively analysis and comparison between rigorous vector method and both simplified methods, respectively, would be shown at late sections.
3.2 The accuracy of scalar diffraction theory
To quantify how element normalized period, element normalized depth, and polarization of incident light affect the validity of the scalar treatment, we present plots of zeroth order and orders transmission efficiencies, respectively, versus period / wavelength () of the surface profile and that versus depth / wavelength () at normal incidence.
Figure 5 shows the comparison of the diffraction efficiencies of both zero order and orders, respectively, versus the normalized period, and those diffraction efficiencies were predicted from scalar transmission theory to exact results calculated with FMM for TE polarization. Since the diffraction efficiencies of the scalar method are independent to the period of surface microstructure, they are shown by straight lines calculated by Eqs. (2) and (3) in Fig. 5. There are two cases of the normalized groove depth to be chosen, and .
In general, when the vector results of transmittances reach to a mostly constant, the scalar diffraction efficiencies are greater than the vector solutions from Fig. 5. In Fig. 5(a), it can seen that the discrepancy of zero order diffraction efficiencies between the result of the scalar theory and that of the FMM trends to a constant of about less than 3% when the value of the normalized period is more than four wavelengths of incident light for the normalized depth 0.5, and the orders diffraction efficiencies predicted by the scalar method is an excellent approximation to the vector results of FMM as shown in Fig. 5(b). Moreover, it is also shown clearly in Fig. 5(c) that the zeroth order diffraction efficiencies calculated by the scalar theory are in agreement with the results calculated by FMM when the value of the normalized period is larger than four wavelengths, which the difference of diffraction efficiencies is less than 3% for the normalized groove depth . Similarly, the error of the orders diffraction efficiencies becomes to the mostly constant of less than 3% when the normalized period is more than four wavelengths as it is shown in Fig. 5(d). Hence, regardless of diffraction order and normalized depth, the validity of the minimum normalized period is about four wavelengths of incident light. This limitation of scalar method is significantly different from that considered by Ref. 11.
To display the accuracy of scalar method for antireflection surface microstructure with respect to the normalized groove depth in detail, the comparison of diffraction efficiencies calculated by the FMM and the scalar theory with discrete values of is shown in Fig. 6 . Figure 6(a) illustrates that the scalar method is valid for zeroth order diffraction efficiencies when the normalized groove depth is less than 1.5 within the error of less than 5% for , but the validity for orders is less than the value of 1.0 for the normalized groove depth. When is 4.0, the zeroth order diffraction efficiencies calculated by scalar theory is in agreement well with the results estimated by FMM for the range of , but the diffraction efficiencies of the orders from scalar method agrees well with that simulated by FMM for as it is shown in Fig. 6(b). In addition, the validity of the scalar method expands sequentially for in Fig. 6(c). At in Fig. 6(d), the results of diffraction efficiencies calculated by the scalar theory are in agreement well with that of FMM for both zeroth order and orders in the region of , but beyond this range the scalar diffraction theory is less valid. Therefore, it is can be concluded that the accuracy of the scalar theory with respect to normalized groove depth increases as the normalized period of surface microstructure increases, and the greater the normalized period is and the better the transmittances from scalar method agree with that from rigorous vector method. This phenomenon is similar to rectangular grating previously published studies [26,27] and results from the scalar theory assumption that an antireflection diffraction element is an infinitely thin mask.
In addition, the comparison of diffraction efficiencies predicted by scalar theory and vector theory for TM polarization incidence shows that, in general, the accuracy of the scalar treatment for TM case is comparable with that of the TE case.
Finally, it is intuitively concluded that the accuracy of the scalar treatment for antireflection diffractive optical components ultimately depends on the normalized period and normalized groove depth of surface microstructure at normal incidence. And the validity for the scalar method is more than a minimum normalized period of about four wavelengths of incident light. This minimum normalized period ensures accurate (within the error of about 5%) results. In other words, when an antireflection surface structure period is more than four wavelengths of illuminating light, scalar method can be used for analyzing and designing effectively the performance of optical elements.
3.3 The accuracy of effective medium theory
When the period of an antireflective surface microstructure is much smaller than the wavelength of the incident light, the EMT is generally used to predict its optical performance. In order to apply effectively EMT for designing and analyzing an antireflection optical component, we would evaluate quantitatively the accuracy of the zeroth order and second order of the EMT at normal incidence. The refractive index, , of surface material is chosen, which may be adequate for optical elements of transmission type.
Figure 7 shows the transmittances calculated by vector theory and EMT theory of both zeroth order and second order at the fixed groove normalized depth, 0.5, and 1.0, as a function of the normalized period of surface profile for TE and TM waves, respectively. The red curves ‘FMM(0th)’ are the transmittance of zeroth order predicted by the rigorous treatment, and ‘EMT(0th)’ and ‘EMT(2nd)’ are results with zeroth-order and second-order effective refractive indices, respectively.
In Fig. 7(a) and 7(b), the normalized groove depth is fixed at . Figure 7(a) shows that the transmittances for zeroth order EMT and second order EMT are extremely approximate to the results derived by FMM in the range of which is quasi-static limit. As the normalized period increases to non-quasi-static limit, the error of diffraction efficiencies between EMT and FMM increases. When is 0.6, the difference of transmittances calculated by zeroth order EMT and FMM increases approximately to 0.4%, but the discrepancy of the results estimated by second-order EMT and FMM is about less than 0.2% for TE polarization. For TM polarization in Fig. 7(b), there is in agreement well with the results calculated by both EMT and FMM when in which the maximum error is about less than 0.1%. But, beyond the diffraction efficiencies calculated by FMM sharply drops, and it is clear that at this position the higher-order diffraction waves begins to propagate according to Eq. (13). Hence it is much inappropriate to use the EMT for estimating the performance of optical elements since the EMT is based on the premise that only the zeroth diffraction order is propagating. Finally, it is noted that in the refractive index, 1.5, and the fixed normalized groove depth, , the result calculated by EMT for TM wave versus the normalized period are in more agreement with that of vector theory than the result of TE wave in the range of .
As the normalized groove depth is shown in Fig. 7(c) and 7(d), for both TE and TM polarizations the diffraction efficiencies calculated by EMT for both zeroth order and second order are extremely approximate to that obtained from FMM not only in the quasi-static limit but also in the non-quasi-static limit in the range of . It seems that the larger the normalized groove depth is, the better the result from EMT agrees well with that of FMM.
In order to further determine the effect of groove depth for the limitation of EMT, Fig. 8(a) and 8(b) show the variation of transmittance as a function of the normalized groove depth for TE and TM waves, respectively. The antireflective microstructure period is fixed at . Then the transmittances with both zeroth order and second order EMT agree well with that with the rigorous treatment except for as it is shown in Fig. 8(a). And the maximum deviation of diffraction efficiencies between zeroth order EMT and FMM for TE wave reaches to 1% at the position of about . For TM polarization in Fig. 8(b), it is illustrated that the result from EMT agrees well with that of FMM regardless of zeroth order and second order EMT. Finally, it is clear that the application of EMT for TM wave is preferable to that for TE wave.
On the basis of these comparative results, we will describe the ability and limitation of EMT from the view point of design of the antireflection optical elements.
It is found that the EMT is helpful to estimate the transmittance of an antireflection structure, especially, for TM wave. However, the transmittances calculated with the EMT strikingly differ from those with the rigorous treatment when the higher-order diffraction waves are to propagate with respect to the normalized period. When only the zeroth order diffraction wave is to propagate, the groove depth also affects the accuracy of EMT. For an example, as the refractive index of material is 1.5 and is 0.65 which is only zeroth order wave is to propagate, the diffraction efficiency calculated by EMT is in agreement well with the result from FMM except for the region of for TE wave incidence. In the region of the maximum error between the zeroth order EMT and FMM reaches to 1%. And it is noted clearly that the second order EMT agree with the rigorous method more than zero order EMT, especially, in the range of . For TM wave incidence, the transmission efficiencies from both zeroth order and second order EMT are agreement well with that of FMM. Consequently, these conclusions are significant for analyzing and designing an antireflection optical component by using EMT.
For analyzing and designing easily an antireflection surface microstructure, We have calculated quantitatively the accuracy of both SDT and EMT for the two range of with the refractive index of surface material, , at normal incidence. The effects of the surface profile’s parameters (period and depth) on the accuracy of both simplified methods have been determined. In addition, the effect of refractive index of surface material on the error of both SDT and EMT is directly attributed to the depth of the surface microstructure, which is not discussed in this paper.
According to the calculated results, the scalar method is useful when the normalized period is more than four wavelengths of incident light within the error of less than 5% regardless of the effect of polarization. And the error of the scalar theory at the groove depth direction decreases as the normalized period increases. Besides, when, which only zeroth order diffraction wave is to propagate, the EMT is valid to estimate the diffraction efficiency within the error of less than 1%. For TE polarization, the diffraction efficiencies of both zeroth order and second order EMT agree well with that of FMM at groove depth direction except for in which the maximum error reaches to mostly 1%. However, for TM wave the result from EMT is in agreement well with that from FMM regardless of the normalized groove depth.
This work is supported by the National Natural Science Foundation of China (Grant No. 10704079).
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