Improvement of the validity of the simplified modal method for designing a subwavelength dielectric transmission grating Xufeng Jing,1,* Junchao Zhang,2 Ying Tian,3 and Shangzhong Jin1 1
Institute of Optoelectronic Technology, China Jiliang University, Hangzhou 310018, China 2
Optic and Laser Division, National Institute of Metrology, Beijing 100013, China
3
College of Materials Science and Engineering, China Jiliang University, Hangzhou 310018, China *Corresponding author: [email protected] Received 9 October 2013; revised 28 November 2013; accepted 6 December 2013; posted 11 December 2013 (Doc. ID 199131); published 9 January 2014
To accurately and easily design the diffraction characteristics of a rectangular transmission grating under the illumination of Littrow mounting, the validity and limitation of the simplified modal method is evaluated by a comparison of diffraction efficiencies predicted by the modal approach to exact results calculated with rigorous coupled-wave analysis. The influence of the grating normalized period, the normalized groove depth, and the fill factor on the accuracy of the modal method is quantitatively determined. More importantly, the reflection effect of two propagating grating modes with the optical thin-film model and the nonsymmetrical Fabry–Perot model is proposed and applied in the modal method to improve the accuracy of the calculated diffraction efficiencies. Generally, it is found that the thin-film model of reflection loss is valid at the smaller normalized period, but the Fabry–Perot model can exactly calculate the reflection loss of grating modes at the larger normalized period. Based on the fact that the validity of the modal approach is determined independently of the incident wavelength, the exact design and analysis of grating diffraction elements can be implemented at different wavelengths by simply scaling the grating parameters. Moreover, the polarization effect of diffraction properties on the limitation of the modal method without and with the reflection loss of grating modes is clearly demonstrated. © 2014 Optical Society of America OCIS codes: (050.0050) Diffraction and gratings; (050.1380) Binary optics; (050.1950) Diffraction gratings; (050.1960) Diffraction theory; (050.2065) Effective medium theory. http://dx.doi.org/10.1364/AO.53.000259
1. Introduction
Diffraction transmission gratings with rectangular grooves etched into a monolithic dielectric material are interesting and applicable in a large number of different applications [1,2], such as high-power laser systems, integrated optics, holography, optical information processing, and fiber Bragg grating (FBG) fabrication using the phase mask method [3–5]. There are some studies where theoretical and experimental analyses of diffraction characteristics, 1559-128X/14/020259-10$15.00/0 © 2014 Optical Society of America
especially field distribution behind the phase mask [3], are presented. For example, the scalar diffraction theory can be used and is good enough to estimate the diffraction performance in FBG fabrication, although the normalized grating pitch in the case of the phase mask is 4 [3–5]. Particularly, binary diffraction gratings in the resonance region with the period comparable to the incident wavelength have been widely investigated theoretically and experimentally [6]. These subwavelength gratings can realize various optical functions by simply optimizing their profile parameters, e.g., polarizing beam splitters [7]. In general, for the gratings in the resonance region, the rigorous vector 10 January 2014 / Vol. 53, No. 2 / APPLIED OPTICS
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methods that directly solve Maxwell’s equations have to be applied for calculating exact diffraction characteristics. For example, rigorous coupled-wave analysis (RCWA) is frequently used to calculate exactly the optical properties of gratings [8,9]. It gives us an accurate solution of Maxwell’s equations with mathematical expansion of the field inside the grating region. However, RCWA is more computationally intensive and cannot be easily inverted to solve the inverse diffraction problem [10]. Furthermore, this numerical treatment can not give much physical insight into the diffraction processes inside the grating region. Fortunately, there is a simple method, i.e., the modal method [11–13], that has been developed and was applied to dielectric gratings with the feature size in the resonance region. In this scale of grating period, it was reported that the scalar diffraction theory and the effective medium theory cannot be used to effectively analyze the diffraction characteristics of gratings [14]. Thus, the modal method fills the blank of simple analysis methods with the grating period comparable to the incident wavelength. This approach can present a clear physical diffraction process by taking the modes of a periodic planar waveguide as the grating modes [6]. Recently, Clausnitzer et al. proposed a simplified modal method that reduces the difficult diffraction process to an easy and intelligible modal interference mechanism [10]. This simplified method enables a grating design without complicated numerical calculations, and it has been warmly embraced because it is easy to use and lacks computational strain. Although the modal method is widely used to approximately design and to analyze diffractive phase elements in the literature [15], no one has fully studied quantitatively its accuracy in the design and analysis of transmission dielectric gratings, resulting in erroneous element designs and disappointing experimental diffraction results. Therefore, it is important and necessary to determine at what normalized period and at what normalized groove depth the modal method will yield inaccurate results. The quantitative error between the modal method and the rigorous treatment should be clearly demonstrated for the accurate design of diffraction characteristics by using the simple method. Importantly, in the case of the design and analysis of a rectangular transmission grating, the reflection losses at grating interfaces are often neglected by using the simplified modal method [6,7,10,16], leading to the outcome that the calculated diffraction efficiencies are always obviously different from the real results. In particular, the reflection of the incident wavelength increases significantly when the grating period approaches λ∕2 [17]. Therefore, the reflection loss of the grating modes in the modal method must be considered to enhance the accuracy of analyzed results. In this present paper, the reflection effect of the effective index modes at the grating interfaces will be clearly demonstrated by using the effective thin-film theory and nonsymmetrical Fabry–Perot resonator model, respectively. 260
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2. Simplified Modal Method
The binary rectangular grating is shown in Fig. 1. A plane wave is incident from air with the incident angle of Littrow configuration θi arcsinλ∕2n0 Λ, where λ is the incident wavelength, n0 is the refractive index of air with the value of 1.0, and Λ is the grating period. The fill factor f of the grating is defined as the ratio of the grating ridge to the period, and the depth of the grating is h. The refractive index ns of the substrate material of fused silica is 1.45 at the incident wavelength 1064 nm. The refractive index ng is the same as that of the substrate. The grating material of fused silica is chosen based on its excellent transparency, with little dispersion change from deep UV to far IR for the design of transmission gratings, and its higher laser damage threshold. It has extensive potential applications not only in laser communication but also in high-power laser systems [1]. In this paper, only two transmitted propagating orders, the zeroth order and the −1st order, are restricted by the grating profile fulfilling the condition of λ∕2 < Λ < 3λ∕2ns. The diffraction angles are θ0 and θ−1 , respectively. The highest diffraction efficiency in the zeroth order or the −1st order can be achieved by optimizing grating parameters. According to the modal theory [6,10–12,18], there are only two excited propagating grating modes that dominate the diffraction process when the grating periods are small enough and comparable to the incident wavelength. Each of the propagating modes carries nearly half of the energy of the incident wave [19]. These two equally excited modes can be named the 0th mode and 1st mode. The detailed modal method has been described in the relevant literature [10–12]. On the basis of the two-beam mode interference mechanism in the simplified modal method [10], the diffraction efficiency can be mainly determined by the accumulated phase difference Δφ of the two excited modes. Generally, many relevant reports ignored any reflection losses of the rectangular grating interfaces based on the low index contrast [6,7,10,16]. However, it will be shown in later sections that the reflection effect of the grating propagating modes is significant and should be considered to accurately analyze the diffraction characteristics of rectangular gratings. First, the reflection effect at the grating interfaces is neglected to quantify the error of the
Fig. 1. Schematic of a rectangular grating under the Littrow configuration illumination.
calculated diffraction efficiency between the modal method and RCWA. The efficiencies of the transmitted zeroth diffraction order and −1st diffraction order can be represented as [19] η0T cos2 Δφ∕2;
(1a)
η−1T sin2 Δφ∕2
(1b)
Δφ k0 n0eff − n1eff h 2πn0eff − n1eff q;
(2)
with
where ηmT is the diffraction efficiency of the mthorder (m 0, −1), k0 is the wavenumber in vacuum k0 2π∕λ, and n0eff and n1eff are the corresponding effective indices of the 0th propagating mode and the 1st propagating mode, respectively. The normalized groove depth is also defined by q h∕λ in Eq. (2). The corresponding effective mode indices neff depend on the grating period, the fill factor, the incident wavelength, and the polarization of incident light upon the grating. They can be determined by solving the following eigenfunction equation for TE polarization (electric field vector perpendicular to the incident plane) as [15] Fn2eff cos2πp sin θi ;
(3)
Fn2eff cos2πβpf cos2πγp1 − f β2 γ 2 sin2πβpf sin2πγp1 − f ; (4) 2βγ q q where β n2g − n2eff , γ n20 − n2eff , and p Λ∕λ. The right-hand part of Eq. (3) indicates the incidence condition, where cos2πp sin θi −1 is the Littrow mounting incidence and cos2πp sin θi 1 is under normal incidence. The normalized period is defined as p Λ∕λ. Moreover, the effective modes index with n2eff > 0 is propagating through the grating, while those with n2eff < 0 are evanescent. For TM polarization (magnetic field vector perpendicular to the incident plane), the dispersion equation can be expressed as −
Fn2eff cos2πβpf cos2πγp1−f −
n4g β2 n40 γ 2 sin2πβpf sin2πγp1−f : (5) 2n2g n20 βγ
Based on the definition of the normalized period p, the effective indices of the grating modes for both TE and TM polarizations do not depend on the value of Λ or λ individually but depend on the value of p. And the accumulated phase difference in Eq. (2) between the 0th mode and the 1st mode depend on q rather than the individual value of h or λ. Therefore, in the design and analysis of gratings, the diffraction characteristics are independent of the real grating period,
the groove depth, and the incident wavelength, but they are dependent on the normalized period and the normalized groove depth. So the same diffraction properties at different wavelengths can be obtained by simply scaling the grating parameters without the consideration of material dispersion. To quantify how element period, element groove depth, and polarization of incident light affect the validity of the simplified modal method without the consideration of the reflection effect of the grating modes, we present plots of zeroth-order and −1st-order transmission efficiencies, respectively, versus the normalized period and the normalized groove depth at the Littrow mounting. The fill factor of the grating is set to be 0.5, at which the gratings are often fabricated easily by holographic lithography. It is well known that RCWA is rigorous and accurate in the calculation of the diffraction efficiency of grating elements [8]. So the results from RCWA are regarded as a comparison criterion to determine the validity of the modal method in this paper. Figure 2 shows the comparison of the diffraction efficiencies on the normalized period between the simplified modal method based on Eq. (1) and RCWA for both TE and TM polarizations. The normalized period is confined to the range of 0.5 < p < 1.0, in which only two propagating grating modes play a dominant role at the Littrow configuration, and other grating modes are evanescent. It should be noted that beyond this normalized period the multiple-mode interference mechanism can also be used to analyze the diffraction process of gratings [6,20,21]. In this paper, we concentrate on the twobeam mode interference mechanism. In Fig. 2, the normalized groove depth q 1.0 is chosen. It can be seen in Fig. 2(a) for TE polarization that although the zeroth-order diffraction efficiencies predicted by the modal method are in agreement with the results from RCWA, with the error less than about 2%, the maximum difference of transmittance can reach more than 40% at about p 0.51 for the −1st-order diffraction. Also, in Fig. 2(a) the discrepancy of the −1storder diffraction efficiencies between RCWA and the modal method trends to a constant of about 6% when the normalized period is more than about 0.6. It is interesting for TM polarization in Fig. 2(b) that the −1st-order diffraction efficiency from the modal method agrees well with that of RCWA, but the maximum error of transmittances for the zeroth-order diffraction reaches about 20%. When the normalized period is beyond about 0.65, the zeroth-order diffraction efficiencies estimated by the simplified modal method are in good agreement with those of RCWA. Thus, it is clear that the results estimated by the modal approach are obviously larger than those from the rigorous treatment. This illustrates distinctly the existence of reflection loss of the grating modes, and the reflection effect should be included in the modal method to enhance its accuracy of the calculated diffraction efficiencies. To display the limitation of the simplified modal method for a rectangular grating without the 10 January 2014 / Vol. 53, No. 2 / APPLIED OPTICS
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Fig. 2. Comparison of diffraction efficiencies between the modal method and RCWA versus the normalized grating period with a fill factor of 0.5. (a) TE polarization and (b) TM polarization.
consideration of reflection effect with respect to the normalized groove depth in detail, the comparison of diffraction efficiencies calculated by the modal method and RCWA is shown in Fig. 3. The fill factor and the normalized period are set to be 0.5 and 0.7, respectively. For TE polarization in Fig. 3(a), the zeroth-order diffraction efficiencies calculated by the modal method are in agreement well with that of RCWA at about 0.3 ≤ h∕λ ≤ 1.5, beyond which the maximum error can reach more than 10%. The estimated results from the modal method can agree well with those from the rigorous vector method at about h∕λ ≤ 0.5 for zeroth-order diffraction. The larger error between both methods is shown at a deeper groove depth. For TM polarization in Fig. 3(b) the validity of the simplified modal method on the normalized groove depth is clearly shown. From Fig. 3, it can be derived that not only can the simplified modal method be utilized in the design and analysis of a deeper diffraction grating [10], but also at the smaller groove depth it can be used to approximately calculate the diffraction efficiency. Therefore, without consideration of the reflection losses at the interface of the grating, the calculated diffraction efficiencies by the modal method are clearly different from those by the rigorous vector theory, especially at the smaller normalized period and the larger normalized groove depth. These larger errors would result in inaccuracy in the design and analysis of gratings by using the modal method. In order to improve the validity of the simplified modal method, the reflection effect of the propagating modes at the grating interfaces must be strictly considered and quantitatively investigated.
3. Consideration of Mode Reflection
According to Eq. (1), the diffraction efficiencies in the zeroth order or −1st order can achieve 100%. However, the maximum diffraction efficiency of the −1st order with the optimal grating parameters in the experiment measurement is much less than 100% [22]. In addition, based on the comparison of diffraction efficiencies between RCWA and the modal method without the consideration of reflection loss, as shown in above section, the existence of the reflection effect of propagating modes for the rectangular transmission grating is implied. Also, Clausnitzer et al. reported that the reflection losses of rectangular transmission gratings increase rapidly as the ratio between the period and the wavelength is decreased [17]. In order to precisely use the simple modal method to accurately design and analyze the diffraction efficiencies of a grating, the reflection loss of the effective index modes must be quantitatively determined. Based on Ref. [19], it is known that the effective index nair eff of the incident light in air is cos θ i , and the effective index nsub eff of the substrate material in the modal method is ns cos θ0i, where θ0i is the refracted angle. The grating structure shown in Fig. 1 can be equivalent to a thin film according to the effective medium theory. The reflection loss of the effective index modes can be calculated analogously to Fresnel reflection. At the air–grating interface, two propagating grating modes, 0th mode and 1st mode, can be excited equally by the incident wave. Both modes couple to the two transmitted diffraction orders at the grating–substrate interface. Because
Fig. 3. Comparison of transmittance characteristics between the modal method and RCWA as a function of the normalized groove depth at a normalized period of 0.7. (a) TE polarization and (b) TM polarization. 262
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the diffraction process in the grating region is very similar to a Mach–Zehnder interferometer [10], the reflection effect of the two grating modes can be considered separately. Therefore, according to the Fresnel reflection, the reflection of each of the grating propagating modes at the air–grating interface m Rm air or at the grating–substrate interface Rsub can be represented by [17] Rm air∕sub
air∕sub nm eff − neff
2 ;
air∕sub nm eff − neff
(6)
where m indicates the number of the mode (m 0, m 1). Figure 4 shows the calculated reflectivity of the 0th and the 1st propagating grating modes as a function of the fill factor at the air–grating and the grating–substrate interfaces for TE and TM polarizations, respectively. The 800 nm grating period with the illumination of a 1064 nm wavelength is assumed. It is shown for both polarization states that at the air– grating interface the reflection losses of the 0th mode are higher than those of the 1st mode, but at the grating–substrate interface the situation of the reflection effect for both modes is the opposite. Although the reflections of the 1st mode at the air–grating interface for about f > 0.4 and the 0th mode at the grating– substrate interface for about f < 0.4 can be neglected for TE polarization, as shown in Fig. 4(a), both grating propagating modes in the analysis of diffraction characteristics of the whole grating element including the air–grating and grating–substrate interfaces should be considered. In order to precisely calculate the reflection loss of the grating modes for the whole grating device as shown in Fig. 1, the thin-film model or the nonsymmetrical Fabry–Perot resonator model can be used. First, the subwavelength diffraction grating can be approximated as a lamellar thin-film layer. According to the matrix method in the film theory [23], the characteristic matrix of this equivalent layers is
B cos δ C iη sin δ
i sin δ∕η cos δ
1 ; ηs
(7)
where δ 2πhnm eff ∕λ is the effective phase for the 0 1 equivalent thin-film layer, and nm eff is neff or neff .
η η0 nm eff , η0 is the optical admittance in free space (η0 ε0 ∕μ0 1∕2 2.6544 × 10−3 S), and ηs η0 nsub eff is the optical admittance of the effective index of substrate. Letting Y be C∕B, the reflectance of the grating propagating modes on the whole grating system including the air–grating and the grating–substrate interfaces is [23] sub sub η0 nair η0 nair 1 eff − Yneff eff − Yneff m : (8) RTH · sub sub 2 η0 nair η0 nair eff Yneff eff Yneff Rm TH is the reflection of the corresponding grating modes in the thin-film model. The asterisk in Eq. (8) indicates the conjugate complex number. Due to the fact that each of both propagating grating modes carries separately half of the incident energy, the multiplier 1∕2 in Eq. (8) is added. Second, the reflection of the excited modes also can be described by a nonsymmetrical Fabry–Perot resonator that is, respectively, filled with the medium of effective refractive indices n0eff and n1eff . The reflectivity Rm FP of both grating modes can be represented as [17] p p2 p 2 2π m m Rm 4 Rm Rm air − air Rsub sin sub λ neff h 1 : · Rm p p FP 2 1 − Rm Rm 2 4 Rm Rm sin2 2π nm h air
sub
air
sub
λ
eff
(9) Rm FP indicates the reflectivity of the 0th grating mode or the 1st mode in the nonsymmetrical Fabry–Perot resonator model. In Fig. 5, the reflectivity of grating modes according to Eq. (8) of the thin-film model is, respectively, plotted as a function of the groove depth and the grating period. Here, the fill factor 0.5 of the rectangular transmission grating is assumed. TE polarization is considered. It is shown that these reflection losses of grating modes increase rapidly as the normalized period is decreased, especially at the grating period approaching λ∕2. Also, the reflection of both grating modes at a fixed grating period indicates the behavior of slight oscillation with respect to the groove depth. Initial results show that the reflection characteristics of grating modes for the TM case are comparable with those of the TE case. Also, the reflectivity properties calculated by Eq. (9) of
Fig. 4. Reflection of the two propagating grating modes at the air–grating and the grating–substrate interfaces with a grating period of 800 nm. (a) TE polarization and (b) TM polarization. 10 January 2014 / Vol. 53, No. 2 / APPLIED OPTICS
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Fig. 5. Simulation of the reflection losses based on Eq. (8) as a function of the groove depth and the normalized period for TE polarization with a fill factor of 0.5. (a) The zeroth mode and (b) the first mode.
Fabry–Perot resonator model on the normalized period and the groove depth are similar to those of the thin-film model. Figure 6 shows a numerical simulation of the reflection losses of the grating modes as a function of the groove depth and the fill factor by using the thin-film model of Eq. (8). A grating period of 800 nm and incident wavelength of 1064 nm are assumed. The plots reveal an increase of the reflectivity with the increased grating fill factor. Thus, in order to realize a highly efficient transmission grating, the smaller fill factor of the grating should be fulfilled. The obvious oscillation of the reflection of the grating modes as a function of the groove depth is demonstrated at the smaller fill factor of the grating. This phenomenon can be attributed to the interference effect of the grating mode itself. However, the reflection characteristics of the modes from the Fabry–Perot resonator model, as shown in Fig. 7, are clearly different from those of the thin-film model.
The incident wavelength in Fig. 7 is 1064 nm. It indicates that these theoretical models of the reflection effect of grating modes would be valid at the specific condition of grating parameters. Based on Eq. (1) and the reflection losses of the grating modes, the diffraction efficiencies of the whole grating element without the absorption loss of the material can be expressed as η0TH∕FP η0T · 1 − R0TH∕FP · 1 − R1TH∕FP ;
(10a)
TH∕FP η−1T · 1 − R0TH∕FP · 1 − R1TH∕FP ; η−1
(10b)
where η0TH∕FP is the transmittance of the thin-film model or the Fabry–Perot resonator model for the TH∕FP shows the transzeroth-order diffraction, and η−1 mittance of the −1st-order diffraction of the rectangular grating. According to Eq. (10), the diffraction efficiency, comprising the reflection losses of grating modes, is
Fig. 6. Reflection of grating modes calculated by Eq. (8) with a grating period of 800 nm for (a), (b) TE polarization and (c), (d) TM polarization. 264
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Fig. 7. Reflection of grating modes calculated by Eq. (9) with a grating period of 800 nm for (a), (b) TE polarization and (c), (d) TM polarization.
calculated as a function of the normalized period for both mode reflection models, as shown in Fig. 8. A normalized groove depth of 1.0 is assumed. For TE polarization in Fig. 8(a), the transmittances calculated by the thin-film model for the zeroth diffraction order and −1 st diffraction order are in agreement well with those from RCWA at about 0.5 < Λ∕λ < 0.7. Beyond this range the nonsymmetrical Fabry–Perot resonator model is precise and valid. Compared with Fig. 2(a) without the consideration of the reflection effect of grating modes, the calculated diffraction efficiencies with the reflection losses of the thin film model are clearly accurate, especially at the smaller normalized period for the −1st-order diffraction. For TM polarization in Fig. 8(b), the estimated transmittances by the thin film modal model agree well with the rigorous vector results at about 0.5 < Λ∕λ < 0.55 for both diffraction orders. When the normalized period is more than about 0.55, the transmittance error for the −1 st-order diffraction remains nearly constant and is less than 10%. As the normalized period is
more than about 0.65 for the nonsymmetrical Fabry– Perot resonator model, the diffraction efficiencies are coincident with the results from RCWA. Thus, the consideration of the reflection effect of grating propagating modes makes the simplified modal method more accurate in the analysis of diffraction characteristics. Generally, the reflection losses calculated by the optical thin-film model are precise and valid at the smaller normalized period. This approximately corresponds to the validity range of effective medium theory, in which the binary grating structure can be exactly equivalent to one kind of medium at the smaller grating period [14]. However, the nonsymmetrical Fabry–Perot resonator model can accurately calculate the reflection losses at the larger normalized period of rectangular transmission gratings. Furthermore, the comparison of diffraction efficiencies from the rigorous method and the simplified modal method with the reflection losses of the grating modes is demonstrated as a function of the normalized groove depth at a normalized period of 0.7 in
Fig. 8. Comparison of transmittance characteristics between the modal method with the consideration of the reflection loss of grating modes and RCWA. (a) TE polarization and (b) TM polarization. 10 January 2014 / Vol. 53, No. 2 / APPLIED OPTICS
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Fig. 9. Comparison of diffraction efficiencies between RCWA and the modal method at a normalized period of 0.7. The key for the lines is the same as in Fig. 8. (a) TE polarization and (b) TM polarization.
Fig. 9. For TE polarization in Fig. 9(a), the results calculated by the modal method almost completely coincide with that of RCWA for −1st-order diffraction at 0 ≤ h∕λ ≤ 1.0 regardless of the reflection model of the grating modes. When the normalized groove depth is less than about 0.2 for the zeroth-order diffraction, the modal method is invalid. In contrast to Fig. 2(a) without the reflection effect, the diffraction efficiencies obtained by the modal method with the reflection losses of the grating modes is distinctly closer to the results of the rigorous method, especially at the maximum extrema of transmittances of the zeroth-order or the −1st-order diffraction. For TM polarization in Fig. 9(b), the slight fluctuation of diffraction efficiencies obtained from the modal method is clearly displayed due to the interference effect of the grating modes. Figure 10 demonstrates quantitatively the error of diffraction efficiencies for RCWA and the simplified modal method with and without the consideration of reflection loss of the grating modes as a function of the normalized period. In the simplified modal method with the reflection loss of the grating modes, the thin-film reflection model and Fabry– Perot reflection model are used. The difference of transmittances between RCWA and the modal method without the reflection loss is also included in Fig. 10 for reference. The error of diffraction effim m ciency is defined as ηm TF∕FP − ηRCWA , in which ηTF∕FP indicates the mth-order transmittance of the thinfilm model or Fabry–Perot model, and ηm RCWA denotes the mth-order transmitted diffraction efficiency of RCWA. For TE polarization in Fig. 10(a), the error between the thin-film model and RCWA can be
decreased slightly compared with the difference of the simplified modal method without reflection loss and RCWA for zeroth-order transmittance at the smaller normalized period. For the thin-film model and Fabry–Perot model, the difference is significantly decreased for the −1st-order diffraction at the smaller period compared with the modal method without the reflection loss of the grating modes. In Fig. 10(b) for TM polarization, it can be seen that the Fabry–Perot model can be applied in the larger normalized period to accurately calculate diffraction efficiencies. Thus, the accuracy of calculated diffraction efficiency by the simplified modal method can be generally enhanced by the consideration of the reflection loss of the grating modes. Furthermore, the quantitative difference between RCWA and the simplified modal method with the reflection loss of the grating modes with respect to the normalized groove depth is demonstrated in Fig. 11. It can be seen that the difference of diffraction efficiency between RCWA and the simplified modal method exhibits clear oscillating behavior as a function of the normalized grating depth. The error of diffraction efficiency between the rigorous vector method and the simplified modal method with the reflection loss of the grating modes can obviously be decreased in −1st-order diffraction for both TE polarization and TM polarization. In order to check the accuracy of the simple modal approach on the fill factor, the diffraction efficiencies are numerically simulated as a function of the fill factor and the groove depth for TE and TM polarization in Fig. 12. The transmittance images of −1st-order diffraction with a grating period of 600 nm under the illumination of a 1064 nm wavelength are
Fig. 10. Error of diffraction efficiency comparison between the simplified modal method with the consideration of reflection loss by the thin-film model and the Fabry–Perot model and RCWA versus the normalized period. (a) TE polarization and (b) TM polarization. 266
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Fig. 11. Error of diffraction efficiency comparison between the simplified modal method with the consideration of reflection loss of the thin-film model and Fabry–Perot model and RCWA versus the normalized groove depth. (a) TE polarization and (b) TM polarization.
calculated by RCWA and the simple modal method, respectively. Figures 12(a) and 12(b) show the results of the rigorous vector method. The optical thin-film model of grating mode reflection in the simplified modal method is used in Figs. 12(c) and 12(d) for both polarization cases. It is shown in Figs. 12(e) and 12(f) that the diffraction efficiencies are calculated by the modal method with the nonsymmetrical Fabry–Perot resonator model of reflection loss of the grating modes. In Fig. 12, the images demonstrate that the
diffraction behavior of the rectangular transmission gratings can be described well by the simple modal method with the consideration of the reflection effect of the grating modes. It is interesting for TM polarization in Fig. 12(b) that the transmittances of the −1st order almost approaches zero as the fill factor nears 0.5 regardless of the groove depth, while the TE polarization in Fig. 12(a) exhibits a dependence of groove depth in −1st-order diffraction. By using this surprising effect,
Fig. 12. Simulation of the diffraction efficiencies of a rectangular transmission grating with a 600 nm period as a function of the groove depth and the fill factor for TE and TM polarization. (a), (b) The numerical calculation by RCWA; (c), (d) the results according to the optical thin-film model of Eq. (10); (e), (f) the results according to the nonsymmetrical Fabry–Perot resonator model of Eq. (10). 10 January 2014 / Vol. 53, No. 2 / APPLIED OPTICS
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a highly efficient polarization beam splitter can be easily designed with the optimal grating groove depth [19]. More importantly, the simple modal method with the consideration of the reflection effect of the grating modes can precisely estimate this phenomenon, as shown in Figs. 12(d) and 12(f). Also, this effect can be convincingly explained based on the modal method with the mode interference mechanism [10]. 4. Summary
The validity and accuracy of the simple modal method without and with the reflection effect of the grating propagating modes for a rectangular transmission grating have been evaluated by a comparison of diffraction efficiencies predicted from the modal approach with exact results calculated by RCWA. The effect of grating parameters, including the normalized period, the normalized groove depth, and the fill factor, on the accuracy of the simplified modal method have been quantitatively determined. Based on the analogy of the optical thin film and the nonsymmetrical Fabry–Perot resonator, the reflection effect of two propagating grating modes has been clearly demonstrated. The physical explanation of the diffraction process is offered by a two-beam mode interference mechanism with the consideration of reflection loss using the thin-film model and Fabry– Perot model. It is found that the accuracy of the simple modal method is obviously improved by the consideration of the mode reflection effect. In general, the thin-film model for the reflection effect of grating modes is accurate at the smaller normalized period for both TE and TM polarizations. However, the Fabry–Perot model of mode reflection can calculate exactly the diffraction efficiencies for the larger normalized period. Therefore, although RCWA always provides the exact solution for any polarization and any element profile in the analysis of gratings, the simple modal method considering the reflection effect of grating modes can also be easily and accurately used to design a grating device and give a clear understanding of the physical mechanism of the diffraction process. The authors acknowledge the support from the Zhejiang Provincial National Natural Science Foundation of China (LY13F050003), the National Natural Science Foundation of China (NSFC) (No. 61308090), overseas students preferred funding of activities of science and technology project, the LED lighting new technological innovation team of Zhejiang Province of China (No. 2010R50020), and the Plan of the National Science and Technology (No. 2011BAF06B02-1). References 1. T. Clausnitzer, J. Limpert, K. Zöllner, H. Zellmer, H.-J. Fuchs, E.-B. Kley, A. Tünnermann, M. Jupé, and D. Ristau, “Highly-efficient transmission gratings in fused silica
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for chirped pulse amplification systems,” Appl. Opt. 42, 6934–6938 (2003). F. Zhu, D. Zhang, P. Fan, L. Li, and Y. Guo, “Non-uniform strain measurement along a fiber Bragg grating using optical frequency domain reflectometry,” Chin. Opt. Lett. 11, 100603 (2013). T. Osuch and Z. Jaroszewicz, “Numerical analysis of apodized fiber Bragg gratings formation using phase mask with variable diffraction efficiency,” Opt. Commun. 284, 567–572 (2011). P. E. Dyer, R. J. Farley, and R. Giedl, “Analysis of grating formation with excimer laser irradiated phase masks,” Opt. Commun. 115, 327–334 (1995). T. Osuch and Z. Jaroszewicz, “Analysis of the Talbot effect in apodized diffractive optical elements,” Photonics Lett. Pol. 1, 190–192 (2009). J. Zheng, C. Zhou, B. Wang, and J. Feng, “Beam splitting of low-contrast binary gratings under second Bragg angle incidence,” J. Opt. Soc. Am. A 25, 1075–1083 (2008). J. Feng, C. Zhou, J. Zheng, H. Cao, and P. Lv, “Design and fabrication of a polarization-independent two-port beam splitter,” Appl. Opt. 48, 5636–5641 (2009). M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982). J. Ma, S. Liu, Y. Jin, C. Xu, J. Shao, and Z. Fan, “Novel method for design of surface relief guided-mode resonant gratings at normal incidence,” Opt. Commun. 281, 3295–3300 (2008). T. Clausnitzer, T. Kämpfe, E.-B. Kley, A. Tünnermann, U. Peschel, A. V. Tishchenko, and O. Parriaux, “An intelligible explanation of highly-efficient diffraction in deep dielectric rectangular transmission gratings,” Opt. Express 13, 10448–10456 (2005). I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981). A. V. Tishchenko, “Phenomenological representation of deep and high contrast lamellar gratings by means of the modal method,” Opt. Quantum Electron. 37, 309–330 (2005). J. Wu, C. Zhou, H. Cao, A. Hu, W. Sun, and W. Jia, “Simplified mode analysis of guided mode resonance gratings with asymmetric coatings,” Chin. Opt. Lett. 11, 060501 (2013). X. Jing and Y. Jin, “Transmittance analysis of diffraction phase grating,” Appl. Opt. 50, C11–C18 (2011). J. Feng, C. Zhou, J. Zheng, and B. Wang, “Modal analysis of deep-etched low-contrast two-port beam splitter grating,” Opt. Commun. 281, 5298–5301 (2008). B. Wang, C. Zhou, J. Feng, H. Ru, and J. Zheng, “Wideband two port beam splitter of a binary fused silica phase grating,” Appl. Opt. 47, 4004–4008 (2008). T. Clausnitzer, T. Kampfe, E. B. Kley, A. Tunnermann, A. V. Tishchenko, and O. Parriaux, “Highly-dispersive dielectric transmission gratings with 100% diffraction efficiency,” Opt. Express 16, 5577–5584 (2008). I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981). T. Clausnitzer, T. Kämpfe, E.-B. Kley, A. Tünnermann, A. Tishchenko, and O. Parriaux, “Investigation of the polarization-dependent diffraction of deep dielectric rectangular gratings illuminated in Littrow mounting,” Appl. Opt. 46, 819–826 (2007). J. Feng, C. Zhou, B. Wang, J. Zheng, W. Jia, H. Cao, and P. Lv, “Three port beam splitter of a binary fused silica grating,” Appl. Opt. 47, 6638–6643 (2008). K. R. Catchpole, “A conceptual model of the diffuse transmittance of lamellar diffraction gratings on solar cells,” J. Appl. Phys. 102, 013102 (2007). H. T. Nguyen, B. W. Shore, S. J. Bryan, J. A. Britten, R. D. Boyd, and M. D. Perry, “High efficiency fused silica transmission gratings,” Opt. Lett. 22, 142–144 (1997). H. A. Macleod, Thin-Film Optical Filters (Institute of Physics, 2001), p. 41.
Institute of Optoelectronic Technology, China Jiliang University, Hangzhou 310018, China 2
Optic and Laser Division, National Institute of Metrology, Beijing 100013, China
3
College of Materials Science and Engineering, China Jiliang University, Hangzhou 310018, China *Corresponding author: [email protected] Received 9 October 2013; revised 28 November 2013; accepted 6 December 2013; posted 11 December 2013 (Doc. ID 199131); published 9 January 2014
To accurately and easily design the diffraction characteristics of a rectangular transmission grating under the illumination of Littrow mounting, the validity and limitation of the simplified modal method is evaluated by a comparison of diffraction efficiencies predicted by the modal approach to exact results calculated with rigorous coupled-wave analysis. The influence of the grating normalized period, the normalized groove depth, and the fill factor on the accuracy of the modal method is quantitatively determined. More importantly, the reflection effect of two propagating grating modes with the optical thin-film model and the nonsymmetrical Fabry–Perot model is proposed and applied in the modal method to improve the accuracy of the calculated diffraction efficiencies. Generally, it is found that the thin-film model of reflection loss is valid at the smaller normalized period, but the Fabry–Perot model can exactly calculate the reflection loss of grating modes at the larger normalized period. Based on the fact that the validity of the modal approach is determined independently of the incident wavelength, the exact design and analysis of grating diffraction elements can be implemented at different wavelengths by simply scaling the grating parameters. Moreover, the polarization effect of diffraction properties on the limitation of the modal method without and with the reflection loss of grating modes is clearly demonstrated. © 2014 Optical Society of America OCIS codes: (050.0050) Diffraction and gratings; (050.1380) Binary optics; (050.1950) Diffraction gratings; (050.1960) Diffraction theory; (050.2065) Effective medium theory. http://dx.doi.org/10.1364/AO.53.000259
1. Introduction
Diffraction transmission gratings with rectangular grooves etched into a monolithic dielectric material are interesting and applicable in a large number of different applications [1,2], such as high-power laser systems, integrated optics, holography, optical information processing, and fiber Bragg grating (FBG) fabrication using the phase mask method [3–5]. There are some studies where theoretical and experimental analyses of diffraction characteristics, 1559-128X/14/020259-10$15.00/0 © 2014 Optical Society of America
especially field distribution behind the phase mask [3], are presented. For example, the scalar diffraction theory can be used and is good enough to estimate the diffraction performance in FBG fabrication, although the normalized grating pitch in the case of the phase mask is 4 [3–5]. Particularly, binary diffraction gratings in the resonance region with the period comparable to the incident wavelength have been widely investigated theoretically and experimentally [6]. These subwavelength gratings can realize various optical functions by simply optimizing their profile parameters, e.g., polarizing beam splitters [7]. In general, for the gratings in the resonance region, the rigorous vector 10 January 2014 / Vol. 53, No. 2 / APPLIED OPTICS
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methods that directly solve Maxwell’s equations have to be applied for calculating exact diffraction characteristics. For example, rigorous coupled-wave analysis (RCWA) is frequently used to calculate exactly the optical properties of gratings [8,9]. It gives us an accurate solution of Maxwell’s equations with mathematical expansion of the field inside the grating region. However, RCWA is more computationally intensive and cannot be easily inverted to solve the inverse diffraction problem [10]. Furthermore, this numerical treatment can not give much physical insight into the diffraction processes inside the grating region. Fortunately, there is a simple method, i.e., the modal method [11–13], that has been developed and was applied to dielectric gratings with the feature size in the resonance region. In this scale of grating period, it was reported that the scalar diffraction theory and the effective medium theory cannot be used to effectively analyze the diffraction characteristics of gratings [14]. Thus, the modal method fills the blank of simple analysis methods with the grating period comparable to the incident wavelength. This approach can present a clear physical diffraction process by taking the modes of a periodic planar waveguide as the grating modes [6]. Recently, Clausnitzer et al. proposed a simplified modal method that reduces the difficult diffraction process to an easy and intelligible modal interference mechanism [10]. This simplified method enables a grating design without complicated numerical calculations, and it has been warmly embraced because it is easy to use and lacks computational strain. Although the modal method is widely used to approximately design and to analyze diffractive phase elements in the literature [15], no one has fully studied quantitatively its accuracy in the design and analysis of transmission dielectric gratings, resulting in erroneous element designs and disappointing experimental diffraction results. Therefore, it is important and necessary to determine at what normalized period and at what normalized groove depth the modal method will yield inaccurate results. The quantitative error between the modal method and the rigorous treatment should be clearly demonstrated for the accurate design of diffraction characteristics by using the simple method. Importantly, in the case of the design and analysis of a rectangular transmission grating, the reflection losses at grating interfaces are often neglected by using the simplified modal method [6,7,10,16], leading to the outcome that the calculated diffraction efficiencies are always obviously different from the real results. In particular, the reflection of the incident wavelength increases significantly when the grating period approaches λ∕2 [17]. Therefore, the reflection loss of the grating modes in the modal method must be considered to enhance the accuracy of analyzed results. In this present paper, the reflection effect of the effective index modes at the grating interfaces will be clearly demonstrated by using the effective thin-film theory and nonsymmetrical Fabry–Perot resonator model, respectively. 260
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2. Simplified Modal Method
The binary rectangular grating is shown in Fig. 1. A plane wave is incident from air with the incident angle of Littrow configuration θi arcsinλ∕2n0 Λ, where λ is the incident wavelength, n0 is the refractive index of air with the value of 1.0, and Λ is the grating period. The fill factor f of the grating is defined as the ratio of the grating ridge to the period, and the depth of the grating is h. The refractive index ns of the substrate material of fused silica is 1.45 at the incident wavelength 1064 nm. The refractive index ng is the same as that of the substrate. The grating material of fused silica is chosen based on its excellent transparency, with little dispersion change from deep UV to far IR for the design of transmission gratings, and its higher laser damage threshold. It has extensive potential applications not only in laser communication but also in high-power laser systems [1]. In this paper, only two transmitted propagating orders, the zeroth order and the −1st order, are restricted by the grating profile fulfilling the condition of λ∕2 < Λ < 3λ∕2ns. The diffraction angles are θ0 and θ−1 , respectively. The highest diffraction efficiency in the zeroth order or the −1st order can be achieved by optimizing grating parameters. According to the modal theory [6,10–12,18], there are only two excited propagating grating modes that dominate the diffraction process when the grating periods are small enough and comparable to the incident wavelength. Each of the propagating modes carries nearly half of the energy of the incident wave [19]. These two equally excited modes can be named the 0th mode and 1st mode. The detailed modal method has been described in the relevant literature [10–12]. On the basis of the two-beam mode interference mechanism in the simplified modal method [10], the diffraction efficiency can be mainly determined by the accumulated phase difference Δφ of the two excited modes. Generally, many relevant reports ignored any reflection losses of the rectangular grating interfaces based on the low index contrast [6,7,10,16]. However, it will be shown in later sections that the reflection effect of the grating propagating modes is significant and should be considered to accurately analyze the diffraction characteristics of rectangular gratings. First, the reflection effect at the grating interfaces is neglected to quantify the error of the
Fig. 1. Schematic of a rectangular grating under the Littrow configuration illumination.
calculated diffraction efficiency between the modal method and RCWA. The efficiencies of the transmitted zeroth diffraction order and −1st diffraction order can be represented as [19] η0T cos2 Δφ∕2;
(1a)
η−1T sin2 Δφ∕2
(1b)
Δφ k0 n0eff − n1eff h 2πn0eff − n1eff q;
(2)
with
where ηmT is the diffraction efficiency of the mthorder (m 0, −1), k0 is the wavenumber in vacuum k0 2π∕λ, and n0eff and n1eff are the corresponding effective indices of the 0th propagating mode and the 1st propagating mode, respectively. The normalized groove depth is also defined by q h∕λ in Eq. (2). The corresponding effective mode indices neff depend on the grating period, the fill factor, the incident wavelength, and the polarization of incident light upon the grating. They can be determined by solving the following eigenfunction equation for TE polarization (electric field vector perpendicular to the incident plane) as [15] Fn2eff cos2πp sin θi ;
(3)
Fn2eff cos2πβpf cos2πγp1 − f β2 γ 2 sin2πβpf sin2πγp1 − f ; (4) 2βγ q q where β n2g − n2eff , γ n20 − n2eff , and p Λ∕λ. The right-hand part of Eq. (3) indicates the incidence condition, where cos2πp sin θi −1 is the Littrow mounting incidence and cos2πp sin θi 1 is under normal incidence. The normalized period is defined as p Λ∕λ. Moreover, the effective modes index with n2eff > 0 is propagating through the grating, while those with n2eff < 0 are evanescent. For TM polarization (magnetic field vector perpendicular to the incident plane), the dispersion equation can be expressed as −
Fn2eff cos2πβpf cos2πγp1−f −
n4g β2 n40 γ 2 sin2πβpf sin2πγp1−f : (5) 2n2g n20 βγ
Based on the definition of the normalized period p, the effective indices of the grating modes for both TE and TM polarizations do not depend on the value of Λ or λ individually but depend on the value of p. And the accumulated phase difference in Eq. (2) between the 0th mode and the 1st mode depend on q rather than the individual value of h or λ. Therefore, in the design and analysis of gratings, the diffraction characteristics are independent of the real grating period,
the groove depth, and the incident wavelength, but they are dependent on the normalized period and the normalized groove depth. So the same diffraction properties at different wavelengths can be obtained by simply scaling the grating parameters without the consideration of material dispersion. To quantify how element period, element groove depth, and polarization of incident light affect the validity of the simplified modal method without the consideration of the reflection effect of the grating modes, we present plots of zeroth-order and −1st-order transmission efficiencies, respectively, versus the normalized period and the normalized groove depth at the Littrow mounting. The fill factor of the grating is set to be 0.5, at which the gratings are often fabricated easily by holographic lithography. It is well known that RCWA is rigorous and accurate in the calculation of the diffraction efficiency of grating elements [8]. So the results from RCWA are regarded as a comparison criterion to determine the validity of the modal method in this paper. Figure 2 shows the comparison of the diffraction efficiencies on the normalized period between the simplified modal method based on Eq. (1) and RCWA for both TE and TM polarizations. The normalized period is confined to the range of 0.5 < p < 1.0, in which only two propagating grating modes play a dominant role at the Littrow configuration, and other grating modes are evanescent. It should be noted that beyond this normalized period the multiple-mode interference mechanism can also be used to analyze the diffraction process of gratings [6,20,21]. In this paper, we concentrate on the twobeam mode interference mechanism. In Fig. 2, the normalized groove depth q 1.0 is chosen. It can be seen in Fig. 2(a) for TE polarization that although the zeroth-order diffraction efficiencies predicted by the modal method are in agreement with the results from RCWA, with the error less than about 2%, the maximum difference of transmittance can reach more than 40% at about p 0.51 for the −1st-order diffraction. Also, in Fig. 2(a) the discrepancy of the −1storder diffraction efficiencies between RCWA and the modal method trends to a constant of about 6% when the normalized period is more than about 0.6. It is interesting for TM polarization in Fig. 2(b) that the −1st-order diffraction efficiency from the modal method agrees well with that of RCWA, but the maximum error of transmittances for the zeroth-order diffraction reaches about 20%. When the normalized period is beyond about 0.65, the zeroth-order diffraction efficiencies estimated by the simplified modal method are in good agreement with those of RCWA. Thus, it is clear that the results estimated by the modal approach are obviously larger than those from the rigorous treatment. This illustrates distinctly the existence of reflection loss of the grating modes, and the reflection effect should be included in the modal method to enhance its accuracy of the calculated diffraction efficiencies. To display the limitation of the simplified modal method for a rectangular grating without the 10 January 2014 / Vol. 53, No. 2 / APPLIED OPTICS
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Fig. 2. Comparison of diffraction efficiencies between the modal method and RCWA versus the normalized grating period with a fill factor of 0.5. (a) TE polarization and (b) TM polarization.
consideration of reflection effect with respect to the normalized groove depth in detail, the comparison of diffraction efficiencies calculated by the modal method and RCWA is shown in Fig. 3. The fill factor and the normalized period are set to be 0.5 and 0.7, respectively. For TE polarization in Fig. 3(a), the zeroth-order diffraction efficiencies calculated by the modal method are in agreement well with that of RCWA at about 0.3 ≤ h∕λ ≤ 1.5, beyond which the maximum error can reach more than 10%. The estimated results from the modal method can agree well with those from the rigorous vector method at about h∕λ ≤ 0.5 for zeroth-order diffraction. The larger error between both methods is shown at a deeper groove depth. For TM polarization in Fig. 3(b) the validity of the simplified modal method on the normalized groove depth is clearly shown. From Fig. 3, it can be derived that not only can the simplified modal method be utilized in the design and analysis of a deeper diffraction grating [10], but also at the smaller groove depth it can be used to approximately calculate the diffraction efficiency. Therefore, without consideration of the reflection losses at the interface of the grating, the calculated diffraction efficiencies by the modal method are clearly different from those by the rigorous vector theory, especially at the smaller normalized period and the larger normalized groove depth. These larger errors would result in inaccuracy in the design and analysis of gratings by using the modal method. In order to improve the validity of the simplified modal method, the reflection effect of the propagating modes at the grating interfaces must be strictly considered and quantitatively investigated.
3. Consideration of Mode Reflection
According to Eq. (1), the diffraction efficiencies in the zeroth order or −1st order can achieve 100%. However, the maximum diffraction efficiency of the −1st order with the optimal grating parameters in the experiment measurement is much less than 100% [22]. In addition, based on the comparison of diffraction efficiencies between RCWA and the modal method without the consideration of reflection loss, as shown in above section, the existence of the reflection effect of propagating modes for the rectangular transmission grating is implied. Also, Clausnitzer et al. reported that the reflection losses of rectangular transmission gratings increase rapidly as the ratio between the period and the wavelength is decreased [17]. In order to precisely use the simple modal method to accurately design and analyze the diffraction efficiencies of a grating, the reflection loss of the effective index modes must be quantitatively determined. Based on Ref. [19], it is known that the effective index nair eff of the incident light in air is cos θ i , and the effective index nsub eff of the substrate material in the modal method is ns cos θ0i, where θ0i is the refracted angle. The grating structure shown in Fig. 1 can be equivalent to a thin film according to the effective medium theory. The reflection loss of the effective index modes can be calculated analogously to Fresnel reflection. At the air–grating interface, two propagating grating modes, 0th mode and 1st mode, can be excited equally by the incident wave. Both modes couple to the two transmitted diffraction orders at the grating–substrate interface. Because
Fig. 3. Comparison of transmittance characteristics between the modal method and RCWA as a function of the normalized groove depth at a normalized period of 0.7. (a) TE polarization and (b) TM polarization. 262
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the diffraction process in the grating region is very similar to a Mach–Zehnder interferometer [10], the reflection effect of the two grating modes can be considered separately. Therefore, according to the Fresnel reflection, the reflection of each of the grating propagating modes at the air–grating interface m Rm air or at the grating–substrate interface Rsub can be represented by [17] Rm air∕sub
air∕sub nm eff − neff
2 ;
air∕sub nm eff − neff
(6)
where m indicates the number of the mode (m 0, m 1). Figure 4 shows the calculated reflectivity of the 0th and the 1st propagating grating modes as a function of the fill factor at the air–grating and the grating–substrate interfaces for TE and TM polarizations, respectively. The 800 nm grating period with the illumination of a 1064 nm wavelength is assumed. It is shown for both polarization states that at the air– grating interface the reflection losses of the 0th mode are higher than those of the 1st mode, but at the grating–substrate interface the situation of the reflection effect for both modes is the opposite. Although the reflections of the 1st mode at the air–grating interface for about f > 0.4 and the 0th mode at the grating– substrate interface for about f < 0.4 can be neglected for TE polarization, as shown in Fig. 4(a), both grating propagating modes in the analysis of diffraction characteristics of the whole grating element including the air–grating and grating–substrate interfaces should be considered. In order to precisely calculate the reflection loss of the grating modes for the whole grating device as shown in Fig. 1, the thin-film model or the nonsymmetrical Fabry–Perot resonator model can be used. First, the subwavelength diffraction grating can be approximated as a lamellar thin-film layer. According to the matrix method in the film theory [23], the characteristic matrix of this equivalent layers is
B cos δ C iη sin δ
i sin δ∕η cos δ
1 ; ηs
(7)
where δ 2πhnm eff ∕λ is the effective phase for the 0 1 equivalent thin-film layer, and nm eff is neff or neff .
η η0 nm eff , η0 is the optical admittance in free space (η0 ε0 ∕μ0 1∕2 2.6544 × 10−3 S), and ηs η0 nsub eff is the optical admittance of the effective index of substrate. Letting Y be C∕B, the reflectance of the grating propagating modes on the whole grating system including the air–grating and the grating–substrate interfaces is [23] sub sub η0 nair η0 nair 1 eff − Yneff eff − Yneff m : (8) RTH · sub sub 2 η0 nair η0 nair eff Yneff eff Yneff Rm TH is the reflection of the corresponding grating modes in the thin-film model. The asterisk in Eq. (8) indicates the conjugate complex number. Due to the fact that each of both propagating grating modes carries separately half of the incident energy, the multiplier 1∕2 in Eq. (8) is added. Second, the reflection of the excited modes also can be described by a nonsymmetrical Fabry–Perot resonator that is, respectively, filled with the medium of effective refractive indices n0eff and n1eff . The reflectivity Rm FP of both grating modes can be represented as [17] p p2 p 2 2π m m Rm 4 Rm Rm air − air Rsub sin sub λ neff h 1 : · Rm p p FP 2 1 − Rm Rm 2 4 Rm Rm sin2 2π nm h air
sub
air
sub
λ
eff
(9) Rm FP indicates the reflectivity of the 0th grating mode or the 1st mode in the nonsymmetrical Fabry–Perot resonator model. In Fig. 5, the reflectivity of grating modes according to Eq. (8) of the thin-film model is, respectively, plotted as a function of the groove depth and the grating period. Here, the fill factor 0.5 of the rectangular transmission grating is assumed. TE polarization is considered. It is shown that these reflection losses of grating modes increase rapidly as the normalized period is decreased, especially at the grating period approaching λ∕2. Also, the reflection of both grating modes at a fixed grating period indicates the behavior of slight oscillation with respect to the groove depth. Initial results show that the reflection characteristics of grating modes for the TM case are comparable with those of the TE case. Also, the reflectivity properties calculated by Eq. (9) of
Fig. 4. Reflection of the two propagating grating modes at the air–grating and the grating–substrate interfaces with a grating period of 800 nm. (a) TE polarization and (b) TM polarization. 10 January 2014 / Vol. 53, No. 2 / APPLIED OPTICS
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Fig. 5. Simulation of the reflection losses based on Eq. (8) as a function of the groove depth and the normalized period for TE polarization with a fill factor of 0.5. (a) The zeroth mode and (b) the first mode.
Fabry–Perot resonator model on the normalized period and the groove depth are similar to those of the thin-film model. Figure 6 shows a numerical simulation of the reflection losses of the grating modes as a function of the groove depth and the fill factor by using the thin-film model of Eq. (8). A grating period of 800 nm and incident wavelength of 1064 nm are assumed. The plots reveal an increase of the reflectivity with the increased grating fill factor. Thus, in order to realize a highly efficient transmission grating, the smaller fill factor of the grating should be fulfilled. The obvious oscillation of the reflection of the grating modes as a function of the groove depth is demonstrated at the smaller fill factor of the grating. This phenomenon can be attributed to the interference effect of the grating mode itself. However, the reflection characteristics of the modes from the Fabry–Perot resonator model, as shown in Fig. 7, are clearly different from those of the thin-film model.
The incident wavelength in Fig. 7 is 1064 nm. It indicates that these theoretical models of the reflection effect of grating modes would be valid at the specific condition of grating parameters. Based on Eq. (1) and the reflection losses of the grating modes, the diffraction efficiencies of the whole grating element without the absorption loss of the material can be expressed as η0TH∕FP η0T · 1 − R0TH∕FP · 1 − R1TH∕FP ;
(10a)
TH∕FP η−1T · 1 − R0TH∕FP · 1 − R1TH∕FP ; η−1
(10b)
where η0TH∕FP is the transmittance of the thin-film model or the Fabry–Perot resonator model for the TH∕FP shows the transzeroth-order diffraction, and η−1 mittance of the −1st-order diffraction of the rectangular grating. According to Eq. (10), the diffraction efficiency, comprising the reflection losses of grating modes, is
Fig. 6. Reflection of grating modes calculated by Eq. (8) with a grating period of 800 nm for (a), (b) TE polarization and (c), (d) TM polarization. 264
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Fig. 7. Reflection of grating modes calculated by Eq. (9) with a grating period of 800 nm for (a), (b) TE polarization and (c), (d) TM polarization.
calculated as a function of the normalized period for both mode reflection models, as shown in Fig. 8. A normalized groove depth of 1.0 is assumed. For TE polarization in Fig. 8(a), the transmittances calculated by the thin-film model for the zeroth diffraction order and −1 st diffraction order are in agreement well with those from RCWA at about 0.5 < Λ∕λ < 0.7. Beyond this range the nonsymmetrical Fabry–Perot resonator model is precise and valid. Compared with Fig. 2(a) without the consideration of the reflection effect of grating modes, the calculated diffraction efficiencies with the reflection losses of the thin film model are clearly accurate, especially at the smaller normalized period for the −1st-order diffraction. For TM polarization in Fig. 8(b), the estimated transmittances by the thin film modal model agree well with the rigorous vector results at about 0.5 < Λ∕λ < 0.55 for both diffraction orders. When the normalized period is more than about 0.55, the transmittance error for the −1 st-order diffraction remains nearly constant and is less than 10%. As the normalized period is
more than about 0.65 for the nonsymmetrical Fabry– Perot resonator model, the diffraction efficiencies are coincident with the results from RCWA. Thus, the consideration of the reflection effect of grating propagating modes makes the simplified modal method more accurate in the analysis of diffraction characteristics. Generally, the reflection losses calculated by the optical thin-film model are precise and valid at the smaller normalized period. This approximately corresponds to the validity range of effective medium theory, in which the binary grating structure can be exactly equivalent to one kind of medium at the smaller grating period [14]. However, the nonsymmetrical Fabry–Perot resonator model can accurately calculate the reflection losses at the larger normalized period of rectangular transmission gratings. Furthermore, the comparison of diffraction efficiencies from the rigorous method and the simplified modal method with the reflection losses of the grating modes is demonstrated as a function of the normalized groove depth at a normalized period of 0.7 in
Fig. 8. Comparison of transmittance characteristics between the modal method with the consideration of the reflection loss of grating modes and RCWA. (a) TE polarization and (b) TM polarization. 10 January 2014 / Vol. 53, No. 2 / APPLIED OPTICS
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Fig. 9. Comparison of diffraction efficiencies between RCWA and the modal method at a normalized period of 0.7. The key for the lines is the same as in Fig. 8. (a) TE polarization and (b) TM polarization.
Fig. 9. For TE polarization in Fig. 9(a), the results calculated by the modal method almost completely coincide with that of RCWA for −1st-order diffraction at 0 ≤ h∕λ ≤ 1.0 regardless of the reflection model of the grating modes. When the normalized groove depth is less than about 0.2 for the zeroth-order diffraction, the modal method is invalid. In contrast to Fig. 2(a) without the reflection effect, the diffraction efficiencies obtained by the modal method with the reflection losses of the grating modes is distinctly closer to the results of the rigorous method, especially at the maximum extrema of transmittances of the zeroth-order or the −1st-order diffraction. For TM polarization in Fig. 9(b), the slight fluctuation of diffraction efficiencies obtained from the modal method is clearly displayed due to the interference effect of the grating modes. Figure 10 demonstrates quantitatively the error of diffraction efficiencies for RCWA and the simplified modal method with and without the consideration of reflection loss of the grating modes as a function of the normalized period. In the simplified modal method with the reflection loss of the grating modes, the thin-film reflection model and Fabry– Perot reflection model are used. The difference of transmittances between RCWA and the modal method without the reflection loss is also included in Fig. 10 for reference. The error of diffraction effim m ciency is defined as ηm TF∕FP − ηRCWA , in which ηTF∕FP indicates the mth-order transmittance of the thinfilm model or Fabry–Perot model, and ηm RCWA denotes the mth-order transmitted diffraction efficiency of RCWA. For TE polarization in Fig. 10(a), the error between the thin-film model and RCWA can be
decreased slightly compared with the difference of the simplified modal method without reflection loss and RCWA for zeroth-order transmittance at the smaller normalized period. For the thin-film model and Fabry–Perot model, the difference is significantly decreased for the −1st-order diffraction at the smaller period compared with the modal method without the reflection loss of the grating modes. In Fig. 10(b) for TM polarization, it can be seen that the Fabry–Perot model can be applied in the larger normalized period to accurately calculate diffraction efficiencies. Thus, the accuracy of calculated diffraction efficiency by the simplified modal method can be generally enhanced by the consideration of the reflection loss of the grating modes. Furthermore, the quantitative difference between RCWA and the simplified modal method with the reflection loss of the grating modes with respect to the normalized groove depth is demonstrated in Fig. 11. It can be seen that the difference of diffraction efficiency between RCWA and the simplified modal method exhibits clear oscillating behavior as a function of the normalized grating depth. The error of diffraction efficiency between the rigorous vector method and the simplified modal method with the reflection loss of the grating modes can obviously be decreased in −1st-order diffraction for both TE polarization and TM polarization. In order to check the accuracy of the simple modal approach on the fill factor, the diffraction efficiencies are numerically simulated as a function of the fill factor and the groove depth for TE and TM polarization in Fig. 12. The transmittance images of −1st-order diffraction with a grating period of 600 nm under the illumination of a 1064 nm wavelength are
Fig. 10. Error of diffraction efficiency comparison between the simplified modal method with the consideration of reflection loss by the thin-film model and the Fabry–Perot model and RCWA versus the normalized period. (a) TE polarization and (b) TM polarization. 266
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Fig. 11. Error of diffraction efficiency comparison between the simplified modal method with the consideration of reflection loss of the thin-film model and Fabry–Perot model and RCWA versus the normalized groove depth. (a) TE polarization and (b) TM polarization.
calculated by RCWA and the simple modal method, respectively. Figures 12(a) and 12(b) show the results of the rigorous vector method. The optical thin-film model of grating mode reflection in the simplified modal method is used in Figs. 12(c) and 12(d) for both polarization cases. It is shown in Figs. 12(e) and 12(f) that the diffraction efficiencies are calculated by the modal method with the nonsymmetrical Fabry–Perot resonator model of reflection loss of the grating modes. In Fig. 12, the images demonstrate that the
diffraction behavior of the rectangular transmission gratings can be described well by the simple modal method with the consideration of the reflection effect of the grating modes. It is interesting for TM polarization in Fig. 12(b) that the transmittances of the −1st order almost approaches zero as the fill factor nears 0.5 regardless of the groove depth, while the TE polarization in Fig. 12(a) exhibits a dependence of groove depth in −1st-order diffraction. By using this surprising effect,
Fig. 12. Simulation of the diffraction efficiencies of a rectangular transmission grating with a 600 nm period as a function of the groove depth and the fill factor for TE and TM polarization. (a), (b) The numerical calculation by RCWA; (c), (d) the results according to the optical thin-film model of Eq. (10); (e), (f) the results according to the nonsymmetrical Fabry–Perot resonator model of Eq. (10). 10 January 2014 / Vol. 53, No. 2 / APPLIED OPTICS
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a highly efficient polarization beam splitter can be easily designed with the optimal grating groove depth [19]. More importantly, the simple modal method with the consideration of the reflection effect of the grating modes can precisely estimate this phenomenon, as shown in Figs. 12(d) and 12(f). Also, this effect can be convincingly explained based on the modal method with the mode interference mechanism [10]. 4. Summary
The validity and accuracy of the simple modal method without and with the reflection effect of the grating propagating modes for a rectangular transmission grating have been evaluated by a comparison of diffraction efficiencies predicted from the modal approach with exact results calculated by RCWA. The effect of grating parameters, including the normalized period, the normalized groove depth, and the fill factor, on the accuracy of the simplified modal method have been quantitatively determined. Based on the analogy of the optical thin film and the nonsymmetrical Fabry–Perot resonator, the reflection effect of two propagating grating modes has been clearly demonstrated. The physical explanation of the diffraction process is offered by a two-beam mode interference mechanism with the consideration of reflection loss using the thin-film model and Fabry– Perot model. It is found that the accuracy of the simple modal method is obviously improved by the consideration of the mode reflection effect. In general, the thin-film model for the reflection effect of grating modes is accurate at the smaller normalized period for both TE and TM polarizations. However, the Fabry–Perot model of mode reflection can calculate exactly the diffraction efficiencies for the larger normalized period. Therefore, although RCWA always provides the exact solution for any polarization and any element profile in the analysis of gratings, the simple modal method considering the reflection effect of grating modes can also be easily and accurately used to design a grating device and give a clear understanding of the physical mechanism of the diffraction process. The authors acknowledge the support from the Zhejiang Provincial National Natural Science Foundation of China (LY13F050003), the National Natural Science Foundation of China (NSFC) (No. 61308090), overseas students preferred funding of activities of science and technology project, the LED lighting new technological innovation team of Zhejiang Province of China (No. 2010R50020), and the Plan of the National Science and Technology (No. 2011BAF06B02-1). References 1. T. Clausnitzer, J. Limpert, K. Zöllner, H. Zellmer, H.-J. Fuchs, E.-B. Kley, A. Tünnermann, M. Jupé, and D. Ristau, “Highly-efficient transmission gratings in fused silica
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