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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 5 — Mar. 2, 2009
  • pp: 3063–3077
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Broadband antireflective structures applied to high resistive float zone silicon in the THz spectral range

Claudia Brückner, Thomas Käsebier, Boris Pradarutti, Stefan Riehemann, Gunther Notni, Ernst-Bernhard Kley, and Andreas Tünnermann  »View Author Affiliations


Optics Express, Vol. 17, Issue 5, pp. 3063-3077 (2009)
http://dx.doi.org/10.1364/OE.17.003063


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Abstract

The optimal structural parameters for an antireflective structure in high resistive float zone silicon are deduced for a rectangular and a hexagonal structure. For this the dependence of the effective index from the filling factor was calculated for both grating types. The structures were manufactured by the Bosch®-process. The required structural parameters for a continuous profile require an adaption of the fabrication process. Challenges are the depth and the slight positive slope of the structures. Starting point for the realization of the antireflective structures was the manufacturing of deep binary gratings. A rectangular structure and a hexagonal structure with period 50 μm and depth 500 μm were realized. Measurements with a THz time domain spectroscopy setup show an increase of the electric field amplitude of 15.2% for the rectangular grating and 21.76% for the hexagonal grating. The spectral analysis shows that the bandwidth of the hexagonal grating reaches from 0.1 to 2 THz.

© 2009 Optical Society of America

1. Introduction

High-resistive float-zone (HRFZ) silicon is used for lenses and beam splitters in the THz spectral range. In [1

1. D. Grischkowsky, S. Keiding, M. van Exter, and C. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am . B 7, 2006–2015 (1990). [CrossRef]

] the refractive index was given as 3.418 for the spectral range of 0.2-2 THz. The absorption coefficient was determined to be less than 0.05 cm-1 for the range of 0.2-2 THz. A very precise measurement of the refractive index and absorption coefficient is given in [2

2. J. Dai, J. Zhang, W. Zhang, and D. Grischkowsky, “Terahertz time-domain spectroscopy characterization of the far-infrared absorption and index of refraction of high-resistivity, float-zone silicon,” J. Opt. Soc. Am . B 21, 1379–1386 (2004). [CrossRef]

]. There, the refractive index was determined to 3.4175 for the spectral range of 0.5-4.5 THz and the power absorption coefficient was determined to be below 0.01 cm-1 for 0.2-1 THz and below 0.025 cm-1 for 0.2-2 THz. Weak absorptions bands are at 3.66 THz and 4.68 THz. A strong absorption band is at 18 THz. With this low absorption, HRFZ silicon is the dielectric material with the highest transparency in the THz spectral range [2

2. J. Dai, J. Zhang, W. Zhang, and D. Grischkowsky, “Terahertz time-domain spectroscopy characterization of the far-infrared absorption and index of refraction of high-resistivity, float-zone silicon,” J. Opt. Soc. Am . B 21, 1379–1386 (2004). [CrossRef]

]. For the considerations in this paper an accuracy of the refractive index of 3.42 is sufficient. Up to now silicon lenses are only used as substrate lenses to couple radiation out of the emitter and into the detector [3

3. M. T. Reiten, S. A. Harmon, and R. A. Cheville, “Terahertz beam propagation measured through three-dimensional amplitude profile determination,” J. Opt. Soc. Am . B 20, 2215–2225 (2003). [CrossRef]

]. Silicon exhibits a similar refractive index as other semiconductors in the THz spectral range. For example galliumarsenide has a refractive index of about 3.6 [1

1. D. Grischkowsky, S. Keiding, M. van Exter, and C. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am . B 7, 2006–2015 (1990). [CrossRef]

]. Because of the low index step at the boundary between the antenna substrate material and the silicon coupling lens the Fresnel losses are negligible. But at the transition from silicon to air, the high index step causes high Fresnel losses (30% of the incident intensity on normal incidence).

The high refractive index of HRFZ silicon would enable lenses with a high numerical aperture, which cannot be realized with plastic lenses, which have a refractive index about 1.5 in the THz spectral range [4

4. C. Brückner, B. Pradarutti, R. Müller, S. Riehemann, G. Notni, and A. Tünnermann, “Design and evaluation of a THz time domain imaging system using standard optical design software,” Appl. Opt . 47, 4994–5006 (2008). [CrossRef] [PubMed]

]. Thus, lenses from HRFZ silicon are imaginable on every arbitrary position in quasi-optical THz systems [5

5. C. Brückner, B. Pradarutti, R. Müller, S. Riehemann, G. Notni, and A. Tϼnnermann, “Design and analysis of quasioptical THz time domain imaging systems,” Proc. SPIE 7100, 71000S (2008). [CrossRef]

].

The structures are intended for optical components in THz time domain imaging systems. There, the THz pulses have a typical bandwidth from 0.1-3 THz (wavelengths: 100–3000 μm). Thus, a broadband antireflective treatment is required.

Although some research to antireflection coatings in the THz spectral region was done, an adequate (broadband) antireflective coating for silicon does not still exist.

For silicon optics in the THz spectral region a plastic layer of parylene was developed [6

6. A. J. Gatesman, J. Waldman, M. Ji, C. Musante, and S. Yngvesson, “An anti-reflection coating for silicon optics at terahertz frequencies,” IEEE Microwave Guided Wave Lett . 10, 264–266 (2000). [CrossRef]

]. But this material has only a refractive index of 1.62 in the THz spectral range. The optimal refractive index for an index matching layer is 1.85. In [7

7. I. Hosako, “Multilayer optical thin films for use at terahertz frequencies: method of fabrication,” Appl. Opt . 44, 3769–3773 (2005). [CrossRef] [PubMed]

] a multilevel antireflection coating was described for germanium. In [8

8. J. Kröll, J. Darmo, and K. Unterrainer, “Metallic wave-impedance matching layers for broadband terahertz optical systems,” Opt. Express 15, 6552–6560 (2007). [CrossRef] [PubMed]

] very thin metallic layers are used for the suppression of the pulse echoes in electro-optic sampling systems. These metallic layers suppress the pulse echoes in THz time domain systems by absorption. The main pulse is damped by the same amount that is required to damp the pulse echo completely. Every coated surface is connected with an additional power loss by the metal coating that adds to the Fresnel loss. In [9

9. W. Withayachumnankul, B. M. Fischer, S. P. Mickan, and D. Abbott, “Retrofittable antireflection coatings for T-rays,” Microwave Opt. Technol. Lett . 49, 2267–2270 (2007). [CrossRef]

] a polyethylene film was used as antireflective coating with a refractive index of 1.5 in the THz spectral range.

In this paper another possibility for an antireflective treatment is investigated. An antireflective effect can also be generated by a surface-relief structure [10

10. P. B. Clapham and M. C. Hutley, “Reduction of lens reflection by the ‘Moth Eye” principle,” Nature 244, 281–282 (1973). [CrossRef]

]. Instead of antireflective layers, an antireflective structure is always a bandpass filter. The lower wavelength limit is determined by the grating period and the upper wavelength limit is determined by the structural depth. Such structures are already realized in silicon for the solar spectral region [11

11. A. Gombert, B. Bläsi, C. Bühler, and P. Nitz, “Some application cases and related manufacturing techniques for optically functional microstructures on large areas,” Opt. Eng . 43, 2525–2533 (2004). [CrossRef]

]. But the manufacturing methods developed for the solar spectral region cannot be applied to antireflective structures in the THz range because of the required structural parameters. A broadband antireflective structure for plastic quasi-optical components in the THz spectral range is already realized [12

12. C. Brückner, B. Pradarutti, O. Stenzel, R. Steinkopf, S. Riehemann, G. Notni, and A. Tünnermann, “Broadband antireflective surface-relief structure for THz optics,” Opt. Express 15, 779–789 (2007). [CrossRef] [PubMed]

]. There the structures were manufactured by a single-point diamond turning process. This fabrication method cannot be easily transferred to silicon. Silicon is a very hard material and causes high material ablation of the diamond tool. But the main point is that due to the tool geometry only an aspect ratio of grating period to depth of 1:2 can be manufactured. An antireflective structure in silicon that covers the same bandwidth like a structure in a plastic material requires a much higher aspect ratio. This is due to the much higher refractive index of silicon.

Broadband antireflective structures can be realized as deep binary, continuously or stepped structures. The binary profile works as effective medium for a certain spectral range. With the filling factor the optimal refractive index of 1.85 can be adjusted. With the continuously profile, a gradient of the refractive index can be generated such that no discontinuity in the index profile appears. Section 2 contains theoretical considerations with respect to grating period, grating type, filling factor and grating depth. The determination of the effective index in dependence on the filling factor for 2D structures is described. The optimal parameters for a continuous profile are deduced. The antireflective structures in silicon were manufactured by deep reactive ion etching (DRIE). Because of the required structural parameters for the continuous profile an adaption of the DRIE process is necessary. Challenges are the intended depth and the low positive slope of the structures. Thus, the starting point was to manufacture deep binary structures. The effect of deep binary subwavelength gratings is investigated in Section 2 as well. In Section 3 the manufacturing process is described. In Section 4 the measuring results of a deep binary rectangular and a deep binary hexagonal structure are presented. An outlook for the fabrication of stable broadband continuous profiles is given.

2. Theory

A surface-relief grating can be used as antireflective structure if two conditions are obeyed: First the period of the structure has to be so small compared to the wavelength that only the zeroth diffraction order is able to propagate. Second the depth of the grating must be in the order of magnitude of the wavelength. If the depth is to low, in extreme case the structure works like a surface roughness and is not recognized by the radiation.

2.1 Grating period

For antireflective structures, several grating types are possible: linear, rectangular and hexagonal grating. Figure 1(a) shows a rectangular grating and Fig. 1(b) shows a hexagonal grating. Figures 1(c) and (d) show the corresponding top view to the gratings. The linear grating can be considered as special case of the rectangular grating with a period approaching infinity (Λy → ∞).

With the reciprocal grating, diffraction phenomena can easily be treated. The basis vectors of the reciprocal grating can be calculated from the basis vectors of the given Bravais grating [13

13. H. Ibach and H. Lüth, Festkörperphysik (Springer, 2002).

]. The reciprocal grating is again a Bravais grating with unit of an inverse length.

The basis vectors of the considered rectangular grating with equal grating period in x- and y-dircetion (Λ = Λx = Λy) are a 1 = (Λ,0,0)T and a 2 = (0,Λ,0)T. The corresponding basis vectors of the reciprocal grating are g 1 = (2π/Λ)u x and g 2 = (2π/Λ)u y, with u x, u y - unit vectors in x-and y-direction. The basis vectors of the hexagonal grating are a 1 = (Λ,0,0)T and a 2 = Λ·(0.5,sin(60°)0,0)T. The corresponding basis vectors of the reciprocal grating are: g 1 = 2π/(Λ sin(60°)) · (sin(60°),-0.5,0)T and g 2 = 2π/(Λ sin(60°))u y. The basis vectors of the reciprocal grating are drawn in Fig. 1(c) and (d) for the rectangular and the hexagonal grating, respectively. The reciprocal grating of the rectangular grating is again a rectangular grating with same axes alignment. The reciprocal grating of the hexagonal grating is again a hexagonal grating but with the axes rotated -30° about the z-axis (Fig. 1(d), red dots). The condition for constructive interference at a certain angle of incidence is:

ki,xykuv,xy=guv,
(1)

with k i,xy - the projection of the incident wave vector onto the xy-plane, k uv,xy - the projection of the diffracted wave vector onto the xy-plane, and g uv = u g 1 + v g 2 - the grating vector belonging to the diffraction order u,v. From Eq. (1) a condition can be deduced such that the z-component of the diffracted wave vector is imaginary, i.e. the diffracted wave is evanescent. For a zeroth-diffraction-order grating following condition must hold:

Λλfgn1sinθcos(φδi)+n22n12sin2θsin(φδi),
(2)

with λ – wavelength in vacuum, n 1 – refractive index of the incident medium (air), n 2 – refractive index of the substrate, θi - the polar angle of incidence and φ - the azimuthal angle of incidence. The factor fg depends on the grating type. For the rectangular grating the factor is fg = 1, and for the hexagonal grating the it is: fg = 1/sin(60°) = 1/0.866 = 1.155. The phase angle δi is the angle between a vector of the reciprocal grating and the basis vector g 1. For the rectangular grating, the phase angles for the two basis vectors are δ 1 = 0° and δ2 = 90°. For the hexagonal grating, the phase angles are δ1 = 0° and δ2 = 120°. For a linear grating, only one basis vector of the reciprocal grating exists with δ 1 = 0°.

From Eq. (2) it is visible that the normalized grating period Λ/λ, depends on the angle of incidence, the refractive index and the grating type.

For normal incidence (θ = 0°), it holds:

Λλfgn2=fg3.42
(3)

For arbitrary angle of incidence (θ = 90°), it holds:

Λλfgn1+n2=fg4.42
(4)

For a certain angle of incidence θ independent of the azimuthal angle φ the maximum ratio appears for (φ - δi) = 0:

Λλfgn1sinθ+n2
(5)
Fig. 1. (a) Binary rectangular grating and top view (c) with incident wave vector k i, and grating vectors a 1 and a 2. The vectors of the reciprocal grating are g 1 and g 2. (b) Binary hexagonal grating and top view (d) with grating vectors a 1 and a 2 and the vectors of the reciprocal grating g 1 and g 2. The red dots indicate the grating points of the reciprocal grating. The hexagonal grating can also be described as a rectangular grating with basis vectors a 1 and a 2 * with different grating periods in x- and y-direction. For both grating types the TE polarization is perpendicular to the plane and the TM polarization lies in the plane spanned by the grating vector a1 and the normal to the surface (z-direction). The grating depth is d, and the widths of the pillars of the rectangular grating in x- and y-direction are wx and wy, respectively. In case of the hexagonal grating the diameter of the pillar is D. The effective indices in x-, y- and z-direction are neff,x, neff,y and neff,z, respectively.

In Table 1 the normalized grating period Λ/λ is given for several angles of incidence. The cutoff wavelength λc is the wavelength for which condition (2) is just obeyed. The relation between the cutoff wavelength at a hexagonal grating and at a rectangular grating is:

λc,hexagonal=0.866λc,rectangular
(6)

Table 1:. Normalized grating period Γ/λ dependent on angle of incidence and grating type

table-icon
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2.2 Effective index

The determination of the effective index is important in the design of the antireflective structures [14

14. E. B. Grann, M. G. Moharam, and D. A. Pommet, “Optimal design for antireflective tapered two-dimensional subwavelength grating structures,” J. Opt. Soc. Am . A 12, 333–339 (1995). [CrossRef]

]. With the effective indices, conventional layer design software can be used.

The effective index of a binary structure depends on the normalized grating period Λ/λ [15

15. E. B. Grann, M. G. Moharam, and D. A. Pommet, “Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings,” J. Opt. Soc. Am . A 11, 2695–2703 (1994). [CrossRef]

] as well as on the normalized grating depth d/λ [16

16. P. Lalanne and D. Lemercier-Lalanne, “Depth dependence of the effective properties of subwavelength gratings,” J. Opt. Soc. Am . A 14, 450–458 (1997). [CrossRef]

].

Several methods for the estimation of the effective refractive index are described in literature. In [17

17. S. Rytov, “Electromagnetic Properties of a Finely Stratified Medium,” Soviet Physics JETP 2, 466–475 (1956).

] for a 1D grating, a closed form for the effective indices for the polarization perpendicular and parallel to the grooves was derived for the case that the two stratified media are infinitely deep. These transcendental equations can be developed into a Taylor series. The truncation of the series after the first term leads to the zeroth-order effective medium theory (EMT), where the exponent of the normalized grating period is zero, i.e. the dependency of the effective index on the normalized grating period is neglected. With this theory, the effective indices in the quasi-static limit (Λ/λ → 0) can be determined:

neff,(0)=(1f)n12+fn22andneff,(0)=n1n2(1f)n12+fn22,
(7)

with n eff,⊥ (0). – the zeroth-order effective index perpendicular to the grating vector, n eff,∥ (0) – the zeroth-order effective index parallel to the grating vector. The effective medium theory of second-order takes the normalized grating period Λ/λ, with the second potency into account:

neff,(2)=neff,(0)2+π23(Λλ)2f2(1f)2(n22n12)2and
neff,(2)=neff,(0)2+π23(Λλ)2f2(1f)2(1n221n12)2(neff,(0))6(neff,(0))2
(8)

with n eff,⊥ – the second-order effective index perpendicular to the grating vector, n eff,∥ (2) – the second-order effective index parallel to the grating vector. With these effective indices a 1D grating can be described as a uniaxial medium with the extraordinary axis parallel to the grating vector.

A 2D structure has a modulation of the refractive index in two dimensions. In [15

15. E. B. Grann, M. G. Moharam, and D. A. Pommet, “Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings,” J. Opt. Soc. Am . A 11, 2695–2703 (1994). [CrossRef]

] it is shown that such a structure can be described with the effective medium theory as a biaxial medium. For the special case that both grating vectors (a 1 and a 2) have the same length (Λx = Λy) the 2D rectangular structure becomes a uniaxial structure with the optic axis perpendicular to the surface, i.e. neff,x = neff,y (Fig. 1(c)). A hexagonal grating can also be described as a rectangular grating with two different grating periods in x and y direction (Fig. 1(d)). Although the grating vectors a 1 and a 2 * have a different length, the effective indices in x and y direction are also equal. Thus, the hexagonal structure works also as a uniaxial medium with the optic axis perpendicular to the surface.

The method for estimating the effective indices of the 2D gratings is based on the method described in [15

15. E. B. Grann, M. G. Moharam, and D. A. Pommet, “Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings,” J. Opt. Soc. Am . A 11, 2695–2703 (1994). [CrossRef]

]. The effective index is determined by a comparison of the reflectance from a structured layer calculated by RCWA (rigorous coupled wave analysis) at variable filling factor and the reflectance from a homogeneous layer at variable refractive index. Thus, the accuracy is determined by the number of the Rayleigh orders at the RCWA-calculation. For the calculation the ±10th diffraction order was taken into account.

The depth dependence of the effective index cannot be taken into account. But for a first approximation of the gradient of the effective index, this method appeared to be sufficient.

In [18

18. P. Lalanne and D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt . 43, 2063–2085 (1996). [CrossRef]

] and [19

19. H. Kikuta, Y. Ohira, H. Kubo, and K. Iwata, “Effective medium theory of two-dimensional subwavelength gratings in the non-quasi-static limit,” J. Opt. Soc. Am . A 15, 1577–1585 (1998). [CrossRef]

] the effective indices are determined by a rigorous approach. With these methods the dependence of the refractive index on the depth of the structure can be taken into account. But these methods require a high program technical effort: The structure has to be developed into its Fourier components in the same way like it is implemented in the RCWA algorithm. For the determination of the effective indices the Fourier series have to be truncated. Thus, the accuracy of the effective index at these methods is determined by the number of Rayleigh orders, as well.

The structures are intended for a bandwidth from 0.1-2 THz (wavelengths 150-3000 μm). Thus, normalized grating period and depth are one order of magnitude larger for the small-wavelength end than for the long-wavelength end. The long-wavelength end of the spectrum is in the quasi-static limit whereas the small-wavelength end is not.

Hence, for a fixed filling factor of a binary grating, the effective index will vary over the spectral range and a binary grating will not exhibit exactly the same properties like a homogeneous layer. The effect of deep binary gratings is investigated at the end of this section.

For the different grating types investigated in this paper, the filling factor has to be defined appropriately. For the square cuboids in a rectangular grating, the filling factor is the ratio of the width of pillars to the grating period fi = wii- with i = {x,y} (Fig. 1(c)). For cylinders in a rectangular grating with same period in x- and y-direction, the filling factor is the diameter of the pillars to the grating period f = D/Λ. For cylinders in a hexagonal grating, the filling factor is the ratio of the diameter of the pillars to the grating period in x direction f = Dx (Fig. 1(d)). Here, a filling factor of one means hexagonal densest packing.

Instead of solving the transcendental equation as described in [15

15. E. B. Grann, M. G. Moharam, and D. A. Pommet, “Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings,” J. Opt. Soc. Am . A 11, 2695–2703 (1994). [CrossRef]

], the effective indices were determined graphically. The reflectance was plotted against the refractive index and against the filling factor (Fig. 2).

The curve in Fig. 2(a) was determined for a quarter-wave layer at the wavelength 375 μm (frequency 0.8 THz) with the matrix formalism used for the determination of the reflectance at thin films [20

20. H. A. MacleodThin-film optical filters (Institute of Physics Publ, 2002).

]. The refractive index of the substrate material was 3.42. For a quarter-wave layer, the conditions n = √n 1 n 2 and d = λ/4n must hold. Thus, the thickness for a quarter-wave layer is 50.694 μm. This thickness was constant at the calculations. For smaller depths, reflectance would not reach zero. For greater depths, there would be several minima such that the curve cannot be used for a definite determination. At the optimal refractive index of 1.85, reflectance is zero.

The curve in Fig. 2(b) was determined by RCWA. It shows the progression of the reflectance in dependence of the filling factor at the example of a rectangular grating in the quasi-static limit (Λ = 0.375 μm). Wavelength, layer thickness and refractive index of the substrate material are the same as before. At f = 0 and f = 1, reflectance is that of the uncoated boundary. At a certain filling factor (here f = 0.8), reflectance is zero. This filling factor corresponds to the optimal refractive index for a quarter-wave layer (n = 1.85). By comparison of both curves, the filling factor can be assigned to the refractive index with the same reflectivity because the effective index is the one that has the same effect like a homogeneous medium.

Fig. 2. (a) Reflectance in dependence of the refractive index for wavelength 375 μm, layer thickness 50.694 μm, refractive index of the substrate material 3.42 and refractive index of the layer 1.85, determined by thin film matrix formalism. (b) Reflectance in dependence on the filling factor for the rectangular grating in the quasi-static limit (wavelength 375 μm, grating period 0.375 μm), grating depth 50.694 μm and refractive index of the substrate material 3.42, determined by RCWA.

With this method the refractive indices normal to the propagation direction can be determined. At normal incidence and polarization in x direction (TE) the wave experiences the refractive index in x direction. For polarization in y direction (TM) the wave experiences the refractive index in y direction. In [15

15. E. B. Grann, M. G. Moharam, and D. A. Pommet, “Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings,” J. Opt. Soc. Am . A 11, 2695–2703 (1994). [CrossRef]

] it is shown how the effective index normal to the surface, i.e. in z-direction, can be determined in the quasi-static limit.

The curves in Fig. 3 for the 2D gratings were determined with this method. Figure 3(a) shows the dependence of the effective index on the filling factor for a rectangular grating at different normalized grating periods. For the ratio Λ/λ = 0.001 the grating is in the quasi-static limit and for Λ/λ = 0.2 the zeroth-order condition is just obeyed. Figure 3(b) shows the dependence of the effective index on the filling factor for different 2D gratings in the quasi-static limit (Λ/λ = 0.001). Figure 3(b) shows that in the quasi-static limit, the filling factor that corresponds to the optimal for a quarter-wave layer is 0.8 for square cuboids in a rectangular grating, 0.84 for cylinders in a hexagonal grating, and 0.9 for cylinders in a rectangular grating. It turned out that for the optimal refractive index, the area filling factors are nearly the same for all grating types: 0.64 for square cuboids in rectangular grating, 0.64 for cylinders in hexagonal grating, and 0.636 the cylinders in rectangular grating. These curves can be used as syntheses curves for a certain refractive index.

Figure 3(b) shows also the effective refractive index in dependence on the filling factor for a 1D grating. The curves were determined with the zeroth and second order EMT according to Eq. (7) and (8), respectively. For the second order effective indices, the normalized grating period Λ/λ was 0.2. The curves show that the high refractive index of silicon causes extreme polarization dependence. This is why 2D structures are preferred as antireflective structures for HRFZ silicon.

Fig. 3. (a) Dependence of the effective index on the filling factor for a rectangular grating at different normalized grating periods. For the ratio Λ/λ = 0.001 the grating is in the quasi-static limit and for Λ/λ = 0.2 the zeroth-order condition is just obeyed. (b) Dependence of the effective index on the filling factor for different grating types. The curves for the 2D gratings and the zeroth-order EMT-curves for the 1D grating are valid in the quasi-static limit (Λ/λ = 0.001). The second-order EMT-curves for the 1D grating are calculated at Λ/λ = 0.2.

2.3 Continuous structures

For a grating with a continual profile, the optimal taper is the one that requires the smallest depth at a given reflectance level. The reflectance depends on the gradient of the refractive index. In [14

14. E. B. Grann, M. G. Moharam, and D. A. Pommet, “Optimal design for antireflective tapered two-dimensional subwavelength grating structures,” J. Opt. Soc. Am . A 12, 333–339 (1995). [CrossRef]

] reflectance is investigated for the linear, exponential, Gaussian, quintic, the Klopfenstein taper, and also for a pyramidal profile.

For a structure with a triangular profile, i.e. for a pyramidal structure and cones in a rectangular or hexagonal grating, the normalized thickness z/d is equal to the filling factor. Thus, Fig. 3 shows at the same time the gradient of the refractive index from the surface (f = 0) to the substrate material (f = 1).

For cones in a rectangular grating there is an index step from 2.5 to 3.42 at the boundary from the structured region to the substrate material. For cones in a hexagonal grating the index step is from 2.9 to 3.42.

With the curves, reflectance in dependence on the normalized depth can be determined with effective medium theory in the quasi-static limit. For this, the structure is divided into thin slices. To each slice an effective index is assigned. The reflectance is then calculated by the matrix methods used for homogeneous layers. Here, a number of 100 slices was required to avoid bandpass effects. For such a high number of slices, the computation time at RCWA was much too long [21

21. J. Bischoff and R. Brunner, “Numerical investigation of the resolution in solid immersion lens systems,” Proc. SPIE 4099, 1–11 (2000).

]. The results of these calculations are shown in Fig. 4. It shows the dependence of the reflectance on the normalized grating depth d/λ. For d/λ ≥ 0.3, reflectance is below 7.1% for cones in a rectangular grating reflectance, below 5.6% for cones in a hexagonal grating reflectance, and below 4.3% for pyramids. Because of the index step a remaining reflectance of about 2% occurs for cones in a rectangular grating and about 0.5% for cones in a hexagonal grating. The minimum normalized grating depth of 0.3 applied to the long-wavelength end (3000 μm) yields a minimum structural depth of 900 μm.

The maximum grating period for normal incidence and a hexagonal grating for the small-wavelength end (150 μm) is about 50 μm (see Table 1). Thus structures with an aspect ratio of 18:1 are required. But this aspect ratio is not very stable. Thus, as a compromise a structural depth of 500 μm and a structural period of 50 μm are intended, i.e. the aspect ratio is 10:1.

In summary, a rectangular or hexagonal grating with period 50 μm and depth 500 μm are a good compromise for an antireflection treatment of HRFZ silicon components at setups with broadband THz radiation with a range of 0.1-2 THz.

Fig. 4. Reflectance in dependence on the normalized depth d/λ for different grating types calculated by effective medium theory in the quasi-static limit.

2.4 Starting point: deep binary structures

The starting point for the fabrication of the antireflective structures in HRFZ silicon were deep binary structures. These are investigated theoretically in this paragraph.

The considerations are confined to normal incidence. Then both polarization directions experience the same refractive index at the rectangular and the hexagonal structure. In this case the binary structure can be compared with a homogeneous layer. But the calculations below show that there are some crucial differences in the non-quasi-static limit.

The transmittance of the binary structures was calculated with RCWA where the ±10th diffraction order was taken into account. The transmittance of the homogeneous layer was calculated with effective medium theory.

Figure 5(a) shows the transmittance from a rectangular grating with filling factor 0.79, grating period 50 μm, and grating depth 500 μm. The filling factor corresponds to an effective index of 1.85. For comparison, the transmittance at a homogeneous layer with this refractive index and same depth is shown. The graphs show that the grating is in the quasi-static limit up to a frequency of about 0.6 THz (wavelength 500 μm). Up to this frequency, the grating behaves like a homogeneous layer. Above, constructive interference appears for frequencies with a smaller ratio to each other, and transmittance does neither reach 100 percent nor the value without a coating. There is no homogeneous layer, which shows such a transmission behavior. Thus, above the quasi-static limit, the behavior of this deep binary structure cannot be predicted by effective medium theory. The vertical black dashed line shows the limit up to which only the zeroth order is able to propagate. Above, some intensity is diffracted into higher diffraction orders. This limit confirms Eq. (2

2. J. Dai, J. Zhang, W. Zhang, and D. Grischkowsky, “Terahertz time-domain spectroscopy characterization of the far-infrared absorption and index of refraction of high-resistivity, float-zone silicon,” J. Opt. Soc. Am . B 21, 1379–1386 (2004). [CrossRef]

): The boundary frequency calculated with this equation is 1.752 THz. The boundary frequency determined by RCWA is 1.7544 THz.

Fig. 5. Comparison of the transmittance at a surface with a binary subwavelength structure and a homogeneous layer. The respective refractive index of the homogeneous layer is given in the plots. The black dashed vertical line is the zeroth-diffraction-order limit. Above, some energy is diffracted into higher diffraction orders. The period is 50 μm and the depth is 500 μm for all calculations. (a) Rectangular grating with filling factor 0.79. (b) Hexagonal grating with filling factor 0.84. (c) Hexagonal grating with filling factor 0.7361.

Figure 5(b) shows the transmittance at a hexagonal grating with filling factor 0.84, grating period 50 μm, and grating depth 500 μm. The filling factor corresponds also to the optimal refractive index of 1.85. The zeroth-order limit is 2 THz. This is also in good agreement with the one predicted with Eq. (2): 2.028 THz. Transmittance is similarly to the one of the rectangular grating. Differences appear in the modulation above 1.2 THz. But, as before, transmittance does neither reach 100 percent nor the value of the uncoated surface.

Figure 5(c) shows the transmittance for a smaller filling factor (f = 0.7361). This corresponds to an effective index of 1.555 in the quasi-static limit. All other parameters are the same as before. The binary grating shows the same behavior as a homogeneous layer up to a frequency of about 0.5 THz. For a homogeneous layer which has not the optimal refractive index, transmittance does not reach one. Constructive interference occurs for all odd multiples of the frequency ν = c/(4n1d1), with c – velocity of light in vacuum, n1 – the refractive index of the layer, and d1 – the layer thickness. Thus, for a lower refractive index than the optimal, the fundamental frequency is higher, and the distance of the odd multiples of this frequency is larger than at the optimal refractive index. Above the quasi-static limit the modulation is higher, transmittance of 100 percent is reached several times within in range of 1.3-1.7 THz, and for frequencies above 0.5 THz transmittance does not drop to the value of the uncoated surface. This behavior cannot be achieved with a homogeneous layer, as well. Thus, the behavior of deep binary structures can only be predicted in the quasi-static limit by effective media theory. Above the behavior can only be predicted by RCWA, i.e. it is not possible to assign a certain refractive index to those deep structures.

3. Manufacturing of the structures by deep reactive ion etching (DRIE)

The intended structures for manufacturing were a rectangular grating with period 50 μm, depth 500 μm and filling factor 0.79 (pillar width 39.5 μm) and a hexagonal grating with period 50 μm, depth 500 μm and filling factor 0.7361 (pillar width 36.8 μm). The structures were applied to 1 mm thick 4-inch HRFZ silicon wafers. The structures were manufactured at the Institute of Applied Physics, FSU Jena. At first the corresponding chromium photo masks were manufactured by electron beam lithography. With the help of these photo masks the chromium etching mask for deep Si-etching was assembled. The Si-wafers with deposited 500 nm chromium layer were spin coated with approximately 1 μm AZ1505 (standard photo resist), and exposed in mask-aligner AL6-2C (Electronic Vision Austria and developed in AZ400K (diluted 1:4 with DI water).

Fig. 6. SEM-images of the rectangular structure. (a) Cross section. The depth of the structure is 462.5 μm, the grating period is 52.7 μm, and the width of the pillars in the upper region of the structure is 36.8 μm (filling factor 0.7). (b) Structure under 15° tilting angle.
Fig. 7. SEM-images of the hexagonal structure. (a) Cross section, the structural depth is 500 μm, the period is 50 μm, and the diameter of the pillars in the upper region of the structure is 37.5 μm (filling factor 0.75). The etching mask was removed completely. (b) Structure under 15° tilting angle. Nearly the whole 4-inch Si-wafer consists of the shown pillars.

Then the chromium was etched in a parallel-plate-plasma reactor (SI-591, Sentech Instruments Berlin). Such obtained binary chromium structures were used for the deep Si-etching with the help of the so called BOSCH®-process. This is a gas chopping method, where passivation and isotropic etching cycles alternate. This etching process was done in an ICP-RIE-plasma reactor SI-500 C (Sentech Instruments Berlin). Figure 6 shows a manufactured rectangular structure from a prior test. The structural depth of this wafer (462.5 μm) was achieved by 900 Bosch®-cycles (7.5h process time). For the intended structural depth of 500 μm 1400 Bosch®-cycles (nearly 9h process time) were used. The realized structural period is 52.7 μm. This is slightly larger than intended and shifts the zeroth-diffraction-order limit to a slightly lower frequency (1.67 THz). The filling factor varies slightly with the depth. The constriction in the middle of the structure is due to a break of the etching process to control the achieved etching depth. The etching process was broken each time after 100 Bosch®-cycles to control the etching depth and the temperature of the wafer. Figure 7 shows the manufactured structure based on the etching mask for the hexagonal structure. Structural period and depth are as intended but as before the pillar diameter varies with the depth, i.e. it grows from the middle of the structure to the bottom. This is also due to breaks of the etching process. The manufacturing process contained also 1400 Bosch®-cycles (nearly 9h process time). After the etching process the etching mask was removed completely by RIE.

4. Measurements and results

For the measurements the samples were placed in the collimated THz beam between two off-axis parabolic mirrors of a standard THz time domain spectroscopy setup (Fig. 8). Thus, the angle of incidence should be approximately 0°. The THz path was flooded with nitrogen to suppress reflections from the water-vapor molecules of the surrounding air. The fs-laser was a MaiTai from Spectra Physics with optical output power of 1 W, pulse duration of 100 fs and repetition rate of 80 MHz. The time constant at the lock-in amplifier was 300 ms. The signal-to-noise ratio at this time constant is 65.5 dB. The velocity of the delay line was 0.05 ps/s. The measurement time for one waveform was 26.7 min. The spectrum reaches from 0.1-2.5 THz. The maximum amplitude is at 0.41 THz. For each sample 3 waveforms were measured to estimate the temporal stability of the system.

Fig. 8. THz time domain spectroscopy system (principle). The sample was placed in the collimated beam between the second set of off-axis parabolic mirrors.

Figure 9 shows the mean value of the measured pulses of the sample with the rectangular structure and the unstructured sample. The pulse maximum of the unstructured sample fluctuated about ±0.88% over all measurement. The main pulse at the structured sample appears 2.63 ps before the main pulse of the unstructured sample. This is due to the reduced optical path length of the structured sample. Theoretically, with an effective index of 1.85 and a depth of 500 μm of the structured region the main pulse should appear at ∆t = ∆n·d/c = (3.42-1.85)·500μm/3.108 m/s = 2.62 ps, with ∆n – change of the refractive index, d – layer depth, and c – velocity of light in vacuum. This is a first indicator that the intended structural parameters are realized. The curve of the unstructured sample shows pulse echoes at 39.13 ps and 60.93 ps. These are due to Fresnel reflections within the silicon sample. The small pulse echo around 23.3 ps (5.9 ps after the main pulse) is due to the alignment of the THz system and appeared also without any sample in the setup. The inset in Fig. 9(a) shows the theoretical delay times of the secondary pulses for the structured sample, if the structured layer has an effective index of 1.85. The secondary pulses should appear at 6.2 ps, 11.4 ps, and 17.6 ps. The second pulse at the structured sample is and overlay of the pulse copy caused by internal reflections and the amplified pulse present in the system 5.9 ps after the main pulse. The maximum of the second pulse appears 4.42 ps after the main pulse. This pulse maximum has a negative sign, because of the phase shift of 180° on reflection from an optical denser medium. Further pulse copies appear at 11.8 ps and 16.44 ps after the main pulse (measured from the position of the pulse maximum). Additionally to the pulse copies, the rising edge of the main pulse shows fast oscillations (see arrow in Fig. 9). These are due to little index steps within the structured region. The index steps are due to the breaks in the etching process (see Fig. 6 and Fig. 7). The pulse maximum at the sample with the rectangular structure increases by 15.2% with respect to the unstructured sample.

The temporal analysis of the hexagonal structure shows similarly effects (Fig. 10(a)). The secondary pulses appear 4.13 ps (with negative maximum), 12.38 ps and 16.53 ps after the main pulse. The oscillations in the rising edge of the main pulse from the structured sample can be observed as well. The second pulse copy at 12.38 ps after the main pulse is more pronounced than at the rectangular structure. This is the reflection from the structured region within the sample. This indicates that the transition from the structured area to the homogeneous medium is sharper than at the rectangular structure. The pulse maximum at the structured sample increases by 21.76% with respect to the unstructured sample. The differences from the theoretical prediction with the homogeneous layer can be explained by considering the spectral analysis.

Figure 9(b) shows the spectral analysis for the rectangular structure. The figure shows the ratio of the electric field amplitudes of the structured and the unstructured sample. The blue curve shows the mean value. The red curves show the standard deviation. The green dashed line shows the theoretical prediction for a binary structure with filling factor of 0.79. For the comparison with the theoretical prediction the pulses of the structured and unstructured sample were cut 9 ps after the maximum of the respective maximum of the main pulse. Thus, the reflection within the structured region is taken into account. This simulates the effect of an integrating detector and allows the comparison with the simulations. The upper black line shows the maximum possible transmission with an antireflective coating. The lower black line shows the limit without any coating. Typical oscillations can be observed but the modulation is strongly reduced towards higher frequencies. The amplitude of the structured sample is increased up to a frequency of about 1.1 THz. The black dashed line at 1.7 THz shows zeroth-diffraction-order limit for normal incidence. Thus the bandwidth of the realized structure is smaller than the theoretical prediction. The low bandwidth of the realized rectangular structure can be due to disadvantageous steps in the structured region. If the structure is realized as step profile the bandwidth is strongly dependent on the depth and width of the single pillars. Because the spectral region where the structure causes an increase of the electric field amplitude coincides with the region of maximum amplitude of the THz spectrum, the pulse maximum of the structured sample is increased, as well. The spectral performance of the hexagonal structure is shown in Fig. 10(b). There the ratio of the electric field amplitudes of the structured and the unstructured sample are shown. Again the green dashed line shows the theoretical prediction for a hexagonal binary structure with filling factor 0.7361. The black dashed line at 2 THz shows the zeroth-diffraction-order limit. The red curves show the standard deviation. Above 1.5 THz the standard deviation increases strongly because of the lower signal-to-noise ratio. But the spectral analysis shows clearly that the structure works up to frequencies of 2 THz. Thus, the theoretical zeroth-order-diffraction limit is preserved. As before, nearly full modulation can only be observed for the low frequencies. With increasing frequency the modulation decreases. Above 2.1 THz the signal-to-noise ratio is too low such that the effect of the structure cannot be determined anymore. Because the bandwidth of the hexagonal structure is much broader than that of the rectangular structure, the increase of the pulse maximum at the temporal analysis is higher.

The modulation is not as high as at the theoretical prediction for a binary grating. This behavior can be explained if one takes a closer look to the filling factor of the realized grating (Fig. 7(a)). Down to a depth of about 242 μm the filling factor is nearly the intended 0.74. From this depth on, the filling factor increases continuously to the bottom of the structure to 0.97. This means the refractive index increases in this area from 1.56 to 2.62 (see Fig. 3(b)).

Fig. 9. (a) Measured waveform of the unstructured sample and the sample with the rectangular structure. The inset shows theoretical delay times of the secondary pulses with respect to the main pulse of the layer system with a layer with an effective index of 1.85 and a layer with a refractive index of 3.42. Both layers have a depth of 500 μm. (b) Spectral analysis: Ratio of the transmitted electric field amplitudes of structured and unstructured sample.
Fig. 10. (a) Measured waveform of the unstructured sample and the sample with the hexagonal structure. (b) Spectral analysis: Ratio of the transmitted electric field amplitudes of structured and unstructured sample.

Figure 11(a) shows the simulation of the reflectance with such a gradient of the refractive index with effective medium theory in the quasi-static limit. The gradient of the refractive index from the middle of the structure to the substrate material causes the reduced modulation. Thus, the break of the etching process led already to the intended gradient of the refractive index in the lower region of the hexagonal structure. For low filling factors the increase of the refractive index with the filling factor is only low (Fig. 3). This behavior can be exploited to manufacture mechanically stable antireflective structures. A filling factor of 0.3 corresponds to a refractive index of 1.1. The reflectance for a hexagonal structure with a step of the effective index of 1.1 at the top of the structure is shown in Fig. 11(b). The remaining intensity reflectance is 1.1%. A filling factor of 0.3 at the top and 0.97 at the bottom of the structure would result in a remaining reflectance of 2.5%.

Fig. 11. (a) Simulation for the hexagonal structure with effective medium theory. (b) Simulation for conic sections in a hexagonal grating with filling factor 0.3 at the top of the structure. The remaining intensity reflectance is 1%. The dashed line shows the zeroth-diffraction-order limit for a grating period of 50 μm. The inset shows the corresponding hexagonal structure with filling factor 0.3 at the top of the structure.

5. Conclusion

Acknowledgment

This research was supported by the Fraunhofer Gesellschaft Internal Programs under Grant No. MAVO 813907.

References and links

1.

D. Grischkowsky, S. Keiding, M. van Exter, and C. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am . B 7, 2006–2015 (1990). [CrossRef]

2.

J. Dai, J. Zhang, W. Zhang, and D. Grischkowsky, “Terahertz time-domain spectroscopy characterization of the far-infrared absorption and index of refraction of high-resistivity, float-zone silicon,” J. Opt. Soc. Am . B 21, 1379–1386 (2004). [CrossRef]

3.

M. T. Reiten, S. A. Harmon, and R. A. Cheville, “Terahertz beam propagation measured through three-dimensional amplitude profile determination,” J. Opt. Soc. Am . B 20, 2215–2225 (2003). [CrossRef]

4.

C. Brückner, B. Pradarutti, R. Müller, S. Riehemann, G. Notni, and A. Tünnermann, “Design and evaluation of a THz time domain imaging system using standard optical design software,” Appl. Opt . 47, 4994–5006 (2008). [CrossRef] [PubMed]

5.

C. Brückner, B. Pradarutti, R. Müller, S. Riehemann, G. Notni, and A. Tϼnnermann, “Design and analysis of quasioptical THz time domain imaging systems,” Proc. SPIE 7100, 71000S (2008). [CrossRef]

6.

A. J. Gatesman, J. Waldman, M. Ji, C. Musante, and S. Yngvesson, “An anti-reflection coating for silicon optics at terahertz frequencies,” IEEE Microwave Guided Wave Lett . 10, 264–266 (2000). [CrossRef]

7.

I. Hosako, “Multilayer optical thin films for use at terahertz frequencies: method of fabrication,” Appl. Opt . 44, 3769–3773 (2005). [CrossRef] [PubMed]

8.

J. Kröll, J. Darmo, and K. Unterrainer, “Metallic wave-impedance matching layers for broadband terahertz optical systems,” Opt. Express 15, 6552–6560 (2007). [CrossRef] [PubMed]

9.

W. Withayachumnankul, B. M. Fischer, S. P. Mickan, and D. Abbott, “Retrofittable antireflection coatings for T-rays,” Microwave Opt. Technol. Lett . 49, 2267–2270 (2007). [CrossRef]

10.

P. B. Clapham and M. C. Hutley, “Reduction of lens reflection by the ‘Moth Eye” principle,” Nature 244, 281–282 (1973). [CrossRef]

11.

A. Gombert, B. Bläsi, C. Bühler, and P. Nitz, “Some application cases and related manufacturing techniques for optically functional microstructures on large areas,” Opt. Eng . 43, 2525–2533 (2004). [CrossRef]

12.

C. Brückner, B. Pradarutti, O. Stenzel, R. Steinkopf, S. Riehemann, G. Notni, and A. Tünnermann, “Broadband antireflective surface-relief structure for THz optics,” Opt. Express 15, 779–789 (2007). [CrossRef] [PubMed]

13.

H. Ibach and H. Lüth, Festkörperphysik (Springer, 2002).

14.

E. B. Grann, M. G. Moharam, and D. A. Pommet, “Optimal design for antireflective tapered two-dimensional subwavelength grating structures,” J. Opt. Soc. Am . A 12, 333–339 (1995). [CrossRef]

15.

E. B. Grann, M. G. Moharam, and D. A. Pommet, “Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings,” J. Opt. Soc. Am . A 11, 2695–2703 (1994). [CrossRef]

16.

P. Lalanne and D. Lemercier-Lalanne, “Depth dependence of the effective properties of subwavelength gratings,” J. Opt. Soc. Am . A 14, 450–458 (1997). [CrossRef]

17.

S. Rytov, “Electromagnetic Properties of a Finely Stratified Medium,” Soviet Physics JETP 2, 466–475 (1956).

18.

P. Lalanne and D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt . 43, 2063–2085 (1996). [CrossRef]

19.

H. Kikuta, Y. Ohira, H. Kubo, and K. Iwata, “Effective medium theory of two-dimensional subwavelength gratings in the non-quasi-static limit,” J. Opt. Soc. Am . A 15, 1577–1585 (1998). [CrossRef]

20.

H. A. MacleodThin-film optical filters (Institute of Physics Publ, 2002).

21.

J. Bischoff and R. Brunner, “Numerical investigation of the resolution in solid immersion lens systems,” Proc. SPIE 4099, 1–11 (2000).

OCIS Codes
(310.1210) Thin films : Antireflection coatings
(050.6624) Diffraction and gratings : Subwavelength structures

ToC Category:
Diffraction and Gratings

History
Original Manuscript: November 13, 2008
Revised Manuscript: December 15, 2008
Manuscript Accepted: January 14, 2009
Published: February 17, 2009

Citation
Claudia Brückner, Thomas Käsebier, Boris Pradarutti, Stefan Riehemann, Gunther Notni, Ernst-Bernhard Kley, and Andreas Tünnermann, "Broadband antireflective structures applied to high resistive float zone silicon in the THz spectral range," Opt. Express 17, 3063-3077 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-5-3063


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References

  1. D. Grischkowsky, S. Keiding, M. van Exter, and C. Fattinger, "Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors," J. Opt. Soc. Am. B 7, 2006-2015 (1990). [CrossRef]
  2. J. Dai, J. Zhang, W. Zhang, and D. Grischkowsky, "Terahertz time-domain spectroscopy characterization of the far-infrared absorption and index of refraction of high-resistivity, float-zone silicon," J. Opt. Soc. Am. B 21, 1379-1386 (2004). [CrossRef]
  3. M. T. Reiten, S. A. Harmon, and R. A. Cheville, "Terahertz beam propagation measured through three-dimensional amplitude profile determination," J. Opt. Soc. Am. B 20, 2215-2225 (2003). [CrossRef]
  4. C. Brückner, B. Pradarutti, R. Müller, S. Riehemann, G. Notni, and A. Tünnermann, "Design and evaluation of a THz time domain imaging system using standard optical design software," Appl. Opt. 47, 4994-5006 (2008). [CrossRef] [PubMed]
  5. C. Brückner, B. Pradarutti, R. Müller, S. Riehemann, G. Notni, and A. Tünnermann, "Design and analysis of quasioptical THz time domain imaging systems," Proc. SPIE 7100, 71000S (2008). [CrossRef]
  6. A. J. Gatesman, J. Waldman, M. Ji, C. Musante, and S. Yngvesson, "An anti-reflection coating for silicon optics at terahertz frequencies," IEEE Microwave Guided Wave Lett. 10, 264-266 (2000). [CrossRef]
  7. I. Hosako, "Multilayer optical thin films for use at terahertz frequencies: method of fabrication," Appl. Opt. 44, 3769-3773 (2005). [CrossRef] [PubMed]
  8. J. Kröll, J. Darmo, and K. Unterrainer, "Metallic wave-impedance matching layers for broadband terahertz optical systems," Opt. Express 15, 6552-6560 (2007). [CrossRef] [PubMed]
  9. W. Withayachumnankul, B. M. Fischer, S. P. Mickan, and D. Abbott, "Retrofittable antireflection coatings for T-rays," Microwave Opt. Technol. Lett. 49, 2267-2270 (2007). [CrossRef]
  10. P. B. Clapham and M. C. Hutley, "Reduction of lens reflection by the 'Moth Eye' principle," Nature 244, 281-282 (1973). [CrossRef]
  11. A. Gombert, B. Bläsi, C. Bühler, and P. Nitz, "Some application cases and related manufacturing techniques for optically functional microstructures on large areas," Opt. Eng. 43, 2525-2533 (2004). [CrossRef]
  12. C. Brückner, B. Pradarutti, O. Stenzel, R. Steinkopf, S. Riehemann, G. Notni, and A. Tünnermann, "Broadband antireflective surface-relief structure for THz optics," Opt. Express 15, 779-789 (2007). [CrossRef] [PubMed]
  13. H. Ibach and H. Lüth, Festkörperphysik (Springer, 2002).
  14. E. B. Grann, M. G. Moharam, and D. A. Pommet, "Optimal design for antireflective tapered two-dimensional subwavelength grating structures," J. Opt. Soc. Am. A 12, 333-339 (1995). [CrossRef]
  15. E. B. Grann, M. G. Moharam, and D. A. Pommet, "Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings," J. Opt. Soc. Am. A 11, 2695-2703 (1994). [CrossRef]
  16. P. Lalanne, and D. Lemercier-Lalanne, "Depth dependence of the effective properties of subwavelength gratings," J. Opt. Soc. Am. A 14, 450-458 (1997). [CrossRef]
  17. S. Rytov, "Electromagnetic Properties of a Finely Stratified Medium," Soviet Physics JETP 2, 466-475 (1956).
  18. P. Lalanne and D. Lemercier-Lalanne, "On the effective medium theory of subwavelength periodic structures," J. Mod. Opt. 43, 2063-2085 (1996). [CrossRef]
  19. H. Kikuta, Y. Ohira, H. Kubo, and K. Iwata, "Effective medium theory of two-dimensional subwavelength gratings in the non-quasi-static limit," J. Opt. Soc. Am. A 15, 1577-1585 (1998). [CrossRef]
  20. H. A. Macleod, Thin-film optical filters (Institute of Physics Publ, 2002).
  21. J. Bischoff and R. Brunner, "Numerical investigation of the resolution in solid immersion lens systems," Proc. SPIE 4099, 1-11 (2000).

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