July 1, 2005 / Vol. 30, No. 13 / OPTICS LETTERS
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Area-coded effective medium structures, a new type of grating design Bernd H. Kleemann and Johannes Ruoff Carl Zeiss AG, Corporate Research & Technology, Optical Design, 73446 Oberkochen, Germany
Ralf Arnold Carl Zeiss SMT AG, Lithography Optics Division, 73446 Oberkochen, Germany Received January 13, 2005 We propose a new way to design gratings with desired diffraction properties by using subwavelength feature sizes perpendicular to the ordinary superwavelength grating period. This is different from well-known onedimensional binary-blazed gratings that use a structuring along the grating period and thus opens new flexibility in generating arbitrary effective-index distributions in the direction of the grating period. Since the subwavelength features form contiguous areas, they are called area-coded effective medium structures (ACES). Compared with well-known binary subwavelength structures in two-dimensional arrangements consisting of pillars, ACES are more stable and have comparable efficiency properties. As an example we show how to design in principle a four-level area-coded effective medium grating, compare the efficiency of ACES with binary-blazed and échelette gratings, and optimize the subwavelength period of ACES. © 2005 Optical Society of America OCIS codes: 050.1950, 050.1960, 050.1970.
Diffractive optics benefits more and more from the progress and availability of lithographic and other microfabrication techniques. Diffractive optical elements have a variety of applications in micro-optical systems for beam shaping, collimation, imaging, and deflection. In some applications such as DVD pickup systems they are introduced for further miniaturization and simplification. To achieve the necessary high efficiency in a specific diffraction order, the blazing effect is one of the most important properties being used. Blazed diffractive elements are currently fabricated by, e.g.,1 (i) direct writing techniques such as single-point diamond turning (for large periods), single-point laser beam writing, or single-point electron-beam writing for small periods or sizes; (ii) a series of photolithographic processes that approximate the continuous surface profile by a staircase or multilevel structure; or (iii) continuous-relief techniques such as halftone mask lithography. For all these techniques it is necessary to control the profile depth variation within the period. For multilevel profiles several masks are needed to create the desired profile. In these processes one has to cope with depth and mask aligning errors and other deficiencies.1 On the other hand, one can easily and exactly control the thickness of coating layers. Additionally, in the actual fabrication process, controlling position in the grating plane is much easier than controlling depth variations. Thus, using, e.g., electronbeam writing, one is able to finely structure this layer. Therefore a method that substitutes a variable depth by means of a variable width of a structure in a single layer should have advantages over the abovementioned fabrication techniques. This holds especially for the manufacture of multilevel gratings by microlithographic structuring because a width control technique would reduce a series of two, three, 0146-9592/05/131617-3/$15.00
or four mask exposures to a single step. In the following we describe this new approach to the so-called area-coded effective medium structures (ACES), which are created by means of a width-control technique. It is based on application of the onedimensional (1D) effective medium theory2 of connected area structures of constant height having two significantly differing extension lengths: one dimension of the structures, sampling period w, is subwavelength, whereas the other, ordinary grating period g, is superwavelength in a wide range that may span from two to hundreds of wavelengths. As an example, in Fig. 1 we show blazed ACES (BLACES) with the subwavelength width of the structure coding a blaze profile while the profile height remains at a constant level. In the early 1990s it was realized that dielectric binary subwavelength structures can be used to generate gratings with arbitrary artificial index distributions within the grating period.3,4 In the 1D case with ridges and gaps, one can define the fill factor as the ratio between the width of the ridge and the sam-
Fig. 1. BLACES with grating period g several wavelengths long and sampling period w ⬍ / 2. © 2005 Optical Society of America
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OPTICS LETTERS / Vol. 30, No. 13 / July 1, 2005
hACE =
Fig. 2. Top view of a sample configuration for multilevel ACES with four levels.
pling period. Because of the subwavelength width of the microstructures, incident light cannot resolve them and therefore averages the permittivity of the ridge material and the gap. The light thus sees an effective mean permittivity, which depends on the fill factor and, especially in the 1D case, on its polarization state. Hence a local variation of the fill factor is related to a local variation of the effective index, and thus arbitrary refractive-index distributions can be synthesized. In this way many types of grating can in principle be designed with binary structures only. Binary-blazed (BB) gratings for the visible range were manufactured and studied both in one dimension and two dimensions in the late 1990s.5–8 The two-dimensional (2D) BB gratings have the advantage of being almost polarization independent. Whereas 1D BB gratings use a subwavelength sampling period along grating period g, ACE gratings use a subwavelength sampling period w perpendicular to grating period g, which is a new design concept. This approach yields connected area structures along the grating period as can be recognized by the samples in Figs. 1 and 2. Because of subwavelength period w, only the zeroth-order diffraction mode in the transverse direction can propagate, and therefore the diffraction spectrum of this 2D grating is the same as that of a 1D grating with grating period g. If an ACE grating as shown in Fig. 1 has a large period g, then the ratio of g / w becomes large and can easily reach 100, 1000, or even more. Since the electron beam usually follows a Cartesian writing grid, it may become difficult to write triangular patterns with oblique boundaries. In this case one would have to resort to some kind of binarization of the profile boundaries. A result of such a binarization process is shown in Fig. 2. The design procedure for binarized ACES is in close analogy to the binarization process of a continuous profile grating resulting in a grating with a multilevel profile. For an m-step multilevel profile of a dielectric material with refractive index n the total profile height is given by hML = 关共m − 1兲 / m兴 ⫻关0 / 共n − 1兲兴, with 0 being the design wavelength and n being the refractive index of the profile. The height of each step is then 共1 / m兲关0 / 共n − 1兲兴. In contrast, ACES have a constant height of
0 max共neff兲 − min共neff兲
共1兲
,
with neff being the local effective index within the grating period. The boundary values min共neff兲 and max共neff兲 have to satisfy 1 艋 min共neff兲 ⬍ max共neff兲 艋 n but otherwise can be chosen arbitrarily. However, the larger the difference between the two values, the smaller hACE becomes, which is advantageous for both the efficiency performance and the fabrication process. For a four-level ACE grating as depicted in Fig. 2 the steps are realized by controlling the fill factors fi = wi / w, i = 1 . . . 4 of the subwavelength structures perpendicular to grating period g in regions 1–4. The fill factors have to be fixed in such a way that effective indices of min共neff兲 + 兵0 , 1 / 4 , 2 / 4 , 3 / 4其共0 / hACE兲 are obtained in regions 1–4. The effective indices of binary gratings with structures much smaller than consisting of ridges of material n1 with material n2 in between can be obtained through2 TE neff =
关fn12
+ 共1 −
f兲n22兴1/2,
TM 1/neff =
冋
f n12
+
共1 − f兲 n22
册
1/2
共2兲 for TE and TM polarization, respectively. From these relations it is obvious that ACE gratings exhibit a certain polarization dependence, which leads to a decrease in the efficiency when illuminated by unpolarized light. Moreover, because of practical reasons, the ratio w / will not be small enough for Eqs. (2) to hold; therefore one would have to either include higher-order terms or use rigorous methods to find the optimal values for the required fill factors for the binarized ACE grating for maximum efficiency. The same procedure can also be applied to realize many other types of grating profile such as sinusoidal or trapezoidal ACES. Since BLACES are an important type of grating, in the following their performance is compared with BB and conventional blazed (CB) gratings (échelettes). Since the grating periods under consideration are close to the wavelength, rigorous methods are required for the correct simulation. All the calculations
Fig. 3. Number of ACE subwavelength periods w per wavelength: g = 5, nACE = 2.3, nSubstrate = 1.46.
July 1, 2005 / Vol. 30, No. 13 / OPTICS LETTERS
Fig. 4. Unpolarized efficiency of BLACES according to Fig. 1 over the grating period in normal incidence compared with 1D BB and CB gratings: = 633 and 817 nm deep TiO2 structures.
Fig. 5. Efficiency of fixed period BLACES compared with 2D BB structures over the incidence angle in air. Unpolarized light of = 633 nm, with g = 1900 nm, and the height of the TiO2 structures is 817 nm.
for BLACES and BB gratings were made with an implementation of the 2D Fourier modal method according to Ref. 9. The accuracy was validated by comparing the results with calculations of a threedimensional finite-difference time-domain method. The results of the Fourier modal method with 11 modes in each dimension were found to be accurate enough as long as g 艋 5. Because the required depth of a blaze profile is inversely proportional to the refractive index, the use of a high-index material results in a lower profile depth. These circumstances and the fact that a lower profile depth is easier to manufacture motivated the authors of Ref. 5 to fabricate their BB gratings in TiO2, which has a refractive index of approximately 2.3 in the visible spectrum. Therefore we use the same advantageous material combination for efficiency optimization of blazed ACE gratings: They are assumed to be TiO2, and the substrate is assumed to be glass. By choosing the appropriate transverse subwavelength period w, one can optimize the structures for high efficiency. Figure 3 shows the efficiency for a fixed grating period of g = 5 as a function of the number of subwavelength periods w per wavelength. For-
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tunately, best efficiency is obtained for w ⬇ / 4 and not for much smaller widths. Further investigations show that a value of w between / 3 and / 4 is optimal for other grating periods and incidence angles, as well. Figure 4 shows the efficiencies of BLACES, 1D BB, and CB gratings as a function of grating period for unpolarized light under normal incidence. For a realistic comparison, in all three cases the blazing structure is assumed to be of TiO2. Clearly the 1D BB gratings show the best performance, which is probably due to a waveguiding effect within the binary ridges as explained in Ref. 10. Additionally we have to note that their performance is optimized for unpolarized light and for each grating period separately without assuming any fabrication constraints. Therefore the resulting binary structures have small ridges and gaps, which are far from being producible. Nevertheless, the BLACES show significantly higher efficiencies than the CB gratings (échelettes) although they have not been especially optimized for unpolarized light but assumed to consist of simple triangular structures with linear boundaries (see Fig. 1). Finally, in Fig. 5 we compare the angular efficiency performance of BLACES and 2D BB gratings for the grating period g = 3. The results for 2D BB gratings are taken from Ref. 8. The incidence is from the substrate with nSubstrate = 1.5, but the angles belong to incidence from air. Although not as high as for 2D BB gratings, the efficiency curve of BLACES remains at a rather constant level for a broad range of incidence angles. Together with the higher stability of ACES owing to the connected area and a smaller aspect ratio, BLACES represent an interesting alternative to échelette and 2D BB gratings, especially in diffractive optics, micro-optics, and integrated optics applications. B. H. Kleemann’s
[email protected].
e-mail
address
is
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