Recent research revealed that using the effective medium approach to generate arbitrary multi-phase level computer-generated holograms is a promising alternative to the conventional multi-height level approach. Although this method reduces the fabrication effort using one-step binary lithography, the subwavelength patterning process remains a huge challenge, particularly for large-scale applications. To reduce the writing time on variable shaped electron beam writing systems, an optimized strategy based on an appropriate reshaping of the binary subwavelength structures is illustrated. This strategy was applied to fabricate a three-phase level CGH in the visible range, showing promising experimental results.
©2011 Optical Society of America
Phase-only computer-generated holograms (CGHs) modify the phase of an incident optical field to tailor the intensity distribution of the wave-front in a certain propagation distance behind the element or in the far field. This feature offers many applications in modern optics like diverse tasks of beam shaping, beam splitting and testing of optical surfaces. Commonly, the required phase change is realized by a surface relief profile of a higher index material which causes a spatially varying phase delay usually between 0 and 2π. Standard binary phase elements are often not applicable because of their inherent property of twin-image generation which causes huge losses in the general case where one wants to create off-axis, asymmetric images. In this case, multi-level or continuous-level CGHs are desirable which can be created using conventional fabrication processes like gray-tone lithography , direct laser beam writing  or variable dose electron beam writing . However, this is a challenging task since the used resist material has to be very sensitive to small dose changes. This leads to a lack of resolution, creating smeared and rounded surface profiles, which is especially problematic at 2π phase jumps that most diffractive elements contain. Another fabrication possibility is the use of multiple exposure and etching steps, using a plurality of exposure masks . This allows the creation of 2n phase levels with n exposure steps, keeping a high resolution and enabling sharp edges such as 2π phase jumps. However, the resulting diffractive elements often suffer from misalignment of the different exposure masks. The complexity and time of the fabrication process quickly increases with an increasing number of phase levels and often allows only a small number of phase steps. It becomes even more challenging when high diffraction angles which require small feature sizes are requested. Typically, in the visible range, more than two binary masks, i.e. four phase levels, already offer only minor improvements . To overcome those technological problems a binary approach to generate two-dimensional multi-phase level elements would be desirable. In this case, the fabrication process can be simplified significantly to one lithographic step followed by one etching step. One common binary approach is to use a pulse-position-modulated high-frequency carrier grating in the resonance domain under Bragg incidence to tailor the phase distribution of the first transmitted diffraction order [6,7]. The aspect ratio of such a modulated binary grating is moderate in the order of unity which eases fabrication issues. However, this design approach imposes significant limitations on the achievable angular size of the signal window (due to necessary separation from the carrier frequency) and the reconstruction geometry (off-axis illumination required).
A further promising alternative is based on a binary effective medium approach which could already be demonstrated for periodic diffractive elements like blazed gratings, diffractive lenses and Fourier-plane array illuminators [7–11]. There the analogy between binary subwavelength periodic structures and homogeneous media with an artificial effective index is used. The phase delay can be controlled by the fraction of material within the unit cell of the periodic structure. Yu et al.  used this approach for non-deterministic CGH phase structures. A periodic repetition of six identical hole and pillar structures was implemented to generate a four-level phase-only CGH in fused silica. In , we could demonstrate that in fact this internal repetition is not necessary and that the phase delay of one CGH pixel can be controlled using only one binary subwavelength structure (BSWS). An arbitrary and thus non-periodic positioning of the BSWS is able to generate any desired phase function. This offers a much more flexible and efficient CGH design even for high diffraction angles and small wavelengths, as shown in . To fabricate such binary subwavelength CGHs with small feature sizes in the nanoscale range, we used the high-resolution electron beam writer Vistec SB350OS which performs a highly sophisticated variable shaped beam writing strategy during exposure. This enables a high writing speed which brings even large-scale applications within reach .
In this paper, we demonstrate a promising design concept to further decrease the required writing time on the electron beam writer which is optimized with respect to the variable shaped beam writing strategy. In Section 2, basic considerations about the used effective medium approach for CGH applications are introduced. The third section describes the design strategy to improve the writing speed by using an appropriate pixel reshaping, preceded by a short introduction of the used variable shaped beam writing method. Since the exact saving in writing time depends on the specific CGH phase pattern, the design of a binary subwavelength CGH for visible light operation is demonstrated in Section 4. Different BSWS are compared with regard to technical feasibility and writing time. The performance of the structure is studied experimentally in Section 5 and compared with theoretical predictions.
2. Effective medium approach for CGH structures
As introduced in , an arbitrary CGH phase function can be realized by non-repeating BSWS. Hence, every CGH pixel corresponds to one BSWS which causes a certain phase delay to the incident wave. This approach offers the possibility to generate arbitrary phase distributions with a very high resolution. We distribute the different subwavelength structures among a 2D square grid where each grid element represents one pixel of a conventional CGH. In principal this periodic pixelation is not required, but it is well adapted to the fabrication and allows the theoretical treatment to be based on the analysis of conventional CGHs.
As illustrated in Fig. 1 , the CGH phase function with values between 0 and 2π can be transferred into a BSWS pattern using a calibration curve which connects the phase delay with the fraction of material inside the unit cell. A variety of different kinds of BSWS are applicable like pillar, hole or ridge structures or even a combination of them. Although the BSWS are sufficiently small to allow only zeroth order diffraction, the dimensions of the structures are close to the incident wavelength. For that reason, the calibration curve is calculated numerically using the rigorous coupled wave analysis (RCWA) [16,17]. Therefore, the subwavelength structures are placed in a square cell with periodic boundary conditions like required for the RCWA calculation. Thus, the influence of the phase caused by the adjacent BSWS cannot be considered and is neglected in the demonstrated design approach.
3. Improvement in electron beam writing speed
Fabricating effective CGHs on a large scale is a demanding process, especially with respect to the required writing time. In order to optimize the writing speed, it is useful to take a closer look at the exposure strategy of the used electron beam writer. In general, there are two main strategies: Gaussian or shaped beam writing.
Whilst Gaussian beam writing provides a very high resolution which only depends on the electron beam diameter, it is limited by a slow writing speed. On the contrary, the shaped beam strategy can illuminate larger, connected areas within the element in one shot and is thus significantly faster, especially in case of pixelated or grating-like structures, while still providing a sufficient resolution. Therefore, the shaped beam strategy is preferable for BSWS elements. As common for variable shaped beam writers, the electron beam is formed by two apertures as schematically illustrated in Fig. 2 .
The positioning of the apertures allows a set of different simple geometries like triangles and rectangles. For our application the rectangular shape is essential which can be adjusted up to a maximum feature size of 2.5 µm. With one electron beam shot, one rectangular shape can be exposed into the resist. Provided that the same writing regime of the electron beam writer is used and similar CGH functions and areas are considered, the writing time is nearly directly proportional to the number of electron beam shots. Hence, our aim is to increase the writing speed by decreasing the number of shots with an appropriate shape design of the BSWS. In the case of positive resist material, as used for our application, the exposed area is removed during the resist development step. For a continuous-phase level hole pattern like demonstrated in  and illustrated schematically in Fig. 3 on the left side, every hole represents one electron beam shot. Because of the huge amount of different hole sizes, several pixels can rarely be connected to one electron beam shot. Therefore, nearly every CGH pixel needs one corresponding exposure shot. The same CGH structure converted with a pillar design would even take a much longer writing time because of the internal partitioning of the grooves surrounding the pillars. The discretization to three levels leads to a reduction to three different kinds of pixels. By an appropriate adjustment of the structural depth of the CGH pattern, it is possible to implement two of the three phase steps with a fully exposed and a completely unexposed pixel. Adjacent pixels which have to be fully exposed can be unified and exposed simultaneously with one electron beam shot. Regarding a 300 nm sampling grid up to 8x8 CGH pixels can be united. Consequently, the huge amount of fully exposed and not exposed pixels leads to a substantial reduction of writing time. A further reduction of electron beam shots and thus writing time can be achieved by reshaping the pixel responsible for the intermediate phase step. Replacing the hole or pillar structure by an appropriate ridge or groove structure (see Fig. 3 on the right side) leads to a further unification of electron beam shots. The groove and the ridge patterns differ from each other considering the required minimal feature size. In the case of a groove structure, the grooves are in a centered position within the CGH pixel, while for a ridge pattern this position is occupied by a ridge structure. The choice between these both structures particularly depends on the feasibility and stability of the pattern required for the intended filling factor. The use of groove or ridge patterns for effective CGHs allow a very high unification rate and are thus convenient for large-scale applications. The exact saving in writing time depends on the required CGH phase pattern which will be discussed in more detail in the next section with regard to a specific CGH phase function.
4. Binary subwavelength CGH design
The CGH phase function is calculated by a iterative Fourier transform algorithm (IFTA) . With an overall resolution of 2048x2048 pixels, the CGH phase pattern generates an off-axis 256x256 pixel matrix containing the desired far field distribution.
To improve the signal-to-noise ratio within the desired signal window, amplitude freedom outside the signal window is used applying an appropriate scale factor inside the signal window  which leads to higher noise in the outer region as visible in Fig. 4(b) . The calculated continuous-phase level distribution provides a theoretical diffraction efficiency in the signal window of about 70%. After discretization to three phase levels, the efficiency is reduced to 48.5%. These theoretical values are only valid for diffractive elements which can be assumed as thin elements. Using the thin element approximation (TEA) is only accurate when diffraction within the structure is neglectable. Hence, we always expect a lower efficiency regarding real diffractive elements. The CGH phase distribution enables diffraction angles up to α = 62.5° for a CGH pixel size of P = 300 nm, according to the grating equation sinα = m∙λ/p with a grating period of p = 2P considering first-order diffraction (m = 1). The signal window ranges from 15° to 29° in x-direction and ±6.5° in y-direction. Hence, we expect a preferred orientation in the y-direction with a blaze-like structure as illustrated in Fig. 4(a). For optimal unification of the electron beam shots during exposure, the pixels are united along this orientation. After discretization in N levels, the highest phase step is given by 2π(N-1)/N. For a three-phase level element, the maximum phase step is thus 4/3π. The depth of the discretized element is adjusted in such a way that the maximum and the minimum phase step can be realized by a fully exposed and a completely unexposed pixel. Because the required aspect ratio of the ridges and grooves can be problematic during the fabrication, we use an element operating in reflection, reducing the required depth by a factor of two since the incident wave passes the structured layer twice. We apply a 265 nm resist layer (FEP 171, Fuji Film, n = 1.58, k = 2e-6) on top of a sputtered 80 nm Cr layer (n = 5.26, k = 4.36) on a standard Si substrate [see Fig. 5(a) ]. The exact shape of the mid-level pixel (2/3π) can be obtained using the RCWA calibration curve [see Fig. 5(b)].
In principle, different kinds of BSWS are possible to generate a binary subwavelength CGH. In the chosen approximation, 2D hole and pillar arrays are polarization independent, whereas the state of polarization has to be considered for a 1D stripe structure. Furthermore, since the fabrication of such nanopatterns is a challenging process, some technological constraints have to be considered. First, the aspect ratio and the resolution of the resist are limited. Secondly, as discussed in the third section, the electron beam writing time strongly depends on the used BSWS design. Both constraints are evaluated and compared for this special layer system for different BSWS, introduced in the first section (see Fig. 3), like square holes, square pillars, groove and ridge structures. According to Table 1 , ridge and groove patterns with a filling factor of 0.475 for TE polarization and a groove pattern with a filling factor of 0.65 for TM polarization seem to be mostly relaxed considering the minimal feature sizes. Due to the mechanical stability of the nanostructure, a ridge structure is preferable because of the lower aspect ratio of the corresponding resist ridges compared to a centered groove structure. For this specific CGH phase pattern the calculated writing time for a 5 mm x 5 mm CGH area is illustrated in Table 2 .
Considering the identical CGH phase function and exposed area, applying the same electron beam writing regime, the writing time scales linearly with the number of electron beam shots. As expected, the writing time for a pillar pattern is about six and for a hole pattern about three times larger than for a groove or ridge pattern. Compared to a multi-phase level hole pattern, only one-eighth of the writing time is required with a stripe design.
Although using a groove pattern enables the fastest exposure, the ridge pattern is chosen because of the stability issues discussed before and since the difference in writing time is marginal. Thus, the exposed structure contains minimal resist ridges of 142.5 nm and minimal groves of 79 nm which is within technical feasibility. It is obvious that the use of such a ridge pattern causes slight polarization effects which are discussed in more detail in the next section. In Fig. 6 the resist pattern after the exposure, tempering and development process is shown. The structural parameters with a filling factor of about 0.47 and a resist depth of about 268 nm are very close to the required ones. Although the stripe pattern is slightly smeared due to the proximity effect and affected by the line edge roughness of the FEP 171 resist layer [see Fig. 6(b)], promising experimental results are obtained as demonstrated in the next section.
5. Experimental results
The measurements are performed using a 532 nm diode laser at normal incidence. Although the element is designed for TE polarization, TM polarized light was also used to determine the polarization dependence of the CGH structure. The measured diffraction efficiencies within the regarded regions correspond to the fraction of the incident light that is reflected and not absorbed by the Cr layer. To calculate the theoretical efficiency within the signal window for TM polarized light, the mid-level phase step is reduced from 2/3π to 0.426π using the phase delay given by the corresponding RCWA calibration curve at a filling factor of 0.475.
The diffractive element is strongly non-paraxial which leads to a slight geometrical distortion of the measured image with increasing angles [see Fig. 7 ], because the coordinates of the pixilated intensity distribution on the screen are not linearly dependent on the spatial frequencies used in the IFTA design. In Table 3 , the theoretical values based on TEA and the measured efficiencies are compared. For TE polarization, an efficiency of 33% is measured compared to 48.5% predicted theoretically. Only 1.5% gets lost in the zeroth order. The twin-image suppression is very good with only 2.5% in the corresponding region, thus it is hardly distinguishable from the stray light. An ideal three-phase level CGH fully suppresses the twin-image. The theoretically predicted loss of 1.6% is noise in the relevant region. In the case of TM polarization the losses in the twin-image region (5%) and in the zeroth order (11%) are much higher due to the mismatch of the intermediate phase step. Only 25% of the incident light is measured in the desired signal window. These measurements are in good agreement with theoretical data which show a similar trend. The mismatch between experimental and theoretical results is, on the one hand, caused by technological reasons, particularly by a slight deviation of the exposed resist structure from the desired one due to the proximity effect and the line edge roughness. On the other hand, the demonstrated RCWA-based design approach neglects the influence of adjacent BSWS calculating the phase delay under the assumption of periodic boundary conditions.
Since the arrangement of different kinds of BSWS, as required for CGH applications, leads to an interaction between them, a deviation of the realized phase delay from the predicted one is caused associated with a loss in diffraction efficiency which is difficult to estimate. The experimental results show that this interaction is reasonably weak, since no strong distortions in the created image can be observed. The rigorous design and optimization of a whole CGH, having extremely small feature sizes combined with a very large overall extent as employed for the example in this paper, is still not possible in a reasonable time with currently available simulation methods and computer hardware, even considering recently developed large scale stitching methods . An improved approximation for the calculation of the phase shift of one BSWS, considering the influence of a limited amount of adjacent pixels during the RCWA calculation, would be a viable next step to improve the diffraction efficiency within the presented framework.
The possibility to use BSWS for multi-phase level CGH applications in the visible range is demonstrated. An effective medium approach is exploited, based on the use of an RCWA calibration curve to transform the CGH phase function into a BSWS pattern. The use of BSWS leads to a reduction of the fabrication complexity and to a more flexible CGH design. Since the exposure process is very time consuming, even when using an advanced electron beam writer based on a variable shaped beam writing strategy, different kinds of BSWS were examined and an optimized exposure strategy could be developed. As demonstrated in this paper, the reduction to three phase levels applying an appropriate ridge pattern design leads to a substantial improvement in writing speed. The developed exposure strategy is further featured by having relaxed fabrication requirements with regard to issues of feasibility and stability of the BSWS pattern. For demonstration, a reflective binary subwavelength CGH was fabricated and evaluated. The results of the polarization-dependent measurement performed with a 532 nm diode laser are in accordance with theoretical predictions. Although the used RCWA-based design requires periodic boundary conditions of the BSWS und hence neglects the influence of adjacent pixels, promising experimental results could be achieved. Further design approaches should consider this influence to improve the diffraction efficiency.
The authors would like to acknowledge the financial support of the Bundesministerium für Bildung und Forschung (BMBF) (projects FKZ 13N9712 and 13N9433). Furthermore, W. Gräf, S. Drovs, H. Schmidt, M. Heusinger and M. Banasch are highly acknowledged for their cooperation in fabricating and evaluating the effective medium CGH.
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