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A periodic grating with an integer number of periods is fabricated at the resist-coated wall of a cylinder by exposing a circularly symmetrical planar high index phase mask to a cylindrical wave. This extends the spatial coherence features easily achievable in a planar 2D space to the 3D space of cylindrical waves and elements.
©2012 Optical Society of America
1. Introduction
The geometry of the overwhelming majority of diffractive optical microstructures is planar as their modus operandi relies upon achieving spatially resolved phaseshifts across an incident light beam. The enabling technology has been inherited from the formidable manufacturing potential of microelectronics. The model wave is a plane wave from which any beam can be easily represented by Fourier composition. In comparison with the planar paradigm, the field of cylindrical waves and the corresponding ensemble of optical elements is much less advanced. High aperture lenses and cones have been developed to transform a collimated beam into cylindrical waves as well as cylindrical lenses, rods and mirrors. However, cylindrical grating elements are essentially lacking, even though they could play an important role in angular rotation velocity sensors and biosensors, and may give rise to new resonance phenomena. Cylindrical gratings for rotation sensor applications exist, but they typically have a large period (several microns), enabling the grating to be fabricated by means of a high precision lathe and an ad hoc writing head locally mimicking a cylindrical symmetry [1,2]. However, such a scheme cannot achieve high resolution rotation sensing. Beyond rotation sensor applications, in the field of Fiber Bragg Gratings where phase masks have been used first, Kashyap [3,4] disclosed a circularly cylindrical phase mask projection technique permitting the deep UV writing of long gratings of short or long period in a fiber wound up around a rotating cylindrical grating or translating synchronously under a rotating radial phase mask. The best possible matching between the geometries of the fiber, of the DUV wave front and of the cylindrical or radial phase mask is designed to achieve the desired spatial frequency distribution along the fiber. The LIGA process can be used to fabricate submicron cylindrical gratings. The depth over which the grating lines remain separated before they merge is however limited to below 0.5 mm [5] because of the limited spatial coherence of the X-ray beam.
The approach described in the present article pertains to optical lithography and involves the holistic printing of a full cylindrical grating by means of a single exposure without mechanical movement. It uses a radial phase mask addressed by a conical cylindrical wave, which together accomplish a coordinate transform between a planar phase mask of radial symmetry and the resulting interferogram printed at the wall of a cylinder. As the planar phase mask can easily be fabricated with an integer number of submicron periods using an e-beam pattern generator, the coordinate transform also ensures that the number of periods at the cylinder wall is an integer.
The fabrication method described here provides a master template, from which an injection molding or embossing technique could be used to replicate a slave mask.
2. The coordinate transform
The geometrical arrangement for creating a cylindrical grating at the wall of a cylinder with lines parallel to the cylinder axis is shown in Fig. 1 . A collimated laser beam, parallel to the cylinder axis, impinges onto a centered reflective cone, which transforms it into a cylindrical wave with a non-zero axial k-vector component. The latter irradiates a phase mask comprising radial lines and grooves defined in a ringed area of inner and outer radii Ri and Re, respectively. The ring is centered on the cylinder axis, and the prolongation of every grating line intersects the axis. The phase-mask grating is characterized by an angular period Λϕ equal to 2π/N, where N is the integer number of periods. It diffracts in transmission a number of orders which propagate with a non-zero radial k-vector component in the transparent cylinder placed under the phase mask, where they mutually interfere and create interferograms, forming dark and bright fringes parallel to the axis. In particular, if only the + and −1st orders are propagating, a cylindrical latent grating of period RΛϕ/2 is formed at the resist-coated wall of a cylinder of radius R.
3. The phase mask design and fabrication
The oblique incidence on the radial phase-mask grating allows an interferogram to be projected onto the cylinder wall, unlike currently used phase masks where the interferogram lines are formed in a plane parallel to the phase-mask plane. The diffraction conditions are purely conical, and they are not constant radially since the period Λ(r) = rΛϕ increases proportionally with r between Ri and Re. The conditions which must prevail in the diffraction event are as follows: sufficient damping of the 0th transmitted order, which would otherwise superpose an interferogram of period RΛϕ onto the desired one; cutoff of the + and −3rd orders, which in a binary grating are always present; and damping of the + and −2nd orders. The 2nd orders are difficult to suppress under conical incidence. Therefore, we chose to use a phase-mask grating which gives rise to solely + and −1st propagating orders, corresponding to a maximum period ΛM at r = Re: ΛM < 2λ/(ns2-sin2θi)1/2, where ns is the cylinder material refractive index and θi is the incidence angle on the phase-mask grating. The condition of the minimum period Λm for the propagation of at least the 1st orders at r = Ri is Λm > λ/(ns2-sin2θi)1/2. One might have expected that the interferograms between the ( + &-1st) and the (1st and 0th) orders form on the cylinder wall at different distances from the phase-mask plane since the diffraction angle of the 0th and 1st orders relative to the cylinder axis are different. This however is not the case because the 1st orders are common to both interferograms. Therefore, damping of the 0th transmitted order must somehow be ensured. This can be achieved in wavelength-scale period gratings by making the corrugation in a high index layer [6], in the present case, LPCVD Si3N4 of refractive index nf = 2, deposited on a low index substrate, here fused quartz. Although the scaling rules of scalar optics for the adjustment of the depth of a corrugation are not valid when the period is of the order of the exposure wavelength, they can be used to obtain a starting point for an optimizing code using the RCWA or true-mode method [7]. Whereas the depth dn required for the cancellation of the 0th order is dn = λ/(2(nf – 1)) under normal incidence in air, it is dc = λ/(2(nfcos(arcsin(sinθi/nf)) – cosθi)) under pure conical incidence of angle θi.
A future application of the first cylindrical grating fabricated by the present phase-mask method is as a submicron grating of a miniature holistic rotation encoder placed on a 8 mm diameter shaft and having an integer number of 2N = 215 periods. This novel concept of an incremental optical rotation encoder simultaneously provides miniaturization, high angular resolution, low sensitivity to eccentricity and a high signal-to-noise ratio. The patented concept uses diffractive interferometry in a circularly cylindrical geometry in which two concentric submicron diffraction gratings with lines parallel to the axis of rotation rotate around each other [8].
The desired length of the cylindrical grating lines is about 1 mm. A fixed incidence angle of 60 degrees in air was chosen and Ri and Re of the phase mask radial grating were 1.2 and 2 mm, respectively. The number N = 214 of radial periods defines an angular period Λϕ = 383.5 μrad, which in turn defines the azimuthal periods Λi = 460 nm and Λe = 776 nm at radii Ri and Re, respectively. These data fit within the diffraction conditions detailed above and the optimization code is entrusted with the task of defining the duty cycle and silicon nitride thickness of the radial grating so as to best damp the 0th transmitted order under 60 degrees incidence at 442 nm wavelength over the range of periods from Λi to Λe. The polarization was chosen to ensure the largest interference contrast in the + and −1st order interferogram at the cylinder wall. Thus the axial collimated incident beam was radially polarized.
Figure 2 shows the transmitted 1st and 0th efficiency versus the phase-mask grating silicon nitride thickness for the range of periods considered. The power diffraction efficiency of the 0th order refracted wave was selected to be less than 8%, corresponding to around 160 nm Si3N4 thickness.
Fig. 2 Efficiency of the transmitted orders versus the silicon nitride thickness for different grating periods.
Figure 3(a) shows a sketch of the geometry of one radial period with detailed data and Fig. 3(b) shows SEM images of a number of periods written on and then etched in a Si3N4 layer of 166 nm thickness; several phase masks with slightly different duty cycle were fabricated to enable the exposure conditions to be adjusted experimentally. The writing time for the phase mask was 10.5 hours. The beam current was 1 nA, resolution 1.25 nm and beam step size 5nm. The applied exposure dose was 170 µC/cm2.
4. The exposure setup
The exposure bench comprises a single-beam coaxial setup as illustrated in Fig. 4 . A linearly polarized beam of a HeCd laser at 442 nm wavelength was coupled into a polarization-maintaining fiber single mode at 442 nm. At the fiber end, the exiting beam is collimated and circularly polarized. A radial polarizer based on resonant reflection from a circular line grating reflected the local TE polarization and transmitted close to 100% of the local TM polarization. Its circular groove corrugation covers an area of 6 mm diameter; the radial period is 256 nm, which can only be obtained with an e-beam. The grating was written and also etched in fused silica by Fraunhofer IOF Jena. After etching, a ZnS waveguide layer was deposited at a thickness of 60 nm, which was selected to produce resonant reflection via its fundamental TE0 mode excitation in the neighborhood of 442 nm.
Figure 5 is an image of the beam exiting the radial polarizer taken through a linear polarizer, showing the characteristic bow-tie intensity distribution.
The radially polarized beam then impinged onto a reflective cone placed at the center of the phase-mask grating ring. The apex angle of the cone was set to 55 degrees to give rise to the designed 60 degree incidence of the cylindrical wave onto the radial grating. The relative positions of the cone and grating ring are shown in Fig. 6 . An 8 mm diameter ring was defined on the mask to facilitate the relative alignment of the phase-mask and photoresist-coated cylinder.
5. The technological steps
Such a cylindrical diffractive element and the proposed method for its fabrication are completely outside the paradigm of planar microstructuring technologies inherited from microelectronics. Fabrication of the phase-mask required the highest precision and resolution that e-beam writing can offer today coupled with the most recent tools (e.g., Vistec EBPG 52000), which are able to control and write feature sizes in the nm range. Once the coordinate transform has been defined, all technological steps have to be adapted or re-invented: substrate holding/handling, photoresist coating, alignment/centering procedures, exposure, and last but not least, characterization. The present paper only reports the preliminary laboratory process steps developed to demonstrate the capability of the method to fabricate a cylindrical grating with an integer number of submicron periods in the photoresist coating at the cylinder wall.
Substrate handling and holding was achieved by means of a dedicated silicone vacuum manipulator, which supported one of the polished ends of the transparent cylinder.
The photoresist coating was applied by controlled pulling under constant vapor pressure. The photoresist was set in its linear or nonlinear regime depending on the desired grating profile, which can thus range between sinusoidal and binary. A very important aspect of the chosen resist photochemistry is its compatibility with a PMMA cylindrical substrate as the described method is not primarily aimed at cylindrical grating manufacturing but at obtaining a master, which then can be replicated at very low cost by embossing or microinjection molding. The 3D geometry of the element and the vulnerability of the microstructure at the element wall requires that the master can be easily dissolved away chemically after the growth of a nickel shim. A complete chain of resist processes has been developed specifically for PMMA substrates to enable this fabrication scheme [9].
The photoresist was subsequently coated by a light absorptive layer: firstly, to prevent the interferogram field from experiencing total internal reflection, and secondly, to prevent the formation of field nodes and antinodes in the resist layer. The resist baking was performed using the vacuum holder to prevent contact of the cylinder wall with any surface.
The resist-coated cylinder was held in place against the phase-mask substrate with an index-matching liquid in-between by means of a gripper, which was attached in a zone of the wall where the grating is not formed. High resolution micropositioners permitted centering of the cylinder by visual inspection from the other polished cylinder end.
6. The printed grating
The characterization of the obtained resist corrugation was challenging.
Only primitive and partially destructive AFM and SEM tests have been made so far. A special holder has been designed for both characterizations, which enables the cylinder to be fixed without damaging the grating. Figure 7(a) is an AFM image and Fig. 7(b) is a SEM view from above, showing the grating lines and a grating depth of about 360 nm. These images confirm that the grating grooves are fully opened throughout the photoresist layer. They also demonstrate that the grating profile is close to binary and rectangular, and the period is, as expected, 776 nm on the flank of the 8 mm diameter cylinder.
Fig. 7 Grating characterization. (a) AFM image of the grating at the cylinder wall and (b) SEM image of the resist grating at the cylinder flank.
The SEM image of the cylinder flank confirms that the resist corrugation has a single spatial frequency corresponding to the +−1st order interferogram. Thus, the weak residual interferogram of the 1st and 0th order was not printed in the resist. There is a small, essentially periodic undulation along the grating lines, whose origin remains to be confirmed. This weak longitudinal modulation is likely to be the effect of waves interfering at the cylinder wall with axial k-vectors parallel and antiparallel to the cylinder axis. The wave with antiparallel k-vector may be due to the reflection of the exposure orders at the bottom of the cylinder, which, unlike the wall, could not be made absorptive to enable optical alignment prior to exposure. Finally, Fig. 8 shows the visual aspect of the resulting resist grating at the cylinder wall. The aspect is very uniform and does not exhibit macroscopic non-uniformities.
Fig. 8 Micrograph of the printed grating (period below 800 nm) on the 8 mm diameter photoresist coated cylinder.
An interesting feature which the visual inspection reveals is a set of discrete, very fine curved dark lines exhibiting some systematic pattern of millimeter period, which, after careful inspection, can also be seen in the phase-mask grating. This feature is almost certainly related to the e-beam writing strategy and is currently being investigated. It was initially assumed that the large-scale accuracy and high resolution features of planar technologies could simply be adapted from 2D space to the 3D space of a cylindrical structure. Evidently, the printing of a submicron 2D structure with radial straight lines of narrowly neighboring orientation does reveal problems which usual straight line patterns crossing at 90 degrees do not. In e-beam writing, the pattern is split into mainfields, which are small enough for beam deflection. An XY-stage is used to move the sample between mainfields. There is inevitably some positioning error when the stage is moved, which may have a significant effect, particularly if the structure is circular. The pattern data conversion software LayoutBeamer from Genisys GmbH is now able to average these flaws by processing the data so that it is written multiple times with differently placed mainfields. This results in, e.g., four less visible borders instead of one very visible grid. This can now be improved as in this multi-pass technique the pattern is written multiple times placing mainfields in different places each time. The sum of exposure doses is equivalent to a one pass exposure. Thus, resulting data has more mainfield borders which are less visible, and the overall stitching error is averaged.
7. Conclusion
When exposed to a cylindrical wave, a circularly symmetrical planar phase-mask comprising radial lines projects a uniform interferogram at the wall of a cylinder with an integer number of dark and bright lines parallel to the cylinder axis. This setup transfers the spatial coherence features of a planar layout into the wrapped 2D space of a cylinder wall. This significantly extends the application field of phase-mask lithography to conical incidence and cylindrical waves, where the use of a phase-mask is the only feasible method for printing such cylindrical gratings. The full chain of related process steps has been defined and implemented, and measures have been developed to permit the downstream processes of mould preparation without damage to the flank corrugation. This work offers new fabrication possibilities for a novel type of 3D diffractive optical elements, whose geometry is well adapted to the processing of circularly symmetrical beams. Further work is required to develop the technological steps for use in manufacturing processes. However, there is also a need for adapting the writing strategy of e-beam pattern generators to the typical geometrical features of a 2D pattern expressed in a polar coordinate system.
Acknowledgments
This work was financed in part by a LST project (Lyon Science Transfert), number LST 566.
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