Parameter-Tolerant Binary Gratings

Transcription

1 Parameter-Tolerant Binary Gratings Thomas Kämpfe * and Olivier Parriaux Université de Lyon, Laboratoire Hubert Curien, UMR CNRS 5516, 18 rue du Professeur Benoît Lauras, F-4000, Saint-Etienne, France * Corresponding author: thomas.kampfe@univ-st-etienne.fr A normalized modal analysis of binary gratings under normal TE-incidence involving the most condensed set of optogeometrical parameters gives the complete solution to the phase mask problem of cancelling the 0 th transmitted order with a large tolerance on the corrugation duty cycle or a large spectral bandwidth. The solution is presented in the form of single normalized 3D charts which shed light on the fulfillment of the 0 th order cancellation condition: balanced excitation and -phase difference between two grating modes. Examples of tolerant gratings are given. OCIS codes: , , , , , Introduction Binary dielectric gratings of high aspect ratio having a restricted number of modes propagating up and down the grooves and walls exhibit interesting and somewhat surprising interference effects which can advantageously be used in novel diffractive elements. A widely used element is the Fiber-Bragg-Grating (FBG) phase mask [1] in the form of a simple binary corrugation of the surface of a fused silica substrate, or of a segmented high index layer in case the exposure wavelength/period ratio is only slightly smaller than one []. The most critical condition which a

2 phase mask under normal incidence must satisfy is the cancellation of the transmitted 0 th order so as to obtain a single spatial freuency 1 st order interferogram of close to 100% contrast. For given ridge and groove refractive index n r and n g, exposure wavelength and grating period, there is in general a groove depth h, a ridge width r, and a groove width g = - r, which essentially cancel the 0 th transmitted order. A standard grating modeling code [3] easily finds the structure parameters cancelling the 0 th order, however, one can usually not predict what the fabrication tolerances will be or how the operation point will behave if a parameter of interest, for example the wavelength, is changed. Identifying the parameter combinations not only cancelling out the 0 th order, but also providing fabrication tolerances or a local independence on a parameter represents the objective of the general design problem. Having realized that this 6- parameter problem can be reduced to a 3-parameter problem by suitably defining normalized variables, we show here that the general design problem can be solved once for all in the form of a normalized 3D-chart, as first attempted in [4]. In the present paper we will develop this approach from a more general standpoint, making it applicable to a broader set of problems and providing a clearer and deeper insight. Beyond the scientific interest of condensing a wide variety of optogeometrical structures having essentially the same optical function into a single general functional structure, and giving a more intelligible modal representation, such a normalized approach does also have a practical significance: this will be shown in the present paper with two distinct objectives. The first objective is the search for the conditions of zero 0 th order transmission with large tolerance on the duty cycle r/. The motivation for this is that dielectric phase-masks for the printing of periods of the order of the exposure wavelength are difficult to fabricate. Whereas the control of the depth is usually not a problem, the duty cycle is difficult to control precisely as

3 the fabrication technology involves two steps where the width control can be lost: resist exposure and development, and reactive ion etching. The second design objective is to obtain zero 0 th order transmission over a relatively wide spectral domain. Whereas phase-masks are mostly used with uasi-monochromatic laser sources, there is a strong interest in using wide-band femtosecond pulses for laser lithography, machining and marking. For instance, the printing of a latent grating in a resist film by two-photon exposure [5] reuires all power to be in the 1 st orders over a spectral range of some tens of nanometers. The present normalized investigation will be carried out for TE polarization since it leads to the formation of a 1 st order interferogram of potentially 100% contrast. However, the general formalism could also be applied to other polarization states.. Modal description of the grating response using normalized parameter The considered structure is composed of three regions as shown in Fig. 1: a corrugated layer, a cover and a substrate. The binary corrugation has a depth h, a period, ridge and groove refractive indices n r and n g. Since the scope of the paper is limited to binary rectangular corrugations, the substrate and the cover also have the refractive index n r and n g, respectively. The incoming beam is a plane wave in the cover propagating in the negative z-direction. The E- field of the TE polarization is oriented in y-direction along the grating lines (TE-polarization).

4 Fig. 1. Binary corrugation grating with ridge and groove refractive index n g and n r, period, line width r, groove width g = - r, and height h under perpendicular TE-illumination. According to a grating mode representation [6] the electromagnetic field in the grating region E g (x,z) can e represented as a sum of modes, characterized by their field distribution E (x) and a propagation constant k z, (or effective index n k z, ) in the z-direction: E g ikz, z ikz, z x, z E xb e b e 0, 0 z h (1) The coefficients / a represent the amplitudes of the th mode in the upward and downward direction, respectively. As described in [7], the modes can be understood as plane waves in the ridges and grooves experiencing reflection and transmission at their vertical boundaries: E x a a r ikx, rx e g ikx, gx e a a r ikx, rx e g ikx, gx e,, nd x nd r nd r x n d 1 r () where r g a / r g a / are the field amplitude constants for the ridge and groove in positive and negative x-direction, respectively. The propagation constants k x and k z in x and z-direction are 0. 5 g / r kz, 1 related via k n x, g / r. The continuity of the electric field and its derivative at the ridge-groove boundaries leads to the known dispersion euation [8]:

5 cos k k, r x, g x, r x, g cos x in (3) kx, g k x, r x, r x, g k rcosk g 0.5 sink rsink g k d x, where k x,in is the x-component of the k-vector of the incoming wave. The number of parameters will now be reduced by introducing the following normalizations: duty cycle: d r (4) refractive index ratio: relative period in ridges: n g c (5) n r p (6) n r Furthermore, the normalized coordinates u = x/, v = y/ and w = z/ are introduced and a relative modal propagation constant is defined: n k z, (7) n n r r Using this new set of parameters and assuming normal incidence, the euations can be significantly simplified. Since r g a / r g = a /, the fields from E. () can be expressed in real terms with cosine or sine functions of even or odd symmetry with respect to the center of ridges or grooves. Under normal incidence the odd modes are not excited, therefore one only keeps the even modes: E even u a a r g cos p cos p 1 c u 0.5d u 0.5d 0.5,, 0 u d d u 1 (8) The modal fields represent an orthogonal and complete functional basis of the electric field inside the grating region [7], with the orthogonality condition:

6 1 0 E * ue pu p ( p : kronecker-delta) (9) which allows, together with the condition of continuity of the field at the groove-ridge boundary, to express the coefficients r a and g a fully analytically: w 1 d sinw cos w d a g sin 1 d 1 d, cos w1 w1 w r cos w a g a cos w (10) 1 using the auxiliary variables w 1 pd 1 and w p 1 d c. The dispersion euation (4) for the case of normal incidence (k x,in =0) then writes: 1 1 cos cos c F w1 w sinw1 sinw 1 0 (11) 1 c The solutions to the implicit euation F() = 0 can be found numerically, resulting in the mode propagation constants, which are either real or imaginary, indicating propagating or evanescent modes, respectively as sketched in Fig. for an arbitrary structure. Fig.. Graph of the dispersion euation versus the suare of the normalized modal effective index in a binary grating under normal incidence.

7 With this set of modes the S-Matrix of the grating can be found [7] and the diffraction spectrum of the grating calculated. The dispersion euation (11) can be solved numerically like the nonnormalized euation (4), however it is now in a form which permits a concise and general analytical analysis disclosing important properties of the solutions, for instance the tolerances on some optogeometrical parameters or the wavelength sensitivity. 3. The two-beam interferometer analogy In most cases of practical interest, only the lowest order modes contribute significantly to the propagation of the light through the grating region, which occurs when the period is at the scale of the illumination wavelength. There are three modes playing the most important role: the TE 0, TE 1 and TE. Under normal incidence the parity of the TE 1 mode is odd which implies that it is not excited. Some care must however be taken in extreme situations: for instance, in structures where the duty cycle d is small the TE 1 and TE may swap parity. Another critical situation is close to the cutoff of the TE 4 mode, or/and in case the corrugation depth is small, where the decay of this first evanescent mode is small enough to lead to a significant power transmission through the corrugation by tunneling. However, the case of shallow corrugations is rarely met in phase masks since the reuirement of destructive interference of the 0 th transmitted order imposes a rather large height h to give rise to a relative phase of the TE 0 and TE modes as large as. Under the above-mentioned assumptions a very simple model of the transmission through the grating is possible (represented symbolically in Fig. 3): The incident field is split at the first interface into the 0 th and nd grating modes. These modes propagate down the grating with their propagation constants, interfere at the second interface, which acts as a beam mixer, and

8 produce transmitted diffraction orders in the substrate. The interference product is solely defined by the accumulated phase difference dif and the amplitude ratio b (hereafter referred to as the balance) between the fields coming from the two branches. For instance, a complete suppression of the transmitted 0 th order reuires a balanced transmission (b = 1) and a phase difference resulting in a destructive interference ( dif = ). Fig. 3 Two-beam interferometer analogy of light transmission through a binary corrugation. The validity of this simple model rests on the further assumption that the impedance mismatch at the grating-cover/substrate interfaces is not so large as to give rise to relatively large reflection and reflective mode coupling which would occur in the case of a large index semiconductor substrate or corrugated layer (3.3 (GaAs) to 4 (Ge)) where it was shown that the hypothesis of reflectionless and couplingless propagation in the grating cannot account for its functional features []. This hypothesis is however essentially satisfied in the domain of phase masks since the refractive index of the transparent materials used in the blue, UV and deep UV range does not exceed.5. This approximation will nevertheless be checked at the end of section 6 for a set of parameter tolerant gratings that will be found using the simple two mode model. As

9 the transmission of the interferometer, as well as its dependence on any parameter, is solely determined by dif and b, it is useful to obtain both uantities in an analytical closed form. The phase difference dif between the two branches is given by: dif hn0 n c0 c c s (1) where ci and is represent the phase of the transmission coefficients t ci and t is at the substrategrating and grating-cover interfaces, respectively, for mode i. Rigorous calculations of these phase jumps in the parameter domain of interest and under normal incidence via the S-Matrix [7] reveal that their absolute value is always close to zero. As a conseuence, and using the normalized parameter, the phase difference accumulated upon transmission through the corrugation writes: (13) dif ph dif with dif = 0 -. For given grating parameters one can therefore always find a grating height h which produces a reuested phase shift, for instance for a destructive interference. This confirms that h is not a parameter involved in the present design problems. The search for the analytical expression of the balance b reuires the understanding of the mechanism whereby the incident plane wave field couples to the grating mode fields which in turn couple to the transmitted diffraction orders. The overall transmission along the path of one of the modes can be calculated by multiplying the transmission coefficients at the two interfaces. These coefficients are elements of the S-Matrix that is calculated by numerical inversion of the T-Matrix using a large number of grating modes [7]. The dependence of these coefficients on the grating parameters must in general be calculated numerically. However, if no additional output

10 ports at the reflection side of the considered interface are present, the calculation of the transmission coefficients t is significantly simplified and remains analytical, since it is in this case solely determined by a multiplication of the overlap integral between the field distributions at an interface () by the impedance mismatch term (), which is in the TE case euivalent to the well known Fresnel coefficient of a plane interface (the validity of this meaningful simplification is ensured if negligible reflection and reflective mode coupling takes place at the interfaces). The field overlap between modal field E and that of a diffraction order E p is given by the integral: 1 *, p ExE pxdx (14) 0 In the present case where one is interested in the suppression of the transmitted 0 th order, the wave outside the grating is a normally incident plane wave with unit amplitude, i.e. E p (x) = 1. Using the normalized parameters, E. (14) reduces to: 1 E udu (15) 0 Inserting the field defined in E. (8), the integral is found analytically, resulting for even modes of order in: r g a d w a 1 1 d w sin sin (16) w1 w The impedance mismatch term is eual to the Fresnel-transmission coefficient between two bulk media with refractive indices n 1 and n for the case of perpendicular incidence: 1, n n n 1 (17) In the grating region, the refractive index in E. (17) will be replaced by the effective index of the mode, which means that in the present case of a binary corrugation grating, the impedance

11 mismatch can be expressed at the cover ( c, ) and substrate (, s ) interfaces by means of the normalized propagation constants (E. (7)):, c c,, s 1 c (18) Now the balance b between the amplitudes at the output of the two interferometer branches can be expressed analytically as: b t0 t,00, c0t0s c s 0 (19) t tcts c,, s Using E. (17) and (19) one can calculate the transmission of the incoming energy to the 0 th transmitted order from the phase difference and the balance. 4. Searching for duty cycle and for wavelength tolerant gratings The normalized expressions (13) and (19) of the two uantities dif and b governing the transmission of the two-branch interferometer now permit to take up the two design problems of the present paper: the determination of the set of all grating structures exhibiting, firstly, a wide transmission tolerance on the corrugation duty cycle d, and, secondly, a locally wavelength independent transmission. In particular we aim at a cancellation of the 0 th transmitted order over a large range of the duty cycle or over a wide spectral range, respectively. Although these two properties are of different nature (the first one is a fabrication tolerance while the second one is a functional feature), the generalized formalism established above permits to treat both analyses similarly The general condition for duty cycle tolerance Before tackling the problem in its generality we give an intuitive insight of why and how the phase difference and the balance can be made independent of d. To this end, Fig. 4 shows the

12 typical duty cycle dependence of the effective indices and transmission t 0 and t of the first two even modes for a typical value of the normalized period p and refractive index ratio c. Varying d from 0.1 to 1 first increases and then decreases dif, which reaches a maximum at d 0.4. At this point corresponding to ( dif )/d = 0 (and thus ( dif )/d = 0 since dif is directly proportional to dif with a d-independent factor of ph/, E. (13) ), dif will exhibit a large tolerance to changes of d. For the transmission t of the modes, Fig. 4 shows that for the cutoffs at d = 0 and d = 1 the 0 th mode has maximal transmission and the nd mode transmission is zero, whereas between these extreme values the t 0 and t reach a minimum and maximum, respectively, around d 0.5. In this domain, which can be found by evaluating b/d = 0, the balance b will be tolerant to a variation of the duty cycle.

13 Fig. 4. Effective indices 0 and (upper plot) and transmission t 0 and t (lower plot) of the first two even grating modes as a function of the duty cycle d for p = and c = 0.5. Shaded region: duty cycle tolerant domain of the phase difference and balance, respectively. Returning now to the general problem, an exhaustive analysis of all possible binary gratings amounts to finding all (d,c,p)-triplets exhibiting b/d 0 and ( dif )/d 0 inside a three-dimensional (d,c,p)-cube bounded by reasonable values of three normalized parameter. For this search the effective index difference and the balance between the 0 th and nd mode are calculated on a three-dimensional, regular grid consisting of 70x70x70 points, limited by the conditions 0.<c<0.8, 0.1<d<0.8 and 1<p<5. Subseuently the reuired derivatives are approximated by finite differences on the (d,c,p)-grid.

14 Fig. 5 shows that the set of structures satisfying the balance tolerance condition b/d 0 represent a surface of very small curvature, while the phase difference tolerance condition ( dif )/d 0 forms a very similar and almost parallel surface, which is located close to the former. This confirms that the general behavior observed in the d-graphs (Fig. 4) is preserved for different values of c and p. The two surfaces do not intersect which in principle means that there are no structures exhibiting a tolerance of both b and dif to the duty cycle d. However, the two surfaces are very close to each other and can be made to merge provided the zero criterions of the derivatives are sufficiently released, causing a wider thickness of the planes. A common plane is formed when the criterions are chosen as b/d <1 and ( dif )/d <0.1 (Fig. 5), still ensuring a relatively small dependence of the 0 th order transmission on d.

15 Fig. 5. (Color online) 3D visualization of (d,c,p)-triplets, fulfilling separately the condition of large phase difference and large balance tolerance with regard to duty cycle variations (upper plot), and fulfilling both conditions at the same time for a weakened zero-criterion, with the color scale indicating the weighted sum of both conditions (lower plot). On the side faces of the cube shadow projections of the (d,c,p)-triplets are depicted. It is worth pointing out that the search for a large 0 th order tolerance amounts to the search for a low d-sensitivity of dif and b separately. An exact calculation of how strongly these two conditions affect the 0 th order amplitude in comparison to each other goes beyond the scope of this paper. However, keeping in mind the analogy of the Mach-Zehnder interferometer tuned

16 to destructive interference, it can be stated that the transmitted power is approx. eually sensitive to a small phase difference detuning dif (i.e. dif = + dif ) and a small balance detuning b (i.e. b = 1+b). Therefore the conditions b/d = 0 and dif /d = 0 are approx. eually important for a large tolerance of the 0 th order. Furthermore, we assume that dif in the interesting parameter region, which means that dif dif for reaching the desired state of destructive interference ( dif =). Therefore, we assume that the influence of dif /d is approx. 10 times greater than the influence of b/d, which is reflected by the choice of the thresholds for displaying the (d,c,p)-triplets in Fig. 5. Having defined in Fig. 5 the locus of all possible grating structures whose 0 th order transmission is locally independent of the duty cycle d, the problem can now be specified further to a phase mask grating where the 0 th order must be cancelled. This only imposes b = 1 since h can be freely chosen so as to give the needed phase shift between the TE 0 and TE modes. To visualize this condition the already calculated 3D scan of the (d,c,p)-space can be used to sort out all triplets satisfying b = 1, defining the subset shown in Fig. 6. It appears in the shape of two connected surfaces, characterized by a strong curvature.

17 Fig. 6. (Color online) 3D visualization of (d,c,p)-triplets fulfilling the balanced transmission ratio condition. Clearly, the surfaces of Fig. 5 and Fig. 6 intersect and define as shown in Fig. 7 the locus of all phase mask structures exhibiting a cancellation of the 0 th transmitted order which is tolerant to the duty cycle d. The colors of the depicted spheres in Fig. 7 represent a weighted average of the second derivative of b and with respect to the duty cycle. A stronger curvature of b(d) and (d) results in a smaller actual width of the tolerant region, thus the second derivatives can be used to compare the expected width of the tolerant region for different parameter choices in the locus of optimal solutions. In general a high refractive index ratio c (i.e. low contrast between ridges and grooves) will lead to the largest tolerance widths. It is remarkable that the line shaped locus contains all possible binary grating structures exhibiting this tolerance characteristic, making the result imprinted in Fig. 7 valid once for all.

18 Fig. 7. (Color online) 3D visualization of (d,c,p)-triplets fulfilling both the tolerance condition with regard to duty cycle changes (Fig. 5) and the balance condition (Fig. 6). 4.. The general condition for wavelength independence In a second step, we will follow the same process for establishing the general condition for a locally wavelength independent 0 th order transmission. This amounts to finding out all (d,c,p)- triplets exhibiting b/ 0 and ( dif )/ 0. These conditions can be rewritten in terms of the normalized parameters d, c and p, keeping in mind that the wavelength is contained in parameter p. Whereas the conditions b/ 0 is identical to b/p 0 (keeping constant) as can be seen from E. (6), ( dif )/ 0 is identical to (p dif )/p 0, since the phase shift between modes contains the wavelength explicitly. As from here the same 3D-scan takes place in the (d,c,p)- space for first sorting out all structures satisfying b/p 0 and (p dif )/p 0 separately as shown in Fig. 8. In contrast to Fig. 5, the two separate subsets exhibit a much stronger curvature, which results in a strong divergence of the two subsets in the range of small p and large d values, meaning that both conditions cannot be satisfied simultaneously. By contrast, in the upper left

19 part of the 3D space (d < 0.6 and p >.5) the surfaces have essentially the same, low-curvature shape, are parallel and very close to each other. This means again that by releasing the zero criterion for the derivatives these two surfaces can be made to merge, indicating that the wavelength independence of both b and dif can be achieved simultaneously for all (d,c,p)- triplets belonging to this merged part. Fig. 8. (Color online) 3D visualization of (d,c,p)-triplets fulfilling separately the conditions of broadband tolerance of the phase difference and the balance These results can now be specified further for a phase mask grating where b = 1 and dif = for the cancellation of the 0 th transmitted order over a wide wavelength range. As in the case of duty cycle independence, h can be freely chosen to achieve the phase shift of. The condition b = 1 is already established in the general 3D space of the normalized parameters d, c and p (Fig. 6). The intersection of the merged surfaces of Fig. 8 with that of Fig. 6 gives in Fig. 9

20 the locus of all phase mask structures exhibiting zero 0 th order transmission over a wide spectral range. Fig. 9. (Color online) 3D visualization of (d,c,p)-triplets showing broadband behavior (Fig. 8) and fulfilling the balance condition (Fig. 6). Interestingly, the subsets depicted in Fig. 7 and Fig. 9 are rather well defined lines which can be given an approximate algebraic formulation. Observing the shadow images of the lines at the side faces of the (d,c,p)-cube, the duty cycle shows minimal variation, and can thus assumed to be constant. The remaining relation between normalized period and refractive index ratio can be approximated by uadratic functions, given below the graphs in Fig. 7 and Fig. 9. If the thresholds for depicting the optimal solutions are weakened, i.e. a slightly degraded balance and an increased sensitivity to parameter variations are allowed, the main effect is a widening of the diameter of the line-shaped subsets. Therefore these analytical approximations of the line shaped

21 subsets are indeed a valid starting point for easily finding the most duty cycle or wavelength tolerant gratings, which will be illustrated in the following section by concrete examples. 5. Application examples In order to create a concrete grating from the 3 normalized parameters, of the 5 optogeometrical parameters n g, n r,, and r (Fig. 1) can be chosen arbitrarily. We will use an air cover (n g = 1.0) and a wavelength in the visible region ( = 53nm). Since according to Fig. 7, the optimal (d,c,p)-triplets cover the whole considered range of the effective index ratio (c = , corresponding to n r = with an air cover), one is also free to chose the refractive index of the ridges. We use n r = 1.45, which is in practice easily realizable by an SiO substrate, thus fixing the refractive index ratio to c = With the polynomial expressions given in Fig. 7, the optimal (d,c,p)-triplet is found to be (0.31, 0.69, 3.00) for duty cycle tolerance and (0.39, 0.69, 3.67) for wavelength tolerance. This corresponds to two gratings with the optogeometrical parameters as given in Fig. 10.

22 Fig. 10. Transmission to 0 th and 1 st order, calculated with RCWA, of a duty cycle (upper plot) and wavelength (lower plot) tolerant grating, using parameters from the line-shaped subsets in Fig. 7 and Fig. 9. The height of the grating is optimized for 0 th -order suppression. The calculation of the 0 th and 1 st order for a variation of the corresponding parameter was done with a rigorous method (rigorous coupled wave approach, RCWA [9]) and is therefore independent of the approximations of the simple two mode model. As predicted, the gratings are capable of suppressing the 0 th order over a large area of the duty cycle (T 0 <1% for d= ) or the wavelength (T 0 <1% for =495nm..586nm), respectively. In order to judge, how this compares with other gratings, we changed a grating parameter so that the corresponding (d,c,p)-triplets are

23 situated outside the line-shaped subset of Fig. 7 and Fig. 9, recalculated the optimal grating height for 0 th order suppression, and plotted the resulting dependence of the 0 th order on the ridge width (Fig. 11) and the wavelength (Fig. 1). For all considered gratings, either the resulting parameter tolerance or the absolute value of the achievable suppression of the 0 th order, or both, were found to be worse for (d,c,p)-triplets away from the line shaped subsets. With the help of Fig. 5 and Fig. 6 it is even partly possible to explain this behavior. For example, in case of the duty cycle tolerant grating, (d,c,p)-triplets corresponding to a reduction of the period (i.e. a reduction of p), are already situated below the lowest point of the subset of the balanced transmission condition, thus no complete suppression of the 0 th order can be achieved. On the contrary, for a larger period an increase of the duty cycle retains the ability of perfect 0 th order suppression, but leads away from the planes in Fig. 5, which means that the achievable duty cycle tolerance decreases. Fig. 11. (Color online) Comparison of the duty cycle tolerant grating from Fig. 10 (solid line) to gratings with different period (dotted lines).

24 Fig. 1. (Color online) Comparison of the wavelength tolerant grating from Fig. 10 (solid line) to gratings with different duty cycle (dotted lines). In Fig. 13 we compare different gratings from the locus of largest tolerance from Fig. 7 and Fig. 9. The observed parameter tolerant behavior for c = allows to make the converse argument that the two mode modal representation is an appropriate model of the grating behavior in this area, since otherwise more complex, multi beam interference behavior would emerge. This can be observed for the case c < 0.5, which proves the increasing influence of neglected effects like multiple reflection and higher propagating orders.

25 Fig. 13: (Color online) Transmission to the 0 th order for optimized duty cycle (upper plot) and wavelength (lower plot) tolerant grating from the line shaped subset of Fig. 7 and Fig. 9, respectively, for different values of the index ratio c. Calculating the reflection and transmission coefficient of the modes rigorously [7] reveals that the main effect not covered by the two mode interference model is a significant back reflection of the modes to themselves due to an increased impedance mismatch term for higher effective index differences. For example in the case of the gratings of Fig. 13, for c = the largest reflection coefficient, occurring for the nd grating mode, allows a maximum of

26 approx 0.3% of the energy reaching the second interface via the nd mode to make a roundtrip back up and down the grating, which is negligible. This value increases to approx. 7% for c = 0.3, which is enough to significantly influence the interference state predicted by the simplified two mode model and destroy the large tolerance behavior, as can be observed for c < 0.5 in Fig Discussion In this work we considered the problem of designing binary gratings that suppress the 0 th transmitted TE diffraction order in order to give an optimal contrast of the interference pattern arising from the 1 st and -1 st transmitted order for use in near field photolithographic applications with the objective of generalizing the phase mask technology. This was undertaken by means of a normalized formalism in terms of the most reduced set of parameters and led to optimized solutions which are given once for all and do not have to be recalculated. Our concern in the choice of the treated design problems was to offer a solution to a fabricability issue which is the most difficult one: the control of the duty cycle d. The normalized analysis reveals that it is possible to closely satisfy the duty cycle independence of the two uantities determining the amplitude of the 0 th transmitted order in the two mode interference model of the phase mask grating over a definite range of normalized parameters given by an algebraic expression. From this expression the designer can immediately know whether a set of refractive indices, wavelength and desired period permit a duty cycle tolerance or what index should be selected to place oneself in a tolerant situation for a given period/wavelength ratio. Our concern was also to offer a universal solution to a functional issue, which is the widening of the spectral width over which the 0 th transmitted order can be cancelled. The solution, which can also be contained in a

27 single algebraic expression, determines all phase mask structures possibly useable for twophoton lithography and, more generally, opens the way to periodic surface and volume structuring by means of femtosecond pulses. The occurrence of the tolerant behavior was explained with the help of a simplified modal model, which takes only the first two important modes into account. The use of generalized parameters and of a vivid 3D-visualization of the tolerance conditions allow an exhaustive investigation of all possible surface relief gratings and results in very easy, analytical design instructions for finding the most tolerant grating structures. The analysis is not limited to duty cycle and wavelength tolerance, since a search for gratings that are tolerant against for example a variation of the period or the refractive index ratio can simply be done by changing the large tolerance conditions on dif and b accordingly and subseuently scanning the (d,c,p)-space as presented in this paper. It would also be possible to search for a combination of conditions, e.g. broadband gratings with large tolerances to critical parameters during fabrication. Furthermore, the generalized approach can be applied to more complex systems, e.g. surface relief gratings having a different refractive index in the substrate or cover region compared to the ridges and grooves, respectively. This will not influence the effective indices of the modes (Fig. 5), but it will change the transmission ratio related subsets (Fig. 6) due to different impedance mismatch terms. Thus the combined subsets (Fig. 7) may shift their position, making different areas of the (d,c,p)-cube accessible and possibly leading to even better solution to the problem of tolerant gratings. Finally it is possible to adapt the approach to different optical setups that have been proven to be explainable by a simplified modal method (e.g. Littrow mounting [10]) and/or to different polarizations. 7. Acknowledgement

28 The postdoc support of the University of Saint-Etienne for T. Kämpfe is gratefully acknowledged. Furthermore, we would like to thank Prof. Alexandre Tishchenko and Dr. Emilie Gamet for fruitful discussions on the subject. 8. References 1. K. Hill, B. Malo, F. Bilodeau, D. Johnson and J. Albert, Bragg gratings fabricated in monomode photosensitive optical fiber by UV exposure through a phase mask, Appl. Phys. Lett. 6, (1993).. E. Gamet, A.V. Tishchenko and O. Parriaux, Cancellation of the zeroth order in a phase mask by mode interplay in a high index contrast binary grating, Appl. Opt., 46, (007) 3 MC-Grating, MC Grating Software Development Company ( 4 E. Gamet, F. Pigeon and O. Parriaux, Duty cycle tolerant binary gratings for fabricable short period phase masks, Journal of the European Optical Society, 4, (009) 5 S. Jeon, V. Malyarchuk, J.A. Rogers and G.P. Wiederrecht, "Fabricating three-dimensional nanostructures using two photon lithography in a single exposure step," Opt. Express, 14, (006) 6 I. Botten, M. Craig, R. McPhedran, J. Adams and J. Andrewartha, The dielectric lamellar diffraction grating, J. Mod. Opt., 8, (1981) 7 A. Tishchenko, A. Phenomenological representation of deep and high contrast lamellar gratings by means of the modal method, Opt. Quantum. Electron., 37, (005) 8 P. Sheng, R. Stepleman and P. Sanda, Exact eigenfunctions for suare-wave gratings: application to diffraction and surface-plasmon calculations, Phys. Rev. B, 6, (198)

29 9 M. Moharam and T. Gaylord, Rigorous coupled-wave analysis of planar-grating diffraction, J. Opt. Soc. Am., 71, (1981) 10 T. Clausnitzer, T. Kämpfe, E.B. Kley, A. Tünnermann, U. Peschel, A. Tishchenko and O. Parriaux, "An intelligible explanation of highly-efficient diffraction in deep dielectric rectangular transmission gratings," Opt. Express 13, (005)

30 List of Figure Captions 1. Binary corrugation grating with ridge and groove refractive index n g and n r, period, line width r, groove width g = - r, and height h under perpendicular TE-illumination.. Graph of the dispersion euation versus the suare of the normalized modal effective index in a binary grating under normal incidence. 3. Two-beam interferometer analogy of light transmission through a binary corrugation. 4. Effective indices 0 and (upper plot) and transmission t 0 and t (lower plot) of the first two even grating modes as a function of the duty cycle d for p = and c = 0.5. Shaded region: duty cycle tolerant domain of the phase difference and balance, respectively. 5. (Color online) 3D visualization of (d,c,p)-triplets, fulfilling separately the condition of large phase difference and large balance tolerance with regard to duty cycle variations (upper plot), and fulfilling both conditions at the same time for a weakened zero-criterion, with the color scale indicating the weighted sum of both conditions (lower plot). On the side faces of the cube shadow projections of the (d,c,p)-triplets are depicted. 6. (Color online) 3D visualization of (d,c,p)-triplets fulfilling the balanced transmission ratio condition. 7. (Color online) 3D visualization of (d,c,p)-triplets fulfilling both the tolerance condition with regard to duty cycle changes (Fig. 5) and the balance condition (Fig. 6). 8. (Color online) 3D visualization of (d,c,p)-triplets fulfilling separately the conditions of broadband tolerance of the phase difference and the balance. 9. (Color online) 3D visualization of (d,c,p)-triplets showing broadband behavior (Fig. 8) and fulfilling the balance condition (Fig. 6).

31 10. Transmission to 0 th and 1 st order, calculated with RCWA, of a duty cycle (upper plot) and wavelength (lower plot) tolerant grating, using parameters from the line-shaped subsets in Fig. 7 and Fig (Color online) Comparison of the duty cycle tolerant grating from Fig. 10 (solid line) to gratings with different period (dotted lines). 1. (Color online) Comparison of the wavelength tolerant grating from Fig. 10 (solid line) to gratings with different duty cycle (dotted lines). 13. (Color online) Transmission to the 0 th order for optimized duty cycle (upper plot) and wavelength (lower plot) tolerant grating from the line shaped subset of Fig. 7 and Fig. 9, respectively, for different values of the index ratio c.