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Abstract
We present a theoretical investigation on laser interference lithography used for the exposure of linear gratings. The focus is on the geometry of the arising interference lines on the substrate, in particular on their period and orientation, depending on the illumination geometry as determined by the setup. The common approach with point sources emitting spherical wavefronts is considered for the illumination. Three different cases are discussed, namely the interference between two point sources with either two convex, two concave or mixed, i.e., convex and concave wavefronts. General equations focusing mainly on the calculation of the period and the orientation of the grating lines are derived for each of the three exposure cases considering arbitrarily positioned point sources and arbitrarily shaped substrates. Additionally, the interference of symmetrically positioned point sources illuminating plane substrates is investigated, as these boundary conditions significantly simplify the derived equations.
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1. INTRODUCTION
Within the last decade, diffraction gratings, i.e., grating waveguide structures, have become essential in laser development. They can be used to serve various purposes, such as intra-cavity stabilization of polarization and wavelength [1,2] or spectral beam combining for power-scaling [3]. Furthermore, they play a particularly important role in the field of chirped pulse amplification (CPA), where they are used in optical pulse compressors. Pulse compressors based on diffraction gratings are state of the art and allow for high dispersions, which in turn form the basis of high-power lasers operating in the megajoule and petawatt range [4–6]. The periodic structures of diffraction gratings are usually produced using well-established technologies in the micro- and nano-electronic industry. The step in the production that mainly defines the geometry of the structures is the lithography, where various techniques are common. Electron-beam lithography, where the structures are directly written into the photo-resist, is very flexible but comparatively slow [7–10]. Nano imprint is faster but less flexible, as the geometry of the structures is fixed by the master stamp [7,11,12]. A technique that is very suitable to produce periodic linear structures is laser interference lithography (LIL), as it is both fast and flexible. The exposure of the photo-resist exploits the interference of two coherent beams, which form a standing intensity pattern inside the photo-resist [13,14]. Two configurations are commonly used for the experimental implementation, which are briefly described in the following. In the first one, the laser beam is split into two beams using a beam splitter. The two coherent beams are then divergently guided toward the substrate, illuminating it with the laser’s far field. The second approach is the so-called Lloyds-mirror approach, where the laser beam is not split up but interferes with itself using a mirror placed perpendicular to the substrate [13–17]. Another approach that cannot be left unmentioned is scanning beam interference lithography (SBIL). However, since its underlying principle is not the one of a one-shot illumination but consists in writing the grating lines similar to electron-beam lithography [18], it is usually not included in classical LIL and, hence, also not in the context of this publication.
On the one hand, the simple generation of the standing intensity pattern and the one-shot illumination is what makes LIL so attractive; on the other hand, it is also its biggest weakness. By principle, only plane wavefronts can create a standing intensity pattern with constant period on a plane substrate. In practice, the incident waves are divergently propagated over long distances, with the planarity being only an approximation [15]. The consequence is called chirp or hyperbolic periodicity, which are both synonyms for the spatially dependent change of the period of the standing intensity pattern on the photo-resist coated substrate.
The term chirp was already used in 1977 by Katzir et al. [19] to describe structures that were produced by interfering plane and convex wavefronts. Since the chirp is usually undesired in the production of periodic structures, various groups have reported on its consequences. For example, Hibino and Hegedus investigated the effects on wavefronts diffracted by period-chirped gratings when modeling the lithographic process with two point sources emitting convex wavefronts that interfere on a plane substrate [20]. Noda et al. [21] presented a theoretical work on holographic and mechanically ruled gratings and considered the exposure of a substrate to the convex waves emitted from two arbitrarily positioned point sources but did neither derive an expression for the two-dimensional distribution of the period and the orientation of the grating lines nor consider the interference patterns generated by combinations with concave spherical waves. In 1996, Ferrera and Schattenburg gave an equation to calculate the grating period across the entire surface of a substrate when the convex wavefronts of two point sources are symmetrically incident on a plane substrate [22]. Kim et al., who deliberately produced chirped gratings, addressed the exposure using mixed wavefronts, i.e., convex and concave, on a plane substrate. The given equations, however, only allow to calculate the period in one axis [23]. In addition to these theoretical investigations on the chirp, various techniques have been developed to detect such periodicity errors during the process [22] or in the final grating [24–26]. Compensation techniques have also been developed, for example, by deliberately bending the substrate during the exposure to reduce the resulting period chirp [27]. The spatially dependent period, however, is not the only consequence when interfering non-planar wavefronts. It also leads to a spatially dependent curvature of the grating lines. Since this effect cannot be separated from the spatially dependent period, the term chirp is typically used to refer to both effects. This spatially dependent curvature has already been investigated for the case of symmetrically positioned point sources with convex wavefronts incident on a plane substrate [28]. The chirp is a prominent problem, but it has only been investigated in part and for specific special cases. Depending on the publications, plane substrates or symmetrically placed point sources were assumed, and equations were often only derived for one axis and based on simplifications. Furthermore, the focus mainly lies on the grating period but hardly on the curvature of the grating lines.
In the present publication, we aim to analyze both the cause and the effects of the chirp in a holistic way, and we present a new mathematical approach, which enables us to incorporate the various cases of exposures with combinations of convex and concave spherical waves into one model. The exposure to aspheric wavefronts as discussed by Noda et al. [29] is beyond the scope of the present report. Considering spherical waves, we derive generally applicable analytical equations, i.e., without any simplifications, to describe the spatial distribution of the period and the orientation of the grating lines on arbitrarily shaped substrates for the general case of arbitrarily positioned point sources. Additionally, a simplified mathematical formalism derived for few special cases (described in Sections 3.C and 5.C) with symmetrically positioned point sources illuminating a plane substrate iso9 described. This holistic mathematical description of LIL is of high interest for many reasons. On the one hand, it serves the manufacturers, who can derive improvement strategies for the fabrication process. On the other hand, it serves the users, who can derive implementation hints from the given equations and models. As we will show, depending on the application, there are preferred areas on chirped gratings.
2. MODELING OF THE SETUPS
The model constituting the basis of all derivations presented in the following is the interference of the spherical wavefronts emitted by two coherent point sources. This model is considered to be valid, since the common LIL setups (not SBIL) use the far field of a laser beam for the exposure of photo-resist-coated substrates, as shown in Fig. 1. A spectrally narrow beam emitted by a fiber-coupled laser is collimated and subsequently evenly divided by a beam splitter. The two replicas of the beam are both focused through an aperture and directed toward the sample, which is to be exposed. The focusing serves both to clean up the beam and to create the divergence required to illuminate the full surface of the sample.
Fig. 1. Simplified sketch of a LIL setup comprising a beam splitter to create two coherent laser beams. The setup is modeled with two coherent point sources $A$ and $B$, located at the foci of the beams. They emit convex spherical wavefronts illuminating the substrate.
Since the distance $z \gg {z_R}$. between the substrate and the focus of the laser beam is much larger than the beam’s Rayleigh length ${z_R}$, the curvature of the spherical wavefronts at the substrate is equal to the distance $z$. The focus of the laser beam can, therefore, be viewed as a point source emitting convex spherical wavefronts. Considering other setups and further implementations of imaging optics, three exposure cases are possible based on the interference of the waves emitted from two coherent point sources. As shown in Fig. 2(a) the first exposure arrangement involves the interference of two waves with convex wavefronts at the substrate. This is considered to be the most common arrangement and is the abstraction of the system shown in Fig. 1.
Fig. 2. Three possible arrangements for the exposure in LIL, based on the model of the interference of spherical waves emitted from two coherent point sources $A$ and $B$. (a) Convex case, direct interference of the waves with convex wavefronts as emitted from the point sources. (b) Concave case, the waves of the two point sources are focused by means of two lenses, thus creating concave wavefronts that converge to the virtual point sources ${A^*}$ and ${B^*}$. (c) Mixed case, inference between a wave with a convex and a wave with a concave wavefront. Only point source $A$ is imaged to ${A^*}$.
The second case, shown in Fig. 2(b), involves the interference of two beams with concave wavefronts. The concave waves are obtained by additional imaging. The third case, shown in Fig. 2(c), corresponds to a combination of the previous two.
3. EXPOSURE WITH TWO CONVEX WAVES
The following derivations for the interference of two convex waves as sketched in Fig. 2(a) are presented in detail as they are the foundation for the two other exposure cases discussed in Sections 4 and 5.
A. Fundamentals
The interference of two convex waves emerging from the point sources $A$ and $B$ is modeled by considering the arrangement shown in Fig. 3. Since the resulting interference pattern is rotationally symmetric around the axis connecting the two point sources, it is sufficient to consider the system in two dimensions to start with. Without loss of generality, a coordinate system is, therefore, chosen in which the two point sources $A$ and $B$ lie on the $x$ axis with the origin exactly in the middle between the two sources. The distance from this axis is denoted by the radial coordinate $r$.
Fig. 3. Cross-sectional (2D) sketch explaining the interference of the two convex spherical waves emitted from the two coherent point sources $A$ and $B$ at a given instant of time $t$. Constructive interference occurs along the curve ${F_{\rm{vex}}}$. The other variables are explained in the text.
It is assumed that, at the time $t = 0$, two waves emitted from the point sources $A$ and $B$ constructively interfere at the point ${P_O}$ on the $x$ axis; the corresponding wavefronts with the radii ${R_A}({t = 0})$ and ${R_B}({t = 0})$ are shown by the solid circles. These wavefronts expand with time $t$, as shown by the dashed circles with the radii ${R_A}(t)$ and ${R_B}(t)$. The point at which these two wavefronts experience constructive interference, therefore, moves from ${P_O} = {P_{I,{\rm vex}}}({t = 0})$ to
Fig. 4. Hyperboloids ${H_n}$ where the waves of the two coherent point sources $A$ and $B$ experience constructive interference. For illustrative purposes they were calculated with $\lambda = 4\;{\rm{mm}}$ and ${x_{\rm{PS}}} = 20\;{\rm{mm}}$. (a) Hyperboloids and their cross sections at $y = 0$ (black lines) as well as their intersections ${G_{{\rm vex,sym},n}}$ (red lines) with a plane substrate ${S_{\rm vex,sym}}$ (gray sheet). (b) Hyperboloids and their intersections ${G_{{\rm vex,gen},n}}$ (red lines) with an arbitrarily shaped substrate ${S_{\rm vex,gen}}$ (gray sheet).
B. General Case of the Convex Arrangement
For the general case, we consider the intersection of the hyperboloids ${H_n}$ with an arbitrarily shaped and arbitrarily positioned substrate as shown in Fig. 4(b). Without loss of generality, we describe its surface ${S_{\rm vex,gen}}$ by defining its $z$ position with the following function,
The grating lines ${G_{{\rm vex,gen},n}}$ arising from the intersection of the hyperboloids ${H_n}$ and the surface ${S_{\rm vex,gen}}$ are then defined by
First, the determination of the period in a two-dimensional system is considered, as shown in Fig. 5(a). The period $\Lambda$ is to be determined at a point $P$ on an axis ${a_S}$, resulting from the interference of the point sources $A$ and $B$. In an infinitesimally small region around the point $P$ the wavefronts of the two point sources can be approximated by plane waves. This assumption is valid for a single point and, therefore, does not contradict the intention of this publication to rigorously consider the curvature of the wavefronts. Without loss of generality, it is furthermore assumed that the wavefronts (represented by the thin red and blue lines) are in phase at point $P$, thus creating a constructive interference pattern as marked by the green lines.
Fig. 5. Model for the derivation of equations describing the grating period and the orientation of the grating lines for the general case. (a) Determination of the period $\Lambda$ at a point $P$ in a two-dimensional system. (b) Transfer of the model shown in (a) into three dimensions.
From the geometrical relations depicted in Fig. 5(a) and as already shown by Kim et al. [23] in a similar way, the period can then be defined as
The orientation of the substrate’s surface, i.e., the tangential plane at the point $P$, is referred to as ${K_S}$ and is defined by its normal vector ${\vec n_S}$. The vector ${\vec{a}_S}$, which describes the orientation of the intersection line between the surface plane ${K_S}$, and the illumination plane ${K_{\rm{IP}}}$ can be calculated by the vectorial product,
This vector corresponds to the axis ${a_S}$ from Fig. 5(a) and, thus, enables the transfer of the three-dimensional system into the two-dimensional system. Using this vector, the angles ${\varphi _A}$ and ${\varphi _B}$ in the illumination plane ${K_{\rm{IP}}}$ can be defined as
By inserting Eqs. (18) and (19) into Eq. (13), the period at the point $P$ with which the vector ${\vec{a}_S}$ intersects the planes ${K_{\rm{CI}}}$ in which constructive interference occurs is given by
This period is illustrated in Fig. 6 where the area around the point $P$ with a top-view on the substrate plane ${K_S}$ and only a small section of the illumination plane ${K_{\rm{IP}}}$ is shown. The three green planes symbolize the planes where constructive interference occurs at and around the point, respectively. They correspond to the three green lines shown in Fig. 5(a), whereby the central plane was also depicted in Fig. 5(b). Applying Eq. (20) to the exposure of a plane substrate yields the same result as published by Noda et al. [21] and by Ferrera [22]. However, the real period ${{{\Lambda}}_{\rm vex,gen}}$ in the substrate plane needs to be measured perpendicular to the grating lines. In order to obtain this period ${{{\Lambda}}_{\rm vex,gen}}$ (shown in orange), the angle $\alpha$ between the vectors ${\vec{a}_S}$ and ${\vec g_{\rm{vex}}}$ defined as
Fig. 6. Close-up on the region near the point $P$ with top-sight on the substrate plane ${K_S}$ showing the difference between the periods ${\Lambda _{{{\vec{a}}_S}}}$ and ${{{\Lambda}}_{\rm vex,gen}}$ as well as the derivation of the latter.
The vector ${\vec g_{\rm{vex}}}$ describes the orientation of the grating line on the substrate and is mathematically given by the intersection line between the substrate plane ${K_S}$ and the plane ${K_{\rm{CI}}}$ where constructive interference occurs. It is, therefore, given by the cross-product of their normal vectors as
By definition, the plane ${K_{\rm{CI}}}$ contains the bisector $\vec b$ between ${\vec{a}_A}{}$ and ${\vec{a}_B}$ and is perpendicular to the illumination plane ${K_{\rm{IP}}}$. The normal vector ${\vec n_{\rm{CI}}}$ of the plane ${K_{\rm{CI}}}$ is, therefore, defined by
withTherefore, the orientation of the grating lines at a point $P$ on an arbitrarily shaped surface can now be determined by combining Eqs. (22)–(24), which yields
With the help of this vector ${\vec g_{\rm{vex}}}$, Eqs. (20) and (21), and the angular relations of the triangle shown in Fig. 6, the period on the substrate is finally found to be
Equations (25) and (26), thus, enable the calculation of the orientation and the period of an interference pattern along an arbitrarily shaped surface created by two arbitrarily placed point sources.
An example is presented in the following, aiming to show both the application as well as the validity of the derived equations. The two point sources $A$ and $B$ are placed at (–340.2 mm, 0 mm, 940.4 mm) and (340.2 mm, 0 mm, 940.4 mm), respectively, illuminating a substrate ${S_{\rm vex,gen}}$, which is arbitrarily defined by
Fig. 7. Example of a wavy surface that is illuminated by the spherical waves of two coherent point sources. (a) Geometry of the surface. The color-scale refers to the $z$ position of the surface. (b) Period of the interference pattern on the illuminated surface as calculated with Eq. (26). The color-scale refers to the value of the period. (c) Orientation of the grating lines as calculated using Eq. (25). The black lines show the local orientation of the grating lines, and the color-scale shows their deviation to the ideal orientation in the $x-y$ plane.
The resulting period calculated with Eq. (26) is shown in Fig. 7(b). The period’s inhomogeneity is a direct consequence of the surface’s waviness. When analyzing it, it becomes apparent that the period is mainly influenced by the orientation ($\partial z/\partial x$) of the surface in $x$ direction rather than its orientation ($\partial z/\partial y$) in the $y$ direction or the overall height $z$. This can be well seen at the two blue areas, for example, where the surface has a different $z$ position and different orientation ($\partial z/\partial y$) in the $y$ direction but a similar orientation ($\partial z/\partial x$) in the $x$ direction. Figure 7(c) shows the orientation of the grating lines as calculated with Eq. (25). The orientation is represented by the black lines on the one hand and by the color-scale on the other hand. While the lines represent the spatially resolved vector ${\vec g_{\rm{vex}}}$ on the surface, the color-scale shows the angular deviation of this vector from the ideal orientation of ${{\vec{g}}_{\rm{ideal}}} = [{0,\;1,\;0}]$ in the $x-y$ plane. It can be seen that the tilt of the lines along the $y$ axis is close to zero since the hyperboloids of constructive interference are almost parallel to the $y-z$ plane here (see Fig. 4). Moreover, the results show that the tilt of the grating lines becomes stronger toward the edges, which is due to an increase of the curvatures of the hyperboloids (see Fig. 4).
C. Symmetric Case of the Convex Arrangement
While the previous section treated the general illumination case, this part is devoted to the examination of the special case of symmetrical illumination of a plane substrate, i.e., where the point sources are aligned symmetrically on the $x$ axis while the plane substrate is orientated parallel to the $x-y$ plane as depicted in Fig. 8(a). In this configuration, the plane substrate’s surface is defined by
where ${z_S}$ is the distance of the substrate’s surface to the origin. Inserting Eq. (28) into Eq. (11) yields the description of the grating lines (shown in red in Fig. 8). Solving this for $x$, the grating lines ${G_{\rm vex,sym}}$ for the symmetric case are found to be given byFig. 8. (a) Symmetric arrangement of the two coherent point sources and a plane substrate. The sources are placed on the $x$ axis with equal distance to the origin while the substrate is orientated parallel to the $x-y$ plane at a distance ${z_S}$. The hyperboloids of constructive interference are shown by the colored surfaces, and their intersections with the substrate are shown in red. (b) Section of (a) showing a small region around the point $P$ serving for the derivation of the period ${{{\Lambda}}_{\rm vex,sym}}$ from the period ${{{\Lambda}}_{{\rm vex,sym},x}}$, which is parallel to the $x$ axis.
In principle, the period of the interference resulting in this symmetric arrangement can of course be calculated with the generally valid Eq. (26). Thanks to the symmetrical configuration, the much simpler equation
The orientation of the grating line ${G_{{\rm vex,sym},n}}$ at the point $P$ is found by differentiating Eq. (29) with respect to $y$, resulting in
Solving Eq. (31) for ${x_{O,n}}$ yields
Inserting this in Eq. (32), one finds
Using this equation for the inclination of the lines and the angle $\epsilon$ from Fig. 8(b), one finds
Combining this with Eq. (30) and applying trigonometrical simplifications to the term $\cos ({\arctan ({{\rm d}{x_{\rm vex,sym}}/dy})})$, the period is found to be
Two illumination examples of this symmetric case are presented in the following to illustrate the application of the derived equations and particularly to show the effects of the chirp. The period, the orientation, and the grating lines themselves were calculated using Eqs. (36), (34), and (29), respectively. For both examples, the same wavelength of $\lambda = 415\;{\rm{nm}}$ and angle of incident of $\theta = 19.89^\circ$ are chosen as already used for the example in the previous section depicted in Fig. 7. The two examples presented in the following only differ in the distance $d$ between the point sources and the center of the substrate, cf. Fig. 8(a). This distance was set to $d\; = 50\;{\rm{mm}}$ for the first example as shown in Fig. 9(a).
Fig. 9. Example of a symmetric illumination with two convex waves. (a) Arrangement of the sources placed at a distance of $d = 50\;{\rm{mm}}$ from the substrate’s center. (b) Calculated spatial distribution of the period on the substrate’s surface. (c) Cross-sectional view of the period distribution shown in (b) including parabolic fits. (d) Graph showing every 5000th grating line (black lines) and the spatial distribution of the absolute deviation of their orientation from the ideal direction (color-coded).
The targeted period ${\Lambda _{\rm{Target}}} = 610\;{\rm{nm}}$ is obtained at the center of the substrate but strongly increases with the distance from the center, as shown in Fig. 9(b). This increase, which is mainly meant by the term chirp, is much stronger in the $x$ direction, which is emphasized by the cross-sectional view in Fig. 9(c). By comparing these cross sections with the additionally shown parabolic fits, it can be concluded that the distribution $\Lambda ({x,y})$ of the period for this case roughly represent a three-dimensional paraboloid. The root mean square error (RMSE) of the fits in the $x$ and $y$ axis are ${{\rm{RMSE}}_x} = \;7.2\;{\rm{nm}}$ and ${{\rm{RMSE}}_y} = 0.32\;{\rm{nm}}$, respectively. Figure 9(d) depicts every 5000th grating line arising from the interference of the two sources. The distances between the lines increase with increasing distance from the center showing the chirp in both the $x$ and $y$ directions. The color-coding in Fig. 9(d) shows the orientation of the grating lines, respectively, and their absolute deviation from the ideal orientation (parallel to the $y$ axis). It is noticeable that the orientation of the grating lines is ideal both along the $x$ axis ($y = 0$) and $y$ axis ($x = 0$), but deteriorates with increasing distance with deviations of up to about 15°. A larger distance of $d = 1000\;{\rm{mm}}$ between the sources and the substrate is applied for the second example shown in Fig. 10.
Fig. 10. Example of a symmetric illumination with two convex waves. (a) Arrangement of the sources placed at a distance of $d = 1000\;{\rm{mm}}$ from the substrate’s center. (b) Calculated spatial distribution of the period on the substrate’s surface. (c) Cross-sectional view of the period distribution shown in (b) including parabolic fits. (d) Graph showing every 5000th grating line (black lines) and the spatial distribution of the absolute deviation of their orientation from the ideal direction (color-coded).
It is noticeable by comparing Figs. 9(b) to 10(b) that the spatial variation of the period looks qualitatively similar but has become smaller by orders of magnitude. While the period of the previous example increased by approximately 500 nm (82% increase) toward the corners, the period of the present example only increases by approximately 1 nm (0.2% increase), due to the larger exposure distance. When looking at the cross-sectional views in Fig. 10(c), it is also noticeable that the parabolic fits match much better than before, only having residual errors of ${{\rm{RMSE}}_x} = 52.4 \cdot {10^{- 6}}\;{\rm{nm}}$ and ${{\rm{RMSE}}_y} = 2.36 \cdot {10^{- 6}}\;{\rm{nm}}$. In Fig. 10(d) it can be seen that the grating lines no longer show any apparent curvature with deviations of the orientation of less than about 0.05°, while the pattern of the tilt is qualitatively similar.
From these two examples, it can already be concluded that gratings produced under such a symmetrical illumination contain an ideal area in the center where both period and curvature are optimal, and that the next best areas are located along the $x$ axis ($y = 0$) and the $y$ axis ($x = 0$). The examples furthermore show that the spatial distribution of the period can be approximated by a paraboloid and that the chirp is reduced by increasing the distance of the point sources.
4. EXPOSURE WITH TWO CONCAVE WAVES
Only the exposure arrangement with interfering convex waves shown in Fig. 2(a) has been considered so far. Interference of concave wavefronts can occur, e.g., when optical imaging is applied as shown in Fig. 2(b). The easiest way to derive the equations for the period and the orientation of the grating lines resulting in this situation is to consider the two virtual point sources ${A^*}$ and ${B^*}$ below the substrate. While the point sources $A$ and $B$ emit the wavefronts, the virtual point sources ${A^*}$ and ${B^*}$ have to be viewed as if they attracted the wavefronts. By means of this observation, the same derivations can be used as above, while only the sign of the velocities of the wavefronts need to be changed. However, since only the time-independent paths along which constructive interferences occurs are considered in the final result, see Appendix A, Eqs. (6) to (9) are not affected by this change of sign. All the equations presented for the convex exposure case can, therefore, also be applied to the concave exposure case just by using the new point sources ${A^*}$ and ${B^*}$ instead of the previously used sources $A$ and $B$.
5. EXPOSURE WITH ONE CONVEX AND ONE CONCAVE WAVE
For the third and final case, the photo-resist is exposed to the interference of convex and concave wavefronts. This mixed exposure arrangement is shown in Fig. 2(c) where the interference between the waves from the real point source $B$ and the virtual point source ${A^*}$ needs to be considered. As shown in the following subsection, this arrangement unfortunately does not allow to use the equations derived for the previous two cases.
A. Fundamentals
In order to consider the interference of one expanding and one contracting spherical wave, we place the two point sources symmetrically around the origin on the $z$ axis, since they are arranged approximately vertically, as shown in Fig. 2(c). This assumption does not limit the generality of the derivation as one may later define an arbitrarily tilted and shaped surface of the illuminated substrate. Due to the rotational symmetry of the system around the chosen $z$ axis, we denote the transversal coordinate(s) with the radial coordinate $r$, as shown in Fig. 11(a). Analogous to the procedure in Section 3.A, we consider a starting point ${P_O}$ on the transversal $r$ axis where two wavefronts (solid lines) experience constructive interference at the time $t = 0$. After a time $t$, this intersection point has moved from ${P_O} = {P_{I,{\rm mix}}}({t = 0})$ to the point
Fig. 11. (a) Cross-sectional (2D) view explaining the interference of a convex and a concave spherical wave. While the wavefronts of the point source $B$ expand with time, the wavefronts of the virtual point source ${A^*}$ contract. Constructive interference arises along the curve ${F_{\rm{mix}}}$. The other variables are explained in the text. (b) Sketch used to determine the origin of the curves ${F_{\rm{mix}}}$ regarding the wavelength of the emitted waves.
By setting ${v_{{A^*}}} = - {v_B}$ and by eliminating the parameterization variable $t$ from Eqs. (38) to (41), as shown in the Appendix B, the curve ${F_{\rm{mix}}}$ is found to be defined by
The location ${r_O}$ marks the starting point ${P_O}$ of the curve ${F_{\rm{mix}}}$. By considering the wavelength $\lambda$ of the point sources, these points are limited to a set of discrete values ${r_{0,n}}$, which can be calculated considering the geometrical relationships shown in Fig. 11(b). Let us assume that the two point sources are in phase in such a way that constructive interference occurs exactly at the origin, which lies in the middle between them. This point is noted by ${r_{O,n = 0}}$. The next point at which constructive interference occurs is denoted as ${r_{O,n = 1}}$, the following one as ${r_{O,n = 2}}$, and so on. The locations of these points can be defined using Pythagoras’ theorem,
where the radius ${R_{\rm{WF}}}(n)$ of the corresponding wavefront $n$ is given byBy combining Eqs. (43) and (44), the origin(s) of the discrete curve(s) ${F_{{\rm mix},n}}$ can be defined as
The presented cross-sectional formulation can be transferred into the Cartesian coordinate system of the three-dimensional space by using
In this system, the curves ${F_{\rm{mix}}}$ are found to be the cross sections of the ellipsoids,
Fig. 12. 3D representation of the ellipsoids ${E_n}$ where the waves of the two coherent point sources ${A^*}$ and $B$ experience constructive interference. For illustrative purposes, the distance of the point sources to the origin was set to ${z_{\rm{PS}}} = \;50\;{\rm{mm}}$, and a wavelength of $\lambda = 8\;{\rm{mm}}$ was chosen. (a) Ellipsoids with an arbitrarily shaped and positioned substrate ${S_{\rm mix,sym}}$ (gray sheet). The curves ${F_{{\rm mix},n}}$ are marked in the cross section at $y = {{0}}$. (b) Ellipsoids and their intersections ${G_{\rm mix,sym}}$ (red lines) with a plane substrate ${S_{\rm mix,sym}}$ (gray sheet) parallel to the $x-y$ plane.
B. General Case of the Mixed Arrangement
As already done for the exposure with two convex waves, a distinction is again made here between the illumination of an arbitrarily shaped surface depicted in Fig. 12(a) and the symmetrical illumination of a plane surface depicted in Fig. 12(b). For the general case, the arbitrarily shaped surface of the illuminated substrate is defined by
The grating lines $G_{{\rm mix,gen},n}$ (still assuming a negative photo-resist) are then given by
As mentioned before, it is mainly the grating period and the local orientation of the grating lines that are of interest. Rather than deriving a new model, we adapt the vectorial model used for the above case of interfering convex waves and consider the real point source $B$, the virtual point source ${A^*}$, and an arbitrarily shaped surface ${S_{\rm mix,gen}}$ as shown in Fig. 13(a).
Fig. 13. Vectors needed for the derivation of the period and the orientation of the grating generated by the interference of a concave and a convex wave on (a) an arbitrarily shaped or (b) a symmetrically arranged plane surface.
The local period and orientation of the grating lines is determined at an arbitrary point $P$ on the illuminated surface. The ray vector of the wave coming from the real point source $B$ is denoted by ${\vec{a}_B},$ and the one of the wave originating from the virtual point source ${A^*}$ is denoted by
The comparison of the current model with that of the two interfering convex waves can be made considering the ray vector,
originating from the real point source $A$. By applying this relation [Eq. (51)] to Eqs. (26) and (25), the period and the orientation of the grating lines resulting from the interference of a convex and a concave spherical wave are found to beC. Symmetrical Case of the Mixed Arrangement
For the consideration of the grating generated on a symmetrically arranged plane surface, we replace the arbitrarily shaped surface ${S_{\rm mix,gen}}$ from Eq. (48) by
cf. Figs. 12(b) and 13(b). The grating linesBy combining Eq. (56) to Eq. (58), the period is found to be
This equation shows that the spatial distribution of the grating’s period is radially symmetric around the $z$ axis, as already seen in Fig. 12(b). Using Eq. (46), the distribution of the period can again be expressed in Cartesian coordinates,
Due to the radial symmetry of the grating lines around the $z$ axis, their local orientation is simply given by
where ${x_{\rm mix,sym}}$ is the $x$ coordinate of a point along one of the grating lines. The following example serves both to understand the interference pattern produced by the interaction of a concave and a convex spherical wave and to compare it to the pattern produced by two convex waves. The same illumination parameters, therefore, are used as above, i.e., a wavelength of $\lambda = 415\;{\rm{nm}}$, a distance between the point sources of $d = 1000\;{\rm{mm}}$, and an angle of incidence of $\theta = 19.89^\circ$. The results are shown in Fig. 14. They can directly be compared to the results obtained for two convex waves shown in Fig. 10. Since the exposure with convex waves is identical to the one with concave waves, which was proven in Section 4, the results presented in Fig. 10 also apply to the latter.Fig. 14. Example of a symmetric illumination of a plane surface by a concave and a convex wave. (a) Arrangement of the sources placed at a distance of $d = 1000\;{\rm{mm}}$ from the substrate’s center. (b) Calculated spatial distribution of the period on the substrate’s surface. (c) Cross-sectional view of the period distribution shown in (b) compared to the cross sections of the period distribution of the interference pattern produced by two convex waves. (d) Graph showing every 5000th grating line (black lines) and the spatial distribution of the absolute deviation of their orientation from their ideal direction (color-coded).
The arrangement of the two point sources ${A^*}$ and $B$ is shown in Fig. 14(a). The distribution of the period of the resulting interference pattern depicted in Fig. 14(b) exhibits a strong gradient in the radial direction (here shown along the $x$ direction). The targeted period is obtained only in the center, and the gradient is much stronger than in the case of two interfering convex waves. This can particularly be seen in Fig. 14(c) giving cross-sectional views of the period’s distributions for the two respective cases. The orientation of the grating lines is shown in Fig. 14(d). The lines show a visible curvature whose center lies at the origin of the underlying coordinate system. It is noticeable that the maximum curvature is an order of magnitude larger than that obtained for the interfering convex waves, cf. Fig. 10(d). In summary, this analysis shows that the interference of a convex with a concave wave produces a much stronger chirp and that it should be avoided if homogeneous gratings with constant period are targeted.
6. CONCLUSION AND OUTLOOK
In summary, we have presented an extensive and general theoretical investigation on the period chirp, which is a prominent problem in the fabrication of linear gratings using LIL. The considered arrangements of point sources are applicable for all setups that are based on far-field illumination of the substrate. Exact analytical equations were derived describing the grating period and the orientation of the grating lines both for a general case having an arbitrarily shaped substrate illuminated by arbitrarily positioned point sources and for a symmetrical case having a plane substrate illuminated by symmetrically positioned point sources. In future publications, we will focus on a modeling of the illumination by Gaussian beams in order to precisely describe the chirp for SBIL as well, and we will address the elimination of the chirp by seizing the idea of Walsh and Smith [27] of bending the substrate for the illumination process.
APPENDIX A: DERIVATION OF EQ. (6)
First, Eqs. (2) and (3) are combined to
Then Eqs. (4) and (5) are inserted into Eq. (A2), resulting in
Using the relationship ${v_{\rm{A}}} = {v_B}$, Eq. (A3) can be transformed to
Inserting Eqs. (4) into (2) yields
By inserting Eq. (A4) into Eq. (A5),
The intersection points, where the two waves experience constructive interference, therefore move along the path ${F_{\rm{vex}}}$ defined by
which defines the geometry of the standing intensity pattern.APPENDIX B: DERIVATION OF EQ. (42)
Equations (38) and (39) are combined to
Then Eqs. (40) and (41) are inserted into Eq. (B2), resulting in
Using the relationship ${v_{{{\rm{A}}^{\rm{*}}}}} = - {v_B}$, Eq. (B3) can be transformed to
Inserting Eqs. (38) into (40) results in
The time and velocity can be removed from the result by inserting Eq. (B4) into Eq. (B5). After simplification, this finally results in
The path ${F_{\rm{mix}}}$ on which the points of constructive interference move along is, therefore, given by
Funding
Horizon 2020 Framework Programme (687880, 825246).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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