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Abstract
We present a new technique to prevent the detrimental period chirp that appears in optical gratings fabricated by laser interference lithography (LIL). The idea is to bend the Lloyd’s mirror in the lithographic setup to eliminate the period chirp already at the step of the grating’s exposure. A new mathematical model was developed to describe the required bending geometry of the mirror. It is shown that this geometry can be described by multiple cross-sections of the mirror, each obtained by the solution of an implicit first-order differential equation. The proposed approach is illustrated on the basis of a concrete example. By slightly bending the Lloyd’s mirror (by ≈ 3.5 mm of maximum deflection over an area of 142 mm × 215 mm) the period chirp of the exposed grating can be eliminated completely.
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1. Introduction
Gratings are essential optical components in the field of laser technology and photonics. In the latter, they are found in various disciplines such as medicine [1–3], metrology [4–6], and telecommunication [7,8]. In the field of lasers, they can be employed intra-cavity for spectral stabilization [9–11] and wavelength selection [10,12] or extra-cavity for multiplexing [13–15] and chirped-pulse amplification (CPA), where they are the main components of pulse stretchers [16–20] and compressors [21–23]. Among the various applications, CPA poses the highest challenges to the gratings, concerning their size [24–27], quality [28–30], and power endurance [26,31–33]. For the production of linear gratings, as they are required for most applications, including CPA, laser interference lithography (LIL), and scanning beam interference lithography (SBIL) are common techniques [27,34–38]. Apart from the period chirp which is a persistent detrimental effect, laser interference lithography (LIL) is an excellent and cheap technique for the exposure of large gratings [39,40]. The period chirp originates from the spherical nature of the interfering waves and causes a spatial dependence of the period and an inclination of the grating lines [41–44]. The spatial dependence of the period is schematically shown in Fig. 1(a), indicated by the yellow curve showing the typical quadratic increase of the period from the substrate’s center [45] towards the edges. Since the chirp’s consequences are extremely detrimental to the application e.g. in pulse compressors, where both the beam quality and the compression of the compressed pulses are reduced [46], its appearance should be avoided. So far only one technique has been proposed to avoid the period chirp (directly in the production). The essence of this idea, proposed by Walsh and Smith, is to bend the substrate during the lithographic exposure [47,48]. After the exposure, when the substrate is flattened again, the chirp should be fully compensated, resulting in parallel grating lines and a homogeneous (i.e. constant) period as schematically shown in Fig. 1(b).
Fig. 1. Schematic depiction of different LIL setups [45]. (a): Classical LIL setup employing the interference between convex spherical wavefronts which creates a spatial distribution of the period $\mathrm{\Lambda }$ that increases quadratically with increasing distance from the substratès center, as indicated with the yellow curve. (b): Setup shown in (a) combined with the technique proposed by Walsh and Smith to bend the substrate during the exposure to eliminate the period chirp [47]. (c): Classical LIL setup comprising a Lloyd’s mirror to create a second, virtual point source B. (d): Setup comprising the technique proposed in this publication, where the Lloyd’s mirror is deliberately bent to eliminate the period chirp.
This idea was proposed in 2001 and although Walsh and Smith could not determine the so-called “zero-chirp” geometry, which would create the full elimination of the period chirp, they could already show its reduction in practice [47]. Only recently we were able to show that the geometry that is necessary to completely compensate for the chirp is given by the rotation of the solution of a first-order differential equation [45]. Using our derived mathematical model we were also able to show first practical results with thin silicon wafers [49]. However, both the theoretical analysis and the practical investigation showed that the bending of the substrates can be a limiting aspect, especially when thick glass substrates are used instead of thin and flexible silicon wafers.
To circumvent this technical constraint, we herewith present a new idea to avoid the period chirp without the requirement to invest in large collimation optics and the operation of the setup in vacuum. Instead, the commonly used setups with divergent beams [35,50] can still be used. We propose to bend the Lloyd’s mirror (LM) in a LIL setup based on a Lloyd’s mirror interferometer [6,51–53]. Like the common LIL setups comprising two-beams, cf. Figure 1(a), also the LIL setups based on a LM lead to a period chirp, cf. Figure 1(c). However, the period chirp can be eliminated by bending the LM, cf. Figure 1(d). This idea is closely related to the bending of the substrate, cf. Figure 1(b), but the bending of a thin metallic LM is much simpler and prevents the requirement of bending thick and brittle substrates.
In the following, we present the first investigation of this idea. We present a mathematical model to determine the zero-chirp geometry of the LM to create a perfectly homogeneous period on the exposed substrate.
Although the approach of bending the LM seems to be similar to the approach of bending the substrate [47], a new mathematical model needs to be derived since the point source model (which was applicable to the method of bending the substrate) cannot be applied for the new approach. While the point-source A emits spherical waves, B* is a virtual source created by the deformed LM which therefore emits correspondingly deformed wavefronts, cf. Figure 1(d), and hence cannot be modeled by a point-source.
2. Determination of the geometry of the Lloyd’s mirror along the central line of symmetry
Figure 2 shows an abstraction of the sketch depicted in Fig. 1(d), rotated by 90°. The surface of the substrate is located on the vertical axis ($y$-axis) shown in magenta. The LM is located at the bottom of the figure shown in green. In the present section we only consider the exposure plane, i.e. the plane comprising the point-source and the substrate’s center [41]. The shape of the LM, i.e. the location of the LM’s surface is defined by the curve yM(x) as given by the intersection of the LM’s surface and the exposure plane. The shape yM(x) of the LM’s surface, which leads to an elimination of the period chirp on the substrate within the exposure plane, can be derived by considering the points S = [0, yS] on the substrate and any point M = [x, yM(x)] on the surface of the LM. At the point S the radiation is incident from two sides. The first part of the radiation is directly coming from the real point source A, while the second part of the radiation (which is also emitted by the real point source A) is reflected by the LM (see blue arrow) at point M before it reaches point S. The period $\mathrm{\Lambda }$ of the interference of the two beams on the substrate at the point S is given by [41]
where $\lambda $ is the wavelength of the laser used for the exposure and φA and φB are the angles of incidence of the two k-vectors ${\overrightarrow k _A}$ (red) and ${\overrightarrow k _{{B^{\ast}}}}$ (blue) of the waves incident on the substrate. In order to achieve a constant period on the substrate the denominator of Eq. (1) must be kept constant. Since φA is directly given by the relation between the locations of the point source A and the point S, the angle φB needs to be adapted to fulfill this condition. The curve yM(x) thus needs to be shaped such, that the radiation originating from A and being reflected at point M on the curve yM(x), reaches point S under such an angle φB that the denominator of Eq. (1) is kept constant. In mathematical terms, this means that the normal (orange dash-dotted line) to the curve yM(x) at point M needs to half the angle 2δ, as it is shown in the sketch.Fig. 2. Abstraction of the schematic model shown in Fig. 1(d), rotated by 90° and used for the derivation of the equations to calculate yM(x). The surface of the substrate is located on the y-axis, the cross-section of the surface of the Lloyd’s mirror (LM) in the exposure plane is given by the green curve yM(x). The point source is indicated by A.
With given locations of the points A and S, the angle φA is given by
Using the location of points M and S the angle φB is given by
Combining Eq. (2) and (3) by eliminating yS one obtains
By replacing φA by means of Eq. (1) this results in
The sum of the angles between the black dashed lines at the left side of point M, cf. Figure 2, yields
and the full sum of the angles at point M, is By combing Eq. (6) and (7) one findsBy relating the angle γ to the slope of the curve yM(x) (see orange line) as
and by using the relationshipEquation (8) can be rewritten in the form
By implementing this into Eq. (5) one finds
The solution of this implicit first-order differential equation describes the curve yM(x) of the LM, which leads to a constant period Λ (zero chirp) on the substrate (at every point S in the exposure plane) by the interference of the waves emitted from A and the waves reflected by the LM, cf. Figure 2, since for all points S on the substrate φB is always adapted such, that the denominator of Eq. (1) is kept constant.
3. Determination of the three-dimensional geometry of the Lloyd’s mirror
In order to find the three-dimensional geometry of the LM, we consider planes that result from a rotation of the exposure plane introduced above by a given angle as sketched in Fig. 3(a).
Fig. 3. Schematic illustrating the technique to determine the three-dimensional geometry of the LM. (a): Depiction of the exposure plane and an additional tilted plane, rotated around the substrate’s normal through the point source at A. (b): Top-view of the system, indicating the geometrical relationship between the exposure plane and the tilted plane.
The geometry of the LM in the exposure plane as determined in the previous section is shown by the green solid curve. The green dots schematically represent the points where constructive interference is created on the substrate along the line of intersection with the exposure plane. As shown in the figure, these points are equidistant and overlap with the grating lines that are to be generated, indicated by the dashed lines on the substrate.
Let us now consider the plane shown in orange which is titled by an angle ɛ around the axis of rotation (dash-dotted line) defined by the normal of the substrate that intersects the point source A. Within this tilted plane, the geometry of the LM can also be determined using the method described in section 2. The only difference is that a stretching of the grating period along the orange dotted line of intersection with the substrate is required to account for the tilt of this plane with respect to the exposure plane. This is further emphasized in Fig. 3(b), showing the top view of Fig. 3(a). The period that needs to be achieved along the line of intersection between the substrate’s surface and the considered tiled plane is thus given by
Furthermore, the distance of the point source to the LM needs to be adapted and is similarly given by
The complete three-dimensional geometry of the LM can now easily be determined by repeatedly solving the implicit differential equation Eq. (12) for different angles ɛ.
4. Determining the Lloyd’s mirror for a specific example
For illustration, the required LM geometry is exemplarily determined for the case of a lithographic setup with similar specifications to the one presented in [45], where the technique of bending the substrate was pursued. This allows for a direct comparison of the proposed technique of bending the LM with the already published one with the bent substrate. Hence a constant period of Λ = 610 nm is targeted on the substrate over a substrate’s length of 100 mm (in the exposure plane). The point source emitting coherent spherical waves with λ = 415 nm is located at A = [940.37 mm, 340.16 mm], yielding a distance of 1 m between the point source and the edge where the substrate meets the LM.
First, the geometry of the LM is determined within the exposure plane. The solution of Eq. (12) with the given parameters was found by using the variable-step, variable-order solver ode15i, provided by MATLAB. The values yM(x = 0 mm) = 0 mm and dyM(x = 0 mm)/dx = 0 were chosen as a boundary condition. The shape yM(x) of the LM was calculated from x = 0 mm to x $= $ 215 mm, with an increment of 0.01 mm.
Figure 4 shows the calculated results obtained for the considered example. The solid black curve yM,num(x) in Fig. 4(a) shows the numerically calculated geometry of the LM in the exposure plane, while the black curve in Fig. 4(b) shows the corresponding grating period Λnum(y) achieved on the substrate. For comparison, the blue dotted curves show the geometry, cf. Figure 4(a), and the resulting period, cf. Figure 4(b), obtained with a plane LM.
Fig. 4. Numerically calculated solution of the implicit differential equation for the given example. (a): The calculated curve yM,num(x) which describes the shape of the LM in the exposure plane is shown in black. For comparison the blue curve yM,plane(x) is added, representing a plane LM. (b): Period Λnum(y) achieved on the substrate when the calculated curve yM,num(x) is used (solid black). The blue dotted line Λplane(y) shows the distribution of the period that is obtained when a plane LM is used.
The full three-dimensional geometry of the curved LM is shown in Fig. 5. Figure 5(a) shows the full LM over a width of 100 mm (z-direction) and a length of 215 mm (x-direction). The colormap indicates the deflection of the mirror in the y-direction. The maximum deflection is less than 3.5 mm. The colored lines in the figure represent the cross-sections of the LM, from which the three-dimensional surface is composed. They were calculated from ɛ = – 12° to ɛ = 12° with 1° increments. The shapes of these cross-sections are shown in more detail in Fig. 5(b). The curve created for ɛ = 0° corresponds to the curve shown in Fig. 4. Figure 5(c) shows the full setup including the substrate, the deflected LM, and the point source. A perfectly homogenous grating period with Λ = 610 nm is achieved within an area of 100 mm × 100 mm. To underline how strong the beneficial influence of the LM’s bending is on the elimination of the period chirp, Fig. 5(d) shows the spatially varying period that would otherwise appear on the substrate when a plane LM would be used. The period was calculated using the model presented in [48].
Fig. 5. Illustration of the approach with a curved LM. (a): Three-dimensional geometry of the LM, determined from the different cross-sections of the LM, indicated by the colored lines. The substrate (not shown) is located in the yz-plane. (b): Cross-sections of the LM used to create the full surface. The blue curve created for ɛ = 0° corresponds to the black curve shown in Fig. 4. (c): Lithographic setup showing the arrangement of the curved LM, the point source A, and the substrate on which a grating with a constant period is generated. (d): Standard arrangement of a lithographic LM-setup with a plane LM and a substrate on which a grating with a period chirp (shown by the color-coded period) is generated.
5. Discussion
As shown in the two simulations in Fig. 5(c) and (d), the presented method enables drastic improvements in the reduction of the detrimental period chirp, even leading to its full elimination. Although this simulative result is already impressive, the presented method needs to prove its capability in practice to have a substantial and beneficial impact on the fabrication of diffraction gratings. Therefore, possible limitations as well as possible advantages, compared to the existing techniques have to be considered and are thus discussed below.
The main challenge in implementing the proposed method in practice is the realization of the LM’s aspherical curvature which eliminates the period chirp. The first step, i.e. the calculation of the required geometry can be done with the equations presented within this publication. Their general formulation, cf. Equation (12), allows for the calculation of the LM in a nearly arbitrary lithographic arrangement. However, when a different period should be obtained on the substrate or when a different exposure wavelength or a different distance between the point source and the substrate is used, the LM needs to be re-calculated and reshaped in practice. This introduces challenges to the lithographic setup. When a mass production of one specific grating is pursued the LM can be fabricated with high precision and stability using diamond machining [54], ion beam figuring [55,56], or additive manufacturing of the mirror’s geometry and subsequent coating with a reflective layer [57]. In order to have more flexibility, deformable mirrors based on the piezo-driven [58,59] or electro-magnetically-driven [60,61] movement of actuators on the mirror’s backside might be suitable. For all of these solutions, the geometry of the LM that is achieved in practice should be as close as possible to the calculated (ideal) geometry of the LM. Any deviations from this will affect the grating period and its spatial homogeneity. The quantitative effects of such deviations must be calculated for the specific lithographic arrangement and the deviation itself. In general, however, it can be stated that, if the LM in practice is less curved than the calculated LM, the resulting grating period will still show something between perfect homogeneity (zero-chirp, cf. Figure 5(c)) and the original chirp (i.e. when assuming a plane LM, cf. Figure 5(d)). Therefore, any small deviations in the curvature of the LM are not expected to deliver worse results than when using the standard setup with a plane LM.
In addition to the shape, also the alignment of the LM is critical. The consequences of misalignment for setups using plane LMs have already been investigated by Walsh [62] and Ma et al. [52]. In general, similar errors are expected for the proposed technique, i.e., that the misalignment of the LM leads to deformation of the grating lines and differences in the grating period from the intended period. However, in contrast to the standard technique, positioning errors of the LM will be more critical for the present technique and will also lead to deformation of the grating lines and differences in the achieved grating period from the intended grating period. Besides this, other common errors, such as substrate waviness, inhomogeneity and varying thickness of the resist, reflections from the substrate’s surface at the resist’s bottom [63], and mechanical vibrations during the exposure [64] are expected to have detrimental influence. However, all these influences are equally detrimental to the performance of the standard setups and are thus no further subject of discussion.
Wavefront deformations will also lead to deviations in the grating period. For the presented derivation, a point source emitting spherical waves was assumed. While this assumption does not hold near the point source, it easily holds for distances r, that are much larger than the Rayleigh length zR, i.e. r > > zR. The validity of this assumption can also be seen in the comparison of the lithographic simulations shown in [48] and [41]. While the sources were simulated by realistic Gaussian beams in [48], the sources in [41] were modeled by a point source, with otherwise identical specifications of the lithographic setup. Although the sources were modeled differently, the two interference patterns on the substrate showed identical geometry and identical period chirp.
Besides the period, other aspects of the grating should be considered as well when comparing the performance of the different setups. This especially concerns the grating’s duty cycle and the shape of the grating lines. Assuming otherwise identical conditions (homogenous resist thickness, flat substrate, etc.) these properties are mainly defined by the contrast and intensity of the interference pattern [64]. For Gaussian intensity distributions of the beams of the lithographic sources, the intensity of the obtained interference pattern on the resist can show strong spatial variations [40,53]. For this reason, special apertures can be introduced to homogenize the intensity of the beam [65], so that a good approximation of a point source with a homogeneous intensity distribution is achieved. Although this drastically improves the contrast, small spatial deviations can still occur when the propagation lengths r of the two beams from the point sources to the substrate are very different for certain points on the substrate. This is due to the inverse proportionality of the beam’s intensities to the square of the propagation lengths r given by I ∼ 1/r2. A mismatch in the propagation lengths thus leads to differences in the intensities of the two beams on the substrate and thus reduces the contrast [66]. When using the commonly employed setups with beam splitters, cf. Figure 1(a), or plane LMs, cf. Figure 1(c), this especially applies to the edges of the gratings. For the setup presented in this paper, this effect is minor because of the LM’s convex shape in the yz-plane. Due to the convergence of the beam reflected by the LM, its intensity is more similar to the intensity of the beam that directly reaches the substrate.
The convex shape of the LM is also very advantageous for the polarization of the beams. For the present setup, the interfering beams and the normal of the substrate are always located in one plane. This is achieved by the rotational approach explained in section 3. Therefore, a polarization state orthogonal to the respective plane can be perfectly maintained for both beams at any point on the substrate, allowing for high contrast. For the commonly used setups, this condition is not given. Only within the plane containing the two point sources and the normal of the substrate (plane of symmetry [41]), the polarization states are identical on the substrate. Thus, only along one line on the substrate, the ideal orientations of the two polarizations can be achieved. At all other points on the substrate projections of the polarization vectors take place and thus reduce the contrast [34].
6. Conclusion and outlook
Stimulated by the original idea of Walsh and Smith who proposed to deform the substrate during the LIL exposure [47], we propose to deform the Lloyd’s mirror (LM) instead, as this eases the exposure of thick and brittle substrates. In order to determine the geometry of the bent LM which is required to achieve a constant period on the substrate we developed a mathematical model from which it follows that the curve describing the geometry of the mirror in a plane normal to the substrate and including the point source of the lithographic setup is given by the solution of an implicit first-order differential equation. The complete surface is obtained by combining multiple cross-sections of the LM in several of such differently tilted planes.
The application of this method was shown on an example in which the complete elimination of the period chirp could be obtained on a substrate with a dimension of 100 mm × 100 mm while the maximum deflection of the LM was less than 3.5 mm within the area of the LM with the dimension 142 mm × 215 mm. The practical implementation of the proposed method will be subject of future research.
Funding
Horizon 2020 Framework Programme (825246).
Acknowledgments
This project is an initiative of the Photonics Public Private Partnership from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 825246.
Disclosures
The authors declare no conflict of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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