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Abstract
We present an experimental and theoretical analysis of chromatic aberration in a monolithic metasurface focusing mirror. The planar focusing mirror is based on a monolithic high contrast grating made from GaAs, designed for a wavelength of 980 nm. Light is focused on the high refractive index side of the mirror. Our measurements, performed between 890 and 1050 nm, indicate a shift of the focal point position that is inversely proportional to the wavelength. The experimental results are in very good agreement with our simulations, in terms of both the position of the focal point and the spectral dependence. Based on our numerical simulations, we show that simply modifying the grating height does not lead to significant alteration of the focal length or to any noticeable reduction in chromatic aberration. Using numerical simulations, we analyze how the height of the stripes, the refractive index of the grating material, and its dispersion combine to influence the chromatic aberration of the mirror.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Chromatic aberration is the inability of an optical element to focus various wavelengths at exactly the same point in space. In other words, if a focusing optical element suffers from chromatic aberration, its focal length depends on the radiation wavelength. It is generally an unwanted property, which leads, for instance to deterioration of the image formed by the optical element. In classical (i.e., refractive) lenses, chromatic aberration is caused by the dispersion in the lens material [1]. In metalenses, which are an example of diffractive lenses [2–5], focusing is obtained by an appropriate forming of the phase of the transmitted light by a metastructure. Typically, the phase profile does not change when the wavelength of the incoming light varies, which implies that the position of the focal point depends on the wavelength. In order to minimize chromatic aberration in metalenses, additional elements which modify the phase appropriately are usually applied [4–6].
Unlike lenses, standard curved mirrors do not suffer from chromatic aberration. However, metamirrors work in a way much more similar to metalenses than to curved mirrors, therefore chromatic aberration may occur. One type of metasurface is a monolithic high contrast grating (MHCG). An MHCG is an arrangement of stripes carved in a substrate material, as shown schematically in Fig. 1(a). A focusing mirror can be designed by arranging the grating stripes in a specific, non-periodic manner [7]. The reflection mechanism in such mirrors [8,9] differs from reflection on an air-metal interface. Whereas light almost does not penetrate a standard metal-covered mirror, in grating mirrors the light does penetrate the entire grating; therefore, chromatic aberration may appear.
Fig. 1. (a) Scheme of a non-periodic monolithic high contrast grating. The grating stripes are parallel to the $x$-axis direction. (b) Microscope photograph of the focusing mirror (indicated by the white box). The purple vertical line shows the focal line of the mirror illuminated by 980 nm laser light. The focal line is visible in the $x$ direction (i.e. along the stripes of the grating). (c) Normalized light intensity profile extracted from the photograph shown in (b). (d) Intensity map of the light reflected by the focusing grating mirror. The map was created based on the profiles (as in (c)) acquired at various positions along $z$. (e) Scheme of the measurement setup.
Recently, a number of studies have described attempts to engineer metalens structures with achromatic properties [5,10–26]. However, there are only a very limited number of reports concerning the wavelength-dependent properties of metasurface-based focusing mirrors [27–29]. Moreover, the existing reports on focusing mirrors remain in some ways contradictory and inconclusive. Investigating a plasmonic metastructure mirror composed of gold nanobricks arranged in a lattice and deposited on a SiO$_2$/Au bilayer, Pors et al. [27] observed a similar inverse relationship between the focal length and the wavelength as in the case of metalenses, between wavelengths of 600 and 1100 nm. In turn, Fang et al. [28], investigating a focusing mirror composed of a Si grating on a SiO$_2$ cladding and Si substrate reported almost wavelength independent focal length of their mirrors between 1530 and 1580 nm. However, due to rather limited resolution of the experimental setup and relatively narrow range of the studied wavelengths one cannot undoubdtly say whether the focal length in their mirror depends or does not depend on the wavelength. To the best of our knowledge, there are no previous reports in the literature concerning monolithic grating mirrors. Therefore, to fill the gap in research on the spectral properties of focusing mirrors based on metasurfaces, we present an experimental demonstration of light focusing by MHCG-based mirrors for wavelengths between 890 and 1050 nm. Our measurements are accompanied by numerical simulations.
2. Results and discussion
2.1 Measurements of the chromatic aberration
In its basic form, an MHCG is a structure consisting of stripes carved in a substrate material (here, GaAs). The focusing mirror discussed in this article has the form of a non-periodic MHCG. As illustrated in Fig. 1(a), the focusing grating mirror is composed of many segments that vary in width $L_i$ [30]. These segments contain stripes of various widths $L_iF_i$ [30], but all stripes have a fixed height $h$. In general, the design of the focusing grating mirror is based on the fact that the interaction of the elements of the grating with the incident light modifies the phase of the reflected light locally, in such a way that it mimics the phase of the focusing mirror (i.e., our MHCG behaves as a parabolic mirror along the $y$-axis for linearly polarized incident light at normal incidence, generating a focal line along the $x$-axis). This results in light being focused at the focal point. A more detailed discussion of this matter has been provided elsewhere [7]. The purpose of the investigated mirror is to focus light of 980 nm wavelength in one dimension. Therefore, as shown by the photograph in Fig. 1(b), instead of a focal point, a focal line occurs. The lateral size of the grating is 300 µm $\times$ 300 µm and the height of the stripes of the grating is $h=0.28$ µm. The mirror was fabricated on top of a 300 µm thick GaAs substrate using electron beam lithography followed by inductively coupled plasma reactive ion etching. More details on the fabrication procedure, including an exemplary scanning electron microscope image showing the very high quality of the mirror, have been published elsewhere [30].
Two aspects of the focusing properties of the MHCG are of particular interest in this study: firstly, how the position of the focal point changes with the wavelength; and secondly, the wavelength range in which the mirror is able to focus light. The focusing properties of the grating mirror were investigated between 890 and 1050 nm at 10 nm intervals, using an NKT Photonics SuperK EXTREME supercontinuum laser and a trinocular Motic PSM 1000 microscope with an ELWD Plan APO objective ($10\times$ magnification, $\mathrm {NA}=0.28$, working distance $w_{\mathrm {D}}=33.5$ mm). The wavelength range was limited by the interband absorption edge in the GaAs on one side and by the sensitivity of the camera matrix on the other side. Only the normal incidence of light was considered. The laser light was directed at the sample through the microscope objective and then the reflected light was collected by a digital camera (Sony $\alpha 6000$, with the infrared filter removed) attached to the microscope head. As indicated schematically in Figs. 1(d) and (e), the light was incident from above and the sample was positioned on a microscope stage with the grating mirror on the underside. Between the microscope objective and the sample, a linear polarizer was inserted which adjusted the transverse-magnetic polarization of light (i.e., as shown in Figs. 1(a) and (e), the non-zero electric field component was perpendicular to the stripes of the grating, commonly denoted as TM polarization). It was assumed that the vertical position of the MHCG was at $z=0$ µm, and that the GaAs substrate extended along the $z$ axis between 0 and 300 µm. On the opposite side to the grating mirror, a Si$_3$N$_4$ anti-reflecting coating matching the 980 nm wavelength was deposited. The experimental analysis was accompanied by simulations. We used the plane-wave reflection transformation method [31] and assumed normal light incidence. In simulations we assumed values of refractive index of GaAs and Si$_3$N$_4$ as reported in [32] and [33], respectively.
To determine the position of the focal point of the mirror, a series of photographs was acquired by moving the microscope head along the $z$ axis with a 2 µm step (i.e., away from the mirror, which remained stationary on the microscope stage). The light intensities $I$ at specific positions $z$ above the grating mirror were determined based on the pixel intensities stored in RAW photographs that were subsequently converted into grayscale intensities. An example photograph is shown in Fig. 1(b). By averaging numerous grayscale intensity profiles crossing the focal line at a right angle (along $x$), a single intensity profile corresponding to a specific vertical position was extracted, as illustrated in Fig. 1(c). After repeating this procedure for a range of photographs corresponding to well-defined vertical positions, the intensity profiles were joined and normalized to the maximum light intensity, forming the map shown in Fig. 1(d). A more extensive description of the measurement setup and of the procedure applied to analyze the acquired photographs is given in Ref. [30].
Figures 2(a)-(e) present light intensity maps acquired for selected wavelengths. The measured maps are accompanied by simulations performed for the same wavelengths (see Figs. 2(f)-(j)). The maps include close-up images of the focal point to ensure a clear view (the rectangular frame around the focal point in Fig. 1(d) can serve as a reference). As indicated both by the white dashed lines that mark the focal point and by the black arrow in Figs. 2(a)-(e), the longer the wavelength, the shorter the focal length. This effect is perfectly in line with the simulation. A small asymmetry visible in the measured light intensity maps is most likely due to the minor manufacturing imperfections described in detail in [30].
Fig. 2. (a-e) Measured and (f-j) simulated maps of the normalized intensity of the light reflected by the focusing grating mirror at various wavelengths; the area of the focal point was indicated by the white square in Fig. 1(d).
Only a few of the measured and the simulated maps are shown in Fig. 2. The others are provided as Visualization 1 (measured maps) and Visualization 2 (simulated maps). A summary of the light intensity profiles along the principal axis of the mirror is provided in Fig. 3. As can be seen, the shift of the focal point occurs uniformly for all the investigated wavelengths, not only those shown in Fig. 2. The black arrow indicates the position of the focal point, which shifts uniformly for all the wavelengths between 890 and 1050 nm.
Fig. 3. Measured normalized profiles of the reflected light intensity at wavelengths between 890 and 1050 nm at 10 nm intervals. The profiles were extracted from the maps shown in Fig. 2(a)-(e) along the principal axis of the mirror. Vertical shift was introduced in the interest of clarity. The black arrow indicates a shift of the focal point with the change of wavelength.
The exact positions of the focal point were determined by fitting the Gaussian distribution to the profiles presented in Fig. 3. The measured data are plotted with red dots in Fig. 4, along with the simulation data drawn with yellow squares. The spectral dependencies of these two sets of data indicate that the focal length is inversely proportional to the wavelength. This was proven by the appropriate fits (i.e. $f_i(\lambda ) = a_i/\lambda + b_i$; a subscript $i$ differentiates fits to the measured and simulated data labeled with $m$ and $s$, respectively), which are plotted with solid lines in colors corresponding to the data points. This type of dependency is typical for both diffractive metalenses and Fresnel lenses [5]. Overall, there is good quantitative agreement between the measured and simulated data.
Fig. 4. (a) Spectral shift of the focal point obtained from simulations (yellow squares) and measurements (red circles). The solid lines present fits $f_i(\lambda ) = a_i/\lambda + b_i$ to the data points. (b) Focal lengths plotted as a function of vacuum wave number $k=2\pi /\lambda$. The solid lines present fits $f_i(k) = a_i \cdot k + b_i$ to the data points. Numbers $a_i$ and $b_i$ are the fitting parameters and subscript $i$ stands for $m$ and $s$ in the case of measured and simulated data, respectively.
The slight difference between the values of the focal lengths obtained in the experiment and from the simulation of the expected structure arises from the fact that, when a laser is used as the light source, it is very difficult to determine the exact position at which the microscope head provides a sharp image of the grating. The discrepancy of roughly 20 µm (corresponding to approximately 4% of the expected focal length) that emerged in this experiment is as little as 1/5 of an entire rotation of a fine microscope focusing knob. Due to this focusing uncertainty, the focal point position may be unintentionally biased. However, once the position of the microscope head yielded a sharp image of the grating mirror, it was maintained as $z=0$ µm for the rest of the measurement. In this way, the slope $df_m/dk$ of the linear fit $f_m(k)$ to the measured data is not subject to this bias.
2.2 Simulations for modifying chromatic aberration
The chromatic aberration observed in our MHCG focusing mirror could, in principle, be either the result of dispersion in the mirror material or an intrinsic property of metasurface mirrors. To analyze the impact of both possible sources of aberration, we performed a series of numerical simulations. We simulated two different sets of theoretical structures:
- ND Made of non-dispersive grating materials ($dn_{\textrm {r}}/dk = 0$; where $n_{\textrm {r}}$ is the refractive index and $k=2\pi /\lambda$ is the vacuum wave number) in the entire analyzed spectral range. In other words, the refractive index of the grating is constant for all considered wavelengths, but we analyze different values.
Fig. 5. (a) Various considered dispersions of the refractive index. (b) Position of the focal point for structures with different linear dispersions. The value of $dn_{\textrm {r}}/dk=0.1$ µm is close to the values observed in typical optoelectronic materials.
As a numerical description of chromatic aberration, we use the slope $df/dk$ of the $f(k)$ function, which in most cases is almost linear in the considered range of wave numbers, as shown in Fig. 5(b). In the case of ND structures, the simulations do not indicate that the refractive index has any significant impact on the chromatic aberration. When the $n_{\textrm {r}}$ of the grating is fixed to a number between 3 and 4, the value of $df/dk$ remains almost constant at approximately 46 µm$^2$, as shown in Fig. 6(a). The error bars are the standard deviations of the slope of $f(k)$ fit.
Fig. 6. The chromatic aberration of (a) focusing mirrors made of non-dispersive material ($dn_{\mathrm {r}}/dk = 0$ µm); and (b) focusing mirrors in which the grating height was modified (the other grating parameters were kept unchanged).
The analysis of the LD structures shows that the material dispersion in the mirror does influence the mirror chromatic aberration (see Fig. 5(b)). In principle, it is possible to eliminate this effect (locally) if a huge negative dispersion of, in our case, $-1.5\,$µm were present in the mirror material. However, if we consider only realistic values, in practice the chromatic aberration of such a mirror cannot be significantly modified by the choice of the material. Table 1 presents the dispersion in widely used materials below their band gaps, whereas Fig. 7 shows that $df/dk$ is a linear function of the material dispersion in a very wide range of $dn_{\textrm {r}}/dk$ for mirrors of various sizes, and thus various numerical apertures. The mirrors narrower than 300 µm were created by trimming the central part of the 300 µm wide grating to a specific width. The slopes of the linear fits to the data shown in Fig. 7 are almost identical and approximately equal to 28 µm for all the mirrors considered here. Due to the blurring of the focal point for narrower mirrors, we do not consider them in our analysis.
Fig. 7. The chromatic aberration of mirrors of various widths as a function of the material dispersion (the slope $df/dk$ was determined based on the data shown in Fig. 5(b)). The solid lines indicate linear fits to the data points. The slope of the linear fit for these mirrors is approximately 28 µm.
Table 1. Dispersion in selected materials used in photonics
We also investigated whether the height of the grating stripes $h$ had a visible impact on chromatic aberration (see Fig. 6(b)). According to the simulations, $df/dk$ increases with increasing $h$, although the increase is very weak—approximately $5.5\%$ when $h$ is increased from 210 nm to 320 nm. We considered a rather narrow range of height variation due to the fact that any larger change of the grating height leads to the decrease of the light intensity at the focal point by more than 50% of the optimized mirror design. In other words, the gratings thinner or thicker than considered here will exhibit significantly poorer focusing properties and eventually, beyond some thickness, the focusing will not take place.
All the simulations presented in this section suggest that the chromatic aberration of such a mirror cannot be significantly modified either by the choice of the grating material or by straightforward modification of the geometry. This suggests that to reduce chromatic aberration of focusing grating mirrors, a more sophisticated mirror design is required. As has been shown for metalenses [5,10–25], it is not sufficient to engineer solely the phase of the transmitted light. Instead, simultaneous engineering of both the phase of the transmitted light and its dispersion seems required to reduce chromatic aberration.
2.3 Analytical description of chromatic aberration in the grating mirror
As briefly mentioned in the Introduction, a grating is able to focus light only if its stripes are arranged in a specific manner. Strictly speaking, the phase profile of the reflected wave must fulfill the following condition [7]
where $k$ and $f$ are the design wave vector and focal length respectively. This formula shows that the phase profile is determined by both the designed focal length and the assumed wave vector of the incoming wave. To build a focusing mirror, the continuous hyperbolic phase given in Eq. (1) must be approximated by the grating stripes, which is accomplished by selecting their appropriate geometrical parameters. A detailed discussion of this process can be found in [7]. In order to analyze the physics of the aberration of the mirror, we calculated the phase profiles of the reflected monochromatic plane waves for one mirror at multiple wavelengths. Figure 8 shows that the phase profile is practically independent from the wavelength apart from the vertical shift by a constant, which does not impact the focal length. Under this assumption, we can differentiate both sides of Eq. (1) with respect to $k$ and we obtain: Because the right-hand-side product is $0$ for all $y$, the second factor must be $0$ everywhere. Hence: Naturally, the focal length should not depend on $y$. The presence of $y$ in this formula stems from the fact, that the phase profile designed for a certain $k$ and $f$ will not match perfectly any other wave vector. However, because the considered mirror widths are very small compared to the focal length, we can neglect $y$ in the above formula. As a result, we get the following formula: In order to compare the result given by the above formula with simulations and experiment, first we have to notice that the Eq. (4) is derived under assumption that the material in which the wave propagates is homogeneous. In the mirror we fabricated, the focal point is outside the GaAs substrate, so we cannot directly compare the measured $\frac {df}{dk}$ with Eq. (4). However, we can compare the value predicted by that formula with appropriate simulations in which GaAs is infinitely thick. For the designed focal length in homogeneous GaAs (with a thickness of 1 mm) at a vacuum wavelength of $980\,$nm, formula (4) predicts $df/dk=$44.7 µm$^2$, while the slope of the simulated function $f(k)$ is 45.2 µm$^2$. Because our simulations of the real device (with the focal point in the air below the substrate) give results very similar to the measurements (see Fig. 4), we think that Eq. (4) is correct at least in the spectral range analyzed in this article.Fig. 8. Reflected phase of the grating stripes for one mirror design at various wavelengths between 900 and 1200 nm at 30 nm interval.
In our mirror, the observed immunity of the reflected phase profile to variations of the wavelength, comes form the fact that all the periodic MHCG mirrors with reflectance above $99\%$ we analyzed react (in terms of the reflected phase) to variations in the wavelength in a very similar way. Namely, they cause almost identical phase increment or decrement with the wavelength change. This suggests that some kind of correction of chromatic aberration by the MHCG itself might be possible at the cost of the total reflectance.
3. Summary
In this study, we investigated the spectral dependence of light focusing enabled by a monolithic high contrast grating mirror. The mirror was investigated at wavelengths of between 890 and 1050 nm at 10 nm intervals. Four main conclusions can be drawn from our results. Firstly, shifts in the focal point were observed with changes of the wavelength – i.e., we observed chromatic aberration. As expected based on simulations, the measured focal length was inversely proportional to the wavelength. Both the experimentally determined position of the focal point and its measured shift, induced by the wavelength variation, were almost in line with the simulation outcomes. Secondly, the focusing grating mirror maintained its focusing abilities across a wide span of wavelengths. This makes it very promising for future applications. Thirdly, the focusing abilities and the focal length of the grating mirror seem to be very resistant to modification of the grating height. Due to their large operation bandwidth and high tolerance for imperfections, grating mirrors have huge potential for focusing purposes. Finally, it is not possible to significantly reduce chromatic aberration either by simply selecting a different material for the mirror, or by modifying the height of the grating. Therefore, a more complex mirror design, similar to those developed for achromatic metalenses, seems to be needed.
Funding
Deutsche Forschungsgemeinschaft (Cooperative Research Center) 787; Fundacja na rzecz Nauki Polskiej (POIR.04.04.00-00-4358/17).
Acknowledgments
Project No. POIR.04.04.00-00-4358/17 is supported by the HOMING programme of the Foundation for Polish Science and co-financed by the European Union under the European Regional Development Fund. The German Research Foundation supports the work at the Technical University of Berlin via the Cooperative Research Center 787. The authors would like to acknowledge Renata Kruszka and Anna Szerling from the Sieć Badawcza Łukasiewicz, Instytut Mikroelektroniki i Fotoniki, for deposition of the anti-reflecting layer, Ronny Schmidt and Stefan Bock from the Technical University Berlin for help with electron beam lithography and inductively coupled plasma reactive ion etching, and Jarosław Lejbrant from VIGO System S. A. for polishing the samples. Finally, we would like to thank Magdalena Kamińska from INTERLAB for providing temporary access to the NKT Photonics SuperK EXTREME supercontinuum laser.
Disclosures
The authors declare no conflicts 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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