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Abstract
Here, we describe in detail a procedure for the numerical design of planar focusing mirrors based on monolithic high contrast gratings. We put a special emphasis on the reconstruction of the hyperbolic phase of these mirrors and we conclude that the phase does not have to be perfectly mimicked to obtain a focusing reflector. We consider here the grating mirrors that focus light not in the air but in the GaAs substrate and we compare them with conventional parabolic reflectors of corresponding dimensions. The light intensity at the focal point of the focusing grating mirrors was found to be comparable to that of the parabolic reflector. Moreover, the reflectivity of the focusing grating mirrors is almost as high as that of parabolic mirrors covered with an additional reflecting structure, if the ratio of the reflector width to the focal length is less than 0.6. Planar focusing grating mirrors offer a good alternative to parabolic mirrors, especially considering the complexity of fabricating three-dimensional structures compared to planar structures.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Spherical and parabolic reflectors are the most popular types of focusing mirrors. If a ray of light propagating along the axis of symmetry impinges on either, it is reflected from the surface and focused at a specific point on the principal axis, called the focal point. However, in the case of spherical mirrors, due to spherical aberration the reflected rays do not strike a single point. Instead, the rays that hit the central region of the mirror are focused further away than those that hit the edges of the reflector, leading to blurring of the focal point. This effect is completely intrinsic to the spherical shape, and has nothing to do with potential manufacturing imperfections. Fortunately, for the approximation of geometrical optics this problem does not affect parabolic reflectors, which are capable of focusing rays traveling parallel to the principal axis at a single point without any intrinsic aberrations. Parabolic reflectors are used in a wide range of applications, including reflecting telescopes [1–3], solar furnaces [4,5], car headlights [6], solar collectors [7–9], solar stoves [10,11], digital holographic microscopy [12], confocal microscopy [13–15], and Göbel Mirrors [16,17].
If a mirror is to act as a parabolic reflector and focus light, either (1) it must have a curved surface in the shape of a paraboloid or (2) the phase of the reflected wave must be modified. Modification can be achieved in two ways:
- 1. by near-field focusing in a customized multilayer structure. The components of the incident plane wave which arrive at different angles reflect at different depths in the structure, and therefore experience different phase delays. These angle-dependent anomalous phase delays contribute to the focusing properties of the mirror [18]. When it comes to lenses instead of focusing mirrors, light focusing may be also obtained in a slab of material with negative refractive index [19,20].
- 2. by far-field focusing, obtained by reflection from flat plasmonic metasurfaces [21–24] or high contrast gratings [25–27]. In both cases, the wave reflects from the surface of the mirror without penetrating the reflector. The relevant phase profile of the reflected wave is obtained by local modulation of the periods and sizes of the constituent elements along the metastructures/gratings. Various types of metasurfaces [22,28,29] and high contrast gratings [27,30–35] are also successfully used to create high-quality planar lenses.
In this article, we compare planar focusing reflectors based on monolithic high contrast gratings (MHCGs) and conventional parabolic mirrors. Focusing grating mirrors have the clear advantages of being flat, with focal lengths in the micrometer range. Their fabrication requires only regular electron beam lithography followed by reactive ion etching [42] or nanoimprint lithography [43,44]. For the fabrication of single-height nanostructures, electron beam lithography is considerably easier and less prone to fabrication errors compared to grayscale lithography.
Recently, subwavelength high contrast gratings (HCGs) have been the subject of extensive research, as one of the mirrors in vertical cavity surface emitting lasers [45–47], as high-Q resonators [48,49], or as broadband reflectors [24,42,50,51]. Despite their small thickness, they can provide reflectivity theoretically reaching 100% through destructive interference of two supermodes inside the grating exited by the incident wave [42]. Recently, we have proven that such mirrors can be integrated with high-refractive-index substrate and still provide 100% reflectivity using the same mechanism [52].
Moreover, there have also been experimental and numerical studies on planar focusing reflectors [24–27,30,53–55] and lenses [27,30–35] based on HCGs. Our focus is on MHCGs instead of gratings suspended in air [24,27,30,54], which suffer from fragility, or HCGs on a cladding layer [25,26,53,55], which are usually electrically non-conductive. The fabrication of such structures also requires additional steps compared to MHCGs. To the best of our knowledge, in all the previous studies on focusing grating mirrors, the grating material had a higher refractive index than the medium in which the incident and reflected waves propagate. In contrast, we consider grating mirrors, where that medium and the grating are made of the same material, namely GaAs. Designing of focusing sub-wavelength grating mirrors is fundamentally different in those two cases. If a wave is reflected in a medium of a lower refractive index than the refractive index of the grating, only the zeroth diffraction order exists. However, in a medium, of the same or a higher refractive index than in the grating, higher diffraction order exist, in principle. However, in a MHCG, if its parameters are carefully chosen, the unwanted in our mirror higher diffraction orders can be eliminated. A detailed discussion on the challenges in designing of such MHCGs is given in Ref. [52].
A planar mirror focusing inside the volume of an optical or optoelectronic device can have interesting applications. Some of them may, possibly the most interesting, be devised in the future, but focusing mirrors have already been used in semiconductor lasers [56,57] and photodetectors [58]. In the VCSELs presented in Ref. [56] (a single-mode GaAs-based IR laser) and [57] (a GaN-based blue laser), the focusing was achieved by concave mirrors formed on the bottom surface of the substrate. In both cases, the presented devices, achieved very high output powers compared to other devices in the same categories. Recently, a photodetector with a focusing grating mirror has been presented [58]. According to the authors, the focusing mirror increased the sensitivity of the device by over 30%. Unlike the mirrors proposed in our paper, the mirror in Ref. [58] was not monolithic (i.e. three different materials were used for its formation).
In section 2.1, we describe the design procedure for planar focusing mirrors in detail. In section 4.1, planar focusing mirrors are compared with conventional curved parabolic reflectors, in terms of their reflectivity and the intensity of light at the focal point. Due to limited computing resources, we consider two-dimensional mirrors that focus light into a line along the $x$ axis, instead of at a single point. Parabolic reflectors are shaped like parabolic cylinders, rather than paraboloids (compare Figs. 1(a) and (b)). The phase profile of the reflected wave of focusing grating mirrors has the form of a hyperbolic cylinder, not a hyperboloid (see Figs. 1(c) and (d)). We will discuss why the phase profile of focusing grating mirrors is hyperbolic in section 2. In section 4.2, we present how the quality of the phase approximation influences the focusing properties of the grating mirrors. Throughout, we will consider reflectors designed for two polarizations, transverse-magnetic and transverse-electric (TM/TE). As shown schematically in Fig. 3, the only non-zero electric field component is perpendicular/parallel to the stripes of the grating.
Fig. 1. Scheme of (a) a parabolic cylinder, (b) a paraboloid, (c) a hyperbolic cylinder, (d) a hyperboloid.
2. Theory of HCG planar focusing reflectors
At the focal point of a parabolic mirror, all the reflected beams have the same phase, provided that the incident wave propagates along the mirror axis. The constructive interference at the focal point creates the characteristic high intensity of the optical field. In the case of a parabolic mirror, the appropriate phase at the focal point is a result of the appropriate distance between the focal point and each of the points on the surface of the mirror. However, if there is the possibility to set the phase of the reflected wave arbitrarily, the focal-point phase condition can be obtained for different shapes of the reflecting surface. Here, we analyze the case of a flat reflecting plane, perpendicular to the incident wave. Figure 2 presents a scheme for such a parabolic mirror. If we assume that the phase of the wave reflected at $y=0$ is 0, then at any arbitrary point $y$ on the mirror the reflected phase can be described by the following formula:
where $f$ is the focal length, $\lambda$ is the wavelength, and $n$ is the refractive index of the medium in which the wave propagates. Here, we use the convention that a plane wave propagates in the direction of the wave vector $\boldsymbol k$.2.1 Design procedure of planar focusing reflectors based on monolithic high contrast gratings
To avoid any confusion in the nomenclature, we will differentiate between two types of mirrors based on MHCGs:
- • by regular grating mirror/reflector we mean a mirror that does not focus light, but only reflects light, and which is constructed from an infinite number of identical grating stripes.
- • by focusing grating mirror/reflector we mean a planar focusing mirror that is based on a particular arrangement of grating bars, which provide phase distribution satisfying Eq. (1). This type of mirror is composed of several grating segments that differ from each other, and is therefore non-periodic.
The design for a focusing grating reflector starts with calculations of the reflectivity of regular grating mirrors, as shown in Fig. 4(a), and the phase shift between the incident and reflected waves (Fig. 4(b)), taking into account a single period of the grating with periodic boundary conditions. As the reflectivity and phase shift depend strongly on the geometrical dimensions of the stripes, we consider wide ranges of periods and fill factors, equal to $0\;<\;L\;<\;1$ µm and $0\;<\;F\;<\;1$, respectively. In all the considered grating mirrors, $h$ is kept constant. This is because the typical processing techniques (e.g. electron beam lithography followed by reactive ion etching, nanoimprint lithography, focused ion beams) enable control of the lateral dimensions while the depth of the etching features remains constant.
Fig. 4. Maps of (a) the reflectivity and (b) the phase shift calculated for regular grating mirrors made of GaAs, TM polarization, $\lambda =980$ nm, and $h=0.28$ µm. The interiors of the white, gray, and black contours indicate phase ranges for which the reflectivity is equal to or larger than 0.8, 0.9, or 0.95, respectively.
After mapping the phase shift for a set of regular grating mirrors, we arrange the appropriate grating stripes along the $y$ axis, to build a focusing grating mirror. The appropriate stripes are those that best approximate the continuous hyperbolic phase given by Eq. (1) (if the phase exceeds $2\pi$, it is mapped to the value modulo $(2\pi )$). These stripes are determined based on the map shown in Fig. 4(b). In other words, from an infinitely long regular grating mirror we extract a single period that provides a suitable value for $\phi (y)$ and align it in a focusing grating mirror at position $y$. It is not obvious that such a strategy will be sufficient to obtain a reflected phase in line with Eq. (1). To prove that the algorithm indeed allows for the reconstruction of the desired phase, in Fig. 5 we compare three sets of data. These are
- • the smooth solid line, which is the desired phase calculated for $n=3.4935$, $\lambda =980$ nm, $f=1000$ µm,
- • data points, which approximate the expected phase (each data point corresponds to one stripe of the grating),
- • the black (slightly jagged) solid line, which presents the reflected phase determined from the electric field of the reflected wave.
Fig. 5. Reflected phase calculated from Eq. (1) (red solid line), approximated by the grating stripes (data points), and calculated from the electric field of the reflected wave (black, slightly jagged solid line). The focusing grating mirror has $f=1000$ µm and $w=300$ µm. As the red-shaded sections become narrower for larger $y$, for clarity only part of the mirror is shown.
The very good alignment of these three sets of data confirms the suitability of the approach presented here. For better clarity, only half of the approximation of the focusing grating reflector is shown in the graph (the part for $y\;<\;0$ µm is symmetrical).
To ensure high reflectivity, we take into account only those regular grating mirrors that provide reflectivity greater than a particular fixed value. In the example shown in Fig. 4(b), these limiting values were set to 0.8, 0.9, and 0.95, marked with white, gray, and black contours, respectively. As can clearly be seen, depending on the contour a wider or narrower range of available phase shift can be found. More precisely, the lower the limiting value for reflectivity, the wider the available range of phase shift. Having the widest possible range of accessible phase shift (i.e. from 0 to $2\pi$) seems to be important, as it affects the possibility of approximating the hyperbolic phase profile. In section 4.2, we discuss the influence of the quality of phase approximation on reflectivity and light intensity at the focal point of focusing grating mirrors.
All the mirrors considered here are two-dimensional. Thus, the wave phase profile is shaped as a hyperbolic cylinder rather than a hyperboloid (see Figs. 1(c) and (d)). As a consequence, the focusing grating mirrors considered here do not focus light into a single point, but instead into a line along the $x$ axis.
3. Simulation details
All the simulation results presented here were acquired using the Photonics Laser Simulation Kit (PLaSK) developed at Lodz University of Technology. The reflectivity and electric field intensity distributions of the MHCGs were investigated using the plane-wave reflection transformation method, which is a modification of the plane-wave admittance method. Details of the model can be found in [59].
Our investigation concentrates on focusing grating mirrors made from GaAs designed for 980 nm. The total width of the reflectors is given as the variable $w$ in each section. In the simulations, we assume that light propagates in the non-absorbing substrate, which is infinitely thick and located at $z\;<\;0$ µm (for the axis orientation see Fig. 3). The bottom of the grating having height $h$ is at $z=0$ µm and the remaining space (i.e. between the grating bars and for $z\;>\;h$ up to infinity) is filled with air. The x-y plane is infinite as well – in the $x$ direction the grating bars extend from minus to plus infinity. Periodic boundary conditions are imposed in the $y$ direction, meaning that the structure is repeated infinitely. Based on our preliminary study, aimed at finding the gating heights that allow for the construction of mirrors exhibiting the highest light intensity at the focal point, the heights of the considered gratings were fixed at 0.24 µm and 0.28 µm for TE and TM polarizations, respectively. All the results discussed in what follows were obtained assuming an incident field in the form of a plane wave approaching the focusing grating reflector from below (i.e. from the substrate side), reflected by the mirror and focused at $z=-f$ (the value of $f$ will be defined in each section).
In particular, we analyze the reflectivity of the focusing grating mirrors and the maps of the electric field intensity distributions in the vicinity of the focal points of the reflectors. On all the light intensity maps ($I_{\mathrm {rel}}$) presented here, the incident plane wave was subtracted from the total electric field intensity distribution to avoid the appearance of Moiré fringes. The units used are such that the intensity of the incident light is 1.
4. Simulation results and discussion
4.1 Parabolic reflector versus planar focusing grating reflector
To compare conventional parabolic mirrors and planar focusing grating mirrors, we consider three types of two-dimensional reflectors: (1) parabolic reflectors (PR) made from GaAs; (2) parabolic reflectors made from GaAs covered with three pairs of SiO$_2$/Si distributed Bragg reflectors (DBR); (3) planar focusing grating reflectors made from GaAs. The expected focal length of the mirrors is $f=30$ µm and their total widths vary. As an example, the maps in Figs. 6(d)-(f) illustrate mirrors with $w=18$ µm. For animations showing other sizes of reflectors, see Visualization 1, Visualization 2, Visualization 3, and Visualization 4. The parabolic shape of the mirrors indicated in Figs. 6(d) and (e) is given by $z(y)=-\frac {y^2}{4f}$. The simulations were performed assuming approximation of this function, with a step function and a step width of 100 nm. The DBR that covers the PR in Fig. 6(e) is marked by yellow lines. The scale of the parabola curvature aspect ratio and the DBR thickness are preserved.
Fig. 6. (a-c) Electric field intensity distribution ($I_{\mathrm {rel}}$) profiles along gray dashed lines (i.e. along the principal axes of the reflectors – $z$ axis), indicated in the maps shown in (d-f). The figures are arranged in columns that correspond (from left to right) to the parabolic reflector, the DBR covered parabolic reflector, and the grating focusing reflector. (g) Normalized electric field intensity distribution profiles along $y$ axis for $z=f$. Calculations were performed for TM polarization and light incident from below.
As depicted in Figs. 6(a) and (d), PRs made from GaAs without any covering material exhibit not one but two focal points. One appears for the reflected light, and the other for the transmitted light, i.e. they also work as focusing lenses. The actual positions of the focal points were determined by fitting the Gaussian distribution to the the peaks in Fig. 6(a). The results of $f^{\mathrm {PR}}_{\mathrm {mirror}}=-29.1$ µm with the standard deviation $\sigma =2.6$ µm and $f_{\mathrm {lens}}^{\mathrm {PR}}=22.9$ µm with $\sigma =5.6$ µm are in good agreement with the anticipated focal lengths of 30 µm in GaAs and $f_{\mathrm {lens}}^{\mathrm {theor}}=\frac {2f}{n_{\mathrm {GaAs}}-1}=24.1$ µm in air. The high degree of agreement between the expected parameters and those obtained via simulations shows the correctness and reliability of the models employed in our software. The slight inaccuracies are caused by the fact that we consider the parabola as a step function and not a smooth function.
When considering light incident perpendicularly from the side of GaAs on the interface between GaAs and the air, the reflectivity is only 0.31. If, on the other hand, GaAs is covered with 3 pairs of SiO$_2$/Si DBR, the reflectivity is 0.99. Therefore, to increase the reflectivity of the PR we cover it with a highly-reflective set of layers. In addition to the expected increase in the reflectivity of the PR, the focal point in the air vanishes (see Figs. 6(b) and (e)). In the case of the focusing grating reflector presented in Figs. 6(c) and (f), no additional layer of material is used. Moreover, the light intensity at the focal point of the focusing grating mirror is comparable that determined for the non-covered PR (see Figs. 6(a) and (c)).
When working with devices that are able to focus light, it is always interesting to know whether they are diffraction limited. According to the Rayleigh criterion, the diffraction limit is defined as the distance between the central maximum and the first minimum of the intensity profile of light diffracted on a single slit. If we consider a single slit that has the width identical to the considered grating and calculate the intensity profile, we obtain the red line shown in Fig. 6(g), which almost overlaps with the normalized electric field intensity distributions for both PR and DBR-covered PR. The first minima of the intensity visible for the single slit, indicated by the red vertical red lines in the inset in Fig. 6(g), determine the diffraction limit. As shown by the dashed blue lines, the grating mirror is not diffraction limited, however, it is very close to that limit. Especially when the full width at half maximum, rather than the distance between the minima, is taken into account.
To estimate focusing efficiency of our mirrors we calculate and compare the following integrals:
where $y_{\mathrm {lm}}$ and $y_{\mathrm {rm}}$ are the lateral position of the left and right minima of the optical field near the focal point; $y_0$ and $y_1$ are the lateral limits of the whole mirror and $z_{\mathrm {f}}$ is the vertical position of the focal point. By calculating the ratio of those two integrals we determine the percentage of the reflected energy which is concentrated at the focal point of the mirror. Our analysis on the mirrors designed for TM-polarization and having $f=30$ µm shows that the focusing efficiency is around 90% for both PRs and DBR-covered PRs regardless of their widths. In turn, for the grating mirrors it decreases from 72% for the 8 µm wide grating to 21% for the 36 µm wide mirror.When designing a focusing grating mirror, it should be kept in mind that the larger the mirror the narrower the sawteeth, which are visible as red shaded sections in Fig. 5. Therefore, the greater $y$ is, the fewer stripes are used to approximate the phase in the whole range from 0 to $2\pi$. For very large $y$, some of the sawteeth are approximated with one or two grating stripes and others are completely non-approximated by grating bars. For a better illustration, see Visualization 5, Visualization 6, Visualization 7, Visualization 8, Visualization 9, and Visualization 10. Therefore, there exists a certain reflector width above which the approximation of the phase is non-effective. To determine this limiting value, we calculated how the light intensity at the focal point changes as the width of the focusing grating mirror increases. In Fig. 7, we summarize the results of simulations obtained for both polarizations (TE and TM) and three fixed values for focal length (30 µm, 50 µm, and 100 µm). As the mirror width increases, the reflecting area of the mirror increases proportionally. Therefore, there is initially a linear increase in light intensity. This linear dependency, shown in Fig. 7 with fitted solid lines, applies consistently to all the considered focal lengths and polarizations, provided that $w/f\;<\;0.6$. If, however, $w/f$ is larger than 0.6, the light intensity at the focal point saturates. In this range, the dashed lines that show linear fits to the data points are almost parallel to the abscissa. The limiting size of the reflector was determined as $w/f=0.6$, what corresponds to the numerical aperture of 1.0. Any increase of the reflector size above this value becomes pointless, due to the fact that a larger mirror does not contribute to greater light intensity, since the effective size of the focusing grating mirror remains unchanged. In other words, the performance of these focusing mirrors is limited by their numerical aperture. The gray-shaded area in Fig. 7 shows the range of $w/f$ that will be used in our comparison of focusing grating mirrors and PRs.
Fig. 7. Maximum value for light intensity at the focal point as a function of the $w/f$ ratio (mirror width/local length) calculated for focusing grating reflectors (MHCG) with the indicated polarization (TE or TM) and focal length. Solid and dashed lines represent linear fits in two ranges for $w/f$ (i.e. $w/f\;<\;0.6$ and $w/f\;>\;0.6$).
In contrast to the linear increase in light intensity observed for the focusing grating mirrors, both types of parabolic reflectors exhibit parabolic increases of max $(I_{\mathrm {rel}})$, associated with the parabolic increase in their area as their width increases. The solid lines in Fig. 8(b) are parabolic fits that perfectly coincide with the calculated data points. In the $w/f$ range under consideration, the light intensity at the focal point of the focusing grating mirrors is slightly larger than the value determined for non-covered PRs. In contrast to the comparable values for max($I_{\mathrm {rel}}$), the reflectivity (see Fig. 8(a)) of all the focusing grating mirrors in the gray-shaded area is definitely larger (around threefold) than the reflectivity of the PRs, and is almost as high as the reflectivity of the DBR-covered PRs. Having DBR on top of a PR significantly increases not only the light intensity in the focal point but also the reflectivity.
Fig. 8. (a) Reflectivity and (b) maximum value for light intensity at the focal point as a function of the $w/f$ ratio (mirror width/local length), calculated for the indicated polarization (TE or TM), for parabolic reflectors (PRs), PRs covered with 3 pairs of SiO$_2$/Si DBR (PR+DBR), and focusing grating reflectors (MHCG). In (a) solid lines are guides to the eye only and in (b) solid lines are fits.
When we examine a wider range of $w/f$ for parabolic reflectors, in Fig. 8(a) we notice a characteristic minimum in the reflectivity for TM polarization which is absent in the case of TE polarization. This can be attributed to the appearance of the Brewster angle. As the parabolic mirror becomes wider, the angle of incidence of the outer rays increases according to the increasing slope of the mirror surface. Assuming that the rays are incoming from the substrate side and are perpendicular to the $y$ axis, the incident angle is equal to $\mathrm {atan}\big (\frac {dz}{dy}\big )$, where $z(y)$ is the function that describes the shape of the parabola. The incident angle of 15.97°, corresponding to the Brewster angle between the GaAs and air, coincides perfectly with $w/f=1.1$. For the DBR covered PRs the situation is more complex, due to the fact that more layers are involved. In 1974, Mahlein [60] determined formulas for the Brewster angle in quarter-wave multilayers, which depended on the number of periods and refractive indices of the layers involved. Based on his calculations, the Brewster angle in our structures does not exist, which is in agreement with the calculation results shown in Fig. 8(a).
Based on the comparison presented in Fig. 8, it seems that if the numerical aperture is smaller than 1 the focusing grating mirror is a very good alternative to both non-covered and DBR covered PRs. This is thanks to the following advantages: (1) focusing grating mirrors allow a single focal point to be obtained without any additional layer covering the grating; (2) the light intensity at the focal point of the focusing grating mirror is comparable to the intensity observed for the PR, and simultaneously the reflectivity of the focusing grating mirror is almost as high as the $R$ of the DBR-covered PR (see Table 1). Importantly, electron beam lithography followed by reactive ion etching is sufficient to fabricate focusing grating mirrors, whereas to produce PRs it is necessary to use a more sophisticated method, namely grayscale lithography. Moreover, to obtain higher reflectivity and light intensity, grayscale lithography must be followed by the growth of additional reflecting layers, which is not required in the case of focusing grating mirrors.
Table 1. Summary of reflectivity and maximum light intensity at the focal points of a parabolic reflector (PR), a PR covered with 3 pairs of SiO$_2$/Si DBR (PR+DBR), and a focusing grating reflector (MHCG) assuming $f=30$ µm, $w=18$ µm, and TM polarization.
Our discussion so far has referred only to the normal incidence, i.e. the situation in which the angle of incidence $\alpha$ has been fixed to 0°. In Fig. 9(a), we present how the maximum light intensity in the focal point varies when the angle of incidence increases, for reflectors with $f=30$ µm, $w=18$ µm, and TM polarization. As is expected and clearly visible in Visualization 11, Visualization 12, and Visualization 13, the greater the angle of incidence, the greater the shift of the focal point away from the principal axis of the focusing mirror. As the angle of incidence increases, the light intensity at the focal point of the DBR-covered PR decreases. Despite this decrease, the light intensity at the focal point of the DBR-covered PR is the highest of all the three reflectors, in the whole $\alpha$ range studied here. The PR and the focusing grating mirror have very similar light intensity at the focal point in the whole $\alpha$ range, and moreover exhibit the minimum light intensity at the focal point at $\alpha =10$°. At $\alpha\;<\;10$°, the focusing grating mirror exhibits slightly higher values of max($I_{\mathrm {rel}}$), whereas for larger angles of incidence PR becomes dominant. The origin of this minimum needs further analysis.
Fig. 9. Maximum value of light intensity at the focal point for variable (a) angle of incidence $\alpha$ and (b) wavelength $\lambda$. Comparison of a parabolic reflector (PR), a PR covered with 3 pairs of SiO$_2$/Si DBR (PR+DBR), and a focusing grating reflector (MHCG), assuming $f=30$ µm, $w=18$ µm, and TM polarization. Solid lines are guides to the eye only.
When we take into account the same three reflectors and now vary the wavelength of the incident wave between 880 and 1300 nm (assuming $\alpha =0$°), we obtain the graph presented in Fig. 9(b). For maps of the electric field intensity distribution see Visualization 14, Visualization 15, and Visualization 16. Both of the parabolic reflectors exhibit decreases in light intensity at the focal point as the wavelength increases. Presumably, this is due to the reduction of the refractive index and so of reflectivity with the wavelength. The light intensity at the focal point of the focusing grating mirror exhibits a wide maximum of around 1030 nm. The value of max($I_{\mathrm {rel}}$) is larger than that for PR if the incident wavelength is between 970 and 1140 nm. The reason for the shift of this maximum from 980 nm, i.e. the wavelength for which the mirrors were designed, is beyond the scope of this article and requires further study.
4.2 Influence of the quality of phase approximation on the focusing properties of grating mirrors
Due to the fact that the hyperbolic phase of focusing grating mirrors is discretized, not mimicked in its continuous form, it is crucial to know how well it must be reproduced for it to act as a focusing reflector. To quantify approximation quality, we will use a parameter called phase coverage (PC). The plots presented in Figs. 10 and 12 will be very helpful in this regard. The dots illustrate the reconstruction of the continuous hyperbolic phase profile (plotted with a solid line), which corresponds to 300 µm wide reflectors with $f=1000$ µm. As indicated with short black horizontal lines near the ordinate axis, we project all the points on the ordinate axis and then calculate the distance between the neighboring points. If this distance is smaller than $\phi _{\mathrm {lim}}$, we assume that this region is well mimicked. To define $\phi _{\mathrm {lim}}$, we first calculate the width of the section no. 1 depicted in Fig. 5 for which Eq. (1) reaches $2\pi$ and we call it $\Delta y_1$. We then determine $\phi _{\mathrm {lim}}$ as
where $N_{\mathrm {seg}}$ is the number of grating segments that are between 0 and $\Delta y_1$. The rectangular shaded areas in Figs. 10(a) and 12(a) represent the well-reproduced regions in which the distance between the neighboring points projected on the ordinate axis is smaller than $\phi _{\mathrm {lim}}$. Subsequently, we calculate the phase coverage as the ratio between the sum of lengths of all the well-reproduced regions and the maximum range of phases (i.e. $2\pi$), following the formula This parameter tells us what percentage of the approximated phase profile coincides with the expected phase profile.Fig. 10. Theoretical (solid line) and reproduced (dots) hyperbolic phase profiles for $f=1000$ µm, $w=300$ µm, and phase coverage equal to (a) 45%, (b) 10%, and (c) 4%. TE polarization of the incident light.
Fig. 11. Maps of the electric field intensity distribution zoomed on the focal point of reflectors with $f=1000$ µm, $w=300$ µm, and phase coverage equal to (a) 45%, (b) 10%, and (c) 4%. TM polarization of the incident light.
We limited the reflectivity value (indicated by the contours in Fig. 4(b)) to values in the range of 0.4-0.95 in the case of TE polarization and to 0.8-0.9995 for TM-polarized reflectors. This corresponds to a PC range of between $66\%-4\%$ and $100\%-4\%$, for reflectors with $f=1000$ µm and $w=300$ µm, respectively. The maps of electric field intensity distribution for three of these reflectors chosen for each polarization are presented in Figs. 11 (TE) and 13 (TM). Visualization 17 and Visualization 18 show more maps. To facilitate comparison, the maps have been zoomed in on the focal point only.
The first observation is that with decrease of the PC, the light intensity in the focal point decreases. However, it is very important to point out that the phase coverage even as small as 4% (see Figs. 10(c)–13(c)) does not prevent obtaining clear focusing properties. The light intensity for $\mathrm {PC}=4\%$ is substantially smaller than that of the other considered reflectors. However, the focal point is clearly visible. To design a mirror with $\mathrm {PC}=4\%$, seven grating segments with various $L$ and $F$ were used. As shown in Fig. 12(c), these seven segments can be divided into two groups providing diverse phase shift, namely $\approx 0.75\pi$ and $\approx 1.36\pi$. Interestingly, if we select one segment from each of these two groups and design a focusing grating mirror, the light intensity at the focal point may be as high as 25. Therefore, one of the most important conclusions is that perfect reproduction of the hyperbolic phase in the whole range between 0 and $2\pi$ is not a mandatory prerequisite for obtaining a focusing reflector. Of course, the larger the coverage of $\phi$, the more efficient the focusing, but some degree of focusing may be obtained if only a small fraction of $\phi$ is covered.
The impact of PC on the reflectivity and maximum light intensity of the mirrors is analyzed in more detail in Figs. 14(a) and (d), respectively, at the focal points ($f=1000$ µm and $w=300$ µm). In Fig. 14, we can see what happens if the PC decreases, i.e. we analyze the graph from left to right. Interestingly, if $\mathrm {PC}\;>\;50\%$, the reflectivity remains approximately constant. If, however, the phase coverage drops below 50%, the reflectivities of all the considered reflectors increase. This increase is step-like in the case of TM-polarized reflectors and linear for the TE-polarized reflectors. Moreover, if several TM-polarized reflectors are considered for which $\mathrm {PC}=100\%$ (where the contours in Fig. 4(b) limit the reflectivity of regular grating mirrors to $0.7\;<\;R\;<\;0.8$), the reflectivity of the focusing grating mirrors varies in the range of between 0.8897 and 0.9007 and their max($I_{\mathrm {rel}}$) is between 151 and 163.
Fig. 12. Theoretical (solid line) and reproduced (dots) hyperbolic phase profiles for $f=1000$ µm, $w=300$ µm, and phase coverage equal to (a) 100%, (b) 38%, and (c) 4%. TM polarization of the incident light.
In contrast to reflectivity, if the phase coverage becomes smaller the light intensity at the focal point decreases for both polarizations, in a step-like fashion. There are, however, two differences between the mirrors considered for different polarizations: (1) the value of PC at which this drop takes place and (2) the magnitude of that drop. The decrease in light intensity from around 140 to roughly 50 at $\mathrm {PC}=45\%$ observed for TM-polarized reflectors is much more subtle compared to the sudden drop from $I_{\mathrm {rel}}\;>\;100$ to values of around 5 at $\mathrm {PC}=10\%$ for TE-polarized mirrors.
When we look at the focusing maps in Figs. 11 and 13, it can be seen that the size of the light spot at the focal point is not affected by the PC. To perform more quantitative analysis, we fitted the Gaussian distribution to the electric field intensity distribution in these reflectors in both directions (namely, along $z$ and $y$). We then compared the the full width-at-half-maximum (FWHM), as depicted in the inset in Fig. 6(c). The FWHM fitted along $y$ for $z=-f$ varies between 0.8 and 0.9 µm, whereas that determined along $z$ for $y=0$ µm is in the range of 21 and 24 µm in the case of all TM reflectors across the whole PC range. For TE-polarized reflectors, the FWHM along $y$ remains the same compared to the TM polarization. The FWHM along $z$ increases from 21 to 24 µm if the PC parameter decreases from 60% to 10% and further to 34 µm for even smaller PC. When we compare the FWHM along $z$ ($\sim 21-34$ µm) to the focal length of the studied reflectors (i.e. 1000 µm), the size of the light spot at the focal point can still be seen to vary very little with the PC.
Fig. 13. Maps of the electric field intensity distribution zoomed on the focal point of reflectors with $f=1000$ µm, $w=300$ µm, and phase coverage equal to (a) 100%, (b) 38%, and (c) 4%. TM polarization of the incident light.
Fig. 14. (a-c) Reflectivity and (d-f) maximum value of the electric field intensity at the focal point as a function of the phase coverage for various sizes of focusing mirrors.
If we consider smaller reflectors designed with less and less PC for TM-polarized light, assuming two ratios $w/f$ (i.e. 0.3 and 0.6), a similar step-like increase in reflectivity (see Figs. 14(b) and(c)) and drop in maximum light intensity at the focal point (see Figs. 14(e) and (f)) occur, provided that $f\geqslant 50$ µm. In all cases, the phase coverage values at which these steps appear are the same as in the case of a mirror with $f=1000$ µm. The magnitude of the drops in light intensity is also very similar, at approximately threefold. For maps of the electric field intensity distribution demonstrating some of these reflectors, see Visualization 19, Visualization 20, and Visualization 21.
The smallest two mirrors considered here have sizes of $f=30$ µm, $w=9$ µm and $f=30$ µm, $w=18$ µm. Compared to all the larger mirrors, these two mirrors do not exhibit such clear step-like behavior. Most likely, this is due to the fact that the smaller reflectors are composed of far fewer stripes. Therefore, it becomes more difficult to identify which regions of the phase profile are well reproduced, and calculating the value of PC is more complicated.
Based on these data, it can be concluded that mirrors with a larger focal length are less sensitive to inaccurate reconstruction of the hyperbolic-wave phase profile. In turn, smaller $f$ reflectors need much more attention and care during their design. To achieve the equivalent of larger PC, Li et al. [31] and Cai et al. [26] introduced a second height of HCG into their structures, for their designs of a lens ($f=7$ µm, $w=26$ µm) and focusing reflector ($f=10$ µm, $\mathrm {radius}=44.11$ µm). Doing so made the structure more complex, but light intensity was improved at the focal point.
5. Summary
In this article, we have compared parabolic reflectors made from GaAs, parabolic reflectors made from GaAs covered with three pairs of SiO$_2$/Si DBR to improve their reflectivity, and planar focusing mirrors constructed based on MHCGs formed with GaAs. We concentrated on two-dimensional focusing mirrors, which focus light into lines along one of the axes and not into a single point. Although the reflectors were designed in GaAs for wavelengths of around 1 µm, the same idea could be easily applied to different materials and, in principle, any arbitrary wavelength.
Unlike conventional focusing mirrors, focusing grating mirrors are planar, can be carved in various semiconductor and dielectric materials, and their focal lengths can be very small – a few tens of micrometers or even less. These properties make them a very attractive option for use in optoelectronic devices, such as light emitters, detectors, and integrated optics in general.
Funding
Fundacja na rzecz Nauki Polskiej (POIR.04.04.00-00-4358/17).
Acknowledgments
The authors would like to acknowledge Prof. Rafał Kotyński for valuable discussions.
Disclosures
The authors declare no conflicts of interest.
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