## Abstract

There has been substantial interest in miniaturizing optical systems by flat optics. However, one essential optical component, free space, fundamentally cannot be substituted with conventional local flat optics with space-dependent transfer functions, since the transfer function of free space is momentum-dependent instead. Overcoming this difficulty is important to achieve the utmost miniaturization of optical systems. In this work, we show that free space can be substituted with nonlocal flat optics operating directly in the momentum domain. We derive the general criteria for an optical device to replace free space and provide two concrete designs of photonic crystal slab devices. Such devices can substitute much thicker free space. Our work paves the way for the utmost miniaturization of optical systems using a combination of local and nonlocal flat optics.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Recently, there has been significant progress in flat optics aiming to miniaturize optical systems by replacing conventional optical components [1–4]. Such flat optics typically achieve wavefront shaping through local control of phase, amplitude, and polarization in the spatial domain. This approach enables compact devices including metalenses, beam deflectors, and holograms, with potential applications such as flat displays, wearable optics, and lightweight imaging systems [5].

However, there is one essential optical component that has been long overlooked in flat optics: free space. Free space is an essential part of most optical systems and usually constitutes a great portion of the system volume. The utmost miniaturization of optical systems therefore requires significant reduction of free space: a truly miniaturized imaging system needs not only compact flat lenses, but also squeezed free space.

Free space fundamentally cannot be substituted with conventional local flat optics characterized by space-dependent transfer functions, since the free space propagation has a momentum-dependent transfer function instead. On the other hand, recently there has been significant interest in lensless Fourier optics using nonlocal photonic nanostructures with transfer functions in the momentum domain. This approach enables important functionalities in optical analog processing including optical differentiation, filtering, and phase mining using compact devices [6–16].

In this work, we show that free space can be substituted with nonlocal flat optics. We derive the general criteria for a device to replace free space and provide two concrete designs of such a device utilizing photonic Fano resonances. The device can substitute much thicker free space with a compression ratio as high as 144. The device can be designed to adapt to different numerical apertures (NAs). It can be placed anywhere in the optical path to reduce free space as needed. Our work therefore opens up significant opportunities in miniaturizing optical systems by squeezing free space with nonlocal flat optics.

The rest of this paper is organized as follows. In Section 2, we provide a theoretical analysis of replacing free space with flat optics using Fano resonances. In Section 3, we provide numerical demonstration of two concrete photonic crystal slab designs to realize such functionality. We conclude in Section 4.

## 2. THEORETICAL ANALYSIS

In this section, we provide a theoretical analysis on substituting free space with flat optics.

First, we briefly examine the propagation of light in free space. Consider a monochromatic optical wave of wavelength $\lambda$ and complex amplitude $U(x,y,z)$ in the free space between the input plane $z = 0$ and the output plane $z = d$. The free space propagation that maps $U(x,y,0)$ to $U(x,y,d)$ is a shift-invariant linear system, since the Helmholtz equation that governs $U(x,y,z)$ is linear, and free space is invariant under spatial translation in the $xy$ plane. Such a shift-invariant linear system is characterized by its transfer function [17]:

where ${\boldsymbol k} = ({k_x},{k_y})$ is the in-plane wavevector, and ${k_0} = 2\pi\! /\lambda$ is the angular wavenumber in free space. $H({k_x},{k_y})$ is a circularly symmetric complex function of ${k_x}$ and ${k_y}$. A plane wave component with $k_x^2 + k_y^2 \le k_0^2$ is propagating. For such a wave, the magnitude $|H({k_x},{k_y})| = 1$, and the phase $\text{arg}\{H({k_x},{k_y})\}$ is wavevector dependent. A plane wave component with $k_x^2 + k_y^2 \gt k_0^2$ is evanescent. For such a wave, $H({k_x},{k_y})$ is exponentially decaying.Given the input field $U(x,y,0)$, the output field $U(x,y,d)$ is determined as

In many cases of interest, the optical waves are paraxial where the input field $U(x,y,0)$ contains only wavevector components for which $k_x^2 + k_y^2 \ll k_0^2$. The transfer function in Eq. (1) then can be simplified as

where ${H_0} = \exp (- i{k_0}d)$ is a global phase. This is the well-known Fresnel approximation [18]. This approximation is applicable ifIn order to squeeze free space of a propagation distance $d$, our objective will be to create an optical device with the same transfer function of Eq. (4), but with a physical thickness that is much lesser than $d$. Since the transfer function of Eq. (4) describes a wavevector-dependent phase shift, with no mixing between different wavevector components, the required optical device must be periodic in order to preserve the wavevector as light transmits through the device. This is in contrast with standard metasurfaces, which are characterized by a space-dependent phase shift, and hence do not preserve the wavevector in the transmission process [11,19]. In addition, to achieve Eq. (4), ideally we will need an all-pass filter with unity amplitude transmission coefficient for all wavevector components.

In this work, we show that the transfer function of Eq. (4) can be achieved utilizing the phase response of Fano resonances [20–22]. We consider a single band of guided resonances in a two-dimensional (2D) photonic crystal slab. Assuming that the slab has mirror symmetry in the vertical direction ($z$ direction), the transmitted and reflected amplitudes’ near-resonant frequencies can be expressed as [23]

For such a system, energy conservation requires [23,24]

Denote dimensionless frequency detuning as

Now, we show that the band dispersion of guided resonances can be used to realize the quadratic phase response as described by Eq. (4). We consider a specific band of guided resonances near ${\boldsymbol k} = {\bf 0}$ with isotropic band dispersion,

where $\alpha$ describes the curvature of the band dispersion, and near-zero dispersion of linewidthsThen, Eq. (11) becomes

Assuming $\Omega (\omega ,{\boldsymbol k}) \ne \mp q$ for all relevant ${\boldsymbol k}$, we substitute Eq. (16) into Eq. (13) and perform a Taylor series expansion to obtain

Finally, we note that by choosing suitable $q$ and operating frequency $\omega \approx \pm 1/q$ correspondingly, in general, one can achieve near-unity transmission $|t(\Omega (\omega ,{\boldsymbol k}))| \approx 1$ for all the relevant ${\boldsymbol k}$. Therefore, the transfer function for monochromatic waves with frequency $\omega$ becomes

With a choice of small ${\gamma _0}$, (i.e., using guided resonances with high-quality factors), one can achieve a large ${d_{\text{eff}}}$ using a photonic crystal slab with a physical thickness that is far smaller than ${d_{\text{eff}}}$. Therefore, we have shown that the use of a properly designed photonic crystal slab can achieve the squeezing of free space.

As was noted in Ref. [10], in a typical photonic crystal slab, due to the photonic spin-orbit coupling, the band structure near ${\boldsymbol k} = {\bf 0}$ is not isotropic. However, one can use the specific design procedure as discussed in Ref. [10] to design a photonic crystal slab with an isotropic band structure of the form of Eq. (14). In the next section, we will follow the similar design procedure of Ref. [10] in our numerical design.

## 3. NUMERICAL DEMONSTRATION

Based on the theoretical considerations above, we provide two concrete designs of such a photonic crystal slab device. Our first device consists of three layers, as shown in Fig. 1(c). The middle layer is a photonic crystal slab with a lattice constant $a$. It has a thickness of $d = 0.55a$ and contains a square array of circular holes with radius $r = 0.111a$. Two homogeneous slabs with thickness ${d_s} = 0.07a$ are placed symmetrically besides the photonic crystal slab. The air gaps between the middle slab and the homogeneous slabs are ${d_g} = 0.94a$. We choose suitable $d$ and $r$ to achieve the isotropic band structure and ${d_g}$ and ${d_s}$ to realize the required background transmission ${t_d}$. The total thickness of such a device is ${d_T} = d + 2{d_s} + 2{d_g} = 2.57a$. All slabs are made of materials with a permittivity $\varepsilon = 12$, which approximates that of Si or GaAs in the infrared wavelength range.

Such a photonic crystal slab device hosts a pair of guided resonances that are doubly degenerate at the $\Gamma$ point (${\boldsymbol k} = {\bf 0}$) with the frequency ${\omega _0} = 0.47656 \times 2\pi\! c/a$. (Such a two-fold degeneracy is required in order for the guided resonance to couple to normally incident light [23].) As we mentioned above, in general, the band structure of the guided resonances is anisotropic around ${\boldsymbol k} = {\bf 0}$ for a photonic crystal slab with the ${C_{4v}}$ symmetry. However, with the geometry parameters chosen above, which are obtained following the same procedure as in Ref. [10], we show the corresponding band structure as obtained using the guided mode expansion method [26,27] in Fig. 2. Figure 2(a) shows that for each band the dispersion along the $\Gamma X$ coincides with that along the $\Gamma M$ direction. Figure 2(b) shows that the isofrequency contours for the upper band are almost circular. Both bands become almost completely isotropic:

where ${\alpha _1} = 1.79 ca/(2\pi\!)$, ${\alpha _2} = - 0.43 ca/(2\pi\!)$, and the 1,2 subscripts correspond to the upper and lower band, respectively [cf. Eq. (14)]. Also, we note that the dispersion of radiative linewidths ${\gamma _i}({\boldsymbol k})$ is anisotropic. Nonetheless, the dispersion of ${\gamma _i}({\boldsymbol k})$ is much smaller than that of ${\omega _i}({\boldsymbol k})$ so that its effect on transfer function is negligible: ${\gamma _i}({\boldsymbol k}) \approx {\gamma _0}$ [cf. Eq. (15)].As was noted in Ref. [10], the isotropic band structure leads to the remarkable effect of single-band excitation: $s\!$/$\!p$-polarized light only couples to the upper/lower band, respectively, for every direction of incidence [10]. Consequently, the $s$ or $p$ polarization is preserved upon transmission through the device (${t_{\textit{ps}}}(\omega ,{\boldsymbol k}) = 0$). We calculate the transmittance of $s$-polarized light ${t_{\textit{ss}}}(\omega ,{\boldsymbol k})$ by the Fourier modal method using a freely available software package [28]. Figures 2(c) and 2(d) depict the magnitude $|{t_{\textit{ss}}}|$ and phase $\text{arg}({t_{\textit{ss}}})$, respectively, at a general azimuthal angle $\phi = \arctan ({k_y}/{k_x}{\!) = 14^ \circ}$. Due to the isotropic band structure, the results are essentially the same for any other $\phi$. The plots clearly show that $s$-polarized light only excites the upper band. The transmission exhibits a sharp dip in magnitude and rapid variation in phase near the band dispersion of the guided resonances.

Figures 2(e) and 2(f) plot the spectra of the transmission magnitude and phase at incident angles $\theta {= 0^ \circ}$ and $\theta {= 1^ \circ}$, respectively. For both incident angles, both the amplitude and phase follows the Fano lineshape formula, as described by Eqs. (12) and (13). As $\theta$ increases, the resonance shifts to higher frequencies, consistent with the band structure. Based on Figs. 2(e) and 2(f), we choose the operating frequency at ${\omega _{\text{op}}} = 0.47640 \times 2\pi\! c\!/a$ as indicated by the green dashed line. At this frequency, the transmission coefficient magnitude stays close to unity, while the phase varies substantially for varying incident angles, as is required for Eq. (19). Also, we note that at this frequency the corresponding wavelength is greater than the lattice constant such that there is no diffraction due to the slab.

At the operating frequency ${\omega _{\text{op}}}$, Figs. 3(a) and 3(b) show the magnitude and phase of the transmission ${t_{\textit{ss}}}({k_x},{k_y})$ in the ${k_x}$ and ${k_y}$ plane. Figures 3(c) and 3(d) show the magnitude and phase of the transmission along the $\Gamma X$ direction. The transmission has a magnitude of unity over the entire wavevector range considered [Fig. 3(a)]. Hence, the structure behaves as an all-pass filter in this wavevector range. The phase is isotropic and shows a quadratic dependency of $|{\boldsymbol k}|$. The magnitude and phase response therefore agrees with Eq. (19).

By fitting the phase response with a quadratic function in $|{\boldsymbol k}|$, as shown in Fig. 3(d), we obtain the parameter $\alpha /[{\gamma _0}(1 + \Omega _0^2)] = 61.97 {a^2}$ for Eq. (19), thus the device is equivalent to free space of a thickness ${d_{\text{eff}}} = 371 a$ as determined from Eq. (20). Since the device has a thickness of ${d_T} = 2.57 a$, by replacing the free space with this device, we have preserved the angle-dependent phase response of the free space while reducing the required physical thickness by a compression ratio of ${d_{\text{eff}}}/{d_T} = 144$. Moreover, since the device behaves as an all-pass filter, a longer effective propagation distance can be achieved by simply cascading multiple devices together. A cascade of $N$ devices results in an effective propagating distance of $N{d_{\text{eff}}}$, whereas the compression ratio is unchanged from a single device.

In Fig. 4, we provide a numerical demonstration of the performance of our device [29]. We consider a converging azimuthally ($s$) polarized cylindrical vector beam [30] at the operating frequency ${\omega _{\text{op}}}$. Its intensity distribution at the $z = 0$ plane is shown in Fig. 4(a). For demonstration, we directly simulate a cascade of $N = 40$ devices with an air gap of thickness $0.5a$ between every two neighboring devices. The total thickness of the 40 cascaded devices including the air gaps is $123 a$. We numerically calculate the intensity distribution of the transmitted beam after the cascaded devices, as shown in Fig. 4(c). The beam size significantly reduces after passing through the devices. We also numerically verify that the result for 40 devices is indeed the same as that for a single device repeated 40 times. As comparison, in Fig. 4(d), we plot the intensity distribution of the original beam after the propagation in free space by $40 {d_{\text{eff}}} = 14823 a$. Figures 4(c) and 4(d) agree very well. In Fig. 4(b), we plot the radial profile for the input beam and the aforementioned two output beams. We see that our device can substitute free space propagation and reproduce the beam width and beam shape very well. The reduced peak intensity is caused by the deviation of the transfer function at large wavevectors [Fig. 3(d)].

Our first device above has $\text{NA} \approx 0.01$. It can be used to miniaturize low-NA optical systems. Such low-NA systems can provide unique functionalities that cannot be achieved by high-NA systems, but they are usually bulky. One example is telecentric lenses, which feature a constant field of view and have important applications in machine vision. They typically exhibit small $\text{NA} \approx 0.01$ comparable to our device. Our device can significantly reduce the size of these bulky low-NA systems.

Our method can be readily extended to higher-NA systems by choosing suitable design parameters. As a demonstration, we provide another design with a higher NA. Different from the previous one, this device is a single-layer hexagonal photonic crystal slab, as shown in Fig. 5. The slab is also made of materials with a permittivity $\varepsilon = 12$. It has a thickness of $d = 0.33a$ and contains a hexagonal array of circular holes with radius $r = 0.15a$, where $a$ is the lattice constant. Such a slab hosts a pair of guided resonances that are doubly degenerate at the $\Gamma$ point with the frequency ${\omega _0} = 0.47556 \times 2\pi\! c\!/a$. Interestingly, for such a photonic crystal slab with the ${C_{6v}}$ symmetry, it can be proved that the band structure is isotropic near the $\Gamma$ point. (See Supplement 1 for a proof.) We show the corresponding band structure in Figs. 5(b) and 5(c). Figure 5(b) shows that for each band the dispersion along the $\Gamma K$ direction coincides with that along the $\Gamma M$ direction. Figure 5(c) shows that the isofrequency contours for the lower band are almost circular. Again, the isotropic band structure leads to the effect of single-band excitation: $s\!$/$\!p$-polarized light only couples to the upper/lower band, respectively, for every direction of incidence. (See Supplement 1 for details). As the isotropic band structure is guaranteed by ${C_{6v}}$ symmetry, we choose suitable geometry parameters $d$ and $r$ to realize the required quality factor $Q$ and background transmission ${t_d}$. We consider the light of $P$ polarization at the operating frequency ${\omega _{\text{op}}} = 0.471 \times 2\pi\! c\!/a$. Figures 5(d) and 5(e) show the magnitude and phase of the transmission ${t_{\textit{pp}}}({k_x},{k_y}\!)$ in the ${k_x}$ and ${k_y}$ plane. Figure 5(f) shows the phase of the transmission along the $\Gamma K$ direction. The transmission has a magnitude of unity over the entire wavevector range considered. The phase is isotropic and exhibits a quadratic dependency of $|{\boldsymbol k}|$. From the fitting, we obtain the equivalent thickness ${d_{\text{eff}}} = 3.70a$ and a compression ratio ${d_{\text{eff}}}/d = 11.2$. The working wavevector range of this device is around 10 times larger than the previous one (cf. Fig. 3), which leads to a much larger $\text{NA} \approx 0.11$. (See Supplement 1 for a demonstration of the device performance.) The resonance has a lower quality factor $Q$ than the previous design. Low-$Q$ resonances can be preferable for applications that require a larger spectral bandwidth.

## 4. DISCUSSION AND CONCLUSION

We note that our device, like free space, is invariant under both translations along the longitudinal direction (i.e., the propagation direction or $z$ direction in Fig. 1) and transverse (i.e., within the $x-y$ plane) directions. Therefore, the device can be inserted anywhere along the optical path, and the effect is independent of the location of insertion. Our device works for both spatially coherent and incoherent optical waves. We also note that our device can work at different frequencies as well, though the equivalent propagation distance can be different. Correction of such a chromatic aberration is of interest for future research.

In the results of the paper above, for illustration purposes, we have only presented results for $s$- or $p$-polarized incident beams. Our device can also operate with beams with left/right circular polarizations. When operating with either $s$ or $p$ polarizations, the incident beam corresponds to a cylindrical vector beam. When operating with the circular polarizations, the incident beam needs not be restricted to a cylindrical vector beam but can be other types including Laguerre–Gaussian beams. In our example, the $s$ and $p$ polarizations have different compression ratios. The compression ratios for the left and right circular polarizations are the same and are between those of the $s$ and $p$ polarizations. The difference in the compression ratio between the $s$ and $p$ polarizations is caused by the different dispersions of the two bands. It is possible to achieve polarization-independent compression if one further designs the structure to enforce the two bands having the same effective mass, which will be explored in the future. (See Supplement 1 for a detailed discussion on the polarization response.)

In conclusion, we have shown that free space can be substituted with nonlocal flat optics. We derive the general criteria for replacing free space with an optical device and provide two concrete designs of photonic crystal slab devices. Such a device can substitute much thicker free space. Our work provides an important complement to local flat optics and may prove important for miniaturization of optical systems.

## Funding

Air Force Office of Scientific Research (FA9550-17-1-0002); U.S. Department of Defense (N00014-17-1-3030).

## Acknowledgment

We acknowledge Dr. Meng Xiao for helpful discussions. This work is supported by the U. S. Air Force Office of Scientific Research (AFOSR) MURI and by the Vannevar Bush Faculty Fellowship from the U. S. Department of Defense. While we were finalizing this paper, a similar recent preprint was brought to our attention (Ref. [31]). Our design here may result in a more compact design as compared with the multilayer configuration shown in Ref. [31].

## Disclosures

The authors declare no conflicts of interest.

See Supplement 1 for supporting content.

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