Nonlinear analog processing with anisotropic nonlinear films

2.1 Unpatterned flat optics with NN response

To elucidate the potential of combining nonlinearity and nonlocality, we discuss the basic scenario of a nonlinear thin anisotropic film for analog image processing [24]. The geometry in Figure 2 refers to an unpatterned thin film made of a material with second-order nonlinear susceptibility, sufficiently large that we can neglect higher-order nonlinear processes. We consider a two-dimensional problem, with translational invariance of the fields and optical properties along the y direction. In addition, we assume that the nonlinear film has a thickness t smaller than the wavelength of the excitation fields t ≪ λ 0 and it is located at z = 0, so that the system can be approximated by a 2D sheet. The nonlinear response of such thin film is mainly dictated by the structure of the second-order nonlinear susceptibility tensor χ 2 , and thus by the chosen material and its crystallographic orientation. Previous works [24] have considered materials for which the only nonzero component of the χ 2 tensor is χ zzz 2 . Here, we consider a broader and more realistic class of materials, characterized by a nonlinear susceptibility tensor for which only the elements χ ijk 2 with i ≠ j ≠ k are nonzero. This is the case of zincblende-type crystals, like gallium arsenide (GaAs) (assuming that the Cartesian axis are oriented along the [100], [010] and [001] axis of the crystal), a material platform with large quadratic nonlinearity that has garnered interest in the field of nonlinear photonics for the development of nonlinear metasurfaces for efficient harmonic generation [25], [26], [27], [28]. The film is excited by two coherent fields with same frequency: a signal and a control field, both p-polarized (electric fields oscillating in the plane xz), so that the second-harmonic field is y-polarized. Hence, second-harmonic generation is mediated by the χ yxz 2 and χ yzx 2 elements of the nonlinear susceptibility tensor. In this illustrative example, the control field C ω r takes the form of a Gaussian beam with a large diameter (≫λ ₀) impinging at oblique incidence with angle θ _c , imparting a linear phase profile across the flat-optics element. The signal field is a small-divergent and normally incident image modulated in amplitude along the x direction by an arbitrary function s x . For illustrative purposes, we consider a signal made by two adjacent Gaussian profiles. The signal and the control fields on the flat-optics element can be written as

where k ̂ S = z ̂ , k ̂ C ⋅ x ̂ = sin θ C , and θ ̂ c = cos θ C x ̂ − sin θ C z ̂ . Importantly, the signal field carries a component along the longitudinal direction z, associated with the field gradients introduced by the spatial modulation s x . Indeed, by applying Gauss’s law in the absence of charges [29], and adopting the small-divergence approximation (k _z ≈ k ₀), we find S z ω ≈ − i S ω k 0 ∂ s x ∂ x . Thus, the z-component of the signal field is proportional to the first derivative of s x . The nonlinear film mixes signal and control fields, resulting in a second-harmonic polarization current, in the plane z = 0, given by

which in turn generates a y-polarized second-harmonic field E ^2ω = G _yy P ^2ω , where G y y = 2 π i ε 0 λ 0 is the Green’s function of free space. The second-harmonic field contains four terms,

which correspond to different nonlinear mixing products of the signal and control fields. Starting from this general result, we can now analyze different special scenarios: first, let us consider the case in which the control field is small or absent (i.e., C ω ≪ S ω ), as in Figure 2a. The first term in Eq. (4), proportional to s x S ω S z ω , is dominant in this case, and it is the only one that emerges perpendicularly with respect to the film. The corresponding SHG field, emerging from the right side (green pattern), contains four peaks, corresponding to the points where the two input Gaussian peaks have the largest slopes. Since the SHG field amplitude is proportional to the signal gradients, this operation can be used to realize a nonlinear form of edge detection on the input function s x .

Figure 2:

Functionalities of a nonlinear nonlocal (NN) flat-optics element made of a thin film of gallium arsenide. The system is illuminated by two coherent beams: a control (C) and a signal (S), reported in orange. (a) When the amplitude of the control is much smaller than the amplitude of the signal, nonlinear edge detection is performed at the SH (the green beam transmitted is the “edge” of the signal, i.e., it is proportional to the derivative of the signal). (b) When C ω = 0.1 S ω , nonlinear beam deflection is dominant. (c) When C ω = 0.1 S ω , the generated SH signal performs simultaneous nonlinear edge detection and nonlinear beam deflection of the signal. The deflection angle is determined by the angle of incidence of the control signal.

We emphasize that the edge detection obtained here (Figure 2a) is not only based on a different mathematical operation, nonlinear in nature, than in the case of Fourier optical elements, but it is also based on a fundamentally different phenomenon than the one obtained in linear metasurfaces [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]. Here, edge detection is obtained by exploiting the anisotropy of the χ 2 tensor of GaAs and the corresponding polarization- and angle-selective response with respect to free-space radiations. In the case of GaAs, the SH field is proportional to the longitudinal component of the input field, which in turn carries information about the spatial derivative of the image. This is in contrast to linear approaches to edge detection, which rely on Fourier filtering performed by linear metasurfaces [3], [5], [12], [14], [19]. In those systems, a dispersive resonance is engineered in order to induce a strong variation of the linear transmission with angle of incidence, which then leads to a Laplacian-type operator in the Fourier domain. Because of this mechanism and the need for high-Q resonances, these linear devices are inherently narrowband. In contrast, the output SH field generated by the NN response of the thin GaAs film contains the product of the signal by its first derivative – a nonlinear operation, which cannot be obtained with any linear system. In addition, the approach demonstrated here (Eqs. (1)–(4)) does not require a resonance or tailored dispersion, and thus it inherently benefits from a broad operational spectral bandwidth. The bandwidth is only limited by the possible onset of absorption inside the thin layer for photons with energy above the material bandgap, as we report experimentally in the next section.

The second and third terms in Eq. (4) are cross-terms produced by the spatial-frequency mixing of control and signal fields. They form a beam that propagates in the direction θ C S = asin sin θ C 2 . In the small divergence approximation k z ≈ k 0 ≫ ∂ ∂ x , the main contribution contained in this beam is the third term in Eq. (4), proportional to − s x S ω C ω ⁡ sin θ c e − i k 0 ⁡ sin θ c x . This term is a scaled version of the input signal deflected toward θ _CS . Importantly, the deflection angle θ _CS is solely determined by the incident angle of the control signal, whereas the amplitude of the control signal may be tuned to make the beam-deflection effect dominant. For example, in Figure 2b, a nonlinear beam deflection can be clearly seen for C ω = 0.1 S ω .

Another interesting scenario is displayed in Figure 2c for the case C ω = 0.01 S ω . Here, the nonlinear edge-detection in the normal direction and the deflection of the input image at the angle θ _CS happen simultaneously. Such a regime may be of interest for applications where simultaneous access to the unprocessed and processed images is required. The fourth term in Eq. (4) is transversely phase-matched to the control plane wave, and therefore, it is the only one that emerges at the angle θ _C . This term is not particularly interesting, and it becomes dominant only when C ω ≫ S ω .

The examples discussed in Figure 2 showcase various intriguing capabilities of NN systems, even in the most basic scenario of an unpatterned thin film: (i) image processing functionalities that typically require patterned metasurfaces in the linear regime, such as beam deflection (or anomalous refraction) and edge detection, may be achieved by leveraging the nonlinear and nonlocal response of unpatterned thin films; (ii) the response of an NN system is intensity and control-beam dependent, hence tunable – in the example above one can switch the functionality from beam defection to edge detection by varying the ratio of signal and control amplitudes; (iii) the resulting operations are inherently nonlinear – for instance, edge detection here is nonlinear and fundamentally different from the Laplacian produced by a linear optical system, as it results from the multiplication of two fields, specifically, the signal itself multiplied by its derivative; (iv) since this approach does not rely on a resonance, it is also free from fundamental constraints on the operational spectral bandwidth, as it happens instead for linear devices based on resonances or tailored dispersion; (v) more complex nonlinear functionalities may be realized by leveraging the control function as a means to write functionalities – while in the reported example, the control is only used to achieve nonlinear beam deflection, for more sophisticated functionalities, such as nonlinear holography, the control may be a structured beam with arbitrary amplitude and phase profile; (vi) by locally structuring the metasurface, we may induce engineered resonances that trade bandwidth for efficiency, and we may engineer the nonlocality of the χ 2 tensor, opening the design space to more complex nonlinear operators.

2.2 Nonlinear edge detection in nonlinear unpatterned films

Edge detection with NN flat-optics has been theoretically proposed [24] based on a thin layer of nonlinear material with susceptibility tensor featuring only the χ zzz 2 element. This type of response can be achieved, for example, at interfaces [30], in nanolaminate films [31], and in quantum well systems [32]. Here, based on the general formalism described above, we demonstrate theoretically and experimentally that nonlinear edge detection is not limited to materials for which the nonlinear permittivity is dominated by χ zzz 2 . By contrast, we show that broadband edge detection arises in unpatterned flat-optics with any type of nonlinear susceptibility tensor that involves at least one longitudinal component of the input pump field. Building upon the example outlined in the previous section, we consider a NN system comprising an unpatterned thin film of (001) GaAs (Figure 3a). For nonlinear edge detection (Figure 2a), the control signal is not necessary. To understand the mechanism of nonlinear edge detection in the GaAs flat-optics system, we assume that the input image, carried by a wave at the fundamental frequency (FF), is projected on the nonlinear film. Using the same approach outlined in the previous section, the FF pump can be written as: E ω = E ⊥ ω + E z ω z ̂ , where the longitudinal component is E z ω ≈ i k ∂ E x ω ∂ x + ∂ E y ω ∂ y [29]. It follows that any nonlinear flat-optics system that is sensitive to E z ω – i.e., which can generate a nonlinear polarization when a z-polarized pump field is present – will produce a nonlinear signal proportional to the spatial derivative of the pump field. One of these systems is the unpatterned GaAs film considered here. Indeed, owing to its zincblende crystal structure, the nonlinear susceptibility tensor of GaAs obeys t − 1 χ 2 ≡ χ xyz 2 = χ yxz 2 = χ zxy 2 = χ xzy 2 = χ yxz 2 = χ zyz 2 ≠ 0 , while all the other components vanish [31].

Figure 3:

Sample and preliminary characterization. (a) Schematic of the sample and measurement. A thin layer of GaAs is bonded on a quartz substrate. The axes x _s , y _s , and z _s are aligned to the [100], [010], and [001] crystal directions of the sample, respectively. The sample is excited by a pump beam (orange arrow) at the fundamental frequency (FF) impinging along a direction defined by the polar (θ) and azimuthal (ϕ) angle, and with either s or p polarization. The second-harmonic emission (green arrow) generated along a direction parallel to the pump is collected on the opposite side of the sample. (b–c) Measured SHG efficiency (arbitrary units) for pump wavelength λ _FF = 1,550 nm, versus θ and ϕ, for a p-polarized (panel b) and s-polarized (panel c) pump. (d) Measured SHG efficiency versus θ, for pump wavelength λ _FF = 1,550 nm and ϕ = 45°. (e) Measured SHG efficiency versus pump wavelength for θ = 40°, ϕ = 45° and p-polarized pump.

Using the Green’s function approach [24], [33], we can retrieve the relationship between the transverse part of the second-harmonic (SH) field and the transverse part of the FF field. The SH field, evaluated at the sheet position, reads

The expressions in Eqs. (5) and (6), derived in the Methods, unveil the nonlinear edge detection operation of the NN film. All the terms in the SH fields are proportional to the first-order spatial derivatives of the pump signal multiplied by a component of the signal itself, confirming the nonlinear high-pass filtering operation illustrated in the previous section. An interesting feature of the nonlinear edge detection operation is the role of the polarization state of the pump, which may be used as a control knob to highlight selectively edges with different orientation. To show this, let us first assume that the pump signal is linearly polarized along x, so that the transverse FF signal is E ⊥ ω = E x ω x , y x ̂ . Since E y ω = 0 , from Eqs. (5) and (6), it follows that the resulting SH field is y-polarized and equal to E y 2 ω = − χ 2 E x ω ∂ E x ω ∂ x . Therefore, large SH signals will be generated by features of the input image with nonzero values of the spatial derivative ∂ E x ω ∂ x ; such scenario is obtained, for example, when the input image contains edges that are parallel to y. Instead, edges that are parallel to x will lead to nonzero values of only the derivative ∂ E x ω ∂ y , which in this case does not contribute to the SHG efficiency. Conversely, when the pump signal is y-polarized, the SH field is x-polarized and equal to E x 2 ω = − χ 2 E y ω ∂ E y ω ∂ y . In this scenario, only horizontal edges (parallel to x) will give rise to high SH signals. For circular polarization E ⊥ ω = E 0 x , y x ̂ ± i y ̂ / 2 , with the plus (minus) sign indicating right (left) handedness, the SH field is

Therefore, all edges of the FF image are uniformly detected in the generated output image. It is important to emphasize once again the relevant differences between the results obtained here and the edge detection obtained in linear metasurfaces by engineering dispersive optical modes. First, the nonlinear spatial differentiation in Eqs. (7) and (8) (i.e., under circularly polarized pump) is isotropic with respect to azimuthal rotations of the input image – i.e., if the input image is rotated within the xy plane, the output image will be rotated as well, without any image distortion (see also Supplementary Information, Section S4). By contrast, edge detection obtained via Fourier filtering in linear metasurfaces often suffers from poor isotropy [10], [12], due to the fact that a metasurface made of periodic patterns cannot be invariant under arbitrary rotations. The most isotropic responses are obtained with metasurfaces with C₆ rotational symmetry [14], within moderately large ranges of impinging polar angles. Yet, high anisotropies are inevitable for larger impinging angles [14]. Second, the output fields in Eqs. (5)–(8) are proportional to the first-order derivative of the input field. First-order (or, in general, odd-order) differentiation is challenging to achieve in linear patterned metasurfaces, because it requires a unit cell that simultaneously breaks vertical and horizontal symmetry [5], introducing significant fabrication challenges. Instead, the proposed NN thin film provides first-order differentiation naturally, without requiring any symmetry breaking and challenging fabrication processes. Finally, while in the experimental results discussed later the practical effect of the mathematical operations in Eqs. (5)–(8) is to enhance the image edges, the underlying mathematical operation is very different from the linear differential operations demonstrated with linear meta-optic devices [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]. Here, we induce a nonlinear operation (multiplication of an image by its gradient), which is not achievable with any linear device.

We experimentally demonstrate this nonlinear image processing operation based on the sample shown in Figure 3a, consisting of a 480-nm-thick flat slab of (001) GaAs bonded on a thick quartz substrate. To verify the capability of this thin, unpatterned slab to provide the desired nonlinear image processing operation, we first measured how the SHG efficiency depends on the pump direction, polarization, and wavelength. The sample was illuminated with a quasi-plane-wave pump excitation at the fundamental frequency (wavelength λ _FF ∈ [1,400, 1,750] nm), provided by a tunable pulsed laser (pulse duration 2 ps, repetition rate 80 MHz). The generated SH signal was collected along the same direction as the pump (see Methods for details on the setup). No polarization selection was performed on the collected SH signal. In Figure 3b and c, we show the measured SHG efficiency as a function of the polar (θ) and azimuthal (ϕ) angles of the pump direction (see Figure 3a for the definition of the angles), for a fixed pump wavelength λ _FF = 1,550 nm and for p-polarization (Figure 3b) and s-polarization (Figure 3c) (see Section 4 for additional details on the measurement protocol). The measured efficiency follows the expected patterns for both polarizations (see Supplementary Information, Section S3, for a comparison between these experiments and numerically calculated data). Figure 3d shows a sliced 1D data for ϕ = 45° for both polarizations. The measurements confirm that pumps propagating at normal incidence (θ = 0°) – which in the imaging experiment carry the information on the DC components of the input image – do not lead to any SH generation. As the excitation polar angle θ increases, the SH signal increases monotonically. For s polarization, no SH emission is observed when ϕ = nπ/2, as expected from the symmetry of the χ 2 tensor. For p-polarization, instead, the SH emission is nonzero for any azimuthal angle, albeit with some large modulation.

Importantly, the SHG efficiency depends weakly on the pump wavelength, as shown in Figure 3e. Here, we measured the SHG efficiency versus pump wavelength for fixed θ = 40°, ϕ = 45° and p-polarized pump. The SHG efficiency remains above 50 % of its peak value for any λ _FF ∈ [1,400, 1,750] nm. The weak modulation of SHG efficiency as a function of the pump wavelength is due to a combination of the frequency dependence of the χ 2 of GaAs [34], [35] and the weak Fabry–Perot resonances created by the finite slab, as confirmed by numerical simulations (see Supplementary Information). The decrease in efficiency for λ _FF < 1,450 nm is attributed to the onset of absorption of the corresponding SH signal in GaAs. In the Supplementary Information (Section S3), we show numerically calculated data corresponding to the measurements in Figure 3b–d, which display excellent agreement with the experiments.

After having verified that our sample can provide the expected nonlinear transfer function, we tested its behavior as a flat optics for nonlinear image processing by using the setup schematized in Figure 4 and discussed in more detail in the Methods. The pump signal was provided by the same tunable pulsed laser used for the measurements in Figure 3. The power and polarization states of the pump were controlled by a series of polarization optics. The collimated laser was then used to illuminate a mask (labeled “target” in Figure 4), which contains different test images. The typical average power on the target was of the order of 1 W, corresponding to a pulse energy of the order of ∼10 nJ. The image created by the target was relayed onto the GaAs slab with a projection system (red-shaded box) formed by the lens L _p and the objective O _p (20×, NA = 0.4). The projection system shrinks the image size by a factor of approximately 10×, thus increasing the local pump intensity. A long pass filter (LPF) inside the projection system rejected any SH signal generated by the material forming the mask. The image at the fundamental frequency impinged on the GaAs slab from one side, and the signal emerging from the other side of the slab (containing both the FF and the SH images) was collected by another objective (O _c , 50×, NA = 0.42). This signal was then redirected either to a NIR camera to record the FF image or to a visible camera to record the SH image. In the second case, the strong FF signal was filtered out by two short-pass filters (SPFs). In this experiment, the GaAs slab was oriented such that the [110] direction of GaAs was parallel to the y-direction of the lab reference frame (defined in Figure 4), while the [−110] direction of GaAs was parallel to the x-direction defined in Figure 4.

Figure 4:

Setup for nonlinear image processing measurements. A pulsed laser with wavelength λ _FF is prepared in a certain polarization state and then used to illuminate a mask (“Target”) to create an optical image. A projection system formed by a lens (L _p ) and an objective (O _p , 20×, NA = 0.4) relays the image on the GaAs slab. A long pass filter (LPF) inside the projection system rejects any SH signal generated by the target material. The signal propagating on the other side of the GaAs slab (which contains both the FF and the SHG image) is collected by a second objective (O _c , 50×, NA = 0.42) and redirected either to a near-infrared camera or to a visible camera. LP = linear polarizer, QWP = quarter wave plate, LPF = long pass filter, SPFs = short pass filters.

In Figure 5, we show the experimental results with input images at a pump wavelength λ _FF = 1,550 nm. In this figure, the pump electric field is circularly polarized, which, based on Eqs. (7) and (8), ensures the enhancement of all edges, independently of their orientation. Here and in all following results, the spatial coordinates on the axis of all figures correspond to calibrated spatial dimensions measured in the plane of the GaAs slab. In Figure 5a and b, the input image (Figure 5a) is a circle with a diameter of 100 µm. The corresponding SHG image (Figure 5b) shows the expected edge-detection functionality: the SH signal is strongly enhanced along the edges of the input images, while it is almost zero in the spatial regions where the intensity of the input image is constant. Similar results are obtained with a different input image, consisting of the logo of one of our institutions (Figure 5c and d).

Figure 5:

Experimental nonlinear image processing. Nonlinear image processing of different input images carried by a circularly polarized pump with λ _FF = 1,550 nm. (a–b) The input image (panel a) is a circle with a diameter of approximately 100 µm. The image recorded at the SH wavelength (panel b) displays the expected signal processing, i.e., an isotropic enhancement of the edges. (c–d) Same as in panels a–b, but with an input image made of the logo of one of our institutions.

Next, we investigate the dependence of the features of the output image on the polarization of the input image. To this aim, we consider input images with well-defined edge orientations, such as squares and rectangles. In Figure 6a–e, we consider a fixed input FF image formed by a square with a width of about 50 µm (Figure 6a). We recorded the generated SH image for different input polarizations: horizontal (Figure 6b), vertical (Figure 6c), right circularly polarized (Figure 6d), and left circularly polarized (Figure 6e). As expected from the discussion above, when the input signal is linearly polarized (Figure 6b and c), the edges of the input image that are oriented orthogonal to the input polarization are maximally enhanced in the output images, while edges that are parallel to the input polarization are absent in the output images. Instead, when using circular polarization (with either chirality), all edges are uniformly enhanced (Figure 6d and e). The results in Figure 6a–e were obtained with a “positive” image, i.e., a shape whose internal area has a large intensity, while the external area has zero intensity. We have also verified that the same image processing and polarization-dependent behavior is obtained with “negative” images, i.e., images composed of dark shapes surrounded by a high-intensity background (Figure 6f–j).

Figure 6:

Dependence on input polarization of nonlinear image processing. (a–e) An input image (panel a) containing a “positive” square (i.e., high intensity inside the square) is processed by the GaAs slab. We recorded the generated SH image for different polarization of the input image: horizontal (panel b), vertical (panel c), right circularly polarized (panel d), and left circularly polarized (panel e). (f–j) Same as in panels (a–e), but for the case in which the input image consists of three “negative” rectangles (i.e., zero intensity inside the rectangles).

Finally, we show that our sample can perform nonlinear image processing over an extremely broad range of input frequencies. In Figure 7, we used the same input image as in Figure 6f–j, and we fixed the input polarization to circular. We varied the input wavelength from λ _FF = 1,420 nm to λ _FF = 1,680 nm and recorded the corresponding SH output images. As shown by the different panels in Figure 7, all output images display well-defined edge detection. The output image is essentially independent of the FF wavelength (indicated in the bottom-right corner of each panel of Figure 7). To quantitatively compare the output intensities and the edge detection efficiency at different input wavelengths, all SHG images in Figure 7 were acquired in the same experimental conditions, and the pixel intensities of each measurement have been normalized by the square of the corresponding FF power. Furthermore, to aid the comparison between different measurements, we normalized all data with respect to the highest peak intensity among all panels in Figure 7. With this procedure, the intensities of the different output images in Figure 7 can be readily compared by looking at the upper bound of the corresponding colorbar. As expected from the efficiency curve in Figure 3e, the highest peak intensity in Figure 7 occurs for λ _FF = 1,500 nm. Moreover, the variation of the peak intensity with input wavelength is fairly weak, and the peak intensity remains above 0.7 for λ _FF = 1,420 nm and above 0.4 for λ _FF = 1,680 nm. We note that the wavelength range considered in Figure 7 (1,420 nm–1,680 nm) was solely determined by the power available in our laser system, and we expect the operational bandwidth to be even larger, ultimately limited only by the increased absorption of the SH within the GaAs slab as the SH wavelength λ _FF/2 further decreases. In the Supplementary Information (Sections S1–S3), we discuss additional numerical calculations of the SHG efficiency of our GaAs slab versus pump wavelength.

Figure 7:

Nonlinear image processing versus input wavelength. The input image is the same as in Figure 5f, and the input polarization is circular. The seven panels in the figure show the recorded SHG image as the input wavelength was varied between 1,420 nm and 1,680 nm. The input wavelength is denoted in the left-bottom corner of each panel. In each measurement, the intensity recorded by the camera was normalized by the squared of the input power. Then, all data were further normalized with the respect to the maximum peak intensity, which occurs for λ _FF = 1,500 nm.

4.1 Second-harmonic generation from a thin sheet of GaAs

The nonlinear thin film is modeled as a flat-optics element, or sheet, with quadratic nonlinearity. The sheet is located at z = 0. Second-harmonic light emission from the sheet is due to of the interaction of its nonlinear susceptibility with the fundamental-frequency field (or pump field) at z = 0. Let’s analyze the quadratic nonlinear response associated with an arbitrary χ ℓ m n 2 element, where ℓ, m, and n indicate Cartesian coordinates. In the Fourier-space representation, the k-spectrum of the input pump field (at frequency ω) in the plane of the sheet (z = 0) is expressed as a two-dimensional Fourier transform, E ̃ i ω k ⊥ , 0 = ℑ E i ω r ⊥ , 0 , where i = x, y, z, r ⊥ = x , y is the direct space in-plane coordinate and k ⊥ = k x , k y is the spatial-frequency in-plane vector. If the nonlinear susceptibility χ ℓ m n 2 does not introduce angular dispersion, the k-spectrum of the second-harmonic field (at frequency 2ω) at any longitudinal position z can be calculated as a convolution of the pump signal field components [24], [33], which in Cartesian coordinates is written as

where the star symbol is the convolution operator and

indicates the spectral Green’s function. Here, k z = k 2 − k x 2 − k y 2 is the longitudinal component of the wavevector and k is the wavenumber. If the pump signal has a small spatial-frequency spectrum in the transverse plane (corresponding to an input image with features much larger than the free-space wavelength), then k _z ≈ k and the expression in Eq. (M1) simplifies to

In the case of GaAs, the only nonzero χ ℓ m n 2 elements are those for which ℓ ≠ m ≠ n, namely t − 1 χ 2 ≡ χ xyz 2 = χ yxz 2 = χ zxy 2 = χ xzy 2 = χ yxz 2 = χ zyz 2 ,where t ≪ λ ₀ is the thickness of the flat-optics element. Therefore, the transverse components of the transmitted second-harmonic field can be written in Fourier space as:

The transverse components of the second-harmonic field at z = 0⁺, i.e., in transmission, can be retrieved by anti-Fourier transforming the expressions in Eqs. (M4) and (M5):

where the anti-transformations ik _x → ∂/∂x and ik _y → ∂/∂y have been applied. Introducing the expression of the longitudinal field as a function of the derivative of the transverse field, E z ω ≈ i k ∂ E x ω ∂ x + ∂ E y ω ∂ y , in Eqs. (M6) and (M7) leads to the expressions reported in Eqs. (5)–(8).

4.2 Sample fabrication

A 480-nm-thick GaAs layer was epitaxially grown on a (001)-oriented GaAs wafer with stop-etch layers. The stop-etch layers consist of 120-nm-thick Al0.55Ga0.45As, 100-nm-thick GaAs, and 20-nm-thick Al0.55Ga0.45As layers, in that order. The wafer was diced to be 10 × 10 mm and flip-chip bonded onto a quartz substrate using epoxy (353ND, EPO-TEK). Then, the GaAs wafer was thinned to be ∼20 µm using a mechanical lapping process and removed by a wet-etching process using citric acid etchant (citric acid:H2O2 = 5:1) until the wet etch stops at the Al0.55Ga0.45As stop-etch layer. The stop-etch layers were then wet-etched using a phosphoric acid etchant (H3PO4:H2O:H2O2 = 20:200:4), and the GaAs spacer between them was etched using citric acid etchant. Finally, only the 480-nm-thick GaAs layer remained on the quartz substrate.

4.3 Optical measurements

The measurements shown in Figure 3 were performed with a custom-built setup, shown in Figure 8. The pump signal at the FF, ranging between 1,400 nm and 1,750 nm, was a pulsed laser (pulse duration τ = 2 ps, repetition rate f = 80 MHz) provided by the signal of an optical parametric oscillator (APE, Levante IR ps) pumped by an Yb laser (APE, Emerald Engine). The power and polarization of the pump laser were controlled by a half-wave plate (HWP) and a linear polarizer (LP) cascaded in sequence. In all measurements in Figure 3, the average power was kept fixed to about 400 mW (for all impinging angles and all wavelengths), corresponding to a pulse energy of about 5 nJ. The laser was then focused on the sample (from the quartz side) with a long-focal-length lens (focal length = 20 cm), resulting in a spot radius (defined as the distance from the center at which the intensity drops by a factor e ²) of approximately 90 µm. The signal emerging from the opposite side of the sample (containing both the FF and the SH signal) was collected by a second identical lens and recollimated. After removing the FF with a pair of short-pass filters (SPF), the SH signal was redirected either to a visible spectrometer (Ocean Optics, HR4000) or to a photodiode (Thorlabs, DET100A2). The power of the laser (before exciting the sample) was continuously monitored by an additional germanium photodiode (not shown in Figure 8).

Figure 8:

Setup used for preliminary measurements shown in Figure 3. See text for details.

To excite the sample with a pump with a well-defined s or p-polarization, we utilized the configuration shown in Figure 8. The reference frame of the lab (labeled as x, y, z, black arrows in Figure 8) is defined such that z is along the optical axis of the setup, and x and y correspond to the horizontal and vertical polarizations, which can be imparted by LP. We also introduce the reference frame of the sample (x _s , y _s , z _s , gray arrows), shown also in Figure 3, defined such that z _s is always perpendicular to the GaAs layer. The sample was mounted on two different rotation stages: a motorized one (Thorlabs, HDR50) to control the polar angle θ, defined as the angle between the optical axis (z) and z _s (see Figure 8); a manual rotation stage to control the azimuthal angle ϕ, defined as the in-plane rotation of the sample around the axis z _s (see also Figure 3a). The rotational geometry (Figure 8) is chosen such that, when θ = 0 and ϕ = 0, the reference frames of the lab (x, y, z) and of the sample (x _s , y _s , z _s ) coincide (apart from a trivial translation). Moreover, for any value of θ and ϕ, the vertical direction of the lab reference frame (y) is always parallel to the sample plane. Figure 8 shows the particular case of ϕ = 0, for which the directions y and y _s coincide. Thanks to this configuration, when the pump is polarized along the y direction of the lab frame (vertical polarization), the electric field of the pump always lies in the sample plane, for any value of θ and ϕ. This corresponds to s polarization. Conversely, when the pump is polarized along the x direction of the lab frame (horizontal polarization), the magnetic field of the pump is along y, and thus it will always lie in the sample plane, for any value of θ and ϕ. This corresponds to p polarization.

The measurements in Figure 3b–d were performed by keeping the FF wavelength fixed at λ _FF = 1,550 nm and by continuously varying the angles θ and ϕ of the sample while acquiring the SHG spectra with the spectrometer. The signal recorded by the spectrometer in a narrow region around λ _FF/2 was then used to retrieve the SHG efficiency. The procedure was repeated for two different orientations of the linear polarizer (LP), in order to set the pump polarization to either s or p as described above. The measurements in Figure 3e were performed by varying the pump wavelength in the 1,400 nm–1,750 nm range while keeping all other parameters fixed. The SHG signal produced by each pump was recorded by the photodiode, and the SHG efficiency (in arbitrary units) was obtained by dividing the voltage read by the photodiode by the square of the laser power (measured separately for each wavelength). The wavelength dependence of the photodiode efficiency (provided by the vendor) was used to correct the measured voltages.

The measurements in Figures 5–7 were acquired with the setup shown in Figure 4 and described in the main text. The laser source was the same as in the setup described in the previous paragraph.