In this section, the focusing characteristics of the monochromatic meta-lens will be studied through its interaction with a broadband beam in the NIR region. The light source that is used for this purpose is the 14 fs OPCPA pulses, in the frequency range from 130 THz to 200 THz. To conduct the experiment, the meta-lens must be placed in the NIR beam path before the beam combining optic. The generated electric field is retrieved in the detection plane by the varying time delay between the NIR OPCPA laser source and the white-light sampling pulse.
The EOI measurement, demonstrating the electric field pattern in the detection plane, is illustrated in Fig. 4.3. The first row of images represents single field frames at time delays of 0 fs, -1.67 fs and -3.34 fs, which correspond to the maximum positive amplitude, zero crossing and a negative extremum of the electric field. The presence of the multi- ring, out-of-phase pattern around the central spot indicates the generation of some kind of aberration by the metasurface. This manifestation is to be expected, given that the meta- sample is designed for single-wavelength operation. The produced electric field structure becomes even more interesting, exposing its spatio-temporal distribution by fixing one spatial coordinate to zero, E(x = 0, y, t) (Fig. 4.3d). The incident spectral components are distributed throughout the detection plane, forming a complex electric field pattern in time and space, due to the multiple focal planes that correspond to different spectral components. As a result, only a few optical components are focused into the EOS crystal, which leads to a significant bandwidth reduction in the imaged plane. Indeed, analyzing the brightest pixel from Fig. 4.3d and revealing its temporal waveform, it is clear that the pulse duration is dramatically increased (Fig. 4.3e). The spectral intensity derived from this waveform is presented in Fig. 4.3f. The spectrum clearly demonstrates that only the frequencies from 160 THz to 180 THz are focused. This relatively narrow bandwidth
4.2 Extremely Chromatic Meta-lens 65
Figure 4.3: Characterization of the NIR field focused by the meta-lens with strong chro- matic aberration. Electric field at the time of (a) highest field strength, (b) zero crossing and (c) negative maximum. (d) Spatio-temporal electric field distribution in the plane E(x = 0, y, t). (e) Time-domain waveform at the point of the highest field strength E(x = 0, y = 0, t). (f ) Corresponding spectrum obtained by the Fourier transform of (e).
implies that the process of chromatic focusing leads to the stretching of the incident 14 fs pulse to about 48 fs, calculated as the FWHM from the corresponding temporal waveform. As discussed earlier, EOI gives full spatio-temporal information about the electric field in the detection plane, which contains the entire bandwidth of the incident beam. In our case, only a narrowband spectral region is focused, appearing in the image as the most intense part at the center. However, the other spectral components, although out of focus in the detection plane are also present in the image, and this information is available. Consequently, the amplitude and phase of each spectral component within the incident bandwidth can be extracted to implement numerical simulation of their linear propagation. Taking the Fourier transform of the obtained temporal waveforms in each pixel yields the complex spectrum in the frequency domain. The frequencies with known amplitude and phase are associated with the corresponding wave vectors k = nω/c. The 2D Fourier transform in space provides a decomposition of the light field into plane waves, and gives the transverse spatial frequencies and corresponding x and y components of each wave vector, i.e. kx and ky. As a result, each plane wave of the light field in the 3D Fourier-
transformed matrix has a frequency, phase, and orientation. Thus, to calculate the form of the light field in any other plane along z axis for the distance ∆z, a wave-vector-dependent phase shift ∆φ should be added
∆φ(k, kx, ky) = kd = k∆z q 1 − kx2+k2y k2 , (4.1)
followed by the inverse Fourier transform, where d = ∆z/cos(θ) is the effective propagation distance of the individual plane wave and θ is the angle between kz and k. With this
simple Fourier-optic procedure, it is possible to find an exact focal plane for each spectral component within the incident bandwidth through numerical shift of the crystal.
Such calculation is presented in Fig. 4.4, where red, green, and blue are mapped to 145 THz, 168 THz, and 189 THz. The first false-color image (Fig. 4.4a) clearly demonstrates the chromatic aberration produced by the meta-lens along the z axis, while the x coordinate is fixed to zero. The numerical calculation is done for the range of −4000 < z < 4000 µm from the original measurement (z = 0). The focal position for each color can be observed and even precisely determined. Accordingly, the x − y false-color images in the second row represent the beam foci for the mapped narrowband frequency ranges around 145 THz (Fig. 4.4b), 168 THz (Fig. 4.4c), 189 THz (Fig. 4.4d), with the exact distance from the original plane. As a result, the flat lens leads to the spatial separation of the incident low- and high-frequency components along the optical axis on the order of ∼ 2.4 mm.
To verify the feasibility of the Fourier-optic numerical displacement of the EOS crystal, one more measurement is carried out, when a physical shift of the crystal is done by ∼ 1.5 mm towards the metasurface lens. The experimental results are presented in Fig. 4.5. A single frame of the generated field pattern is depicted in the x − y image (Fig. 4.5a). The dynamical spatio-temporal image in Fig. 4.5b shows the modifying wavefront curvature, indicating that some spectral components are about to focus, others are strongly defocused.
4.2 Extremely Chromatic Meta-lens 67
Figure 4.4: Visualisation of the chromatic aberration created by the meta-lens. (a) False- color y − z image of different focus planes shifted numerically, where red, green, and blue are mapped to 145 THz, 168 THz, and 189 THz. (b)-(d) Cross-section images of the focus position for each mapped frequency.
The temporal evolution of the brightest pixel in Fig. 4.5c illustrates the broadening of the generated field in the new detection plane, with a pulse duration of 49 fs. In principle, shifting the detection crystal closer to the lens should make the low-frequency components dominate. The spectrum obtained by the Fourier transform of (c) naturally confirms it. Applying the numerical phase shift through the 3D Fourier transform and moving the detection plane along the z axis, false-color images can be obtained for the same frequency ranges: red, green, and blue represent 145 THz, 168 THz, and 189 THz (Fig. 4.5e-g). The physical crystal position is at z = 0. The focal plane of the red components is determined to be 100 µm away from the real crystal plane. The green components are about 1550 µm, which is in good agreement with the previous results. It has to be noted that the focal plane of the blue components cannot be reliably measured because this information is barely captured by the crystal. However, assuming agreement between this measurement and the one before (Fig. 4.3), an approximate focal plane can be numerically calculated.
It has been demonstrated that a single EOI measurement yields a cube of waveforms that can be exploited to calculate the light field form in any other plane along the propa- gation axis. The spatio-temporal distortions in the case where an incident beam contains a large bandwidth can be easily detected and potentially corrected to design a metasurface lens that maximally confines the light in both time and space.
Figure 4.5: Characterization of the NIR field focused by the meta-lens after moving the detection plane by ∼ 1.5 mm. Measured electric field (a) at the time of positive extremum and (b) along the y axis in dynamics. (c) The temporal evolution of the brightest pixel and (d) its corresponding spectrum. (e)-(g) Foci for each mapped frequency range (red, green, and blue represent 145 THz, 168 THz, and 189 THz), calculated numerically.