The 2D RTSA is a perfect example of combining temporal dispersion of a phaser with the spatial dispersion of an LWA to achieve a larger frequency resolution in a test signal spectrum. With the engineered phase response of the phaser feeding network, the overall scanning path length in this system can be conveniently extended to achieve multiple cycles of frequency scanning in the f = 0° plane so as to maximally populate the (в, f) region of space. While this system requires inter-operability between guided-wave phasers and the spatial dispersion element (here a LWA), a purely spatial 2D RTSA can also be developed based on the principles described in the previous sections. Compared to the 2D RTSA where the test signal is fed at a single point in space, the 2D spatial RTSA decomposes a pulsed wave spectrally in space upon incidence on the dispersive element.

Conventional 2D Spectral Decomposition

A functional schematic of a conventional system for 2D spectral decomposition in a 2D Spatial RTSA is illustrated in Fig. 5.12, which employs two dispersive elements separated in space. Consider a broadband pulse:

where w_{0} is the carrier frequency, wavenumber k = 2p/1 = 2pw/c and Y(x , y; t) is the pulse envelope of spectral bandwidth Aw = (w_{start} - w_{stop}). At the input of the first dispersion element, y(x,y, 0_{-}; t) = Re{Y(x , y ; t)}, where the time dependence е^{1<а}° is dropped for convenience.

This system works in two stages. In the first stage, dispersion element #1 spectrally decomposes y(x,y, 0_{-}; t) along the x-axis first at the first image plane at z = z_{1}. In the second stage, the dispersion element #2 then decomposes the wave y(x, y, z_{1}; t) along the y-axis. The spectrally decomposed wave is then finally focused onto the focal plane of the lens, so that each frequency w (or wavelength A) of the input is focused at a unique point [x_{0}(w), y_{0}(w)] on the output plane. The first dispersion element is typically either an AWG or a VIPA, which includes multiple free-spectral ranges (FSR) A^Fsr within the operation bandwidth Aw of the system, so that x_{0}(w) = x_{0}(w + nA®p_{SR}], where n is an integer. This existence of multiple FSRs within the operating bandwidth leads to the so-called spectral shower, as shown in the right of Fig. 5.12. The total length of this system is Az = (z_{1} + z_{2} + d), which can, in principle, be reduced by placing the lens directly after the second dispersion element [Goodman (2004)], so that Az_{min} = (z_{1} + d).

The question one may ask is: Is it possible to realize a spectral shower using just a single dispersion element, so that the system length is reduced to Az = d, as illustrated in Fig. 5.13? This question is addressed here by proposing a metasurface as a single dispersive element that operates on the broadband signal y(x,y, 0_{-}, t) to produce a unique frequency-to-space decomposition between its frequency components w and spatial coordinates (x_{0}, y_{0}) in the output plane z = d, and thereby acting as a 2D spatial phaser.

Metasurfaces generally consist of non-uniform spatial arrays of subwavelength scattering particles, with sizes much smaller than the operating wavelengths and provide unprecedented flexibility in controlling wavefronts in space, time or both space and time [Kild- ishev et al. (2013); Holloway et al. (2012); Liu and Zhang (2013); Yu and Capasso (2014); Aieta et al. (2014); Minovich et al. (2015)]. The sought metasurface for 2D RTSA is characterized by a complex transmittance function, t_{m}(x, y; w), which directly operates on the input and produces a spectrally resolved output at the image plane z = d.

Figure 5.12 A conventional free-space optical system to spectrally decompose a broadband y(x, y, z; t) in two spatial dimensions, using two dispersive elements.