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Metasurface TransmittanceThe complex transmittance function t_{m}(x, y; w) of a metasurface phaser decomposing the frequency contents of an incident electromagnetic pulsed wave can be established by considering the arrangement of various dispersion elements in Fig. 5.14. It consists of a series cascade of a thin linear wedge with transmittance t_{w}(x, y) and a wavelengthdependent refractive index n(w), a diffraction grating with transmittance t_{g}(x, y), and a focusing lens with transmittance t_{1}(x, y) and focal length d. The thin wedge and the lens are placed directly after the diffraction grating, and any refractions within the wedge are assumed to be negligible. Also, paraxialwave propagation is assumed for simplifying the analytic expressions. It will be shown that the cascade of these three elements is functionally equivalent to the metasurface phaser of Fig. 5.13. Figure 5.13 A metasurface as a spatial phaser spectrally decomposing a broadband pulse y(x, y, z; t) along two dimensions, where each frequency u is mapped to a specific coordinate point [x_{0}(ai), y_{0}(w)] on the image plane. Let us first consider a thin wedge of thickness d(x) = a_{0}(x + w_{0}), where a_{0} and w_{0} are constants. The corresponding transmittance function is given by: Figure 5.14 An equivalent setup for mapping a temporal frequency w to a specific point on the output plane wto [x_{0}(w], y_{0}(w)]. Next, the complex transmittance functions of a thin sinusoidal diffraction grating are given by (5.5):
Consider a plane wave propagating along the zaxis:
so that y(x,y, 0_{}; t) = y(x,y, 0_{}) = 1, where the explicit time dependence e^{jwt} is dropped for convenience. The wave just after the grating can then be expressed as:
The first term inside the brackets on the RH side of (5.21) corresponds to the 0^{th} diffraction order, whose location in the output plane is not sensitive to w along theyaxis of the grating [Goodman (2004)], and thus will be ignored. Considering the lens transmittance function t_{1}(x, y; w), wave output after the lens is given by
which, after freespace propagation, leads to the field intensity:
where only one of the diffraction orders d(y  2nZcd/W) is kept due to symmetry. This equation finally gives the frequencytospace mapping relation of this system where each frequency w is mapped onto a specific point (x_{0}, y_{0}) on the output plane according to the relation:
Equation 5.24b is the conventional 1D scanning achieved using a diffraction grating only, i.e., (5.11). On the other hand, the frequency scanning along the xaxis directly depends on the refractive index, n(w), of the wedge. Therefore, a precise control over the refractive index of the wedge and the spatial period of the diffraction grating provides a mechanism to achieve the specified unique frequency tospace decomposition between frequency and space, thereby enabling the sought 2D spectral decomposition in the output plane. The setup of Fig. 5.14 consisting of a thin dispersive wedge, a diffraction grating, and a focusing lens may be replaced by a single metasurface with a complex transmittance function
where the diffraction grating transmittance function is written in complex exponential form to avoid symmetric diffraction orders. Such a metasurface will focus on the spectral contents of a pulsed wave following the frequencytospace mapping of (5.24) at the focal plane located at z = d, by directly operating on the input field as:
where F_{t}() and F_{t}^{_1}() are the Fourier transform and inverse Fourier transform operators in the time domain. This finally completes the synthesis of the metasurface phaser for 2D spectrum analysis. 
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