# Metasurface Transmittance

The 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*and a wavelength-dependent refractive index n(w), a diffraction grating with transmittance

_{w}(x, y)*t*and a focusing lens with transmittance t

_{g}(x, y),_{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, paraxial-wave 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*and

_{0}*w*are constants. The corresponding transmittance function is given by:

_{0}**Figure 5.14 **An equivalent setup for mapping a temporal frequency *w* to a specific point on the output plane *w*to *[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 z-axis:

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 they-axis 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 free-space 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 frequency-to-space 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 x-axis 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- to-space 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 frequency-to-space 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.