Chapter 2: Polarization and Phase Control
2.2 Simultaneous polarization and phase control
As seen in Fig.2.2a, the metasurface is composed ofα-Si nano-posts with rectangular cross-sections on a low-index fused silica substrate. With a proper choice of theα-Si layer thickness and lattice constant (650 nm and 480 nm respectively for an operating wavelength of 850 nm), the nano-posts can provide full and independent 2π phase control overx- andy-polarized light, wherex- andyare aligned with the nano-post axes (see Fig. 2.A1) [14]. Using the phase versus dimension graphs, one could calculate the nano-post dimensions required to provide a specific pair of phase values, φxandφy, as shown in Fig.2.2b. This allows for designing a metasurface that controls
xandy-polarized light independently. With a simple generalization, the same can be applied to any two orthogonal linear polarizations using nano-posts that are rotated around their optical axis with the correct angle to match the new linear polarizations (e.g., the x0− y0axis in Fig.2.2c). An important and interesting point demonstrated in [14] is that this can be done on a point-by-point manner, where the polarization basis is different for each nano-post. This property allows us to easily design the metasurface PBS for the two linear bases of interest. Moreover, as demonstrated in [14], an even more interesting property of this seemingly simple structure is that
the independent control of orthogonal polarizations can be generalized to elliptical and circular polarizations as well (with a small drawback that the output and input polarizations will have the opposite handedness). To see this, here we reiterate the results presented in the supplementary material of [14], as it is important to make the design process clear.
The operation of a nano-post can be modeled by a Jones [179] matrix relating the input and output electric fields (i.e.,Eout =TEin). For the rotated nano-post shown in Fig.2.2c, the Jones matrix can be written as:
T= " Tx x Txy Tyx Ty y # =R(θ) " eiφx0 0 0 eiφy0 # R(−θ), (2.1) where R(θ) denotes the rotation matrix by an angle θ in the counter-clockwise direction. Here we have assumed a unity transmission since the nano-posts are highly transmissive. We note here that the right hand side of Equation2.1is a unitary and symmetric matrix. Using only these two conditions (i.e., unitarity and symmetry), we findTxy =Tyx, |Tyx| =
p
1− |Tx x|2, andTy y = −exp(i2∠Tyx)Tx x. As one could
expect, these reduce the available number of controllable parameters to three (|Tx x|, ∠Tx x, and∠Tyx), corresponding to the three available physical parameters (φx0,φy0,
andθ). Using these relations to simplifyEout =TEin, we can rewrite it to find the Jones matrix elements in terms of the input and output fields:
" Exout∗ Eouty ∗ Exin Eyin # " Tx x Tyx # = " Exin∗ Exout # , (2.2)
where∗denotes complex conjugation. Equation2.2is important as it shows how one can find the Jones matrix required to transform any input field with a given phase and polarization, to any desired output field with a different phase and polarization. This is the first application of the birefringent meta-atoms, i.e.,complete and independent polarization and phase control. This capability is what allows us to design and implement the vectorial holograms discussed in the following sections.
The Jones matrix is uniquely determined by Eq.2.2, unless the determinant of the coefficients matrix on the left hand side is zero. In this case, the matrix rows (i.e., Eout∗ and Ein) will be proportional. Since the Jones matrix is unitary (i.e., the input and output powers are equal), the proportionality coefficient must have a unit amplitude: Eout∗ = exp(iφ)Ein. This equation means thatEoutandEinhave the same polarization ellipse, but an opposite handedness. Now, this input/output field set imposes only one equation on the Jones matrix elements. To uniquely determine
the Jones matrix, a second equation is required. To get this second equation, we use a second set of input/output fields that satisfy the same condition as the first set: Eout2 ∗ = exp(iφ2)Ein2. Here we are using the numeral subscripts to distinguish between the two input/output field sets. This way, the equation for the first set becomesEout1 ∗ =exp(iφ1)Ein1. Ifφ1andφ2can be independently controlled, one can
see using a conservation of energy argument thatEin1 andEin2 (as well asEout1 and Eout2 ) should be orthogonal to each other. Thus, we can write the final equation as:
" E1in,x E1in,y E2in,x E2in,y # " Tx x Tyx # = " E1out,x E2out,x # = " exp(iφ1)E1in,x∗ exp(iφ2)E2in,x∗ #