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Examining and explaining the “generalized laws of reflection and refraction” at metasurface gratings

Markus Schake

Open Access Open Access

Abstract

The widespread concept of “generalized laws of reflection and refraction” that is commonly applied to wave propagation through metasurfaces is thoroughly explained on the foundation of diffraction theory. This allows definition of strict constraints to the applicability of these generalized laws and highlights the underlying physical effects. A diffraction-based explanation of the reported phenomena is provided that yields a solid theoretical foundation for the prediction of experimental results and that clarifies many of the convoluted explanations found throughout the literature.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. INTRODUCTION

The use of metasurfaces in beam shaping applications is an active field of research [1–15]. The generalized laws of reflection and refraction proposed in [1] are a commonly applied tool to design the propagation behavior of metasurfaces and make predictions about their far field intensity distributions observed in experiments [1–15]. However, there is a problem with the derivation of the generalized laws of reflection and refraction in [1] because it implies a violation of the wave vector conservation at the metasurface interface. This is well illustrated by a quote from [4]: “Note that due to the lack of translational invariance along the interface the tangential wavevector of the incident photon is not conserved; the interface contributes an additional ‘phase matching’ term equal to the phase gradient, which allows one to control the direction of the reflected and refracted beams” [4] (p. 1703). This claim basically proposes the idea that the boundary conditions that apply for the propagation of electromagnetic waves through an interface between two media based on Maxwell’s equations do not apply at the metasurface interface, an idea that contradicts the classical understanding of wave propagation.

There have been critiques of the generalized laws of reflection and refraction [16,17] that discuss the relation of these laws to diffraction theory. However, some points as, for example, the relation between the grating phase profile and the position of the observed interference peaks requires a more detailed treatment to fully unravel their significance in the process.

The idea of the generalization of Snell’s law has inspired multiple works [5–15,18]. A 2019 review summary on metamaterials states: “These advances have led to a relaxation of the fundamental Snell’s law for light refraction […]” [5] (p. 1). Another review from 2017 [19] features a mixture of diffraction and refraction related terminology, but again employs the generalized Snell’s law from [1], implying that it yields the actual physical interpretation of relaxing the constraints of translational invariance in wave propagation: “Physically, one can interpret the phase gradient as an effective wavevector, leading to a generalization of the conservation of the wavevector parallel to the surface” [19] (p. 140). It is important to note that the experimental results and methods presented in [1–15,18,19] propose novel and interesting techniques for beam shaping and wavefront manipulation that contribute considerable advances to the field of metamaterials in optical applications. However, the attribution of the observations to anomalous reflection and refraction that are covered by a generalized Snell’s law, and that oppose the translational invariance of the wave vector, and therefore contradict the principle of wave propagation as described by Maxwell’s equations, is an issue that must be resolved. The correct understanding and description of the physical process at the phase-grating-like metasurfaces employed in [1–15] in terms of diffraction theory can benefit future development in this field of science and is of general interest to the optics community.

In this paper, the experimental results of the far field intensity distributions of metasurface gratings are considered in terms of diffraction theory. It is demonstrated here that the derivation of the generalized Snell’s law in [1], employing Fermat’s principle, requires special constraints and yields incomplete information about the actual far field distribution. An analytical description based on diffraction theory is introduced that agrees with the experimental results of [1] while yielding additional insight into the image formation process at metasurfaces. The diffraction model demonstrates that control of the beam direction in terms of an arbitrary angle of reflection or transmission as proposed in [1,4] by the phase gradient is possible only if the grating period is adjusted.

2. PHASE GRADIENT METASURFACES AND DIFFRACTION THEORY

When an incoming light wave transmits through or gets reflected at a periodic structure close to the dimension of its own wavelength, such as a phase or amplitude grating, a typical diffraction pattern of constructive and destructive interference will occur in the far field. Figure 1(a) illustrates the diffraction orders of a transmitted or reflected incoming plane wave. The intensity distribution observed in the far field is derived employing Fresnel–Kirchhoff diffraction theory and the Fraunhofer approximation [20]. By this theory, the diffraction pattern formed in the image plane in the far field is dependent on the Fourier transform of the diffracting aperture. As pointed out in [16,21], the diffraction pattern of the diffraction grating designed in [1], under consideration of an infinite grating width, may be described as the product of a Dirac comb and the Fourier transform of the transmission function of each grating period, which is referred to as motif. Employing the paraxial approximation $\theta \approx \sin (\theta)$, the spatial representation of the complex amplitude transfer function ${t_A}({x_1})$ of a blazed grating of finite width for a transmitted or reflected plane wave of arbitrary angle of incidence having unit amplitude may be described as

$$\begin{split}{t_A}({x_1})& = \frac{1}{w}{\rm rect}\left({\frac{{{x_1}}}{w}} \right)\left[\frac{1}{\Gamma}{\rm rect}\left({\frac{{{x_1}}}{\Gamma}} \right)\exp \left({- j2\pi \frac{{{p_B}{\lambda _B}}}{{{\lambda _0}\Gamma}}{x_1}} \right) \right.\\&\quad* \left.\frac{1}{\Gamma}{\rm comb}\left({\frac{{{x_1}}}{\Gamma}} \right) \right]\exp \left({- j2\pi \frac{{{\theta _m}}}{{{\lambda _m}}}{x_1}} \right) ,\end{split}$$
where ${x_1}$ is the lateral coordinate in the direction of the grating period, $w$ is the width of the illuminated section of the grating assuming homogeneous illumination, $\Gamma$ is the grating period, ${p_B} \in {\mathbb Q}$ is a design parameter to scale the linear phase profile, ${\lambda _B}$ is the design wavelength, ${\lambda _0}$ is the wavelength in vacuum, ${\lambda _m}$ is the effective wavelength of the illuminating light in the $m$th medium ${\lambda _m} = \frac{{{\lambda _0}}}{{{n_m}}}$ with ${n_m}$ being the refractive index, ${\rm rect}(\frac{{{x_1}}}{w})$ is a rectangle function of width $w$ along ${x_1}$, ${\rm comb}(\frac{{{x_1}}}{\Gamma})$ is a Dirac comb with period $\Gamma$ along ${x_1}$, and $*$ denotes convolution. Equation (1) describes the amplitude and phase transfer behavior of the blazed grating as illustrated in [16] Fig. 3 (p. 2392). The rectangle function ${\rm rect}({x_1}/w)$ is added here to the equation from [16,21] to take account of the finite length of the illuminated grating section. The rectangle function ${\rm rect}({x_1}/\Gamma)$ describes the amplitude transmission behavior of the grating periods of length $\Gamma$, the exponential function $\exp ({- j2\pi \frac{{{p_B}{\lambda _B}}}{{{\lambda _0}\Gamma}}{x_1}})$ describes the phase transmission behavior of the grating periods of length $\Gamma$, and the convolution with the Dirac comb describes the periodic continuation of the grating profile. The complex exponential function implies a $2\pi$ phase ambiguity for the spatial distribution of the phase delay introduced by the grating, such that the phase profile $\Phi ({x_1}) = 2\pi \frac{{{p_B}{\lambda _B}}}{{{\lambda _0}\Gamma}}{x_1}\; \in [{- \pi ,\pi}]$ on each interval of width $\Gamma$. The exponential function $\exp ({- j2\pi \frac{{{\theta _m}}}{{{\lambda _m}}}{x_1}})$ describes the phase gradient distributed along the full grating width along ${x_1}$ due to the inclination angle ${\theta _m}$ of the incoming wave. The Fourier transform of Eq. (1) is derived by employing the frequency shifting and convolution theorem (details of the operation are shown in Supplement 1):
$$\begin{split}{{\cal F}\{{{t_A}({x_1})} \}}&= {\rm sinc}({w\xi} ) * {\rm sinc}\left({\Gamma \left({\xi - \frac{{{p_B}{\lambda _B}}}{{{\lambda _0}\Gamma}} - \frac{{{\theta _m}}}{{{\lambda _m}}}} \right)} \right) \\[-3pt]&\quad\cdot {\rm comb}\left({\frac{{\xi - \frac{{{\theta _m}}}{{{\lambda _m}}}}}{{1/\Gamma}}} \right)\\[-3pt]&= \sum\limits_{k = - \infty}^\infty { \rm sinc}\left({w\left({\xi - \frac{k}{\Gamma} - \frac{{{\theta _m}}}{{{\lambda _m}}}} \right)} \right)\\[-3pt]&\quad\times{\rm sinc}\left({\Gamma \left({\xi - \frac{{{p_B}{\lambda _B}}}{{{\lambda _0}\Gamma}} - \frac{{{\theta _m}}}{{{\lambda _m}}}} \right)} \right).\end{split}$$
In Eq. (2), $\xi = {x_2}/({\lambda}z)$ [21] (pp. 087105-16) denotes the spatial frequency in the observation plane along the direction ${x_2}$, which is parallel to ${x_1}$ at a distance $z$. The spatial frequency $\xi$ and the wavelength $\lambda$ without additional indexing serve as placeholders for either ${\xi _{\rm T}},\;{\xi _{\rm R}}$ or ${\lambda _m},\;{\lambda _{m + 1}}$, which are specific to a transmission or reflection grating, respectively. This equation describes the Fraunhofer diffraction pattern of a periodic blazed grating of finite width $w$. Employing Eq. (2) and an incoming plane wave of electric field amplitude ${E_{0m}}$ yields the irradiance distribution in the observation plane:
$$\begin{split}{I({x_2})}&\propto {{\left| {{E_{0m}} \cdot {\cal F}\{{{t_A}({x_1})} \}{|_{\xi = {x_2}/\lambda z}}} \right|}^2}\\[-3pt]{I({x_2})}&\propto \left[{E_{0m}} \cdot \sum\limits_{k = - \infty}^\infty {\rm sinc}\left({w\left({\xi - \frac{k}{\Gamma} - \frac{{{\theta _m}}}{{{\lambda _m}}}} \right)} \right)\right.\\[-3pt]&\quad\left.{\rm sinc}\left({\Gamma \left({\xi - \frac{{{p_B}{\lambda _B}}}{{{\lambda _0}\Gamma}} - \frac{{{\theta _m}}}{{{\lambda _m}}}} \right)} \right) \right]^2.\end{split}$$
To distinguish between a transmission and reflection grating, the refractive index of the $m$th and $m + 1$th medium needs to be considered. The effective wavelength employed to scale the spatial frequency is different for a transmission or reflection grating:
 figure: Fig. 1.

Fig. 1. (a) Schematic of a metasurface diffraction grating showing an incoming plane wave along the wave vector ${\vec k_m}$ and diffraction orders ${p_T} \in [- 2, - 1,0, + 1, + 2]$ of a hypothetical transmission grating as well as the diffraction orders ${p_R} \in [- 1,0,1]$ of a hypothetical reflection grating. The green circles indicate the position of phase shifting resonators forming the grating’s phase profile along the abscissa (${x_1}$ direction). The depicted grating contains $Q = 5$ resonators on each grating period $\Gamma$ and a total of $P = 2$ periods. The numbering of the quadrants is shifted to align with the depiction of diffraction orders in [21]. (b) Intensity distribution of the normalized interference function $H(P,\Gamma ,\xi ,{\theta _m}) = {[{\sum\nolimits_{k = - \infty}^\infty {\rm sinc}({w({{\xi _{\rm T}} - \frac{k}{\Gamma} - \frac{{{\theta _m}}}{{{\lambda _m}}}})})}]^2}$ (blue solid line) and normalized intensity function of one grating period ${I^{(0)}}(s,\xi ,{p_B},{\theta _m}) = {[{{\rm sinc}({s({{\xi _{\rm T}} - \frac{{{p_B}{\lambda _B}}}{{{\lambda _0}\Gamma}} - \frac{{{\theta _m}}}{{{\lambda _m}}}})})}]^2}$ (black dashed line) for a transmission grating of grating period $\Gamma$ containing $P = 16$ grating periods, transmissive section of each period $s = \Gamma$, and total width $w = P\Gamma$. The product of these functions determines the grating’s intensity distribution [20] (p. 404) as given by Eq. (4). The design parameter ${p_B}$ is related to the linear phase profile of a grating period. The bottom plot shows an example for ${p_B} \notin {\mathbb Z}$, which is usually avoided in the design of blazed gratings.

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  • • in the case of a transmission grating, ${\xi _{\rm T}} = {x_2}/({\lambda _{m + 1}}z)$ with ${\lambda _{m + 1}} = \frac{{{\lambda _0}}}{{{n_{m + 1}}}}$,
  • • in the case of a reflection grating, ${\xi _{\rm R}} = {x_2}/({\lambda _m}z)$ with ${\lambda _m} = \frac{{{\lambda _0}}}{{{n_m}}}$,

and thus results in the analytic solution for the intensity distributions for a transmission and reflection grating:

$$\begin{split}I{({x_2})_{{\rm T,A}}}& \propto \left[{E_{0m}} \cdot \sum\limits_{k = - \infty}^\infty {\rm sinc}\left({w\left({\frac{{{x_2}}}{{\frac{{{\lambda _0}}}{{{n_{m + 1}}}}z}} - \frac{k}{\Gamma} - \frac{{{\theta _m}{n_m}}}{{{\lambda _0}}}} \right)} \right)\right.\\[-3pt]&\quad\times\left.{\rm sinc}\left({\Gamma \left({\frac{{{x_2}}}{{\frac{{{\lambda _0}}}{{{n_{m + 1}}}}z}} - \frac{{{p_B}{\lambda _B}}}{{{\lambda _0}\Gamma}} - \frac{{{\theta _m}{n_m}}}{{{\lambda _0}}}} \right)} \right) \right]^2,\end{split}$$
$$\begin{split}I{({x_2})_{{\rm R,A}}}& \propto \left[{E_{0m}} \cdot \sum\limits_{k = - \infty}^\infty {\rm sinc}\left({w\left({\frac{{{x_2}}}{{\frac{{{\lambda _0}}}{{{n_m}}}z}} - \frac{k}{\Gamma} - \frac{{{\theta _m}{n_m}}}{{{\lambda _0}}}} \right)} \right)\right.\\&\quad\left.{\rm sinc}\left({\Gamma \left({\frac{{{x_2}}}{{\frac{{{\lambda _0}}}{{{n_m}}}z}} - \frac{{{p_B}{\lambda _B}}}{{{\lambda _0}\Gamma}} - \frac{{{\theta _m}{n_m}}}{{{\lambda _0}}}} \right)} \right) \right]^2.\end{split}$$
It should be noted that this analytical solution of the far field distribution is applicable only for metasurface phase gratings employing a linear phase distribution $\frac{{\partial \Phi}}{{\partial {x_1}}}$ such as that used in [1]. $\Phi ({x_1})$ describes the spatial distribution of the phase delay introduced by the metasurface grating to the incoming wave along the interface. In the case of the complex amplitude transfer function ${t_A}({x_1})$ considered in Eq. (1), $\Phi ({x_1})$ refers to the phase transmission behavior of the grating periods $\Phi ({x_1}) = 2\pi \frac{{{p_B}{\lambda _B}}}{{{\lambda _0}\Gamma}}{x_1}$. Some general conclusions may be drawn from Eq. (3) that help us to understand the constraints of the proposed generalized Snell’s law. As described in [20] (p. 401), the intensity distribution of an arbitrary grating $I(\xi)$ may be described as the product of the normalized interference function $H(\xi)$ and the normalized intensity function ${I^{(0)}}(\xi)$ associated with a single period of the grating (motif in [16]). This is a general concept that does not include any assumptions about the amplitude profile or phase profile of the grating periods. Applying this concept to the intensity distribution in Eq. (3) results in
$$I(\xi) = H(P,\Gamma \xi ,{\theta _m}){I^{(0)}}(s,\xi ,{p_B},{\theta _m}),$$
$$H(P,\Gamma ,\xi ,{\theta _m}) = {\left[{\sum\limits_{k = - \infty}^\infty {\rm sinc}\left({w\left({\frac{{{x_2}}}{{\lambda z}} - \frac{k}{\Gamma} - \frac{{{\theta _m}}}{{{\lambda _m}}}} \right)} \right)} \right]^2},$$
$${I^{(0)}}(s,\xi ,{p_B},{\theta _m}) = {\left[{{\rm sinc}\left({s\left({\frac{{{x_2}}}{{\lambda z}} - \frac{{{p_B}{\lambda _B}}}{{{\lambda _0}\Gamma}} - \frac{{{\theta _m}}}{{{\lambda _m}}}} \right)} \right)} \right]^2}.$$
These functions are illustrated for $P = 16$ grating periods of grating period $\Gamma = 15\;{\unicode{x00B5}{\rm m}}$, each containing $Q = 8$ resonators and a transmissive section $s = \Gamma$, wavelength ${\lambda _0} = 8\;{\unicode{x00B5}{\rm m}}$, ${n_1} = 3.4$, ${n_2} = 1$, and different values of ${p_B}$ in Fig. 1(b). The key observation is that the normalized interference function $H(P,\Gamma \xi ,{\theta _m})$, which describes the far field intensity distribution due to the interference of light from different grating periods [20] (p. 403), is dependent only on the total number of grating periods $P$, grating period $\Gamma$, angle of incidence ${\theta _m}$, and phase evolution due to propagation in the medium, which comprises $\xi = \frac{{{x_2}}}{{\lambda z}}$. The number of grating periods $P$ determines the width of the interference maxima, with a higher number of periods resulting in narrower constructive interference peaks. The grating period $\Gamma$ determines the spacing between the constructive interference maxima, which is clearly visible in Fig. 1(b). A change in the angle of incidence ${\theta _m}$ would cause a phase offset and shift the whole comb-like interference function. Due to the paraxial approximation $\sin (\theta) \approx \theta$, the angle from the grating interface ${x_1}$ to a point in the observation plane ${x_2}$ at a distance $z$ may be described as ${\theta _{m + 1}}({x_2}) = {\rm{atan}}({\frac{{{x_2}}}{z}}) \approx \frac{{{x_2}}}{z}$ for a transmission grating and as $\grave{\theta}_m(x_2)={\rm{atan}}(\frac{x_2}{z})\approx \frac{x_2}{z}$ for a reflection grating. This allows writing the normalized interference function given by Eq. (7) in a different form that is closer to the notation in [17,20]. In Eq. (9), the case of a transmission grating is considered (the same can be done for a reflection grating):
$$\begin{split}&H(P,\Gamma ,\xi ,{\theta _m})\\&= {{{\left[{\sum\limits_{k = - \infty}^\infty {\rm sinc}\left({w\left({\frac{{{\theta _{m + 1}}{n_{m + 1}}}}{{{\lambda _0}}} - \frac{k}{\Gamma} - \frac{{{\theta _m}{n_m}}}{{{\lambda _0}}}} \right)} \right)} \right]}^2}}\\ &= {{\left[{\sum\limits_{k = - \infty}^\infty {\rm sinc}\left({w\left({\frac{{{n_{m + 1}}\sin ({\theta _{m + 1}}) - {n_m}\sin ({\theta _m})}}{{{\lambda _0}}} - \frac{k}{\Gamma}} \right)} \right)} \!\right]}^2}\!.\end{split}$$
The formula in Eq. (9) is the link between the normalized interference function and the grating Eq. (10). The grating equation of a transmission grating predicts the position of the constructive interference maxima. The grating equation results from the condition that the argument of the ${\rm sinc}()$ function in Eq. (9) vanishes and is formulated as
$${n_{m + 1}}\sin ({\theta _{m + 1}}) - {n_m}\sin ({\theta _m}) = {p_d}\frac{{{\lambda _0}}}{\Gamma},$$
with diffraction order ${p_d} \in {\mathbb Z}$. The grating equations of both transmission and reflection gratings are thus derived from the normalized interference function, which notably is a general function with no dependence on the amplitude or phase profile of the individual grating periods.

The normalized intensity function of a single period of the grating ${I^{(0)}}(s,\xi ,{p_B},{\theta _m})$ is the link to the typical characteristic of different grating profiles. In general, a grating may show an arbitrary amplitude and phase profile within a single grating period. The argument of the ${\rm sinc}()$ function in Eq. (8) is scaled with the width $s \in [{0,\Gamma}]$ of the transmitting or reflecting part of the grating period. In the case of a slit grating as described in [20] (p. 404), the width $s$ is typically smaller than the whole grating period $\Gamma$. In our example, we face the special case of a pure phase grating with no opaque sections, which causes the width of the transmitting or reflecting grating section on each period to align with the full grating period length $s = \Gamma$. The width of the interference maximum of the function ${I^{(0)}}(s,\xi ,{p_B},{\theta _m})$ is dependent on $s$, since $s$ determines the spacing of its minima, as can be seen in Fig. 1(b). Like the normalized interference function, the normalized intensity function of a single period of the grating also depends on the grating period $\Gamma$, angle of incidence ${\theta _m}$, and phase evolution due to propagation in the medium comprising $\xi = \frac{{{x_2}}}{{\lambda z}}$. A change in the angle of incidence ${\theta _m}$ would also cause a phase offset shifting the interference maximum of ${I^{(0)}}(s,\xi ,{p_B},{\theta _m})$. The major difference between Eqs. (7) and (8) is that Eq. (8) does not show a periodic repetition and is dependent on the amplitude and phase profile of a single grating period. Writing Eq. (8) in dependence of the angular position, again employing ${\theta _{m + 1}}({x_2}) = {\rm{atan}}({\frac{{{x_2}}}{z}}) \approx \frac{{{x_2}}}{z}$, which results from the paraxial approximation and the assumption of a transmission grating, yields the following:

$$\begin{split}&{I^{(0)}}(s,\xi ,{p_B},{\theta _m})\\& = {\left[{{\rm sinc}\left({s\left({\frac{{{n_{m + 1}}\sin ({\theta _{m + 1}}) - {n_m}\sin ({\theta _m})}}{{{\lambda _0}}} - \frac{{{p_B}{\lambda _B}}}{{{\lambda _0}\Gamma}}} \right)} \right)} \right]^2}.\end{split}$$
Under consideration of Eq. (11), we may think of the normalized intensity function of a single period of the grating ${I^{(0)}}(s,\xi ,{p_B},{\theta _m})$ as a sinc-shaped window that we can arbitrarily place in dependence of our grating’s phase profile, represented by the design parameter ${p_B} \in {\mathbb Q}$, to cut out a specific part of the normalized interference function. The maximum of the sinc-like window function in Eq. (11) lies at the position where the argument of the ${\rm sinc}()$ function vanishes. The position of the window for a transmission grating is thus given by
$${n_{m + 1}}\sin ({\theta _{m + 1}}) - {n_m}\sin ({\theta _m}) = \frac{{{\lambda _B}}}{{{\lambda _0}}}{p_B}\frac{{{\lambda _0}}}{\Gamma}.$$
Here, it is important to realize that a change in the phase profile of the grating periods by adjusting ${p_B}$ or ${\lambda _B}$ will shift our window only ${I^{(0)}}(s,\xi ,{p_B},{\theta _m})$ and not cause any changes to the position of the interference maxima of the normalized interference function $H(P,\Gamma ,\xi ,{\theta _m})$ as illustrated in Fig. 1(b). The combination of Eqs. (9), (10) and (11), (12) provides full coverage of the far field distribution of a metasurface transmission phase grating. The conclusion of Eqs. (9) and (10) is that the position of the observable interference maxima is defined by the grating period $\Gamma$, wavelength ${\lambda _0}$, refractive index ${n_m}$, and angle of incidence ${\theta _m}$. Thus, to steer a transmitted beam towards a specific angle, at least one of these parameters needs to be modified. The conclusion of Eqs. (11) and (12) is that the periodic phase profile $\Phi ({x_1})$ of the single grating periods in dependence of the scaling parameter ${p_B}$ and the design wavelength ${\lambda _B}$, together with the wavelength ${\lambda _0}$, refractive index ${n_m}$, and grating period $\Gamma$ determines which part of the far field interference pattern is visible to the observer. However, the generalized laws of reflection and refraction from [1] apparently yield a prediction of the angular position of the dominant visible interference maximum with a single formula, ignoring diffraction and interference effects. In the next sections, we will discuss which constraints are imposed on the validity of these predictions and why these laws may cause a misleading interpretation of the actual physical process at the metasurface.

3. GENERALIZED SNELL’S LAW AND ITS CONSTRAINTS

The generalized laws of reflection and refraction in [1] Eq. (2) have the form

$${n_{m + 1}}\,\sin \left({{\theta _{m + 1}}} \right) - {n_m}\,\sin \left({{\theta _m}} \right) = \frac{{{\lambda _0}}}{{2\pi}}\frac{{\partial \Phi}}{{\partial {x_1}}} \quad {\rm Transmission,}$$
$${n_m}\sin \left( {\theta _m^\prime} \right) - {n_m}\,\sin \left( {{\theta _m}} \right) = \frac{{{\lambda _0}}}{{2\pi }}\frac{{\partial \Phi }}{{\partial {x_1}}}\quad {\rm Reflection}.$$
The far field intensity distribution of the metasurface grating is described by Eq. (4) or Eq. (5) and illustrated for a transmission grating in Fig. 1(b). For the intensity distribution on the screen to resemble that of a refracted or reflected beam, only a single distinguished interference peak may be visible. This is the case only if for a phase grating with slit width $s = \Gamma$, the sinc-shaped normalized intensity function ${I^{(0)}}(s,\xi ,{p_B},{\theta _m})$ is centered on one of the interference maxima given by the grating Eq. (10) for a transmission grating, and its zero-crossings suppress all the side maxima. To achieve this condition for a transmission grating, the phase gradient $\frac{{\partial \Phi}}{{\partial {x_1}}}$ in the generalized Snell’s law in Eq. (13) has to be chosen such that it solves Eq. (10) for the position of the interference peaks and Eq. (12) for the window position at the same time. This results in the constraint
$$\begin{split}\frac{{{\lambda _0}}}{{2\pi}}\frac{{\partial \Phi}}{{\partial {x_1}}} &= \frac{{{\lambda _B}}}{{{\lambda _0}}}{p_B}\frac{{{\lambda _0}}}{\Gamma} = {p_d}\frac{{{\lambda _0}}}{\Gamma}\quad {\rm with}\quad {p_B} \in {\mathbb Q},\;{p_d} \in {\mathbb Z},\\\frac{{\partial \Phi}}{{\partial {x_1}}}& = \frac{{{\lambda _B}}}{{{\lambda _0}}}{p_B}\frac{{2\pi}}{\Gamma} = {p_d}\frac{{2\pi}}{\Gamma}.\end{split}$$
A correct prediction is therefore possible only if $\frac{{\partial \Phi}}{{\partial {x_1}}} = {p_d}\frac{{2\pi}}{\Gamma}$, which with regard to the design of the phase grating means that the phase profile has to be designed in such a way that the phase changes an exact integer multiple of $2\pi$ on each grating period of width $\Gamma$. This can be demonstrated for the reflection grating in an analog manner. By increasing or decreasing the phase profile on the period by $2\pi$, the window position is switched between the interference maxima. The crucial meaning of this constraint is that the only free parameter to adjust the angle of the observed interference peak is the grating period $\Gamma$. If the phase gradient is adjusted and the predictions of the generalized Snell’s law from [1] yield reasonable results, there are only two allowed manipulations, as noted below.
  • • The phase delay along one grating period $\Gamma$ may be increased by an integer multiple of $2\pi$. This would be possible for a traditional blazed grating, by adjusting the thickness of the sawtooth profile. However, in the case of the experiment presented in [1], it would not be possible, since the employed antenna resonators may cause a phase delay only in the range of $[{0,2\pi}]$. Besides that, as seen in Eq. (1), the phase profile is subject to the $2\pi$ ambiguity of the complex exponential function. Therefore, increasing the phase delay by integer multiples of $2\pi$ has the effect of reducing the effective grating period by the same integer factor.
  • • The grating period $\Gamma$ may be adjusted. If this is done, the phase delay along the grating period needs to be adjusted, such that it again equals an integer multiple of $2\pi$. For a traditional blazed grating, this can be achieved by adjusting the slope of the sawtooth profile. In the case of the antenna metasurface, the spacing or composition of the antennas needs to be adjusted.

In [1], it is claimed that the angles of anomalous reflection and refraction are controlled by the phase gradient $\frac{{\partial \Phi}}{{\partial {x_1}}}$; although this claim is not strictly wrong, it implies that the phase gradient might be changed by introducing a non-integer multiple of $2\pi$ phase shift distributed along one grating period $\Gamma$. However, this would immediately lead to a deviation between the prediction of Eqs. (13) and (14) and the actual observation. This is clearly illustrated in Fig. 1(b), where ${p_B} = 0.5$ associated to a phase gradient of $\frac{{\partial \Phi}}{{\partial {x_1}}} = \frac{\pi}{\Gamma}$ is chosen as a design parameter, moving the sinc window between the zeroth and ${-}1$st diffraction order, while the interference function remains unchanged. The generalized Snell’s law from [1] would predict the anomalous refracted beam at the position of the maximum of the black dashed curve; however, the diffraction pattern with the damped zeroth and ${-}1$st diffraction order peaks instead results as the intensity distribution. Hence it is obvious that the generalized Snell’s law is not related to refraction or reflection in a classical understanding, but rather offers a very specific approximation to the position of the visible interference maximum of a special kind of diffraction pattern and should not be employed to make predictions pertaining to metasurface grating experiments.

4. FERMAT’S PRINCIPLE, WAVE VECTOR CONTINUITY, AND THE GENERALIZED SNELL’S LAW

Moreover, the general question should be considered as to what the relation actually is between the generalized laws of reflection and refraction from [1] in Eqs. (13) and (14) and the far field intensity distribution, Eq. (6), of the metasurface grating. The actual intensity distribution in the far field is described by the combination of the interference pattern of the grating Eq. (7) with the intensity distribution due to the phase and amplitude characteristics of the motif Eq. (8). Therefore, it has to be expected that the generalized laws of reflection and refraction from [1] represent only a part of the information contained in the diffraction-based solution. To figure out which information Eqs. (13) and (14) represent, we revisit their derivation in terms of Fermat’s principle, which is employed in [1]. In its basic form, Fermat’s principle states that the path light takes to travel from point $A$ to point $B$ through an interface and, thus, the refraction or reflection angle of a light wave passing an interface, may be predicted on the assumption that the light will pass the interface at a point ${P_{\textit{AB}}}$, where a small variation in its passing position will cause no change in the travel time or the associated phase $\frac{{\partial t}}{{\partial {x_1}}}{|_{{P_{\textit{AB}}}}} = \frac{{\partial \Phi}}{{\partial {x_1}}}{|_{{P_{\textit{AB}}}}} = 0$ [20,22]. Based on this assumption, Snell’s law may be derived employing the geometric relations among points $A$, ${P_{\textit{AB}}}$, $B$, the velocity of light, and the refractive indices of the media. The zero crossing in the derivative indicates an extremum in the travel time or phase, which may be interpreted to mean that the light will take the fastest or shortest available path (this is why Fermat’s principle is also referred to as principle of least time).

In [23], Fermat’s principle is applied to a diffraction grating to derive the grating equation. The authors of [23] employ Huygen’s principle to state that if a light source at point $A$ illuminates a grating, the field amplitude at an observer point $B$ is given by the sum of the field at point A proceeded along all possible nonintersecting paths connecting $A$ and $B$ along the grating. The grating will modulate the optical path length depending on the point of intersection ${P_{\textit{AB}}}$ along ${x_1}$, and finally only extremal paths will contribute to the field at point $B$ for which the derivative of the accumulated path length difference along ${x_1}$ equals zero, which is the requirement for constructive interference in point $B$. Thus, the task solved in [23] may be described as follows: find the points of constructive interference $B$ that result from diffraction at a periodical grating of period $\Gamma$, which is illuminated from a source in point $A$. The condition for constructive interference between paths intersecting the grating at different grating periods is that the rate of change in their optical path lengths $\overline {A{P_{\textit{AB}}}} ({x_1}) + \overline {{P_{\textit{AB}}}B} ({x_1})$ along ${x_1}$ is equal to an integer multiple of the wavelength ${p_d}{\lambda _0}$:

$$\frac{\partial}{{\partial {x_1}}}\left[{\overline {A{P_{\textit{AB}}}} ({x_1}) + \overline {{P_{\textit{AB}}}B} ({x_1}) \pm {p_d}{\lambda _0}\frac{{{x_1}}}{\Gamma}} \right] = 0.$$
Here, $\overline {A{P_{\textit{AB}}}} ({x_1})$ and $\overline {{P_{\textit{AB}}}B} ({x_1})$ describe the optical path length difference from point $A$ to point ${P_{\textit{AB}}}$ on the interface and from point ${P_{\textit{AB}}}$ to point $B$. From the condition for the position of the constructive interference maxima in Eq. (16), the grating equation is derived in [23], which is equivalent to Eq. (10). It is important to realize that the result considers only the interference of paths between different grating periods, and only the strictly periodic structure of the grating has been considered, while its motif has no impact on the result. It would be possible to employ Fermat’s principle as demonstrated in [23] for a single period of the grating to find the interference maximum in dependence of the motif. In this case, the condition for constructive interference between paths through different grating periods ${p_d}{\lambda _0}\frac{{{x_1}}}{\Gamma}$ is replaced by the phase profile of the grating $\Phi ({x_1})$. The task to solve may then be described as follows: find the point of constructive interference $B$ that results from diffraction at the grating period of width $\Gamma$, which modulates the path length difference according to its phase delay profile $\Phi ({x_1})$, which is illuminated from a source in point $A$:
$$\frac{\partial}{{\partial {x_1}}}\left[{\overline {A{P_{\textit{AB}}}} ({x_1}) + \overline {{P_{\textit{AB}}}B} ({x_1}) + \Phi ({x_1})} \right] = 0.$$
From Eq. (17), the position of the maximum of the normalized intensity function as described in Eq. (12) may be derived. This shows that although Fermat’s principle may be employed to find the geometric relations among a source at point $A$, an observer at point $B$, and the intersecting point ${P_{\textit{AB}}}$ at a grating under the consideration of extremal paths, which lead to constructive interference, the results still do not contain the full information yielded by the far field intensity distribution in Eq. (6). One simple reason for this is that the case of grating interference due to the interaction of multiple different grating periods and the interference conditions due to the motif of the single grating periods cannot be considered simultaneously, since they are different. Another important observation is that Fermat’s principle employs just the positions of points and optical path length differences. Therefore, its solutions have to be understood in the same way. The extremal path conditions from Eqs. (16) and (17) yield the position of constructive interference for the grating and motif interference, respectively. The field at the respective observing points $B$ is acquired by integration over all available paths intersecting the grating and thus results from the superposition of an infinite amount of elementary waves that originated from the diffraction grating. Thus, it is misleading to appoint the vector connecting point ${P_{\textit{AB}}}$ on the grating and point $B$ of the observation the significance of a wave vector. In the case of a diffraction grating, there is no single plane wave emerging from the grating and traveling in the direction of the observing points, which is obvious, given the intensity distribution.

Having reviewed the application of Fermat’s principle to diffraction gratings, we will now move on to the derivation of the generalized Snell’s laws in [1], which is based on [23]. A key problem in the derivation in [1] is that the authors ignore the periodic structure of their grating. Instead, they employ the extremal path condition for the motif of a single grating period as demonstrated in Eq. (17). However, as discussed above, the phase gradient employed in [1] is constrained to $\frac{{\partial \Phi}}{{\partial {x_1}}} = {p_d}\frac{{2\pi}}{\Gamma}$, and thus the extremal path condition for the motif is the same as that of the ${p_d}$th diffraction order of the grating. Yet, the generalized Snell’s laws in Eqs. (14) and (13) are lacking the grating characteristic of periodic interference maxima, which is still visible in [23] and may cause the misleading assumption that a single plane wave will emerge behind the diffraction grating, instead of the complex wave interference pattern, which actually results. The questionable interpretation of the position of the interference maximum in terms of a wave vector leads to the statement about the violation of wave vector conservation at the metasurface which was quoted in the Introduction. “Note that due to the lack of translational invariance along the interface the tangential wavevector of the incident photon is not conserved;…” [4] (p. 1703).

Alternatively, Snell’s law may be derived employing wave vectors based on the boundary conditions of continuous propagation for the tangential field components originating from Maxwell’s equations [24]. The boundary conditions impose the constraint that the wave vectors of the incoming, reflected, and refracted waves propagate continuously through the interface [20,22,25], which may be expressed as a vector product of the wave vector ${\vec k_m}$ and the normal vector of the boundary surface ${\vec u_{\textit{bs}}}$ between the $m$th and $(m + 1)$th medium Eq. (18):

$${\vec k_m} \times {\vec u_{\textit{bs}}} = {\vec k_{m + 1}} \times {\vec u_{\textit{bs}}}.$$
In this derivation, the wave vectors represent actual plane waves. However, this approach is feasible only for continuous interfaces with no lateral structure. It would be possible, however, to apply the boundary conditions within the range of infinitesimal sections along the interface. On each of these sections, the incoming wave vector would describe the propagation of a plane wave that arrives at the intersection between the two media with a phase delay specific to the antenna resonators at its position along ${x_1}$ (this is depicted in [1] Fig. 2G). At the interface, each of the complex plane waves would be refracted or reflected according to Snell’s law as given in Eq. (18), while perfectly conserving its wave vector and maintaining its location specific phase delay. The interference pattern of all these waves in the far field would then form the grating intensity distribution.

With this information, we may solve the initial question of this section as to what the relation is between the generalized laws of reflection and refraction from [1] in Eqs. (13) and (14) and the far field intensity distribution, Eq. (6), of the metasurface grating. Considering the constraints for the generalized Snell’s laws introduced in Section 3, it is feasible to conclude that these laws predict the position of the visible ${p_d}$th interference maximum of a blazed phase grating, if the window function given by its motif is also centered on the ${p_d}$th maximum. Thus, the generalized Snell’s laws yield incomplete information about the far field intensity distribution of a diffraction grating, which might be obtained employing diffraction theory.

Supplement 1 contains multiple examples of the application of the analytical model from Eq. (3) as well as a numerical simulation based on Huygen’s theory to describe wave propagation at metasurfaces. These examples demonstrate that the results of diffraction theory explain the experimental results presented in [1] and may be employed to predict the far field intensity distributions of metasurfaces, for which the generalized Snell’s laws would yield incorrect results.

5. CONCLUSION

This paper provides extensive coverage of grating-like metasurface structures in terms of diffraction theory. Although the resonators that form the phase profile of the metasurface have subwavelength dimensions, the metasurface as a whole emulates a phase grid with a periodic phase profile distributed over the grating periods of width $\Gamma$, which span over multiple resonators. The subwavelength dimensions of the single resonators enable the implementation of a virtually continuous and variable phase gradient, which in turn enables the miniaturization of complex beam shaping applications.

However, there is no “…lack of translational invariance along the interface…” [4] (p. 1703) and also no “…relaxation of the fundamental Snell’s law for light refraction…” [5] (p. 1). These assumptions stem from the infeasible interpretation of the results of Fermat’s principle for the positions of the interference maxima in the observation plane as being connected to the intersection point on the grating interface by a classical wave vector. Instead, the positions of the interference maxima need to be treated as simple points for which the superposition from waves of all available paths through the grating satisfies the extremal path condition.

The “generalized laws of reflection and refraction” in [1] are capable of correct predictions of the position of the constructive interference maxima of the grating equation only if strict conditions apply. The most important constraint is that the phase profile has to be designed in such a way that the phase changes an exact integer multiple of $2\pi$ on each grating period of width $\Gamma$. This implies that the position of the interference maxima may be controlled only by changing the grating period $\Gamma$. The more general statement in [1] that the beam direction is controlled by the phase gradient $\frac{{\partial \Phi}}{{\partial {x_1}}}$ is limited to the case $\frac{{\partial \Phi}}{{\partial {x_1}}} = \frac{{2\pi}}{\Gamma}$ and boils down to the well-known fact that the grating period of a diffraction grating determines the position of its interference maxima.

The physical foundation of the observed effect is diffraction at the periodic structure of the resonator array. The term “anomalous reflection and refraction” employed in [1] is delusive since the effects have to be attributed to diffraction theory and are not related to classical refraction. Thus, the concept of “generalized laws of refraction and reflection” and “anomalous refraction and reflection” as proposed in [1] is applicable only under specific constraints, yields incomplete information compared to the results of diffraction theory, and may cause misleading interpretations of the physical effects occurring at the grating interface. This terminology should be strictly avoided, since it implies a violation of basic axioms of electromagnetic wave propagation found in the quotations above. Instead, the intensity distribution of metasurfaces with linear phase profiles may be predicted employing the analytical model for the far field intensity distribution in Eq. (3) or using numerical approaches based on Huygen’s theory, which may also handle non-linear phase distributions.

Acknowledgment

I kindly thank Dr. Birk Andreas for reading through the manuscript and for his constructive discussion of this paper.

Disclosures

The author declares no conflicts of interest.

Data availability

All data and simulations employed in the paper are available through the open access repository of the Physikalisch-Technische Bundesanstalt [26].

Supplemental document

See Supplement 1 for supporting content.

REFERENCES

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11. D. Zhang, M. Ren, W. Wu, N. Gao, X. Yu, W. Cai, X. Zhang, and J. Xu, “Nanoscale beam splitters based on gradient metasurfaces,” Opt. Lett. 43, 267–270 (2018). [CrossRef]  

12. H. Xie and Z. Hou, “Nonlocal metasurface for acoustic focusing,” Phys. Rev. Appl. 15, 034054 (2021). [CrossRef]  

13. L. Fan and J. Mei, “Multifunctional waterborne acoustic metagratings: From extraordinary transmission to total and abnormal reflection,” Phys. Rev. Appl. 16, 044029 (2021). [CrossRef]  

14. S. W. Lee, Y. J. Shin, H. W. Park, H. M. Seung, and J. H. Oh, “Full-wave tailoring between different elastic media: a double-unit elastic metasurface,” Phys. Rev. Appl. 16, 064013 (2021). [CrossRef]  

15. C. E. Gutiérrez, L. Pallucchini, and E. Stachura, “General refraction problems with phase discontinuities on nonflat metasurfaces,” J. Opt. Soc. Am. A 34, 1160–1172 (2017). [CrossRef]  

16. S. Larouche and D. R. Smith, “Reconciliation of generalized refraction with diffraction theory,” Opt. Lett. 37, 2391–2393 (2012). [CrossRef]  

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21. J. E. Harvey and R. N. Pfisterer, “Understanding diffraction grating behavior: including conical diffraction and Rayleigh anomalies from transmission gratings,” Opt. Eng. 58, 087105 (2019). [CrossRef]  

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26. M. Schake, “Simulation data for “Examining and explaining the “Generalized Laws of Reflection and Refraction” at metasurface gratings”,” Physikalisch-Technische Bundesanstalt, 2020, https://doi.org/10.7795/710.20220601.

Supplementary Material (1)

NameDescription
Supplement 1       Mathematical details and examples.

Data availability

All data and simulations employed in the paper are available through the open access repository of the Physikalisch-Technische Bundesanstalt [26].

26. M. Schake, “Simulation data for “Examining and explaining the “Generalized Laws of Reflection and Refraction” at metasurface gratings”,” Physikalisch-Technische Bundesanstalt, 2020, https://doi.org/10.7795/710.20220601.

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Figures (1)
Fig. 1.
Fig. 1. (a) Schematic of a metasurface diffraction grating showing an incoming plane wave along the wave vector ${\vec k_m}$ and diffraction orders ${p_T} \in [- 2, - 1,0, + 1, + 2]$ of a hypothetical transmission grating as well as the diffraction orders ${p_R} \in [- 1,0,1]$ of a hypothetical reflection grating. The green circles indicate the position of phase shifting resonators forming the grating’s phase profile along the abscissa ( ${x_1}$ direction). The depicted grating contains $Q = 5$ resonators on each grating period $\Gamma$ and a total of $P = 2$ periods. The numbering of the quadrants is shifted to align with the depiction of diffraction orders in [21]. (b) Intensity distribution of the normalized interference function $H(P,\Gamma ,\xi ,{\theta _m}) = {[{\sum\nolimits_{k = - \infty}^\infty {\rm sinc}({w({{\xi _{\rm T}} - \frac{k}{\Gamma} - \frac{{{\theta _m}}}{{{\lambda _m}}}})})}]^2}$ (blue solid line) and normalized intensity function of one grating period ${I^{(0)}}(s,\xi ,{p_B},{\theta _m}) = {[{{\rm sinc}({s({{\xi _{\rm T}} - \frac{{{p_B}{\lambda _B}}}{{{\lambda _0}\Gamma}} - \frac{{{\theta _m}}}{{{\lambda _m}}}})})}]^2}$ (black dashed line) for a transmission grating of grating period $\Gamma$ containing $P = 16$ grating periods, transmissive section of each period $s = \Gamma$ , and total width $w = P\Gamma$ . The product of these functions determines the grating’s intensity distribution [20] (p. 404) as given by Eq. (4). The design parameter ${p_B}$ is related to the linear phase profile of a grating period. The bottom plot shows an example for ${p_B} \notin {\mathbb Z}$ , which is usually avoided in the design of blazed gratings.

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Equations (18)

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t A ( x 1 ) = 1 w r e c t ( x 1 w ) [ 1 Γ r e c t ( x 1 Γ ) exp ⁡ ( − j 2 π p B λ B λ 0 Γ x 1 ) ∗ 1 Γ c o m b ( x 1 Γ ) ] exp ⁡ ( − j 2 π θ m λ m x 1 ) ,
F { t A ( x 1 ) } = s i n c ( w ξ ) ∗ s i n c ( Γ ( ξ − p B λ B λ 0 Γ − θ m λ m ) ) ⋅ c o m b ( ξ − θ m λ m 1 / Γ ) = ∑ k = − ∞ ∞ s i n c ( w ( ξ − k Γ − θ m λ m ) ) × s i n c ( Γ ( ξ − p B λ B λ 0 Γ − θ m λ m ) ) .
I ( x 2 ) ∝ | E 0 m ⋅ F { t A ( x 1 ) } | ξ = x 2 / λ z | 2 I ( x 2 ) ∝ [ E 0 m ⋅ ∑ k = − ∞ ∞ s i n c ( w ( ξ − k Γ − θ m λ m ) ) s i n c ( Γ ( ξ − p B λ B λ 0 Γ − θ m λ m ) ) ] 2 .
I ( x 2 ) T , A ∝ [ E 0 m ⋅ ∑ k = − ∞ ∞ s i n c ( w ( x 2 λ 0 n m + 1 z − k Γ − θ m n m λ 0 ) ) × s i n c ( Γ ( x 2 λ 0 n m + 1 z − p B λ B λ 0 Γ − θ m n m λ 0 ) ) ] 2 ,
I ( x 2 ) R , A ∝ [ E 0 m ⋅ ∑ k = − ∞ ∞ s i n c ( w ( x 2 λ 0 n m z − k Γ − θ m n m λ 0 ) ) s i n c ( Γ ( x 2 λ 0 n m z − p B λ B λ 0 Γ − θ m n m λ 0 ) ) ] 2 .
I ( ξ ) = H ( P , Γ ξ , θ m ) I ( 0 ) ( s , ξ , p B , θ m ) ,
H ( P , Γ , ξ , θ m ) = [ ∑ k = − ∞ ∞ s i n c ( w ( x 2 λ z − k Γ − θ m λ m ) ) ] 2 ,
I ( 0 ) ( s , ξ , p B , θ m ) = [ s i n c ( s ( x 2 λ z − p B λ B λ 0 Γ − θ m λ m ) ) ] 2 .
H ( P , Γ , ξ , θ m ) = [ ∑ k = − ∞ ∞ s i n c ( w ( θ m + 1 n m + 1 λ 0 − k Γ − θ m n m λ 0 ) ) ] 2 = [ ∑ k = − ∞ ∞ s i n c ( w ( n m + 1 sin ⁡ ( θ m + 1 ) − n m sin ⁡ ( θ m ) λ 0 − k Γ ) ) ] 2 .
n m + 1 sin ⁡ ( θ m + 1 ) − n m sin ⁡ ( θ m ) = p d λ 0 Γ ,
I ( 0 ) ( s , ξ , p B , θ m ) = [ s i n c ( s ( n m + 1 sin ⁡ ( θ m + 1 ) − n m sin ⁡ ( θ m ) λ 0 − p B λ B λ 0 Γ ) ) ] 2 .
n m + 1 sin ⁡ ( θ m + 1 ) − n m sin ⁡ ( θ m ) = λ B λ 0 p B λ 0 Γ .
n m + 1 sin ⁡ ( θ m + 1 ) − n m sin ⁡ ( θ m ) = λ 0 2 π ∂ Φ ∂ x 1 T r a n s m i s s i o n ,
n m sin ⁡ ( θ m ′ ) − n m sin ⁡ ( θ m ) = λ 0 2 π ∂ Φ ∂ x 1 R e f l e c t i o n .
λ 0 2 π ∂ Φ ∂ x 1 = λ B λ 0 p B λ 0 Γ = p d λ 0 Γ w i t h p B ∈ Q , p d ∈ Z , ∂ Φ ∂ x 1 = λ B λ 0 p B 2 π Γ = p d 2 π Γ .
∂ ∂ x 1 [ A P AB ¯ ( x 1 ) + P AB B ¯ ( x 1 ) ± p d λ 0 x 1 Γ ] = 0.
∂ ∂ x 1 [ A P AB ¯ ( x 1 ) + P AB B ¯ ( x 1 ) + Φ ( x 1 ) ] = 0.
k → m × u → bs = k → m + 1 × u → bs .
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