Controlling the sign of chromatic dispersion in diffractive optics with dielectric metasurfaces

Diffraction gratings disperse light in a rainbow of colors with the opposite order than refractive prisms, a phenomenon known as negative dispersion. While refractive dispersion can be controlled via material refractive index, diffractive dispersion is fundamentally an interference effect dictated by geometry. Here we show that this fundamental property can be altered using dielectric metasurfaces, and we experimentally demonstrate diffractive gratings and focusing mirrors with positive, zero, and hyper negative dispersion. These optical elements are implemented using a reflective metasurface composed of dielectric nano-posts that provide simultaneous control over phase and its wavelength derivative. In addition, as a first practical application, we demonstrate a focusing mirror that exhibits a five fold reduction in chromatic dispersion, and thus an almost three times increase in operation bandwidth compared to a regular diffractive element. This concept challenges the generally accepted dispersive properties of diffractive optical devices and extends their applications and functionalities.


I. INTRODUCTION
Most optical materials have positive (normal) dispersion, which means that the refractive index decreases at longer wavelengths. As a consequence, blue light is deflected more than red light by dielectric prisms [ Fig. 1 |(a)]. The reason why diffraction gratings are said to have negative dispersion is because they disperse light similar to hypothetical refractive prisms made of a material with negative (anomalous) dispersion [ Fig. 1 |(b)]. For diffractive devices, dispersion is not related to material properties, and it refers to the derivative of a certain device parameter with respect to wavelength. For example, the angular dispersion of a grating that deflects normally incident light by a positive angle θ is given by dθ/dλ = tan(θ)/λ (see [1] and Supplementary Section S2). Similarly, the wavelength dependence of the focal length (f ) of a diffractive lens is given by df /dλ = −f /λ [1,2]. Here we refer to diffractive devices that follow these fundamental chromatic dispersion relations as "regular". Achieving new regimes of dispersion control in diffractive optics is important both at the fundamental level and for numerous practical applications. Several distinct regimes can be differentiated as follows. Diffractive devices are dispersionless when the derivative is zero (i.e. dθ/dλ = 0, df /dλ = 0 shown schematically in Fig. 1 |(c)), have positive dispersion when the derivative has opposite sign compared to a regular diffractive device of the same kind (i.e. dθ/dλ < 0, df /dλ > 0) as shown in Fig. 1 |(d), and are hyper-dispersive when the derivative has a larger absolute value than a regular device (i.e. |dθ/dλ| > | tan(θ)/λ|, |df /dλ| > | − f /λ|) as seen in Fig. 1 |(e). Here we show that these regimes can be achieved in diffractive devices based on optical metasurfaces.
Similar to other diffractive devices, metasurfaces that locally change the propagation direction (e.g. lenses, beam deflectors, holograms) have negative chromatic dispersion [1,2,29,30]. This is because most of these devices are divided in Fresnel zones whose boundaries are designed for a specific wavelength [30,31]. This chromatic dispersion is an important limiting factor in many applications and its control is of great interest. Metasurfaces with zero and positive dispersion would be useful for making achromatic singlet and doublet lenses, and the larger-than-regular dispersion of hyper-dispersive metasurface gratings would enable high resolution spectrometers. We emphasize that the devices with zero chromatic dispersion discussed here are fundamentally different from the multiwavelength metasurface gratings and lenses recently reported [30][31][32][33][34][35][36][37][38][39][40]. Multiwavelength devices have several diffraction orders, which result in lenses (gratings) with the same focal length (deflection angle) at a few discrete wavelengths. However, at each of these focal distances (deflection angles), the multi-wavelength lenses (gratings) exhibit the regular negative diffractive chromatic dispersion (see [30,31], Supplementary Section S3 and Fig. S1).

II. THEORY
Here we argue that simultaneously controlling the phase imparted by the meta-atoms composing the metasurface (φ) and its derivative with respect to frequency ω (φ = ∂φ/∂ω which we refer to as chromatic phase dispersion or dispersion for brevity) makes it possible to dramatically alter the fundamental chromatic dispersion of diffractive components. This, in effect, is equivalent to simultaneously controlling the "effective refractive index" and "chromatic dispersion" of the meta-atoms. We have used this concept to demonstrate metasurface focusing mirrors with zero dispersion [41] in near IR. More recently, the same structure as the one used in [41] (with titanium dioxide replacing α:Si) was used to demonstrate achromatic reflecting mirrors in the visible [42].
Using the concept introduced in [41], here we experimentally show metasurface gratings and focusing mirrors that have positive, zero, and hyper chromatic dispersions. We also demonstrate an achromatic focusing mirror with a highly diminished focal length chromatic dispersion, resulting in an almost three times increase in its operation bandwidth.
First, we consider the case of devices with zero chromatic dispersion. In general for truly frequency independent operation, a device should impart a constant delay for different frequencies (i.e. demonstrate a true time delay behavior), similar to a refractive device made of a nondispersive material [1]. Therefore, the phase profile will be proportional to the frequency: φ(x, y; ω) = ωT (x, y), where ω = 2πc/λ is the angular frequency (λ: wavelength, c: speed of light) and T (x, y) determines the function of the device (for instance T (x, y) = −x sin θ 0 /c for a grating that deflects light by angle θ 0 ; T (x, y) = − x 2 + y 2 + f 2 /c for a spherical-aberration-free lens with a focal distance f ). Since the phase profile is a linear function of ω, it can be realized using a metasurface composed of meta-atoms that control the phase φ(x, y; ω 0 ) = T (x, y)ω 0 and its dispersion φ = ∂φ(x, y; ω)/∂ω = T (x, y). The bandwidth of dispersionless operation corresponds to the frequency interval over which the phase locally imposed by the meta-atoms is linear with frequency ω. For gratings or lenses, a large device size results in a large |T (x, y)|, which means that the meta-atoms should impart a large phase dispersion. Since the phase values at the center wavelength λ 0 = 2πc/ω 0 can be wrapped into the 0 to 2π interval, the meta-atoms only need to cover a rectangular region in the phase-dispersion plane bounded by φ = 0 and 2π lines, and φ = 0 and φ max lines, where φ max is the maximum required dispersion which is related to the device size (see Supplementary Section S5 and Fig. S2). The required phase-dispersion coverage means that, to implement devices with various phase profiles, for each specific value of the phase we need various meta-atoms providing that specific phase, but with different dispersion values.
Considering the simple case of a flat dispersionless lens (or focusing mirror) with radius R, we can get some intuition to the relations found for phase and dispersion. Dispersionless operation over a certain bandwidth ∆ω means that the device should be able to focus a transform limited pulse with bandwidth ∆ω and carrier frequency ω 0 to a single spot located at focal length f [ Fig. 2 |(a)]. To implement this device, part of the pulse hitting the lens at a distance r from its center needs to experience a pulse delay (i.e. group delay t g = ∂φ/∂ω) smaller by ( r 2 + f 2 − f )/c than part of the pulse hitting the lens at its center. This ensures that parts of the pulse hitting the lens at different locations arrive at the focus at the same time. Also, the carrier delay (i.e. phase delay t p = φ(ω 0 )/ω 0 ) should also be adjusted so that all parts of the pulse interfere constructively at the focus. Thus, to implement this phase delay and group delay behavior, the lens needs to be composed of elements, ideally with sub-wavelength size, that can provide the required phase delay and group delay at different locations. For a focusing mirror, these elements can take the form of sub-wavelength one-sided resonators, where the group delay is related to the quality factor Q of the resonator (see Supplementary Section S7) and the phase delay depends on the resonance frequency. We note that larger group delays are required for lenses with larger radius, which means that elements with higher quality factors are needed. If the resonators are single mode, the Q imposes an upper bound on the maximum bandwidth ∆ω of the pulse that needs to be focused.
The operation bandwidth can be expanded by using one-sided resonators with multiple resonances that partially overlap. As we will show later in the paper, these resonators can be implemented using silicon nano-posts backed by a reflective mirror.
For instance, the parameter ξ(ω) can be the deflection angle of a diffraction grating θ(ω) or the focal length of a diffractive lens f (ω). As we show in the Supplementary Section S4, to independently control the parameter ξ(ω) and its chromatic dispersion ∂ξ/∂ω at ω = ω 0 , we need to control the phase dispersion at this frequency in addition to the phase. The required dispersion for a certain parameter value ξ 0 = ξ(ω 0 ), and a certain dispersion ∂ξ/∂ω| ω=ω 0 is given by: This dispersion relation is valid over a bandwidth where a linear approximation of ξ(ω) is valid.
One can also use Fermat's principle to get similar results to Eq. 10 for the local phase gradient and its frequency derivative (see Supplementary Section S6).
We note that discussing these types of devices in terms of phase φ(ω) and phase dispersion ∂φ/∂ω, which we mainly use in this paper, is equivalent to using the terminology of phase delay (t p = φ(ω 0 )/ω 0 ) and group delay (t g = ∂φ/∂ω Assuming hypothetical meta-atoms that provide independent control of phase and dispersion up to a dispersion of −150 Rad/µm (to adhere to the commonly used convention, we report the dispersion in terms of wavelength) at the center wavelength of 1520 nm, we have designed and simulated four gratings with different chromatic dispersions (see Supplementary Section S1 for details). The simulated deflection angles as functions of wavelength are plotted in Fig. 2 |(d). All gratings are 150 µm wide, and have a deflection angle of 10 degrees at their center wavelength of 1520 nm. The positive dispersion grating exhibits a dispersion equal in absolute value to the negative dispersion of a regular grating with the same deflection angle, but with an opposite sign. The hyper-dispersive design is three times more dispersive than the regular grating, and the dispersionless beam deflector shows almost no change in its deflection angle. Besides for all nano-post side lengths. The operation of the nano-post meta-atoms is best intuitively understood as truncated multi-mode waveguides with many resonances in the bandwidth of interest [28,44]. By going through the nano-post twice, light can obtain larger phase shifts compared to the transmissive operation mode of the metasurface (i.e. without the metallic reflector). The metallic reflector keeps the reflection amplitude high for all sizes, which makes the use of high quality factor resonances possible. As discussed in Section 2, high quality factor resonances are necessary for achieving large dispersion values, because, as we have shown in Supplementary Section S7, dispersion is given by φ ≈ −Q/λ 0 , where Q is the quality factor of the resonance.
Using the dispersion-phase parameters provided by this metasurface, we designed four grat- with dispersion twice as large as a regular mirror with the same focal distance, and the hyper-dispersive mirror has a negative dispersion three and a half times larger than a regular one. The zero dispersion mirror shows a significantly reduced dispersion, while the hyper-dispersive one shows a highly enhanced dispersion. The positive mirror shows the expected dispersion in the ∼1470 to 1560 nm range.
As an application of diffractive devices with dispersion control, we demonstrate a sphericalaberration-free focusing mirror with increased operation bandwidth. For brevity, we call this device dispersionless mirror. Since the absolute focal distance change is proportional to the focal distance itself, a relatively long focal distance is helpful for unambiguously observing the change in the device dispersion. Also, a higher NA value is preferred because it results in a shorter depth The dispersionless mirror, however, shows a highly diminished chromatic dispersion. Besides, as seen from the focal plane intensity measurements, while the dispersionless mirrors are in focus in the 850 µm plane throughout the measured bandwidth, the regular mirror is in focus only from 1500 to 1550 nm (see Supplementary Figs. S13 and S14 for complete measurement results, and the Strehl ratios). Focusing efficiencies, defined as the ratio of the optical power focused by the mirrors to the power incident on them, were measured at different wavelengths for the regular and dispersionless mirrors (see Supplementary Section S1 for details). The measured efficiencies were normalized to the efficiency of the regular metasurface mirror at its center wavelength of 1520 nm (which is estimated to be ∼80%-90% based on Fig. 3 |, measured grating efficiencies, and our previous works [16]). The normalized efficiency of the dispersionless mirror is between 50% and 60% in the whole wavelength range and shows no significant reduction in contrast to the regular metasurface mirror.

V. DISCUSSION AND CONCLUSION
The reduction in efficiency compared to a mirror designed only for the center wavelength (i.e. Another method to address this issue is the Euclidean distance minimization method that was used in the design process of the devices presented here. In conclusion, we demonstrated that independent control over phase and dispersion of metaatoms can be used to engineer the chromatic dispersion of diffractive metasurface devices over continuous wavelength regions. This is in effect similar to controlling the "material dispersion" of meta-atoms to compensate, over-compensate, or increase the structural dispersion of diffractive devices. In addition, we developed a reflective dielectric metasurface platform that provides this independent control. Using this platform, we experimentally demonstrated gratings and focusing mirrors exhibiting positive, negative, zero, and enhanced dispersions. We also corrected the chromatic aberrations of a focusing mirror resulting in a ∼3 times bandwidth increase (based on an Strehl ratio > 0.6, see Supplementary Fig. S14). In addition, the introduced concept of metasurface design based on dispersion-phase parameters of the meta-atoms is general and can also be used for developing transmissive dispersion engineered metasurface devices.               (d) 30        The setup used to measure the efficiencies of the gratings. The power meter was placed at a long enough distance such that the other diffraction orders fell safely outside its active aperture area.     Extended Data Figure 17 | Schematic of a generic metasurface. The metasurface is between two uniform materials with wave impedances of η 1 and η 2 , and it is illuminated with a normally incident plane wave from the top side. Virtual planar boundaries Γ 1 and Γ 2 are used for calculating field integrals on each side of the metasurface.

S1. MATERIALS AND METHODS
Simulation and design.
The gratings with different dispersions discussed in Fig. 2(d) were designed using hypothetical meta-atoms that completely cover the required region of the phase-dispersion plane. We assumed that the meta-atoms provide 100 different phase steps from 0 to 2π, and that for each phase, If the actual meta-atoms provided an exactly linear dispersion (i.e. if their phase was exactly linear with frequency over the operation bandwidth), one could use the required values of the phase and dispersion at each lattice site to choose the best meta-atom (knowing the coordinates of one point on a line and its slope would suffice to determine the line exactly). The phases of the actual meta-atoms, however, do not follow an exactly linear curve [ Fig. 3(d)]. Therefore, to minimize the error between the required phases, and the actual ones provided by the meta-atoms, we have used a minimum weighted Euclidean distance method to design the devices fabricated and tested in the manuscript: at each point on the metasurface, we calculate the required complex reflection at eight wavelengths (1450 nm to 1590 nm, at 20 nm distances). We also calculate the complex reflection provided by each nano-post at the same wavelengths. To find the best meta-atom for each position, we calculate the weighted Euclidean distance between the required reflection vector, and the reflection vectors provided by the actual nano-posts. The nano-post with the minimum distance is chosen at each point. As a result, the chromatic dispersion is indirectly taken into account, not directly. The weight function can be used to increase or decrease the importance of each part of the spectrum depending on the specific application. In this work, we have chosen an inverted Gaussian weight function (exp((λ − λ 0 ) 2 /2σ 2 ), λ 0 = 1520 nm, σ = 300 nm) for all the devices to slightly emphasize the importance of wavelengths farther from the center. In addition, we have also Reflection amplitude and phase of the meta-atoms were found using rigorous coupled wave analysis technique [45]. For each meta-atom size, a uniform array on a subwavelength lattice was simulated using a normally incident plane wave. The subwavelength lattice ensures the existence of only one propagating mode which justifies the use of only one amplitude and phase for describing the optical behavior at each wavelength. In the simulations, the amorphous silicon layer was assumed to be 725 nm thick, the SiO 2 layer was 325 nm, and the aluminum layer was 100 nm The measurement setup is shown in Fig. 8 |(a). Light emitted from a tunable laser source (Photonetics TUNICS-Plus) was collimated using a fiber collimation package (Thorlabs F240APC-1550), passed through a 50/50 beamsplitter (Thorlabs BSW06), and illuminated the device. For grating measurements a lens with a 50 mm focal distance was also placed before the grating at a distance of ∼45 mm to partially focus the beam and reduce the beam divergence after being deflected by the grating in order to decrease the measurement error (similar to Fig. 8 |(b)). The light reflected from the device was redirected using the same beamsplitter, and imaged using a custom built microscope. The microscope consists of a 50X objective (Olympus LMPlanFL N, NA=0.5), a tube lens with a 20 cm focal distance (Thorlabs AC254-200-C-ML), and an InGaAs camera (Sensors Unlimited 320HX-1.7RT). The grating deflection angle was found by calculating the center of mass for the deflected beam imaged 3 mm away from the gratings surface. For efficiency measurements of the focusing mirrors, a flip mirror was used to send light towards an iris (2 mm diameter, corresponding to an approximately 40 µm iris in the object plane) and a photodetector (Thorlabs PM100D with a Thorlabs S122C head). The efficiencies were normalized to the efficiency of the regular mirror at its center wavelength by dividing the detected power through the iris by the power measured for the regular mirror at its center wavelength. The measured intensities were up-sampled using their Fourier transforms in order to achieve smooth intensity profiles in the focal and axial planes. To measure the grating efficiencies, the setup shown in Supporting Information Fig. 8 |(b) was used, and the photodetector was placed ∼50 mm away from the grating, such that the other diffraction orders fall outside its active area. The efficiency was found by calculating the ratio of the power deflected by the grating to the power normally reflected by the aluminum reflector in areas of the sample with no grating. The beam-diameter on the grating was calculated using the setup parameters, and it was found that ∼84% of the power was incident on the 90 µm wide gratings. This number was used to correct for the lost power due to the larger size of the beam compared to the grating.

S2. CHROMATIC DISPERSION OF DIFFRACTIVE DEVICES.
Chromatic dispersion of a regular diffractive grating or lens is set by its function. The grating momentum for a given order of a grating with a certain period is constant and does not change with changing the wavelength. If we denote the size of the grating reciprocal lattice vector of interest by k G , we get: where θ is the deflection angle at a wavelength λ for normally incident beam. The chromatic angular dispersion of the grating ( dθ/dλ) is then given by: and in terms of frequency: Therefore, the dispersion of a regular grating only depends on its deflection angle and the wavelength. Similarly, focal distance of one of the focal points of diffractive and metasurface lenses changes as df /dλ = −f /λ (thus df /dω = f /ω ( [1,30,31]).

S3. CHROMATIC DISPERSION OF MULTIWAVELENGTH DIFFRACTIVE DEVICES.
As it is mentioned in the main text, multiwavelength diffractive devices ( [30,31,33]) do not change the dispersion of a given order in a grating or lens. They are essentially multi-order gratings or lenses, where each order has the regular (negative) diffractive chromatic dispersion. These devices are designed such that at certain distinct wavelengths of interest, one of the orders has the desired deflection angle or focal distance. If the blazing of each order at the corresponding wavelength is perfect, all of the power can be directed towards that order at that wavelength.
However, at wavelengths in between the designed wavelengths, where the grating or lens is not corrected, the multiple orders have comparable powers, and show the regular diffractive dispersion.
This is schematically shown in Fig. 1 |(a). Figure 1 |(b) compares the chromatic dispersion of a multi-wavelength diffractive lens to a typical refractive apochromatic lens. Here we present the general form of equations for the dispersion engineered metasurface diffractive devices. We assume that the function of the device is set by a parameter ξ(ω), where we have explicitly shown its frequency dependence. For instance, ξ might denote the deflection angle of a grating or the focal distance of a lens. The phase profile of a device with a desired ξ(ω) is given by φ(x, y, ξ(ω); ω) = ωT (x, y, ξ(ω)), which is the generalized form of the Eq. (1). We are interested in controlling the parameter ξ(ω) and its dispersion (i.e. derivative) at a given frequency ω 0 . ξ(ω) can be approximated as ξ(ω) ≈ ξ 0 + ∂ξ/∂ω| ω=ω 0 (ω − ω 0 ) over a narrow bandwidth around ω 0 . Using this approximation, we can rewrite 7 as At ω 0 , this reduces to φ(x, y; ω)| ω=ω 0 = ω 0 T (x, y, ξ 0 ), (9) and the phase dispersion at ω 0 is given by Based on Eqs. (9) and (10) the values of ξ 0 and ∂ξ/∂ω| ω=ω 0 can be set independently, if the phase φ(x, y, ω 0 ) and its derivative ∂φ/∂ω can be controlled simultaneously and independently.
Therefore, the device function at ω 0 (determined by the value of ξ 0 ) and its dispersion (determined by ∂ξ/∂ω| ω=ω 0 ) will be decoupled. The zero dispersion case is a special case of Eq. (10) with ∂ξ/∂ω| ω=ω 0 = 0. In the following we apply these results to the special cases of blazed gratings and spherical-aberration-free lenses (also correct for spherical-aberration-free focusing mirrors).
For a 1-dimensional conventional blazed grating we have ξ = θ (the deflection angle), and T = −x sin(θ). Therefore the phase profile with a general dispersion is given by: where D = ∂θ/∂ω| ω=ω 0 = νD 0 , and D 0 = − tan(θ 0 )/ω 0 is the angular dispersion of a regular grating with deflection angle θ 0 at the frequency ω 0 . We have chosen to express the generalized dispersion D as a multiple of the regular dispersion D 0 with a real number ν to benchmark the change in dispersion. For instance, ν = 1 corresponds to a regular grating, ν = 0 represents a dispersionless grating, ν = −1 denotes a grating with positive dispersion, and ν = 3 results in a grating three times more dispersive than a regular grating (i.e. hyper-dispersive). Various values of ν can be achieved using the method of simultaneous control of phase and dispersion of the metaatoms, and thus we can break this fundamental relation between the deflection angle and angular dispersion. The phase derivative necessary to achieve a certain value of ν is given by: or in terms of wavelength: For a spherical-aberration-free lens we have ξ = f and T (x, y, f ) = − x 2 + y 2 + f 2 /c.
Again we can approximate f with its linear approximation f (ω) = f 0 + D(ω − ω 0 ), with D = ∂f /∂ω| ω=ω 0 denoting the focal distance dispersion at ω = ω 0 . The regular dispersion for such a lens is given by D 0 = f 0 /ω 0 . Similar to the gratings, we can write the more general form for the focal distance dispersion as D = νD 0 , where ν is some real number. In this case, the required phase dispersion is given by: ∂φ(x, y; ω) ∂ω which can also be expressed in terms of wavelength: Since the maximum achievable dispersion is limited by the meta-atom design, it is important to find a relation between the maximum dispersion required for implementation of a certain metasurface device, and the device parameters (e.g. size, focal distance, deflection angle, etc.). Here we find these maxima for the cases of gratings and lenses with given desired dispersions.
For the grating case, it results from Eq. (13) that the maximum required dispersion is given by where X is the length of the grating, and k 0 = 2π/λ 0 is the wavenumber. It is important to note that based on the value of ν, the sign of the meta-atom dispersion changes. However, in order to ensure a positive group velocity for the meta-atoms, the dispersions should be negative. Thus, if 1 − ν > 0, a term should be added to make the dispersion values negative. We can always add a term of type φ 0 = kL 0 to the phase without changing the function of the device. This term can be used to shift the required region in the phase-dispersion plane. Therefore, it is actually the difference between the minimum and maximum of Eqs. 13 and 15 that sets the maximum required dispersion. Using a similar procedure, we find the maximum necessary dispersion for a spherical-aberration-free lens as where f is the focal distance of the lens, and Θ = (f 2 + R 2 )/f 2 = 1/(1 − NA 2 ) ( R: lens radius, NA: numerical aperture). log [φ max /(−k 0 f /λ 0 )] is plotted in Fig. 2 |(a) as a function of NA and ν. In the simpler case of dispersionless lenses (i.e. ν = 0), Eq. (17) can be further simplified to where R is the lens radius and the approximation is valid for small values of NA. The maximum required dispersion for the dispersionless lens is normalized to −k 0 R/λ 0 and is plotted in Supporting Information Fig. 2 |(b) as a function of NA.

S6. FERMAT'S PRINCIPLE AND THE PHASE DISPERSION RELATION.
Phase only diffractive devices can be characterized by a local grating momentum (or equivalently phase gradient) resulting in a local deflection angle at each point on their surface. Here we consider the case of a 1D element with a given local phase gradient (i.e. φ x = ∂φ/∂x) and use Fermat's principle to connect the frequency derivative of the local deflection angle (i.e. chromatic dispersion) to the frequency derivative of φ x (i.e. ∂φ x /∂ω). For simplicity, we assume that the illumination is close to normal, and that the element phase does not depend on the illumination angle (which is in general correct in local metasurfaces and diffractive devices). Considering Fig.   16 |(a), we can write the phase acquired by a ray going from point A to point B, and passing the interface at x as: Φ(x, ω) = ω c [n 1 x 2 + y A 2 + n 2 (d − x) 2 + y B 2 ] + φ(x, ω) To minimize this phase we need: For this minimum to occur at point O (i.e. x = 0): which is a simple case of the diffraction equation, and where r = d 2 + y B 2 is the OB length. At ω + dω, we get the following phase for the path from A to B' [ Fig. 16 |(b)]: Φ(x, ω + dω) = ω + dω c [n 1 x 2 + y A 2 + n 2 (d − x + dx) 2 + (y B + dy) 2 ] + φ(x, ω + dω) where we have chosen B' such that OB and OB' have equal lengths. Minimizing the path passing through O: φ x (ω + dω) = ω + dω c n 2 (d + dx) r = n 2 (ω + dω) c sin(θ(ω + dω)) subtracting 21 from 23, and setting φ x (ω + dω) − φ x (ω) = ∂φx ∂ω dω, we get: ∂φ x ∂ω = n 2 c sin(θ(ω)) + dθ dω n 2 ω c cos(θ(ω)).
One can easily recognize the similarity between 24 and 10. Here we show that the phase dispersion of a meta-atom is linearly proportional to the stored optical energy in the meta-atoms, or equivalently, to the quality factor of the resonances supported by the mata-atoms. To relate the phase dispersion of transmissive or reflective meta-atoms to the stored optical energy, we follow an approach similar to the one taken in chapter 8 of [47] for finding the dispersion of a single port microwave circuit. We start from the frequency domain Maxwell's equations: and take the derivative of the Eq. 25 with respect to frequency: Multiplying Eq. 26 by H * and the conjugate of Eq. 27 by ∂E/∂ω, and subtracting the two, we Similarly, multiplying Eq. 27 by E * and the conjugate of Eq. 26 by ∂H/∂ω, and subtracting the two we find: Subtracting Eq. 29 from Eq. 28 we get: Integrating both sides of Eq. 30, and using the divergence theorem to convert the left side to a surface integral leads to: where U is the total electromagnetic energy inside the volume V , and ∂V denotes the surrounding surface of the volume. Now we consider a metasurface composed of a subwavelength periodic array of meta-atoms as shown in Fig. 17 |. We also consider two virtual planar boundaries Γ 1 and Γ 2 on both sides on the metasurface (shown with dashed lines in Fig. 17 |). The two virtual boundaries are considered far enough from the metasurface that the metasurface evanescent fields die off before reaching them. Because the metasurface is periodic with a subwavelength period and preserves polarization, we can write the transmitted and reflected fields at the virtual boundaries in terms of only one transmission t and reflection r coefficients. The fields at these two boundaries are given by: where E is the input field, E 1 and E 2 are the total electric fields at Γ 1 and Γ 2 , respectively, and η 1 and η 2 are wave impedances in the materials on the top and bottom of the metasurface.
Inserting fields from Eq. 32 to Eq. 31, and using the uniformity of the fields to perform the integration over one unit of area, we get: whereŨ is the optical energy per unit area that is stored in the metasurface layer. For a loss-less metasurface that is totally reflective (i.e. t = 0 and r = e iφ ), we obtain: where we have used P in = |E| 2 /η 1 to denote the per unit area input power. Finally, the dispersion can be expressed as: We used Eq. 35 throughout the work to calculate the dispersion from solution of the electric and magnetic fields at a single wavelength, which reduced simulation time by a factor of two. In addition, in steady state the input and output powers are equal P out = P in , and therefore we have: where we have assumed that almost all of the stored energy is in one single resonant mode, and Q