Chromatic Aberration in Metalenses

TL;DR

This paper introduces a mathematical framework to analyze chromatic aberration in metalenses, revealing that their dispersion in the visible spectrum is comparable to traditional lenses, which is crucial for optical design.

Contribution

It develops a generalized Snell's law and envelope concept to quantify dispersion in metalenses, providing new insights into their chromatic aberration behavior.

Findings

Chromatic aberration in metalenses is quantitatively similar to standard lenses.

A mathematical approach using generalized Snell's law explains phase discontinuities.

Dispersion magnitude in metalenses is characterized across the visible spectrum.

Abstract

This paper provides a mathematical approach to study chromatic aberration in metalenses. It is shown that radiation of a given wavelength is refracted according to a generalized Snell's law which together with the notion of envelope yields the existence of phase discontinuities. This is then used to establish a quantitative measure of dispersion in metalenses concluding that in the visible spectrum it has the same order of magnitude as for standard lenses.

Equations

$$n=\frac{c}{v}=\frac{2\pi c}{\lambda \omega }=\frac{{\lambda }_0}{\lambda }=\frac{k}{k_0}.$$

$$k_1|P-A|+k_2|B-P|-\varphi (P),$$

$$\nabla \left(k_1|P-A|+k_2|B-P|-\varphi (P)\right)\times \nabla \psi (P)=0.$$

$$\nabla \left(k_1|P-A|+k_2|B-P|-\varphi (P)\right)=k_1\frac{P-A}{|P-A|}-k_2\frac{B-P}{|B-P|}-\nabla \varphi (P),$$

$$\left(k_1\mathbf{x}-k_2\mathbf{m}\right)\times \nu =\nabla \varphi \times \nu .$$

$$k_0(n_1\mathbf{x}-n_2\mathbf{m})\times \nu =\nabla \varphi \times \nu ,$$

$$\left(n_1\mathbf{x}-n_2\mathbf{m}\right)\times \nu =\left(\nabla \left({\lambda }_0\varphi /2\pi \right)\right)\times \nu .$$

$$n_1\mathbf{x}-n_2\mathbf{m}=\mu \nu +\nabla \left({\lambda }_0\varphi /2\pi \right),$$

$$\mu =\left(n_1\mathbf{x}-\nabla \left({\lambda }_0\varphi /2\pi \right)\right)\cdot \nu -\sqrt{n_2^2-\left({\left|n_1\mathbf{x}-\nabla \left({\lambda }_0\varphi /2\pi \right)\right|}^2-{\left[\left(n_1\mathbf{x}-\nabla \left({\lambda }_0\varphi /2\pi \right)\right)\cdot \nu \right]}^2\right)}.$$

$${\left[\left(\mathbf{x}-\frac{\nabla \left({\lambda }_0\varphi /2\pi \right)}{n_1}\right)\cdot \nu \right]}^2\ge {\left|\mathbf{x}-\frac{\nabla \left({\lambda }_0\varphi /2\pi \right)}{n_1}\right|}^2-{\left(\frac{n_2}{n_1}\right)}^2.$$

$$\{\begin{array}{l}{f}_{x_1}(x)=V_1(x,u(x))+V_3(x,u(x))u_{x_1}(x)\\ f_{x_2}(x)=V_2(x,u(x))+V_3(x,u(x))u_{x_2}(x),\end{array}$$

$${\partial }_{x_2}\left(V_1(x,u(x))+V_3(x,u(x))u_{x_1}(x)\right)={\partial }_{x_1}\left(V_2(x,u(x))+V_3(x,u(x))u_{x_2}(x)\right),x\in \Omega .$$

$$V(x,u(x))\cdot (1,0,u_{x_1}(x))$$

$$V(x,u(x))\cdot (0,1,u_{x_2}(x))$$

$$P(x)=(x,u(x),f(x)),x\in B.$$

$$N(x)=\left(V(x,u(x)),-1\right),$$

$${\Pi }_x=\{y=(y_1,y_2,y_3,y_4)\in {\mathbb{R}}^4:\left(y-P(x)\right)\cdot N(x)=0\}.$$

$$F(y,x)=\left(y-P(x)\right)\cdot N(x)$$

$$G(y,x)=\left(F(y,x),\frac{\partial F}{\partial x_1}(y,x),\frac{\partial F}{\partial x_2}(y,x)\right).$$

$$G(y,x)=(0,0,0).$$

$$P_{x_1}(x)\cdot N(x)$$

$$\frac{\partial F}{\partial x_1}$$

$$\frac{\partial F}{\partial x_1}(P(x),x)=0.$$

$$\frac{\partial G}{\partial (y_4,x_1,x_2)}\left(P(x_0),x_0\right)\ne 0,$$

$$G(y_1,y_2,y_3,g_1(y_1,y_2,y_3),g_2(y_1,y_2,y_3),g_3(y_1,y_2,y_3))=(0,0,0),$$

$$F(y_1,y_2,y_3,g_1(y_1,y_2,y_3),g_2(y_1,y_2,y_3),g_3(y_1,y_2,y_3))=0,$$

$$0=F_{y_1}+F_{y_4}(g_1{)}_{y_1}+F_{x_1}(g_2{)}_{y_1}+F_{x_2}(g_3{)}_{y_1}.$$

$$F_{y_1}(y,x)$$

$$F_{y_4}(y,x)$$

$$V_1(x,u(x))=(g_1{)}_{y_1}(x,u(x)),x\in U.$$

$$V_2(x,u(x))=(g_1{)}_{y_2}(x,u(x)),V_3(x,u(x))=(g_1{)}_{y_3}(x,u(x)).$$

Full text

1. 1 Introduction

2. 2 Generalized Snell’s law

3. 3 Construction of gradient fields using the notion of envelope

4. 4 Existence of phase discontinuities focusing a collimated beam into a given set of directions

5. 4.1 Existence of phases when ${\bf m}$ is constant

6. 4.2 Existence of phases focusing all rays into one point

7. 5 Analysis of chromatic dispersion in metalenses

8. 5.1 Dispersion of dichromatic light

9. 5.2 Analysis of the chromatic aberration for a plane metasurface

10. 5.3 Analysis of the chromatic aberration for a general radial function $u$

11. 6 Comparison with chromatic aberration in standard lenses

1. Introduction

The shaping of the light wavefronts with standard lenses relies on gradual phase changes accumulated along the optical path. Metalenses are ultra thin surfaces which use nano structures to focus light in imaging. They introduce abrupt phase shifts over the scale of the wavelength along the optical path to bend light in unusual ways. These nano structures are composed of arrays of tiny pillars, rings, and other arrangements of materials, which work together to manipulate light waves as they pass by. The subject of metalenses is a current important area of research, one of the nine runners-up for Science’s Breakthrough of the Year 2016 [S], and is potentially useful in imaging applications. Metalenses are thinner than a sheet of paper and far lighter than glass, and they could revolutionize optical imaging devices from microscopes to virtual reality displays and cameras, including the ones in smartphones, see [S] and [LSW*+*19].

Polychromatic light is the superposition of radiation with more than one wavelength, and when it passes from an homogeneous medium to another, each wavelength is refracted at a different angle causing the phenomenon of dispersion. The failure to focus radiation composed of several wavelengths into one point is called chromatic aberration. It is our purpose in this paper to analyze this phenomenon in metalenses and give a measure of the dispersion. The design of metalenses so that chromatic aberration is controlled is an active area of research [AKG*+*12],[CZS*+*18], and [GRB*+*18]. As well as for the design of standard lenses, this is an important challenge, see [LC18] and [BMS*+*19]. These thin surfaces are made with sub wavelength phase shifters, creating a phase discontinuity on the interface yielding to a more general law of refraction, see Section 2. These components can be also be arranged to design graphene-based metasurfaces that can be actively tuned between different regimes of operation, see [BGN*+*18]. These challenging tasks have been recently an active area of research in Optical Engineering, see the comprehensive reviews [YC14], [CTY16], and [ZKL*+*17].

Mathematically, a metalens is a pair $(\Gamma,\phi)$ where $\Gamma$ is a surface in 3-d space given by the graph of a $C^{2}$ function $u$, and $\phi$ is a $C^{1}$ function defined in a small neighborhood of $\Gamma$ called phase discontinuity. In this paper, we first study the existence of metalenses refracting collimated monochromatic radiation into an arbitrary field of directions. We next analyze the behavior of those metalenses when the incoming radiation is polychromatic. More precisely, we are given a set $\Omega\subset\mathbb{R}^{2}$, a function $u\in C^{2}(\Omega)$ and a $C^{1}$ field of unit directions ${\bf m}(x)$. Let $\Gamma=\{(x,u(x)),x\in\Omega\}$ be the graph of $u$ and assume it is surrounded by vacuum. Our goal is to find a $C^{1}$ function $\phi$ defined in a neighborhood of $\Gamma$ so that for every $x\in\Omega$, each vertical monochromatic ray with wavelength $\lambda_{0}$ emitted from $x\in\Omega$ is refracted at $(x,u(x))$ into the direction ${\bf m}(x)$; see for example Figure 1(a). We find in Section 4 necessary and sufficient conditions on the field ${\bf m}$ and the function $u$ so that the phase discontinuity $\phi$ exists in a neighborhood of $(x_{0},u(x_{0}))$ for each $x_{0}\in\Omega$, see Theorem 4.1. To do this, using first the generalized Snell law from Section 2 we see that the values $\nabla\phi(x,u(x))$ are determined. We next apply the method from Section 3 to extend $\phi$ to a neighborhood of $(x,u(x))$. This construction is then applied when ${\bf m}$ is a constant field, and when ${\bf m}$ points toward a point $P_{0}=(0,0,p_{0})$, see Sections 4.1, and 4.2. If $\Gamma$ is the graph of any smooth radial function, and ${\bf m}$ points towards $P_{0}=(0,0,p_{0})$, then there exists also a phase $\phi$ in a neighborhood of $(0,u(0))$ such that the metalens $(\Gamma,\phi)$ focuses all vertical rays with wavelength $\lambda_{0}$ into $P_{0}$; see Remark 4.4. We then study the behavior of these metalenses when the incoming rays have wavelength $\lambda$ instead of $\lambda_{0}$. We prove that that if $\Gamma$ is a horizontal plane, Section 5.2, or more generally if $\Gamma$ is the graph of a radial function, Section 5.3, then all vertical rays with wavelength $\lambda$ are focused into the points $P_{\lambda}(x)=(0,0,p_{\lambda}(x))$ depending on $|x|$. We then estimate in both cases the chromatic aberration $\left|P_{0}-P_{\lambda}(x)\right|$; see Theorem 5.1 for the planar case and (5.11) when $u$ is radial.

Finally, in Section 6, we analyze a similar problem with a standard lens. In this case, it is known that the surfaces separating two media $n_{1}$ and $n_{2}$ with $n_{1}>n_{2}$ that focus vertical radiation in medium $n_{1}$ with wavelength $\lambda$ into a fixed point in medium $n_{2}$ are hyperboloids, see [GH09]. It is then shown that the error for chromatic aberration in metalenses for visible wavelengths is of the same order of magnitude as for standard lenses.

Finally, we mention that the phase discontinuity functions needed to design metalenses for various refraction and reflection problems with prescribed distributions of energy satisfy partial differential equations of Monge-Ampère type which are derived and studied in [GP18]. These type of equations appeared also naturally in solving problems involving aspherical lenses, see [GH09], [AGT16], [GS16], [GS18], and references therein.

2. Generalized Snell’s law

The refractive index of a medium corresponding to an electromagnetic wave with frequency $\omega$ is given by $n=\dfrac{c}{v}$, where $c$ is the velocity of the wave in vacuum and $v$ its apparent velocity in the medium. $v$ depends on the wave and on the material [F, Vol. 1, Chap. 31], see formula (31.19) there for $n$ that highlights the dependence of $n$ on $\omega$ and on the material. See also [H, Chap. 3] and [BW59, Sec. 2.3.4]. Note that the frequency $\omega$ is a characteristic of the wave and is independent of the material traversed. The wavelength of the wave in the medium is the distance between two consecutive crests and is given by $\lambda=\dfrac{2\pi}{\omega}v$. The wave number $k=\dfrac{2\pi}{\lambda}$ is the number of waves per unit distance, $\lambda_{0}$ denotes the wavelength of a wave with the same frequency $\omega$ when traveling in vacuum, and $k_{0}=\dfrac{2\pi}{\lambda_{0}}$ is the corresponding wave number. We have $\lambda_{0}=\dfrac{2\pi c}{\omega}$, then

[TABLE]

In [GPS17], Snell’s law has been generalized for metasurfaces using wave fronts introducing a phase shift along the optical path. Using Fermat principle, we next derive here this law in a form needed for our purposes. Assume $\Gamma$ is a surface, separating media $I$ and $II$, and given by the level set $\psi(x,y,z)=0$, with $\psi$ a $C^{1}$ function. Assume a phase discontinuity $\phi$ is defined in a neighborhood of the surface $\Gamma$. A light wave with frequency $\omega$ emitted from a point $A$ in medium $I$ strikes $\Gamma$ at the point $P(x,y,z)$ and is then refracted into the point $B$ in medium $II$. Let $n_{1}$ and $n_{2}$ be the refractive indices of media $I$ and $II$ corresponding to $\omega$, and let $k_{1},k_{2}$ be the corresponding wave numbers. By Fermat’s principle of stationary phase we have that $P(x,y,z)$ is a critical point to the function

[TABLE]

and since $P\in\Gamma$, then we also have that $\psi(P)=0$. Therefore by Lagrange multipliers, we have that

[TABLE]

Since $\nabla\psi$ is parallel to the normal $\nu$ to $\Gamma$ at the point $P$, and

[TABLE]

denoting the unit directions of the incident and refracted rays by ${\bf x}=\dfrac{{P-A}}{|P-A|}$ and ${\bf m}=\dfrac{{B-P}}{|B-P|}$, respectively, we then get the following form of Snell’s law for the metalens $(\Gamma,\phi)$

[TABLE]

From (2.1), $k_{i}=n_{i}k_{0}$ and so (2.2) can be re-written as

[TABLE]

We then obtain

[TABLE]

This is clearly equivalent to

[TABLE]

for some $\mu$. From [GPS17, Formula (11)], the value of $\mu$ can be calculated:

[TABLE]

In order to avoid total internal reflection, the term inside the square root in (2.5) must be non-negative, that is,

[TABLE]

3. Construction of gradient fields using the notion of envelope

Let $\Omega\subset{\mathbb{R}}^{2}$ be an open simply connected domain, $u\in C^{2}(\Omega)$, and let $\Gamma$ be the graph of $u$, i.e., $\Gamma=\{(x,u(x)):x\in\Omega\}$; $x=(x_{1},x_{2})$. A field $V=(V_{1},V_{2},V_{3}):\Gamma\to{\mathbb{R}}^{3}$ is given on $\Gamma$ such that $V(x,u(x))\in C^{1}(\Omega)$. Here we answer the following question: When is it possible to have a $C^{1}$ function $\phi$ defined in a neighborhood $\Gamma$ such that $\nabla\phi=V$ on $\Gamma$?

Suppose such a $\phi$ exists, that is, $\nabla\phi(x,u(x))=V(x,u(x))$, $x=(x_{1},x_{2})\in\Omega$, and consider $f(x)=\phi(x,u(x))$. Since $\phi=\phi(x_{1},x_{2},x_{3})$ is differentiable and defined in a neighborhood of $\Gamma$, $f_{x_{i}}(x)=\phi_{x_{i}}(x,u(x))+\phi_{x_{3}}(x,u(x))\,u_{x_{i}}(x)=V_{i}(x,u(x))+V_{3}(x,u(x))\,u_{x_{i}}(x)$, $i=1,2$, so $f$ satisfies the system

[TABLE]

for $x\in\Omega$. Since $V(x,u(x))$ is $C^{1}$ and $u\in C^{2}$, it follows that $f\in C^{2}$ and so $f_{x_{1}x_{2}}(x)=f_{x_{2}x_{1}}(x)$ which is equivalent to

[TABLE]

Therefore, if such a $\phi$ exists, then $V$ must satisfy (3.2).

Vice versa, if we are now given a field $V$ defined on $\Gamma$ and satisfying (3.2), we want to construct $\phi$ in a neighborhood of $\Gamma$ so that $\nabla\phi=V$ along $\Gamma$. Actually, we will give a sufficient condition on the field $V$ so that $\phi$ exists in a neighborhood of a point $\left(x_{0},u(x_{0})\right)\in\Gamma$. In order to do this, we will apply the implicit function theorem, for which we assume $V(x,u(x))$ to be $C^{2}$ in a neighborhood of $x_{0}$. Indeed, if $V$ satisfies (3.2) in a ball $B=B_{\delta}(x_{0})\subset\Omega$, then the system (3.1) is solvable in $B$, and let $f$ be a solution111There are many solutions to (3.1). However, given a point $x_{0}\in\Omega$ and a number $y_{0}\in{\mathbb{R}}$, there is a unique solution $f$ satisfying (3.1) with $f(x_{0})=y_{0}$.. We then have

[TABLE]

We seek a function $\phi$ defined in a neighborhood of the point $(x_{0},u(x_{0}))$ so that $\phi(x,u(x))=f(x)$, and $\nabla\phi(x,u(x))=V(x,u(x))$ in a neighborhood of $x_{0}$.

Let us then define $\mathcal{S}$ to be the surface in ${\mathbb{R}}^{4}$ given by the vector

[TABLE]

At each point $P(x)$ consider the vector

[TABLE]

and the plane $\Pi_{x}$ passing through $P(x)$ with normal $N(x)$, that is,

[TABLE]

For $x\in B$ and $y\in{\mathbb{R}}^{4}$ define the function

[TABLE]

and the map

[TABLE]

Consider then the system of equations

[TABLE]

We will construct $\phi$ by solving (3.6) using the implicit function theorem.

Claim 3.1**.**

For each $x\in B$, the vector $(P(x),x)$ solves the system (3.6).

Proof.

Clearly, from the definition of $F$, $F(P(x),x)=0$. From (3.3)

[TABLE]

Hence for each $(y,x)\in{\mathbb{R}}^{4}\times B$

[TABLE]

and so

[TABLE]

Similarly, from (3.4) $P_{x_{2}}(x)\cdot N(x)=0$ so we get $\dfrac{\partial F}{\partial x_{2}}(P(x),x)=0$, and the claim follows. ∎

Consider the point $\left(P(x_{0}),x_{0}\right)=\left(x_{0},u(x_{0}),f(x_{0}),x_{0}\right).$ From Claim 3.1, $G\left(P(x_{0}),x_{0}\right)=(0,0,0)$. From the differentiability assumptions on $u$ and $V(x,u(x))$, $G$ has continuous first order partial derivatives in a neighborhood of $(P(x_{0}),x_{0})$. If the Jacobian determinant

[TABLE]

then the implicit function theorem implies that there exist an open neighborhood $O\subseteq B\times{\mathbb{R}}$ of $\left(x_{0},u(x_{0})\right)$, an open neighborhood $W\subseteq{\mathbb{R}}\times B$ of $(f(x_{0}),x_{0})$, and unique $C^{1}$ functions $g_{1},g_{2},g_{3}:O\to W$ solving

[TABLE]

for all $(y_{1},y_{2},y_{3})\in O$.

We shall prove that the desired $\phi$ is $\phi=g_{1}$. Indeed, from the continuity of $u(x)$ and $f(x)$ in $B$ there exists a neighborhood $U\subseteq B$ of $x_{0}$ such that for every $x\in U$, we have $(x,u(x))\in O$, and $(f(x),x)\in W$. Hence, for $x\in U$ we have $(P(x),x)\in O\times W$, and from Claim 3.1 $G(P(x),x)=(0,0,0)$. Therefore, from the uniqueness of the functions $g_{1},g_{2},g_{3}$, it follows that $g_{1}(x,u(x))=f(x)$, $g_{2}(x,u(x))=x_{1},g_{3}(x,u(x))=x_{2}$, for every $x=(x_{1},x_{2})\in U$. Then, it remains to show that $\nabla g_{1}(x,u(x))=V(x,u(x))$ for $x\in U$, i.e., along $\Gamma$. In fact, since for $(y_{1},y_{2},y_{3})\in O$

[TABLE]

differentiating this identity with respect to $y_{1}$ yields

[TABLE]

From (3.8), $F_{x_{1}}=F_{x_{2}}=0$ at $\left(y_{1},y_{2},y_{3},g_{1}(y_{1},y_{2},y_{3}),g_{2}(y_{1},y_{2},y_{3}),g_{3}(y_{1},y_{2},y_{3})\right)$. Moreover for each $(y,x)\in{\mathbb{R}}^{4}\times B$, we have

[TABLE]

Since $g_{2}(x,u(x))=x_{1}$ and $g_{3}(x,u(x))=x_{2}$ for $x=(x_{1},x_{2})\in U$, substituting the obtained values of $F_{x_{1}},F_{x_{2}},F_{y_{1}},F_{y_{4}}$ in (3.10) yields

[TABLE]

Differentiating (3.9) with respect to $y_{2}$ and with respect to $y_{3}$, and proceeding similarly, yields

[TABLE]

Hence $\nabla g_{1}(x,u(x))=V(x,u(x))$ for $x\in U$.

We have then proved the following theorem.

Theorem 3.1**.**

Let $x_{0}\in\Omega$, $u\in C^{2}$ in a neighborhood of $x_{0}$, and let $V(x,u(x))$ be a field that is $C^{2}$ in a neighborhood of $x_{0}$ satisfying (3.2) in that neighborhood. If (3.7) holds, where $G$ is defined by (3.5), then there exists a function $\phi$ defined in a neighborhood of $(x_{0},u(x_{0}))$ such that $V=\nabla\phi$ in that neighborhood.

For our application, let us re write condition (3.7) in simpler terms. We have

[TABLE]

Since $F_{y_{4}}(y,x)=-1$,

[TABLE]

From the proof of Claim 3.1, $F_{x_{i}}(y,x)=(y-P(x))\cdot N_{x_{i}}(x)$, $i=1,2$, and differentiating yields $F_{x_{i}x_{j}}(y,x)=-P_{x_{j}}\cdot N_{x_{i}}+(y-P)\cdot N_{x_{i}x_{j}}$. Therefore (3.7) becomes

[TABLE]

4. Existence of phase discontinuities focusing a collimated beam into a given set of directions

We are given a $C^{1}(\Omega)$ unit field with non negative vertical component,

[TABLE]

corresponding to the set of directions where we want the radiation to be steered. Assume the medium surrounding $\Gamma$ is vacuum; $\Gamma$ is the graph of a function $u\in C^{2}(\Omega)$. We consider a vertical parallel beam of monochromatic light of wavelength $\lambda_{0}$. From each $x\in\Omega$ a ray with direction ${\bf e}=(0,0,1)$ strikes $\Gamma$ at the point $(x,u(x))$. The goal of this section is to apply the results from Section 3 to show existence of a phase discontinuity $\phi\in C^{1}$ defined in a neighborhood of $\Gamma$ so that $\nabla\phi$ is tangential to $\Gamma$, and for each $x\in\Omega$, the vertical ray with direction ${\bf e}$ is refracted by the metalens $(\Gamma,\phi)$ at $(x,u(x))$ into the direction ${\bf m}(x)$; see Figure 1(a).

Let $\nu:=\nu(x)=\dfrac{(-\nabla u,1)}{\sqrt{1+|\nabla u|^{2}}}$ be the outer unit normal to $\Gamma$ at each point $\left(x,u(x)\right)$. Since $m_{3}\geq 0$, it follows that $m(x)\cdot\nu\geq 0.$ A tangential phase discontinuity $\phi$ means $\nabla\phi(x,u(x))\cdot\nu=0$ for each $x\in\Omega$.

We shall prove the following theorem.

Theorem 4.1**.**

If a tangential phase discontinuity $\phi$ solving the problem above exists, then

[TABLE]

and ${\bf m}$ satisfies the condition

[TABLE]

for $x\in\Omega$.

Conversely, suppose ${\bf m}$ is $C^{2}$ and satisfies (4.2), let $h\in C^{2}(\Omega)$ satisfying

[TABLE]

and let $V(x,u(x))$ be given by (4.1) with $u\in C^{3}(\Omega)$. If

[TABLE]

at some $x_{0}\in\Omega$, then there exist a neighborhood $W$ of $\left(x_{0},u(x_{0})\right)$ and a function $\phi$ defined in $W$ satisfying

[TABLE]

for some constant $C$, with $\nabla\phi(x,u(x))=V(x,u(x))$ for $x$ in a neighborhood of $x_{0}$. In addition, for each $\left(x,u(x)\right)\in W$ the metalens $(\Gamma,\phi)$ refracts the ray with direction ${\bf e}$ striking $\Gamma$ at $(x,u(x))$ into the direction ${\bf m}(x)$.

Proof.

To prove the first part of the theorem, from the Snell law (2.3) with $n_{1}=n_{2}=1$, the phase must satisfy

[TABLE]

at all points $(x,u(x))$. Let us calculate $\nabla\phi$ from this expression. Taking cross product again with $\nu$ yields

[TABLE]

Since $\phi$ is tangential to $\Gamma$, $\nabla\phi\cdot\nu=0$, $\nu\cdot\nu=1$, and for all vectors $a,b,c$ we have $a\times(b\times c)=b\,(a\cdot c)-c\,(a\cdot b)$, we obtain (4.1). Hence $V\cdot\left(1,0,u_{x_{1}}\right)=\dfrac{2\pi}{\lambda_{0}}\left(-{\bf m}\cdot\left(1,0,u_{x_{1}}\right)+u_{x_{1}}\right)$ and $V\cdot\left(0,1,u_{x_{2}}\right)=\dfrac{2\pi}{\lambda_{0}}\left(-{\bf m}\cdot\left(0,1,u_{x_{2}}\right)+u_{x_{2}}\right)$. Therefore, from the existence of $\phi$, (3.2) holds and consequently (4.2) also.

To prove the converse, we apply Theorem 3.1 to $V$ given in (4.1). Since $\bf m$ is $C^{2}$ and $u$ is $C^{3}$, $V$ is $C^{2}$. That $V$ satisfies condition (3.2) follows from (4.2). Let us verify (3.7). Let $f=\dfrac{2\pi}{\lambda_{0}}\left(u-h\right)$, so $f$ solves (3.1). From the definition of $G$ in (3.5), the vectors $P$ and $N$ are

[TABLE]

so to show (3.7) it is enough to show (3.11). We have

[TABLE]

Notice that

[TABLE]

Thus

[TABLE]

and in general

[TABLE]

Therefore the determinant in (3.11) is $-\left(\dfrac{2\pi}{\lambda_{0}}\right)^{2}$ times the determinant in (4.4), and consequently if (4.4) holds we obtain (3.7). Consequently, the existence of $\phi$ follows from Theorem 3.1.

With the phase $\phi$ obtained satisfying (4.5), it remains to show that the metalens $(\Gamma,\phi)$ refracts incident rays with direction ${\bf e}$ into the direction ${\bf m}(x)$ at the point $(x,u(x))$ in a neighborhood of $(x_{0},u(x_{0}))$. Since $\nabla\phi$ satisfies (4.1), then ${\bf e}-{\bf m}-\nabla\left(\lambda_{0}\phi/2\pi\right)$ is parallel to $\nu$ and so Snell’s law (2.3) is verified and consequently ${\bf e}$ is refracted into ${\bf m}$. However, to avoid total internal reflection, the ray ${\bf e}$ is refracted into the medium (vacuum) above the metasurface when ${\bf e}$ and $\nabla\phi$ satisfy condition (2.6) with $n_{1}=n_{2}=1$, and ${\bf x}={\bf e}$, that is when

[TABLE]

To verify this, from (4.1) we have

[TABLE]

and so (4.6) follows since $\nabla\phi\cdot\nu=0$.

∎

4.1. Existence of phases when ${\bf m}$ is constant

We apply the converse in Theorem 4.1 with ${\bf m}(x):={\bf m}$ a constant vector for each $x\in\Omega$ and assume $u\in C^{3}$. It is clear that (4.2) holds. To calculate the determinant in (4.4), since ${\bf m}$ is constant, from (4.3) it follows that $h(x)={\bf m}\cdot(x,u(x))+C,$ and $D^{2}h(x)=\left({\bf e}\cdot{\bf m}\right)D^{2}u(x)$. So the determinant in (4.4) equals

[TABLE]

If the last expression is not zero at $x=x_{0}$, then (4.4) holds and therefore a phase discontinuity $\phi$ exists in a neighborhood of $\left(x_{0},u(x_{0})\right)$.

Remark 4.2**.**

We remark that (4.4) is only a sufficient condition for the existence of $\phi$ and it is not necessary. In fact, when $\Gamma$ is a plane, i.e., $u(x)={\bf a}\cdot x+b$ with ${\bf a}\in{\mathbb{R}}^{2}$, $b\in{\mathbb{R}}$, and ${\bf m}$ is a constant field, the determinant in (4.4) equals the determinant in (4.7) which is zero. However, in this particular case we can still find a phase discontinuity $\phi$ in a neighborhood of $\Gamma$. In fact, the normal $\nu=\dfrac{\left(-{\bf a},1\right)}{\sqrt{1+|{\bf a}|^{2}}}$, and from (4.1) if $\phi$ exists it must verify

[TABLE]

Letting $\phi(x_{1},x_{2},x_{3})={\bf v}\cdot(x_{1},x_{2},x_{3})$ yields the desired phase and so the metalens $(\Gamma,\phi)$ refracts all vertical rays into the constant direction ${\bf m}$.

4.2. Existence of phases focusing all rays into one point

Here we show the existence of metalenses $(\Gamma,\phi)$ that refract all vertical rays into a fixed point $P=(0,0,p)$, $p>0$. The set of refracted directions is

[TABLE]

where $u(x)<p$ for all $x$. In this case, to show existence of phase discontinuities will use the following proposition.

Proposition 4.3**.**

For each function $u\in C^{2}$, ${\bf m}$ defined by (4.8) satisfies (4.2). Moreover, the determinant in (4.4) can be written as

[TABLE]

where $\nabla h=\left({\bf m}\cdot\left(1,0,u_{x_{1}}\right),{\bf m}\cdot\left(0,1,u_{x_{2}}\right)\right)$. Therefore, if (4.9) is not zero at $x_{0}$ and $u\in C^{3}$, then from Theorem 4.1 a phase discontinuity $\phi$ exists in a neighborhood of $(x_{0},u(x_{0}))$.

Proof.

We have

[TABLE]

and similarly ${\bf m}(x)\cdot(0,1,u_{x_{2}})=-\left\{\sqrt{|x|^{2}+(p-u(x))^{2}}\right\}_{x_{2}}.$ Hence ${\bf m}(x)$ satisfies (4.2). If we set

[TABLE]

then $h$ satisfies (4.3).

Let us calculate the matrix inside the determinant in (4.4). From the definition of ${\bf m}$

[TABLE]

Let us next calculate $D^{2}h$:

[TABLE]

Then

[TABLE]

Combining these calculations it follows that the determinant in (4.4) equals

[TABLE]

and factoring out $1/h$ yields (4.9). ∎

Remark 4.4**.**

With this proposition we show existence of phases when $\Gamma$ is given by the graph of a radial function $u$. Let us write $u(x):=v(|x|^{2})$ and we have $p>u(x)$ for all $x$. Setting $r=|x|^{2}=x_{1}^{2}+x_{2}^{2}$ it follows that

[TABLE]

Hence

[TABLE]

We now calculate the components of the matrix $\mathcal{A}$ inside the determinant in (4.9):

[TABLE]

Notice that in this case, the function $h$ in (4.10) is radial with $h=h(r)=-\sqrt{r+(p-v(r))^{2}}$. Replacing in the formula for $\mathcal{A}$ we obtain

[TABLE]

Notice that if $x=0$, then $\mathcal{A}=Id$, and therefore from Proposition 4.3 there is a phase discontinuity in a neighborhood of $x=0$.

5. Analysis of chromatic dispersion in metalenses

5.1. Dispersion of dichromatic light

Let $(\Gamma,\phi)$ be a metasurface in ${\mathbb{R}}^{3}$ surrounded by vacuum. A dichromatic ray with unit direction ${\bf x}$, superposition of two wavelengths $\lambda_{1}$ and $\lambda_{2}$, strikes $\Gamma$ at a point $P$. Let $\nu(P)$ be the outer-unit normal to $\Gamma$ at $P$, and let ${\bf m}_{1}$, ${\bf m}_{2}$ be the unit directions of the refracted rays at $P$ corresponding to $\lambda_{1}$ and $\lambda_{2}$. From (2.3), the vectors ${\bf x}-{\bf m_{1}}-\nabla(\lambda_{1}\phi(P)/2\pi)$ and ${\bf x}-{\bf m_{2}}-\nabla(\lambda_{2}\phi(P)/2\pi)$ are parallel to $\nu(P)$. We show that the following statements are equivalent

1. (1)

${\bf m_{1}}={\bf m_{2}}$, 2. (2)

$\nabla\phi(P)$ is colinear to $\nu(P)$, 3. (3)

${\bf m_{1}}={\bf m_{2}}={\bf x}$.

This means that if $\nabla\phi(P)$ is not parallel to $\nu(P)$, then rays composed with different colors will be dispersed by the metalens, i.e., refracted in different directions. And only in case $\nabla\phi(P)$ is parallel to $\nu(P)$, the incident ray propagates through the metalens without changing direction.

Proof.

$(3)\implies(1)$ is trivial, so we will show that $(1)\implies(2)\implies(3).$

$(1)\implies(2)$. Since $n_{1}=n_{2}$, from (2.3)

[TABLE]

implying $\dfrac{\lambda_{1}-\lambda_{2}}{2\pi}\nabla\phi(P)\times\nu(P)=0,$ and since $\lambda_{1}\neq\lambda_{2}$, $(2)$ follows.

$(2)\implies(3)$. If $\nabla\phi(P)$ is parallel to $\nu(P)$, then from (2.3) $\left({\bf x}-{\bf m_{1}}\right)\times\nu(P)=0,$ so there exists $\mu_{1}\in{\mathbb{R}}$ such that

[TABLE]

Dotting (5.1) with ${\bf x}$ and ${\bf m}_{1}$ yields

[TABLE]

and adding these identities, we obtain $\mu_{1}({\bf x}+{\bf m_{1}})\cdot\nu(P)=0.$ Hence $\mu_{1}=0$ or ${\bf x}\cdot\nu(P)={\bf m_{1}}\cdot\nu(P)=0$, since ${\bf x}\cdot\nu(P)$ and ${\bf m_{1}}\cdot\nu(P)$ are both non negative. Dotting (5.1) with $\nu(P)$, if ${\bf x}\cdot\nu(P)={\bf m_{1}}\cdot\nu(P)=0$ then we also get $\mu_{1}=0$. Therefore ${\bf x}={\bf m_{1}},$ and similarly, ${\bf x}={\bf m_{2}},$ concluding $(3)$. ∎

5.2. Analysis of the chromatic aberration for a plane metasurface

Let $\Gamma$ be the horizontal plane $x_{3}=a$ in ${\mathbb{R}}^{3}$. Suppose a phase discontinuity $\phi$ is defined in a neighborhood of $\Gamma$ such that vertical rays having color with wavelength $\lambda_{0}$ and striking $\Gamma$ are refracted into a point $P_{0}=(0,0,p_{0})$ above $\Gamma$ as in Figure 1(a). We then ask how this planar metasurface $(\Gamma,\phi)$ focuses vertical rays with wavelength $\lambda\neq\lambda_{0}$. Assuming that all incoming rays pass through the circular aperture $x_{1}^{2}+x_{2}^{2}\leq R^{2}$, we shall prove that for appropriate values of $\lambda$ each refracted ray with wavelength $\lambda$ intersects the $z$-axis at some point $P_{\lambda}(x)=(0,0,p_{\lambda}(x))$ depending on $x=(x_{1},x_{2})$. In addition, we shall prove an estimate for the distance between the points $P_{\lambda}$ and $P_{0}$ in terms of $p_{0}-a$ and the ratio $\lambda/\lambda_{0}$, see (5.3) and Figure 1(b). We summarize the results of this section in the following theorem.

Theorem 5.1**.**

Let $P_{0}=(0,0,p_{0})$. Consider the metalens $(\Gamma,\phi)$ surrounded by vacuum, with $\Gamma$ the horizontal plane $x_{3}=a$ in ${\mathbb{R}}^{3}$, $a<p_{0}$ and with

[TABLE]

where $g$ is an arbitrary $C^{1}$ function satisfying $g^{\prime}(a)=0$. Suppose vertical rays that are superposition of two colors, with wavelengths $\lambda_{0}$ and $\lambda$, strike $\Gamma$ at $(x_{1},x_{2},a)$, with $x_{1}^{2}+x_{2}^{2}<R^{2}$.

Then each ray splits into two rays one with wavelength $\lambda_{0}$ and another with wavelength $\lambda$. Each ray with wavelength $\lambda_{0}$ is refracted into the point $P_{0}$. And, if $\dfrac{\lambda}{\lambda_{0}}<1$, or $1<\dfrac{\lambda}{\lambda_{0}}\leq\sqrt{1+\dfrac{(p_{0}-a)^{2}}{R^{2}}}$, then each ray with wavelength $\lambda$ is refracted into the point $P_{\lambda}(x)=\left(0,0,\dfrac{\lambda_{0}}{\lambda}\sqrt{\left(1-\left(\lambda/\lambda_{0}\right)^{2}\right)\,|x|^{2}+(p_{0}-a)^{2}}\right).$ We also have

[TABLE]

for each $x=(x_{1},x_{2})$ with $x_{1}^{2}+x_{2}^{2}\leq R^{2}$.

Proof.

Let us first calculate the phase discontinuity focusing rays with color $\lambda_{0}$ into $P_{0}$. The incident ray with direction ${\bf e}=(0,0,1)$ emanating from $x=(x_{1},x_{2})$ with $x_{1}^{2}+x_{2}^{2}\leq R^{2}$ strikes the plane $x_{3}=a$ and then bends to strike the point $P_{0}$. Applying the set up from Section 4 in this case we have ${\bf m}(x)=\dfrac{P_{0}-(x,a)}{|P_{0}-(x,a)|}$ and the normal $\nu={\bf e}$. Since we seek $\phi$ tangential to $\Gamma$, we then have from (4.1) that

[TABLE]

Clearly, the function $\phi$ in (5.2) is the desired phase function. Therefore, the planar metasurface having this phase focuses all rays with color $\lambda_{0}$ into the point $P_{0}$.

Let us now see how the metalens $(\Gamma,\phi)$ focuses rays with wavelength $\lambda$. First, to avoid total reflection, from (2.6), rays with color $\lambda$ are refracted by the metalens $(\Gamma,\phi)$ if

[TABLE]

Since $\nu=(0,0,1)={\bf e},$ and $\nabla\phi\cdot\nu=0$, then (5.5) is equivalent to

[TABLE]

Clearly, this inequality holds for $\lambda<\lambda_{0}$. If $\lambda>\lambda_{0}$, then $\Delta\geq 0$ if and only if $(p_{0}-a)^{2}\geq\left(\left(\lambda/\lambda_{0}\right)^{2}-1\right)\,|x|^{2}$. Since $|x|^{2}\leq R^{2}$, if $\lambda>\lambda_{0}$ satisfies

[TABLE]

then $\Delta\geq 0$ for all $|x|\leq R$. Therefore for these values of $\lambda$ total reflection is avoided.

Next, we need to know the directions of the refracted rays with color $\lambda$. According to (2.4), $(\Gamma,\phi)$ refracts rays with color $\lambda$ into the unit direction ${\bf m^{\prime}}$ with

[TABLE]

where from (2.5)

[TABLE]

Writing (5.6) in coordinates yields

[TABLE]

We now see where the line $t\,{\bf m}^{\prime}+(x,a)$ intersects the vertical line $x_{1}=x_{2}=0$, that is, we find the point on this line that is focused by the refracted ray emanating from $x=(x_{1},x_{2})$ with direction ${\bf e}$ and color $\lambda$. So we need to find $t$ so that $t\,m_{1}^{\prime}+x_{1}=0$ and $t\,m_{2}^{\prime}+x_{2}=0$. This means $t=\dfrac{\lambda_{0}}{\lambda}\,\sqrt{|x|^{2}+(p_{0}-a)^{2}}$. Hence $t\,m_{3}^{\prime}+a=\dfrac{\lambda_{0}}{\lambda}\,\sqrt{\Delta}+a$, and the focused point on the vertical line by the vertical ray emanating from $x$ with color $\lambda$ is $P_{\lambda}(x)=\left(0,0,\dfrac{\lambda_{0}}{\lambda}\,\sqrt{\Delta}+a\right)$.

We now calculate the distance between $P_{0}$ and $P_{\lambda}(x)$. First write

[TABLE]

We then obtain

[TABLE]

since the right hand side is a radial function of $x$, then rays with color $\lambda$ emitted from all points in a circle are focused into the same point. Set $|x|^{2}=s$ and consider the function $F(s)=\dfrac{s+\left(p_{0}-a\right)^{2}}{\dfrac{\lambda_{0}}{\lambda}\,\sqrt{\left(1-\left(\lambda/\lambda_{0}\right)^{2}\right)\,s+(p_{0}-a)^{2}}+\left(p_{0}-a\right)}$. To estimate the error (5.7) and obtain (5.3), we will find the maximum of $F(s)$ for $s\in[0,R^{2}]$. Suppose first that $\lambda<\lambda_{0}$. In this case $\Delta>0$ and it is easy to see that $F^{\prime}(s)>0$ in the interval $[0,R^{2}]$ and so $F$ is increasing. Therefore when $\lambda<\lambda_{0}$

[TABLE]

On the other hand, if $\lambda_{0}\sqrt{1+\dfrac{(p_{0}-a)^{2}}{R^{2}}}>\lambda>\lambda_{0}$, then it is also easy to see that $F$ is increasing in $[0,R^{2}]$ and so the error

[TABLE]

Therefore, we conclude the estimate (5.3). ∎

5.3. Analysis of the chromatic aberration for a general radial function $u$

In this section, the surface $\Gamma$ is given by the graph of a function $u(x)=v(|x|^{2})$. In view of Remark 4.4, a phase $\phi$ exists in a neighborhood of $x=0$ so that the metalens $(\Gamma,\phi)$ refracts all vertical rays with wavelength $\lambda_{0}$ into a point $P_{0}=(0,0,p_{0})$. We analyze chromatic aberration caused by $(\Gamma,\phi)$, that is, how rays with wavelength $\lambda\neq\lambda_{0}$ are focused. We shall prove that the order of magnitude of the focusing error in the radial case is as in the planar case (5.3).

In order to do this, we first find an expression for $\nabla\phi(x,u(x))$ when $u(x)=v(|x|^{2}).$ In this case, the set of transmitted directions for rays with wavelength $\lambda_{0}$ is

[TABLE]

And from Theorem 4.1 the gradient of $\phi$ satisfies (4.1). As in Remark 4.4, we let $h(x)=-\sqrt{|x|^{2}+(p_{0}-u(x))^{2}}$. Set $r=|x|^{2},$ so $\nabla u=2\,v^{\prime}(r)\,x$, and $h(r)=-\sqrt{r+(p_{0}-v(r))^{2}}$. Then

[TABLE]

so from (4.1)

[TABLE]

To simplify the notation set $A(r)=1+2v^{\prime}(r)(v(r)-p_{0}-h)$.

Since the phase $\phi$ is defined in a neighborhood of $x=0$, we assume that $|x|^{2}=r<R^{2}$ for some $R$ small.

As in Section 5.2, we next find conditions on $\lambda$, $\lambda_{0}$ and $R$ so that rays with wavelength $\lambda$ are not totally internally reflected. We have $\nabla\phi\cdot\nu=0$, then from (2.6) with $n_{1}=n_{2}=1$, ${\bf x}={\bf e}$, to avoid total reflection we should verify that

[TABLE]

Indeed, we have $\left({\bf e}\cdot\nu\right)^{2}=\dfrac{1}{1+4v^{\prime}(r)^{2}r}$, and from (5.8)

[TABLE]

Hence (5.9) is satisfied if and only if

[TABLE]

We conclude that taking $R$ small enough, above inequality holds for $r\leq R^{2}$, and total internal reflection is avoided for rays with color $\lambda$.

We next study the direction of the refracted rays with wavelength $\lambda$. From (2.4) the direction of the refracted ray with color $\lambda$ is

[TABLE]

where from (2.5) and the above calculation

[TABLE]

Writing ${\bf m^{\prime}}=\left(m_{1}^{\prime},m_{2}^{\prime},m_{3}^{\prime}\right)$ yields

[TABLE]

Recall that $r$ is chosen small enough so that the term inside the square root in the identity above is positive.

Now we find $t$ such that $t\,\left(m_{1}^{\prime},m_{2}^{\prime}\right)+x=0$, that is,

[TABLE]

obtaining

[TABLE]

Next,

[TABLE]

The ray with color $\lambda$ then focuses at the point $P_{\lambda}=\left(0,0,t\,m_{3}^{\prime}+u(x)\right)$, and we want to see how far is this point from $P_{0}=(0,0,p_{0})$. So we then need to estimate the error

[TABLE]

Taking limits when $r\to 0$ yields

[TABLE]

Therefore

[TABLE]

obtaining in the radial case that the order of magnitude of the error is as in the planar case (5.3).

6. Comparison with chromatic aberration in standard lenses

We analyze here the chromatic aberration in a standard lens sandwiched by a horizontal plane and a hyperboloid, and compare this dispersion with the one obtained for metasurfaces.

It is known that hyperboloids having appropriate eccentricity refract vertical rays into their focus point. More precisely, for a fixed wavelength $\lambda$, we have two materials $I,II$ with corresponding refractive indices $n_{1},n_{2}$, respectively, and a point $Y=(y,y_{3})\in{\mathbb{R}}^{3}$ located in material $II$ to be focused; let $\kappa=n_{1}/n_{2}>1$. Let

[TABLE]

whose graph is a sheet of a hyperboloid with upper focus $Y$ and eccentricity $\kappa$ as shown in Figure 2.

Suppose the material below this hyperboloid is $I$ and the material above is $II$. If a vertical ray with wavelength $\lambda$ strikes the hyperboloid at a point $(x,h(x))$, then this ray is refracted into a direction passing through the point $Y$ as shown in Figure 2. We analyze how a vertical ray having different wavelength would be refracted by the same hyperboloid. In other words, how far is the new refracted point from the focus $Y$?

We assume that $II$ is vacuum, so $n_{2}=1$ for all wavelengths, $n_{1}=n=n(\lambda)$ for the wavelength $\lambda$, and the focus $Y=(0,0,0)$; ${\bf e}=(0,0,1)$. From the standard Snell law, the incoming ray from below the hyperboloid with direction ${\bf e}$ is refracted by the hyperboloid into a ray with unit direction

[TABLE]

where $\nu$ is the outer unit normal to $h$ at the striking point, and

[TABLE]

see [AGT16, Section 2]; with $\kappa=n$. Suppose now that the vertical ray has wavelength $\lambda^{\prime}$ so the refractive index for the material $I$ under this wavelength has value $n^{\prime}=n(\lambda^{\prime})$. From the Snell law such a ray is then refracted by the hyperboloid into a direction

[TABLE]

with

[TABLE]

where $\kappa^{\prime}=n^{\prime}$.

Consider the line through the point $X=(x,h(x))$ with direction $\Lambda^{\prime}$, and we want to determine where this line intersects the vertical line $x=0$. That is, we need to find $t$ such that $X+t\,\Lambda^{\prime}$ intersects the vertical line $x=0$, and see how far this intersection point is from the original focus $(0,0,0)$. To do this we have $\nu=\dfrac{\left(-\nabla h,1\right)}{\sqrt{1+|\nabla h|^{2}}},$ so

[TABLE]

and therefore $t$ must satisfy $x+t\,\delta^{\prime}\left(-\dfrac{\nabla h}{\sqrt{1+|\nabla h|^{2}}}\right)=(0,0)$. Since $\nabla h=\dfrac{-1}{\sqrt{b^{2}+(n^{2}-1)\,|x|^{2}}}\,x$, we get $\sqrt{1+|\nabla h|^{2}}=\dfrac{\sqrt{b^{2}+n^{2}\,|x|^{2}}}{\sqrt{b^{2}+(n^{2}-1)\,|x|^{2}}}$. Therefore

[TABLE]

To calculate $\delta^{\prime}$, we first have ${\bf e}\cdot\nu=\dfrac{\sqrt{b^{2}+(n^{2}-1)\,|x|^{2}}}{\sqrt{b^{2}+n^{2}\,|x|^{2}}},$ hence

[TABLE]

Thus,

[TABLE]

The last component of $X+t\,\Lambda^{\prime}$ equals

[TABLE]

The ray with wavelength $\lambda^{\prime}$ is refracted at $X$ into the point $(0,0,E(x))$. Let

[TABLE]

we assume $r>0$ satisfying $b^{2}+\left(n^{2}-n^{\prime 2}\right)\,r\geq 0$ to avoid total reflection of rays with color $\lambda^{\prime}$. If $n>n^{\prime}$, then $g$ is strictly increasing and we obtain

[TABLE]

for all $x$. And if $n<n^{\prime}$, then $g$ is strictly decreasing and so

[TABLE]

for $x$ satisfying $b^{2}+\left(n^{2}-n^{\prime 2}\right)\,|x|^{2}\geq 0$.

We have

[TABLE]

Let us translate the refractive indices in terms of wavelengths. To see the order of magnitude in the error $E$ when $n$ and $n^{\prime}$ are given in terms of wavelengths, we use Cauchy approximate dispersion formula, see [BW59, Sec. 2.3.4, Formula (43)] or [JW01, Sec. 23.3] (valid only in the visible spectrum)

[TABLE]

Here the terms with powers of $\lambda$ bigger that four in [BW59, Sec. 2.3.4, Formula (41)] have been neglected and $n^{2}$ is replaced by $2(n-1)$ when $n$ takes values for various gases; see discussion in [BW59, Sec. 2.3.4, page 100] and Table 2.6 therein. We set $n=n(\lambda)$ and $n^{\prime}=n(\lambda^{\prime})$, so from Cauchy’s formula

[TABLE]

The order of magnitude of this error, except for a bounded multiplicative constant, and in terms of $\lambda/\lambda^{\prime}$, is similar to the order of magnitude in (5.3) where $\lambda_{0}$ plays the role of $\lambda$ and $\lambda$ the role of $\lambda^{\prime}$. Notice that in the formula above for $E(0)$, the coefficient $b$ can be chosen arbitrarily, in particular, if $b$ is sufficiently small we can control the size of the multiplicative factor in front of $1-\dfrac{\lambda}{\lambda^{\prime}}$. Notice also that using more terms in the full Cauchy dispersion formula $n(\lambda)=A+B/\lambda^{2}+C/\lambda^{4}+D/\lambda^{6}+\cdots$ yields the same order of magnitude in the error $E(0)$.

Acknowledgements

C. E. G. was partially supported by NSF grant DMS–1600578, and A. S. was partially supported by Research Grant 2015/19/P/ST1/02618 from the National Science Centre, Poland, entitled ”Variational Problems in Optical Engineering and Free Material Design”.

This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 665778.