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Ray tracing in metasurfaces

C. P. Castañeda-Almanza and Ivan Moreno

Open Access Open Access

Abstract

Ray tracing in a metasurface is the process to obtain the three-dimensional (3D) light path under reflection and transmission, which may be helpful in the optical design of metalenses and other metaoptical devices. In this work, first we deduce the 3D vector form of Snell's law for metasurfaces by using a geometric approach. And then, we deduce the general equations to calculate the direction of the reflected and refracted beams in any metasurface, and for any incident beam. In other words, we derive vector form equations for the 3D direction of transmitted and reflected beams at a metasurface with arbitrary 2D phase profile, and for any 3D direction of incident light.

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

1. Introduction

Recently, metalens technology based on metasurfaces has shown considerable optical performance while opening up new opportunities to thin, flat, lightweight optical devices [1,2]. Although metalens performance is tested under wave optics, its optical design is primarily performed by ray tracing, in particular for general analyses and for correcting aberrations. Moreover, metasurfaces are finding advanced applications in compound optical systems such as compound metalenses, or new applications as nonimaging metalenses [3], and then analyzing the paths of rays through these optical systems is becoming more and more important. The exact ray tracing schemes in traditional optics are based on vector formulations and computer programs. Classical laws of refraction and reflection for refractive materials are well known and studied in its vector form, and even recent studies are reported [4–6]. Therefore, to formulate ray tracing procedures in metasurfaces, it is convenient to transform the law of refraction and reflection into vector forms.

In the case of metasurfaces, the beam’s refraction and reflection is determined by the phase spatial profile of the metasurface. The laws of reflection and refraction for metasurfaces with 1D phase profile were reported and experimentally tested in a seminal paper of metaoptics [7]. Also, a set of scalar equations of the Snell law for a metasurface with 2D phase profile were developed in [8]. These equations are for the case in which the plane of incidence is along the direction of the phase gradient profile. Later, a vector formulation for metasurfaces with 1D phase spatial profile, for the case that the plane of incidence is along the phase gradient, was reported in [9]. Recently, a set of Snell equations valid for arbitrary direction of the incident light, and for 2D phase profiles, were reported [3,10]. However, ray tracing methods through metasurfaces require a vector form of the direction of the reflected and refracted beams through a metasurface with 2D phase gradient [11], and valid for any direction of the incident light, which is what we report here.

2. Theory

Let us first define the geometry, and consider monochromatic light incident on a metasurface surrounded by homogeneous and isotropic media (Fig. 1(a)). The direction of propagation of the incident light beam and the refracted (or reflected) one are given by the normalized wavevectors ${{\boldsymbol k}_1}$ and ${{\boldsymbol k}_2}$, respectively. Here the subindex “1” represents the incident beam, and the subindex “2” indicates the refracted or reflected beam. Normalized wavevectors may be expressed as function of the angles ${\theta _{1(2 )}}$ (the angle between ${{\boldsymbol k}_{1(2 )}}$ and the $z$-axis), and ${\varphi _{1(2 )}}$ (the angle formed by the $x$-axis and the projection of ${{\boldsymbol k}_{1\left( 2 \right)}}$ on the $xy$ plane). With this choice of angular coordinates, the normalized wavevectors of the obliquely incident beam and refracted (reflected) beam can be represented as follows:

$${{\boldsymbol k}_{1(2 )}}\; \; = \left( {\begin{array}{{c}} {\sin {\theta_{1(2 )}}\cos {\varphi_{1(2 )}}}\\ {\sin {\theta_{1(2 )}}\sin {\varphi_{1(2 )}}}\\ {\cos {\theta_{1(2 )}}} \end{array}} \right){\boldsymbol \; \; }.$$

 figure: Fig. 1.

Fig. 1. (a) Schematics of the generalized refraction at a metasurface with arbitrary phase profile. The incident light beam (green) has a normalized wave-vector ${{\boldsymbol k}_1}$ with arbitrary direction. The refracted beam (orange) acquires a direction given by a unitary vector ${{\boldsymbol k}_2}$. The metasurface is in the xy plane at the interface between two media with refractive indices ${n_1}$ and ${n_2}$. (b) Scheme of the refraction showing the dual phase gradient vector on a metasurface. Also it is shown the unit vectors of incident beam ${{\boldsymbol k}_1}$ and the refracted beam ${{\boldsymbol k}_2}$, and ${\boldsymbol n}$ the normal vector of the surface. The local phase gradient has two components, $\partial \Phi / \partial x$ and $\partial \Phi / \partial y$ at each point $({x,y} )$ on the metasurface.

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The direction of the refracted (or reflected) beam is determined by the phase spatial profile Φ of the metasurface, which is generated by its surface nanostructure [7,8]. For example, a simple metalens has a radial phase profile Φ(r), where the radial coordinate is given by $r = \sqrt {{x^2} + {y^2}} $. In general, a metasurface may have an arbitrary phase profile Φ(x,y), and the generalized refraction (reflection) laws are given by two coupled equations [3,10]:

$$\left\{ {\begin{array}{{c}} {{k_{02}}\sin {\theta_2}\cos {\varphi_2} = {k_{01}}\sin{\theta_1}\cos {\varphi_1} + \; \frac{{\partial \mathrm{\Phi }}}{{\partial x}}}\\ {}\\ {{k_{02}}\sin{\theta_2}\sin {\varphi_2} = {k_{01}}\sin{\theta_1}\sin {\varphi_1} + \; \; \frac{{\partial \mathrm{\Phi }}}{{\partial y}}} \end{array}} \right., $$
where the wave number is ${k_{01(2 )}} = {k_0}{n_{1(2 )}}$. Here ${k_0} = {\raise0.7ex\hbox{${2\pi }$} \!\mathord{/ {\vphantom {{2\pi } {{\lambda_0}}}}}\!\lower0.7ex\hbox{${{\lambda _0}}$}}$ is the free space wave number, and ${n_{1(2 )}}$ is the refractive index of the incident (refracted or reflected) medium. In the following results, we will only refer to the refracted beam for simplicity. However, the reflected light beam may be obtained by making n2 equal to n1.

Equations (2) depend on the $x$-axis and $y$-axis phase profile gradients. We may define, in a vector form, a dual phase gradient $\nabla \mathrm{\Phi }$ in the $xy$ plane of the metasurface (Fig. 1(b)),

$$\nabla \mathrm{\Phi } = \frac{{\partial \mathrm{\Phi }}}{{\partial x}}\hat{{\boldsymbol i}} + \frac{{\partial \mathrm{\Phi }}}{{\partial y}}\hat{{\boldsymbol j}}. $$
And by combining Eq. (2) and Eq. (3) we write:
$$\nabla \mathrm{\Phi } = {k_{02}}({\sin {\phi_2}\cos {\varphi_2}\hat{{\boldsymbol i}} + \sin{\phi_2}\sin {\varphi_2}\hat{{\boldsymbol j}}} )- \; {k_{01}}(\sin{\phi _1}\cos\; {\varphi _1}\hat{{\boldsymbol i}} + \sin{\phi _1}\sin {\varphi _1}\hat{{\boldsymbol j}}). $$
By using the definition of cross product in Eq. (4), the Snell law of metasurfaces may be written in vector form as:
$${k_{02}}({{\boldsymbol n} \times {{\boldsymbol k}_2}} )\times {\boldsymbol n} - \; {k_{01}}({{\boldsymbol n} \times {{\boldsymbol k}_1}} )\times {\boldsymbol n} = \nabla \mathrm{\Phi }, $$
where ${{\boldsymbol k}_1}$ and ${{\boldsymbol k}_2}$ are the unit vectors along the directions of the incident ray and refracted ray, and ${\boldsymbol n}$ represents the unit vector along the normal to the metasurface as shown in Fig. 1(b). Also, Eq. (5) reduces to the vector form of classical Snell law in purely refractive interfaces by doing $\nabla \mathrm{\Phi } = 0$, i.e. ${n_2}({{\boldsymbol n} \times {{\boldsymbol k}_2}} )= \; {n_1}({{\boldsymbol n} \times {{\boldsymbol k}_1}} ).$

Using geometric features, and the definitions of the dot and cross product between any three unit vectors (${{\boldsymbol k}_1}$, ${{\boldsymbol k}_2}$, ${\boldsymbol n}$), the unit vectors of the incident ray and refracted ray may be represented by:

$$\left\{ {\begin{array}{{c}} {{\boldsymbol \; }{{\boldsymbol k}_1} = ({{\boldsymbol n} \cdot {{\boldsymbol k}_1}} ){\boldsymbol n} + ({{\boldsymbol n} \times {{\boldsymbol k}_1}} )\times \boldsymbol{n}}\\ {{\boldsymbol \; }{{\boldsymbol k}_2} = ({{\boldsymbol n} \cdot {{\boldsymbol k}_2}} ){\boldsymbol n} + ({{\boldsymbol n} \times {{\boldsymbol k}_2}} )\times \boldsymbol{n}} \end{array}} \right..$$

Substituting Eq. (6) in Eq. (5):

$${{\boldsymbol k}_2} = ({{\boldsymbol n} \cdot {{\boldsymbol k}_2}} ){\boldsymbol n} + \beta \; \nabla \mathrm{\Phi } + \mathrm{\mu}({{\boldsymbol n} \times {{\boldsymbol k}_1}} )\times {\boldsymbol n},$$
where $\beta = {\raise0.7ex\hbox{$1$} \!\mathord{/ {\vphantom {1 {{k_{02}}}}}}\!\lower0.7ex\hbox{${{k_{02}}}$}} = {\raise0.7ex\hbox{$1$} \!\mathord{/ {\vphantom {1 {{k_0}{n_2}}}}}\!\lower0.7ex\hbox{${{k_0}{n_2}}$}}$, and $\mathrm{\mu} = {\raise0.7ex\hbox{${{n_1}}$} \!\mathord{/ {\vphantom {{{n_1}} {{n_2}}}}}\!\lower0.7ex\hbox{${{n_2}}$}}$.

And because the vector $({{\boldsymbol n} \cdot {{\boldsymbol k}_2}} ){\boldsymbol n}$ is normal to the metasurface plane, it is orthogonal to the vector $\beta \; \nabla \mathrm{\Phi } + \mathrm{\mu}({{\boldsymbol n} \times {{\boldsymbol k}_1}} )\times {\boldsymbol n}$, which lays on the metasurface interface. Therefore, by Pythagoras theorem, the norm of the refracted beam is:

$$1 = {({{\boldsymbol n} \cdot {{\boldsymbol k}_2}} )^2} + {|{\beta \; \nabla \mathrm{\Phi } + \mathrm{\mu}({{\boldsymbol n} \times {{\boldsymbol k}_1}} )\times {\boldsymbol n}} |^2}, $$
and then, by replacing the dot product of Eq. (8) in Eq. (7), the direction of the refracted ray in terms of the direction of incident ray, phase gradient, and the metasurface normal, is:
$${{\boldsymbol k}_2} = \mathrm{\mu}({{\boldsymbol n} \times {{\boldsymbol k}_1}} )\times {\boldsymbol n} + \beta {\boldsymbol \; }\nabla \mathrm{\Phi } \pm \sqrt {1 - {{|{\beta \; \nabla \mathrm{\Phi } + \mathrm{\mu}({{\boldsymbol n} \times {{\boldsymbol k}_1}} )\times {\boldsymbol n}} |}^2}} \; \; {\boldsymbol n}.$$

Equation (9) shows two possible values, which depends if the root is positive or negative. The minus sign is for refraction and the positive sign is for reflection. This is because the refracted beam is propagating away from the other side of the metasurface, or because ${{\boldsymbol k}_2}$ has the same direction along the normal as that of ${{\boldsymbol k}_1}$, i.e. the sign of ${\boldsymbol n} \cdot {{\boldsymbol k}_2}{\boldsymbol \; }$ is equal to the sign of ${\boldsymbol n} \cdot {{\boldsymbol k}_1}$ (see Fig. 1(b)). And for some applications, to automatically calculate the sign in Eq. (9), a function Sign(${\boldsymbol n} \cdot {{\boldsymbol k}_2}$), following a rule of signs [5], can be added instead of ± sign. Therefore, we can write the direction of the refracted light in vector form for a metasurface with 2D phase gradient, valid for arbitrary direction of incident light k1, as cross products:

$${{\boldsymbol k}_2} = \mathrm{\mu}({{\boldsymbol n} \times {{\boldsymbol k}_1}} )\times {\boldsymbol n} + \frac{{{\lambda _0}}}{{2\pi {n_2}}}\; \nabla \mathrm{\Phi } - \sqrt {1 - {{\left|{\frac{{{\lambda_0}}}{{2\pi {n_2}}}\; \nabla \mathrm{\Phi } + \mathrm{\mu}({{\boldsymbol n} \times {{\boldsymbol k}_1}} )\times {\boldsymbol n}} \right|}^2}} \; {\boldsymbol n},$$
where ${\lambda _0}$ is the free space wavelength, µ=n1/n2, ${n_1}$ is the refractive index of the incident medium and ${n_2}$ of the refracted medium. And using the relation between dot and cross products, the direction of the refracted beam may be written as dot products:
$${{\boldsymbol k}_2} = \mathrm{\mu}[{{{\boldsymbol k}_1} - ({{\boldsymbol n} \cdot {{\boldsymbol k}_1}} ){\boldsymbol n}} ]+ \frac{{{\lambda _0}}}{{2\pi {n_2}}}\; \nabla \mathrm{\Phi } - \sqrt {1 - {{\left|{\frac{{{\lambda_0}}}{{2\pi {n_2}}}\; \nabla \mathrm{\Phi } + \mathrm{\mu}({{{\boldsymbol k}_1} - ({{\boldsymbol n} \cdot {{\boldsymbol k}_1}} ){\boldsymbol n}} )} \right|}^2}} {\boldsymbol n}. $$

In case of reflection, the root in Eq. (9) is positive. This is because the reflected beam is propagating in the same side of the metasurface, or because the direction of the reflected beam ${{\boldsymbol k}_{2{\boldsymbol r}}}{\boldsymbol \; }$ has the opposite direction along the normal as that of the incident beam ${{\boldsymbol k}_1}$, i.e. the sign of ${\boldsymbol n} \cdot {{\boldsymbol k}_{2{\boldsymbol r}}}{\boldsymbol \; }$ is opposite to the sign of ${\boldsymbol n} \cdot {{\boldsymbol k}_1}$. Therefore, by considering ${n_2} = {n_1}$ in the Eq. (9), and the positive sign in the root, the expression of the direction of the reflected beam, in vector form as cross products, is:

$${{\boldsymbol k}_{2{\boldsymbol r}}} = ({{\boldsymbol n} \times {{\boldsymbol k}_1}} )\times {\boldsymbol n} + \frac{{{\lambda _0}}}{{2\pi {n_1}}}\; \nabla \mathrm{\Phi } + \sqrt {1 - {{\left|{\frac{{{\lambda_0}}}{{2\pi {n_1}}}\; \nabla \mathrm{\Phi } + ({{\boldsymbol n} \times {{\boldsymbol k}_1}} )\times {\boldsymbol n}} \right|}^2}} \; \; {\boldsymbol n}. $$

And using the relation between dot and cross products, the direction of the reflected beam may be written as dot products:

$${{\boldsymbol k}_{2{\boldsymbol r}}} = [{{{\boldsymbol k}_1} - ({{\boldsymbol n} \cdot {{\boldsymbol k}_1}} ){\boldsymbol n}} ]+ \frac{{{\lambda _0}}}{{2\pi {n_1}}}\; \nabla \mathrm{\Phi } + \sqrt {1 - {{\left|{\frac{{{\lambda_0}}}{{2\pi {n_1}}}\; \nabla \mathrm{\Phi } + ({{{\boldsymbol k}_1} - ({{\boldsymbol n} \cdot {{\boldsymbol k}_1}} ){\boldsymbol n}} )} \right|}^2}} {\boldsymbol n}. $$

Equations (10)–(13) were tested for its ability to describe the transmitted or reflected beams of light through a metasurface. The validity of these equations was checked by comparing its output with the scalar Eq. (2) for several numeral values (see Appendix A), giving the same results. Also, the deduced Eqs. (10)–(13) reduce to the vector form of classical refraction in purely refractive interfaces by doing $\nabla \mathrm{\Phi } = 0$, and by noting that $|{({{\boldsymbol n} \times {{\boldsymbol k}_1}} )\times {\boldsymbol n}} |= |{{\boldsymbol n} \times {{\boldsymbol k}_1}} |$ [4–6]. In addition, Eqs. (11) and (13) coincide with those reported by Gutierrez et. al. [11], which are deduced by a totally different procedure (see their Eq. (11) and that in Remark 2). However, the geometric approach that we carry out makes our deduction simple, easy to understand, and simple to apply.

3. Results

Finally, we apply Eqs. (10)–(13) to visually show the transmitted or reflected beams of light through a refracting or reflecting metasurface, by plotting a 3D vector field in Fig. 2. These plots shows a bunch of 100 light beams. The light has a wavelength of $\lambda = 600\; nm$, the metasurface separates two media with refractive indices, ${n_1} = 1$, ${n_2} = 1.5$. The dual phase gradient is $\nabla \mathrm{\Phi } = ({{\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi {7 \times {{10}^6}}}}}\!\lower0.7ex\hbox{${7 \times {{10}^6}}$}},\; \; {\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi {9 \times {{10}^6}}}}}\!\lower0.7ex\hbox{${9 \times {{10}^6}}$}}} ){m^{ - 1}}$ in Figs. 2(a)-(c). Figure 2 shows the simulation of different types of incidence, i.e. for different incident vector directions ${{\boldsymbol k}_1}$, on a dual phase gradient, constant along the metasurface. Figure 2(a) shows the transmission of light that falls in a fixed direction given by ($\mathrm{\theta } = {\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi 4}}}\!\lower0.7ex\hbox{$4$}}$, $\mathrm{\varphi } = {\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi 3}}}\!\lower0.7ex\hbox{$3$}}$). While Fig. 2(b) shows the transmission of rays with random direction of incidence given by $\mathrm{\theta } \in \; [{0,{\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi 2}}}\!\lower0.7ex\hbox{$2$}}} ]$, and $\mathrm{\varphi } \in \; [{0,\; 2\pi } ]$. Figure 2(c) shows the reflection of incident rays with fixed direction of incidence ($\mathrm{\theta } = {\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi 3}}}\!\lower0.7ex\hbox{$3$}}$, $\mathrm{\varphi } = {\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi 4}}}\!\lower0.7ex\hbox{$4$}}$). Figure 2(d) shows the refraction on a metasurface with a polar phase gradient $\nabla \mathrm{\Phi } = \left( {\frac{{\partial \mathrm{\Phi }}}{{\partial r}} = {\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi {4 \times {{10}^6}}}}}\!\lower0.7ex\hbox{${4 \times {{10}^6}}$}}{m^{ - 1}},\; \frac{{\partial \mathrm{\Phi }}}{{\partial \psi }} = {\raise0.7ex\hbox{${2\pi }$} \!\mathord{/ {\vphantom {{2\pi } {10}}}}\!\lower0.7ex\hbox{${10}$}}ra{d^{ - 1}}} \right),$ and by tracing incident rays with random directions $\mathrm{\theta } \in \; [{0,{\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi 2}}}\!\lower0.7ex\hbox{$2$}}} ]$, $\mathrm{\varphi \;\ } \in \; [{0,\; 2\pi } ]$ (see Appendix B). As shown by Fig. 2, the deduced vector equations allow calculations of any number of rays, at any wavelength incident on a given metasurface with any 2D phase gradient. These equations can be adapted to numerical algorithms to obtain a reliable description of the performance of a given optical system with metasurfaces, for example in the design of hybrid achromatic metalenses [12].

 figure: Fig. 2.

Fig. 2. Schematic of ray tracing for refraction and reflection on a metasurface by using Eqs. (10–13). (a) Refraction of the incident rays with direction fixed. (b) Refraction of the incident rays with random directions. (c) Reflection of the incident rays with direction fixed. (d) Refraction on a metasurface with a polar phase gradient, and for incident rays with random directions.

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Now let us consider a more general phase gradient, the radial phase profile of a metalens, which is given by:

$$\mathrm{\Phi }(r )= \frac{{2\pi }}{\lambda }\left( {\sqrt {{r^2} + {f^2}} - f} \right), $$
where f is the focal length, $\lambda $ is the wavelength, and the radial coordinate is $r = \sqrt {{x^2} + {y^2}} $. We apply Eq. (14) in Eq. (11) to visually show the transmitted beams of light through of a metalens, by plotting a 3D vector field in Fig. 3. These plots shows a bunch of 100 light beams. The focal length is $f = 19mm$, $\lambda = 600\; nm$, and refractive indices ${n_1} = 1$ and ${n_2} = 1.5$. The Fig. 3(a) shows the propagation of light rays with fixed directions from a point light source. And Fig. 3(b) shows the propagation of light rays with random incident directions, given by $\mathrm{\theta } \in \; [{0,\; {\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi 2}}}\!\lower0.7ex\hbox{$2$}}} ]$, and $\mathrm{\varphi } \in \; [{0,\; 2\pi } ]$.

 figure: Fig. 3.

Fig. 3. Schematic of ray tracing for light transmission through a metalens by using Eqs. (11) and (14). (a) Refraction of the rays from a point source. (b) Refraction of the incident rays with random direction.

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4. Conclusions

In summary, we have derived the general equations to calculate the direction vectors of the reflected and refracted light in metasurfaces. In other words, we derived vector form equations for the direction of transmitted and reflected beams at a metasurface with arbitrary 2D phase spatial profile, which are valid for any direction of incident light. These formulas may simplify the 3D ray tracing through systems of metalenses and other devices with metasurfaces. They are simple expressions that contain three known quantities: the unit vector normal to the metasurface, the direction vector of the incident beam, and the gradient of the phase profile. These equations represent a basis for the design and analysis of imaging metalenses [1], metalenses for illumination [3], general metasurface problems [11,13], and other advanced optical devices using metasurfaces. In this sense, further work may include the following: (a) to analyze total internal reflection; and (b) to improve the derived vector equations for high efficient unambiguous computations [5,6].

In addition, the deduced vector formulas have flexibility because they do not depend on the coordinate system, and the metasurface may have any orientation. Finally, the formulas obtained were applied to the visualization of a vector field of incident rays and transmitted rays in a metasurface, including a metalens.

Appendix A

The correspondence of Eqs. (10–13) with their scalar form Eq. (2) can be tested by comparing the components of ${{\boldsymbol k}_2}$ for the case of a metasurface with its normal vector along the z direction.

The x and y components of ${{\boldsymbol k}_2}$, according to scalar Eq. (2), which has been experimentally confirmed [8], are given by

$$\left\{ {\begin{array}{{c}} {{k_{2x}} = \mu \; \sin{\theta_1}\cos {\varphi_1} + \beta \; \frac{{\partial \mathrm{\Phi }}}{{\partial x}}}\\ {}\\ {{k_{2y}} = \; \mu \; \sin{\theta_1}\sin {\varphi_1} + \beta \; \frac{{\partial \mathrm{\Phi }}}{{\partial y}}} \end{array}} \right.\; , $$
and because ${{\boldsymbol k}_2}$ is normalized, the z component is given by,
$${k_{2z}} ={\pm} \; \textrm{cos}{\theta _2} ={\pm} {\boldsymbol \; }\; \sqrt {1 - ({k_{2x}^2 + k_{2y}^2} )\; } ,$$
where ${k_{2z}}$ has negative sign for refraction and positive sing for reflection.

Appendix B

In order to define a polar phase gradient, let us assume that the phase profile $\mathrm{\Phi }({r,\; \psi \; } )$ is a continuous function of the radial position r on the metasurface, and of the polar angle $\psi $. After some mathematical treatments we can write Eq. (2) in polar coordinates as:

$$\left\{ {\begin{array}{{c}} {{k_{02}}\sin {\theta_2}\cos {\varphi_2} = {k_{01}}\sin{\theta_1}\cos {\varphi_1} + \; \cos (\psi )\frac{{\partial \mathrm{\Phi }}}{{\partial r}} - \frac{1}{r}\sin(\psi )\frac{{\partial \mathrm{\Phi }}}{{\partial \psi }}}\\ {}\\ {{k_{02}}\sin{\theta_2}\sin {\varphi_2} = {k_{01}}\sin{\theta_1}\sin {\varphi_1} + \; \; \sin (\psi )\frac{{\partial \mathrm{\Phi }}}{{\partial r}} + \frac{1}{r}\cos (\psi )\frac{{\partial \mathrm{\Phi }}}{{\partial \psi }}} \end{array}} \right.,$$
where $r = \sqrt {{x^2} + {y^2}} $. By using Eqs. (11) and (17) the transmitted rays, incident at a metasurface with polar phase gradient, may be calculated.

Funding

Consejo Nacional de Ciencia y Tecnología (316548).

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

References

1. N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nature Mater 13(2), 139–150 (2014). [CrossRef]  

2. P. Lalanne and P. Chavel, “Metalenses at visible wavelengths: past, present, perspectives,” Laser Photonics Rev. 11(3), 1600295 (2017). [CrossRef]  

3. I. Moreno, M. Avendaño-Alejo, and C. P. Castañeda-Almanza, “Nonimaging metaoptics,” Opt. Lett. 45(10), 2744–2747 (2020). [CrossRef]  

4. E. R. Tkaczyk, “Vectorial laws of refraction and reflection using the cross product and dot product,” Opt. Lett. 37(5), 972–974 (2012). [CrossRef]  

5. J. C. Valencia-Estrada and J. García-Márquez, “Freeform geometrical optics I: principles,” Appl. Opt. 58(34), 9455–9464 (2019). [CrossRef]  

6. A. S. Glassner, editor “An introduction to ray tracing.” Morgan Kaufmann Publisher, (1989).

7. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011). [CrossRef]  

8. F. Aieta, P. Genevet, N. F. Yu, M. A. Kats, Z. Gaburro, and F. Capasso, “Out-of-plane reflection and refraction of light by anisotropic optical antenna metasurfaces with phase discontinuities,” Nano Lett. 12(3), 1702–1706 (2012). [CrossRef]  

9. D. Smith, M. Campbell, and A. Bergmann, “Vectorial approach of determining the wave propagation at metasurfaces,” Ann. Phys. 361, 184–189 (2015). [CrossRef]  

10. J. Hu, J. Xie, S. Tian, H. Guo, and S. Zhuang, “Snell-like and Fresnel-like formulas of the dual-phase-gradient metasurface,” Opt. Lett. 45(8), 2251–2254 (2020). [CrossRef]  

11. C. E. Gutiérrez, L. Pallucchini, and E. Stachura, “General refraction problems with phase discontinuities on nonflat metasurfaces,” J. Opt. Soc. Am. A 34(7), 1160–1172 (2017). [CrossRef]  

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13. C. E. Gutiérrez and L. Pallucchini, “Reflection and refraction problems for metasurfaces related to Monge–Ampère equations,” J. Opt. Soc. Am. A 35(9), 1523–1531 (2018). [CrossRef]  

Data availability

No data were generated or analyzed in the presented research.

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Figures (3)
Fig. 1.
Fig. 1. (a) Schematics of the generalized refraction at a metasurface with arbitrary phase profile. The incident light beam (green) has a normalized wave-vector ${{\boldsymbol k}_1}$ with arbitrary direction. The refracted beam (orange) acquires a direction given by a unitary vector ${{\boldsymbol k}_2}$. The metasurface is in the xy plane at the interface between two media with refractive indices ${n_1}$ and ${n_2}$. (b) Scheme of the refraction showing the dual phase gradient vector on a metasurface. Also it is shown the unit vectors of incident beam ${{\boldsymbol k}_1}$ and the refracted beam ${{\boldsymbol k}_2}$, and ${\boldsymbol n}$ the normal vector of the surface. The local phase gradient has two components, $\partial \Phi / \partial x$ and $\partial \Phi / \partial y$ at each point $({x,y} )$ on the metasurface.

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Fig. 2.
Fig. 2. Schematic of ray tracing for refraction and reflection on a metasurface by using Eqs. (10–13). (a) Refraction of the incident rays with direction fixed. (b) Refraction of the incident rays with random directions. (c) Reflection of the incident rays with direction fixed. (d) Refraction on a metasurface with a polar phase gradient, and for incident rays with random directions.

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Fig. 3.
Fig. 3. Schematic of ray tracing for light transmission through a metalens by using Eqs. (11) and (14). (a) Refraction of the rays from a point source. (b) Refraction of the incident rays with random direction.

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Equations (17)

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k 1 ( 2 ) = ( sin ⁡ θ 1 ( 2 ) cos ⁡ φ 1 ( 2 ) sin ⁡ θ 1 ( 2 ) sin ⁡ φ 1 ( 2 ) cos ⁡ θ 1 ( 2 ) ) .
{ k 02 sin ⁡ θ 2 cos ⁡ φ 2 = k 01 sin ⁡ θ 1 cos ⁡ φ 1 + ∂ Φ ∂ x k 02 sin ⁡ θ 2 sin ⁡ φ 2 = k 01 sin ⁡ θ 1 sin ⁡ φ 1 + ∂ Φ ∂ y ,
∇ Φ = ∂ Φ ∂ x i ^ + ∂ Φ ∂ y j ^ .
∇ Φ = k 02 ( sin ⁡ ϕ 2 cos ⁡ φ 2 i ^ + sin ⁡ ϕ 2 sin ⁡ φ 2 j ^ ) − k 01 ( sin ⁡ ϕ 1 cos φ 1 i ^ + sin ⁡ ϕ 1 sin ⁡ φ 1 j ^ ) .
k 02 ( n × k 2 ) × n − k 01 ( n × k 1 ) × n = ∇ Φ ,
{ k 1 = ( n ⋅ k 1 ) n + ( n × k 1 ) × n k 2 = ( n ⋅ k 2 ) n + ( n × k 2 ) × n .
k 2 = ( n ⋅ k 2 ) n + β ∇ Φ + μ ( n × k 1 ) × n ,
1 = ( n ⋅ k 2 ) 2 + | β ∇ Φ + μ ( n × k 1 ) × n | 2 ,
k 2 = μ ( n × k 1 ) × n + β ∇ Φ ± 1 − | β ∇ Φ + μ ( n × k 1 ) × n | 2 n .
k 2 = μ ( n × k 1 ) × n + λ 0 2 π n 2 ∇ Φ − 1 − | λ 0 2 π n 2 ∇ Φ + μ ( n × k 1 ) × n | 2 n ,
k 2 = μ [ k 1 − ( n ⋅ k 1 ) n ] + λ 0 2 π n 2 ∇ Φ − 1 − | λ 0 2 π n 2 ∇ Φ + μ ( k 1 − ( n ⋅ k 1 ) n ) | 2 n .
k 2 r = ( n × k 1 ) × n + λ 0 2 π n 1 ∇ Φ + 1 − | λ 0 2 π n 1 ∇ Φ + ( n × k 1 ) × n | 2 n .
k 2 r = [ k 1 − ( n ⋅ k 1 ) n ] + λ 0 2 π n 1 ∇ Φ + 1 − | λ 0 2 π n 1 ∇ Φ + ( k 1 − ( n ⋅ k 1 ) n ) | 2 n .
Φ ( r ) = 2 π λ ( r 2 + f 2 − f ) ,
{ k 2 x = μ sin ⁡ θ 1 cos ⁡ φ 1 + β ∂ Φ ∂ x k 2 y = μ sin ⁡ θ 1 sin ⁡ φ 1 + β ∂ Φ ∂ y ,
k 2 z = ± cos θ 2 = ± 1 − ( k 2 x 2 + k 2 y 2 ) ,
{ k 02 sin ⁡ θ 2 cos ⁡ φ 2 = k 01 sin ⁡ θ 1 cos ⁡ φ 1 + cos ⁡ ( ψ ) ∂ Φ ∂ r − 1 r sin ⁡ ( ψ ) ∂ Φ ∂ ψ k 02 sin ⁡ θ 2 sin ⁡ φ 2 = k 01 sin ⁡ θ 1 sin ⁡ φ 1 + sin ⁡ ( ψ ) ∂ Φ ∂ r + 1 r cos ⁡ ( ψ ) ∂ Φ ∂ ψ ,
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