Key Points:
- One way of describing the physics of periodic metasurfaces is via coupled mode theory. Energy within a metasurface grating is confined to certain allowed “modes”, dictated by material properties and geometry, that may couple amongst themselves or back into free space
- Constructive and destructive interference amongst these modes at the top and bottom interfaces of the grating give rise to the characteristic peaks and dips in transmission and reflection coefficients for metasurface unit cell designs. High contrast grating theory and the coupled mode approach provide some intuition about metasurface designs, but many practical metasurfaces are aperiodic. There are methods to treat aperiodic systems as a set of slowly varying periodic gratings
- In practice, it is still necessary to rigorously simulate, or fabricate and test, devices to evaluate their performance more holistically. Having physical intuition about metasurface operation, however, can still be critical for troubleshooting and realizing new designs
Overview
As touched on in a separate article, how one describes the physics of a metasurface will depend on the model used. There are, however, a few standard ways of thinking about how metasurfaces operate that are useful to go through in a bit more detail. This article provides a high-level description and focuses on an intuitive view of metasurface physics based on a coupled mode theory perspective. Please refer to and review in detail the references at the end if you would like to get further into the underlying mathematics and derivations—they are quite useful and worth your time (especially if you’re a PhD student studying metasurfaces).
The coupled mode theory perspective is useful for gaining intuition about the physics of unit cell designs for periodic gratings, and the discussion here is based largely on the body of work on high contrast gratings that largely preceded the metasurface research of the 2010’s (please refer to the references at the end for further reading). Other formalisms and approaches can be used to describe metasurface operation. One notable approach relies on a framework called the Generalized Sheet Transition Conditions and will be the focus of a future article.
High Contrast Gratings
Many dielectric metasurfaces are like other gratings with the distinguishing feature that their pitch (i.e., period) is subwavelength. The relationship between wavelength and period is central to understanding the operation of a metasurface. A useful framework for thinking about this is the body of work on high contrast gratings and coupled mode theory. A high contrast grating is an idealized abstraction of a grating wherein a 1-D periodic array of rectangular grating bars exists within a vacuum—this makes the math simpler, though this can be extended to include a substrate, different interstitial materials, various grating cross sections, as well as 2-D gratings rather than just 1-D. Much of the intuition of the 1-D case still holds though and so it’s useful to walk through it as an example.
In the case of the 1-D high contrast rectangular grating, light couples into the periodic bar array and eventually will either transmit, reflect, or be absorbed. In the simplest model, the dielectric is lossless, so there’s no absorption—so all light is ultimately transmitted or reflected. What do we mean when we say that light “couples into” the grating? What happens? And for what reason do we see resonances or sudden dips in amplitude or phase when we compute either the transmission or reflection coefficient of such a grating as a function of its geometry?
Modes and Coupling Mechanisms
Based on the refractive index and geometry of the grating, this dictates boundary conditions and defines a set of possible solutions to Maxwell’s equations. Any given field distribution that can exist within the grating can be expressed as a linear combination of “modes” of the grating. What is a “mode”? A mode is essentially an allowed field or energy distribution dictated by a system’s boundary conditions and material properties—this is fairly general, and not just for optics and metasurfaces. In this context, it means that when a planewave (i.e., a mode of free space) is incident and “couples into” a grating, it means that a portion of the planewave’s energy is being transferred into the allowed set of modes within the grating. Within the grating, the allowed modes bear resemblance to those of waveguides and high contrast grating theory decomposes the electric and magnetic field distributions within the system into superpositions of these “waveguide” modes.
Each of these modes has a coupling coefficient to free space, but all the modes also have coupling coefficients to one another (and themselves). Think of these coupling coefficients as describing how efficiently the energy from a given mode is converted into another mode. The modal dynamics of the metasurface are thus governed by energy oscillating amongst a set of allowed modes within the grating. The modes propagate up and down within the grating independently of one another, except at the top and bottom interfaces, where they may either “transmit” into free space or “reflect” back into the grating, either into the same mode or potentially coupling into a different grating mode.
In practice, on each roundtrip through the grating, a fraction of a mode’s energy will couple to free space, a fraction will couple back into the same mode, and a fraction will couple into other grating modes. In metasurface design, when one engineers the geometry to achieve a particular phase or amplitude response, they arrive at a particular balance between these couplings. The interference of all these different outcouplings gives rise to the overall transmission and reflection coefficient of a unit cell design via varying degrees of constructive and destructive interference.
Different Mode Regimes
Depending on the size of the grating relative to the wavelength, the refractive index, and geometry, the set of modes will be different. Broadly speaking, there are three distinct operating regimes: 1) the deep subwavelength case, 2) the dual mode regime, and 3) the diffraction regime. In the diffraction regime, the grating period is larger than the wavelength and while a variety of modes may exist in the grating, much of the light is diffracted to higher orders. In this regime, the grating is not considered a metasurface as it is not subwavelength. The deep subwavelength regime has a very fine period relative to the wavelength. In this regime, there is only one allowed mode and you can effectively model the grating as an effective medium with a Fabry-Perot like resonance. It is the second case, the dual mode regime, that is often the most interesting for metasurfaces, as the interference of these modes can give rise to more rich and diverse optical responses.
In the dual mode regime, the modal dynamics are analogous to those of allowed modes in a pair of coupled masses connected by a spring, each of which is attached to a boundary by their own separate spring. This is a canonical example of coupled oscillators. Under certain conditions, given the appropriate masses of the springs and coupling constant (i.e., the spring constant between them), the masses may oscillate in tandem together, completely opposite to one another, or in some mixed state. This can present cases where the masses are strongly or weakly coupled to one another, where the state of one mass affects the state of the other to different degrees.
The same can be said about the modes in a metasurface. Depending on the coupling coefficients amongst the different modes, there can be cases of strong coupling which can lead to high quality factor resonances, where light is confined for many roundtrips within the grating. Situations in which the outcoupled modes constructively interfere with one another can give rise to very high transmission or reflection, depending on whether one’s considering the top or bottom of the grating relative to the source.
In the figure at the beginning of this article, as a function of period, the amplitude of the reflection coefficient of a representative high contrast grating is shown. Three distinct operating regimes are shown, with a checkerboard-like pattern of amplitude at center. This pattern arises from alternating constructive and destructive mode interference based on the grating geometry. When the period is much smaller relative to the wavelength, this behavior disappears and instead the reflected amplitude exhibits a series of peaks akin to those of Fabry-Perot resonances, whereas when the period is larger than the wavelength, the amplitude is significantly reduced owing to a large portion of the light being diffracted to higher orders.
Strong Coupling and Weak Coupling
While the case of strong coupling in periodic dielectric gratings is fascinating from a physical perspective, and while it has applications in high-Q resonant gratings (e.g., for mirrors in VCSELs in place of Distributed Bragg Reflectors), most metasurfaces do not rely on such strongly coupled modes for their operation. Oftentimes, a metasurface design will rely on a geometry that corresponds to a far lower quality factor resonance. This is for a variety of reasons. Firstly, for achieving the necessary phase shifts, such high-quality factor resonances are usually overkill and can actually be limiting from a manufacturing perspective because of how sensitive they are. Secondly, as most metasurfaces are not actually periodic, these highly resonant gratings tend to be far more perturbed by small the geometric changes of neighboring elements, killing the resonant behavior of the nanostructure if it’s designed with a very high-quality factor. As such, most metasurfaces in practice tend to rely on low-quality factor resonances for their operation, which facilitate relaxed manufacturing tolerances while still providing a useful range of phase shifts.
Extending Beyond Periodic Arrays
What we’ve discussed here has focused on periodic grating designs. It’s rare that one would actually design a grating directly using this formalism, and instead one would use methods such as rigorous coupled-wave analysis (RCWA) or finite-difference time-domain (FDTD) to simulate the grating without having the designer examine and understand each individual mode; however, this approach helps to provide intuition about the physics of metasurfaces and how certain phase and amplitude features may arise from constructive and destructive interference amongst modes that are coupling to free space.
One limitation of this formalism is that most metasurface designs are not periodic. That being said, locally it is often reasonable to still describe the operation of a metasurface via coupled mode theory, assuming the grating geometry varies “slowly”. In practice, it is necessary to either simulate an aperiodic design directly or fabricate and test the device as the coupled mode theory perspective, focused on periodic systems, is incomplete. One way of thinking about aperiodic metasurfaces is through decomposing them as a set of many periodic gratings, where the grating design varies across the surface. In this manner, much of the intuition granted via coupled mode theory still holds, but practical calculations are possible with other techniques. In a future article, we’ll discuss further implications of aperiodic designs for higher order diffraction.
References
Lalanne, Philippe, Jean Paul Hugonin, and Pierre Chavel. “Optical properties of deep lamellar gratings: a coupled Bloch-mode insight.” Journal of Lightwave Technology 24.6 (2006): 2442.
Chang-Hasnain, Connie J., and Weijian Yang. “High-contrast gratings for integrated optoelectronics.” Advances in Optics and Photonics 4.3 (2012): 379-440.
Yang, Jianji, and Jonathan A. Fan. “Analysis of material selection on dielectric metasurface performance.” Optics express 25.20 (2017): 23899-23909.
Sandeep, Srikumar, Jian-Ming Jin, and Christophe Caloz. “Finite-element modeling of metasurfaces with generalized sheet transition conditions.” IEEE Transactions on Antennas and Propagation 65.5 (2017): 2413-2420.
Ghaneizadeh, Alireza, and Mojtaba Joodaki. “Generalized sheet transition conditions (GSTCs) in electromagnetic metasurface modeling.” IEEE Access 12 (2024): 74305-74326.
Arbabi, Amir, et al. “Subwavelength-thick lenses with high numerical apertures and large efficiency based on high-contrast transmitarrays.” Nature communications 6.1 (2015): 7069.