Where Does All the Light Go? Metasurface Efficiency and Stray Light as Two Sides of the Same Coin

Key Points:

• Efficiency and stray light are key challenges for metasurface components and in many ways they represent slightly different perspectives of the same effect. Light that doesn’t go into the desired response (i.e., the desired diffraction order) will show up elsewhere as stray light, which can ultimately impact contrast and system performance.
• Stray light in metasurfaces can be understood through familiar concepts like phase errors and diffraction into higher orders. While the details are more complex than in conventional diffractive optics, similar intuition still applies, with unwanted orders showing up as background signal or ghost images.
• As numerical aperture (or required phase gradient) increases, efficiency typically drops due to phase sampling limits and breakdown of local approximations between neighboring meta-atoms. These effects make high NA designs more challenging, even if metasurfaces may still be the only practical way to achieve the high phase gradients beyond the coarse sampling supported by more conventional, multi-level diffractive optics.

Some light is transmitted, some is absorbed, and some is reflected…as with any other optic. Fourier theory applied to diffractive optics, however, enables us to delve further into where and how this modulated light gets distributed

One of the main objectives that we’re aiming to accomplish with this series of learning posts is to dispense with the notion that metasurfaces are inherently “special” or exhibit properties that depart unphysically from those of other optical systems. While we do view metasurfaces as a promising, emerging technology (obvious, given our company’s existence and mission), the nature of these components is often more mundane than some may have you believe—metasurfaces do exhibit niche advantages in the right applications (e.g., high NA, freeform surface engineering, and/or multifunctional applications across lightfield parameters), but they also have their own limitations and notably stray light and inefficiency are commonly characteristic of metasurface components. These components largely overlap with the capabilities of multi-level diffractive optics, but there are conditions under which these two platforms diverge and when metasurfaces can have the upper hand. There has been little formal treatment, however, on the mechanisms and manner in which these two platforms diverge in the context of stray light and diffraction efficiency.

When light is incident on a metasurface, a portion of that will be transmitted, a portion will be reflected, and a portion may also be absorbed depending on the extinction coefficient of the materials in the optical stack. In this manner, metasurfaces are like any other optical component, they must abide by the same energy conservation laws and there is no means around that.

Efficiency, however, comes up very frequently as a talking point in discussions about metasurfaces, whether in relative terms when touting advantages compared to other metasurface designs and fabrication processes, or as a means of showcasing inferiority of metasurfaces as a platform more generally depending on who you’re talking to (e.g., relative to refractive optics or conventional, binary optics). Emphasis on efficiency is important, although we’d argue it sometimes misses the full picture, because ultimately the signal-to-noise ratio itself and the contrast in an imaging or non-imaging system is more critical (rather than a raw efficiency metric per se). In another article we talk about metasurface efficiency and some of the varying definitions and significance of these. In this article, we instead focus on stray light in particular, which is inherently coupled to the problem and challenge of metasurface efficiency. Put in plain terms, the question we are trying to address is:

For light incident on a metasurface, where does the light go that does not go where we want it to go?

The reality is more complicated than in the scalar diffraction regime, but similar intuition still holds and there are analogous effects

In the scalar diffraction regime where conventional, multi-level diffractive optics reside, there are already well-established formalisms and theory that can predict efficiencies of these components based on both quantization effects (i.e., driven by the number of phase levels in a design) as well as phase errors from certain classes of manufacturing defects (e.g., an error in thickness of the material). When designing or building a diffractive optical element, there is usually a target phase and/or amplitude function. In practice, there will be some discrepancy between the target function and the actual realized phase function. In general, one can treat the actual modulated wavefront as arising from application of amplitude and phase error functions to the underlying target function, as below:

Where $\phi$ is the target phase, a is the amplitude modulation function, and p is the phase modulation function, that together constitute a function m that characterizes how a target phase is transformed into a realized wavefront. If p and a are just identity functions, then there is no phase or amplitude error and the target phase is exactly implemented. Moreno and coauthors previously derived and showed that decomposition of this effective wavefront through a Fourier series enables deconstructing the output wavefront into a superposition of diffracted orders.

With the coefficients $G_{\alpha}$ determined via

When the error in this modulation function is 0 (i.e., the imparted phase is exactly equal to the target phase), then all of the incident light will be diffracted into the first order, which mathematically can be seen as there only needing to be the $\alpha = 1$ term in the Fourier series expansion of the modulation function.  When error is nonzero, however, terms in the Fourier series expansion beyond the $\alpha = 1$ term become nonzero as well. When considering the case of a lens, orders outside of +1 effectively become background signal that reduce contrast in the image (i.e., stray light). Nonzero positive orders can be particularly problematic for converging imaging lenses, as these form higher order foci at integer divisions of the target focal length (i.e., f/2, f/3, f/4, etc.). The relative magnitude of these focal spots will depend on the actual target phase profile and the form of the error introduced to it, but in general these higher order foci will be present, resulting in the formation of ghost images along the optical axis. As can been seen in the Fourier series characterizing a lens phase (shown below), there will be a superposition of different lensing terms, including converging and diverging lenses with the $G_{\alpha}$ coefficients determining the relative magnitude of each term, with there being focal lengths that occur at discrete, integer harmonics corresponding to individual $\alpha$ values in the series expansion. For example, $\alpha = 1$ is the target design order, $\alpha = 2$ corresponds to a converging lens with half the focal length, $\alpha = 0$ is zero order light that passes through unmodulated, $\alpha = -1$ is a diverging lens of the same optical power, $\alpha = -2$ is a diverging lens of half the focal length, etc.

For the case of a metasurface, the formalism developed by Moreno and coauthors in its original form does not technically apply, as it was assumed that the components in those cases operated in the scalar diffraction regime where the underlying element period was multiple wavelengths at the minimum. In being in such a regime, one could neglect the effects of different phase elements in the mask coupling to one another.

With a metasurface, the situation is a bit more complicated; however, this formalism still provides very useful intuition and via generalization of the error functions in their modulation function, one could determine the diffracted orders outside of the +1 target order similarly for the case of a metasurface. In general, however, these error functions for the case of a metasurface are nontrivial to determine and heavily depend on the actual target phase profile, the particular meta-atom design, and any additional manufacturing defects.

Phase errors and higher order diffraction represent slightly different perspectives on the same effect

Through the formalism applied by Moreno and coauthors, we can see that the p and a error functions introduced onto the target phase are a means of describing the effective wavefront in the spatial domain. This is perfectly valid. Equivalently, however, when considered from the Fourier domain as shown through their Fourier series expansion, application of these error functions ultimately give rise to a set of discrete orders that depend on both the nature of the error as well as the underlying target function itself. In this manner, wavefront errors give rise to stray light in the form of diffraction into orders outside the +1 design order. And conversely, with metasurface components that exhibit a large degree of zero order light, or higher order focal spots (e.g., occurring at the predicted Fourier harmonics), this stray light can be explained as arising from underlying errors in the phase imparted by the metasurface itself.

Practically speaking, this may or may not matter and will depend on how well you can control and suppress this stray light. We would argue, however, that having intuition and a deeper understanding of this mechanism is necessary (though perhaps not always sufficient) for mitigating stray light in metasurface systems.

As an example, what happens when there is a phase error in the form of a restricted phase range (e.g., only a fraction of the full 0 to 2π range is feasible to achieve with a given design platform)? This can occur with fabrication error if a design is improperly etched or the achieved thickness of the device layer is lower than the target. In the figure below, we show for a metalens phase what the diffracted axial intensity looks like for a few different cases including achieving the full 0 to 2π range and then successively smaller fractions of this by letting the the phase error function p be a scalar multiple that dictates the achievable phase range. As can be observed, harmonics in the form of higher order focal spots are present at the reduced phase range and the relative magnitude of these higher order foci increases as the phase range reduces further.

Phase errors can also occur when the input condition changes and the metasurface itself remains the same. For example, if we have a metasurface that functions well at 532 nm wavelength, but we than change the source wavelength to 525 nm, then there will be an “error” in the phase function at 525 nm relative to the ideal metalens phase at that wavelength. This manifests as chromatic aberration, with there be a longitudinal shift of the +1 order focal plane; however, even if there were no diffraction outside of +1 at the nominal wavelength of 532 nm, there will now be light “lost” to these extra orders at the off-design wavelength.

At low NA, metasurfaces can be as efficient as well-designed, conventional multi-level diffractive optics if designed appropriately, but in this regime a metasurface is usually unnecessary (unless manufacturing and unit economics at scale make it favorable for the given use case)

Metasurfaces are often considered to be inefficient (and they can be), but this is not always the case, but rather it is a function of the scatterer design and the NA (or required phase gradients) for a given application. As the NA or effective phase gradient for a target phase mask increases, there are several effects that ultimately will usually lead to the diffraction efficiency decreasing into the +1 order. One of these is really quite analogous to what happens with multi-level diffractive optics, which is that at a higher phase gradient (e.g., a larger deflection angle for a beam deflection phase) the physical length scale over which a ramp from zero to 2π phase occurs decreases, and given the fixed center-to-center spacing between meta-atoms will mean that there are fewer phase steps that sample this zero to 2π range. It’s already well established that quantization errors in terms of number of phase levels affect efficiency, with fewer steps reducing diffraction efficiency into the +1 order.

There are additional effects that are more complicated for the case of a metasurface. Notably, at a high phase gradient, given the subwavelength scale between nanostructures, the degree to which the local phase approximation is violated is much greater, which practically means there is a larger difference in feature size between immediately adjacent unit cells. While the local phase approximation assumes zero phase error, which via the formalism of Moreno and coauthors would mean 100% efficiency, we know that that is not the case for a metasurface and this approximation successively breaks down as the NA requirement increases. As deflection angles become even higher, there are other effects that may be better understood from antenna theory, wherein at large enough feature sizes certain nanostructures exhibit off-axis nulls in their scattering profile, which can cause large reductions in efficiency.

At higher NA, however, where conventional DOEs cannot meet target specifications owing to inadequate sampling of the phase profile at periods of several to tens of wavelengths, a metasurface (even if “inefficient”) may be the only design platform capable of achieving high enough phase gradients

By definition, conventional multi-level diffractive optics operate in the scalar diffraction regime where their period is at a minimum several multiples of the wavelength, and they are fundamentally limited to supporting small numerical aperture. These components can achieve higher deflection angles by primarily diffracting into the second or third order, but there are still significant limitations. With a metasurface, even if there are efficiency reductions at higher NA, if fine sampling of the phase profile on the order of the wavelength or less is necessary, then by definition it may be the only viable platform for implementing such a profile. This is functionally equivalent to the concept that metasurfaces exhibit higher space bandwidth product for manipulating wavefronts as compared to coarser pitch alternatives.

Stray light modeling (and correction) is a challenge for metasurface R&D and commercialization efforts

Stray light is a frequent issue with metasurface components and systems, both in imaging and display applications, leading to reduced contrast and degraded image quality. As a designer extends to multi-element systems, these challenges can be further exacerbated, with multiplicative losses of light to higher orders. From a design perspective, this can be nontrivial to address, but in practical terms this also constrains commercialization of metasurface systems because it’s challenging to mitigate stray light given the limitations of existing models.

At this time, designers primarily rely on simplistic models (e.g., utilizing the local phase approximation) that poorly capture the nature of stray light in metasurface systems, or at the other extreme rely on full-wave methods (e.g., finite-difference time-domain), which can more accurately capture these effects, but that are infeasible to apply to metasurface components at a commercially relevant aperture size (e.g., millimeters to centimeters) without expending weeks to months of simulation time (if not years), which in compute costs would become expensive quickly when competing for GPU time with AI developers and companies.

There is a middle ground of simulation approaches, such as using overlapping domains, and scatterer proxy functions that extend the local phase approximation to larger neighborhoods of multiple unit cells. These partly address stray light generation, but they offer an incomplete picture of the underlying physics and can demand more computational resources than the standard local phase approximation for any given simulation run, depending on settings and certain model assumptions.

EdgeDyne is developing a fast and accurate stray light model amenable to multi-element systems

EdgeDyne is developing an accelerated stray light model that is amenable to multi-element systems for accurately computing both transmission and focusing/diffraction efficiencies and stray light without requiring computationally intensive full-wave simulations of large metalens apertures; the aim is to reduce computational requirements without commensurate reductions in simulation accuracy relative to full-wave methods. This requires accurate treatment of coupling mechanisms between meta-atoms and generalization to calculations for arbitrary target phase profiles. The developed module will be offered as part of EdgeDyne’s software for holistic metasurface simulation, design, and optimization.

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