Phase Profile Sampling in Metasurfaces: When Does Aliasing Matter?

Key Points:

• Metasurface phase profiles are implemented on a discrete spatial lattice. If the sampling period is too coarse, the realized optic can deviate significantly from an intended continuous phase profile because of aliasing effects
• The relatively small periods available with appropriately designed metasurfaces can provide an advantage over coarser diffractive optics when implementing rapidly varying phase profiles
• Coarse sampling does not necessarily preclude achieving the desired optical function. Instead, it can redistribute power into unwanted propagating orders, reducing intensity in the intended field and increasing stray light
• Finer spatial sampling alone does not guarantee better performance. The meta-atom response, fabrication limits, and other physical constraints still need to support the intended phase profile.

Overview

Metasurfaces are useful platforms for implementing optical phase masks and, in many cases, can provide much finer spatial sampling than conventional diffractive optics. This can be an important advantage when the target phase varies rapidly across an aperture, since a smaller sampling period provides more spatial samples over the same phase variation and can better represent steep phase gradients.

Reducing the period does not automatically produce a better optic. Meta-atom transmission, coupling, fabrication limits, incidence-angle dependence, polarization, and other electromagnetic and stray light effects still matter. Here, however, we focus on a simpler question: is the desired phase profile being sampled finely enough in the first place?

Local Spatial Frequency and Fourier Spatial Frequency Are Not The Same

For a general complex optical function,

$$g(x,y)=a(x,y)e^{i\phi(x,y)},$$

the local spatial frequencies can be defined from the phase gradients,

$$f_{x,\mathrm{local}} = \frac{1}{2\pi} \frac{\partial \phi}{\partial x}, \qquad f_{y,\mathrm{local}} = \frac{1}{2\pi} \frac{\partial \phi}{\partial y}.$$

For a constant-amplitude linear phase ramp,

$$g(x)=e^{i2\pi f_0x},$$

the local spatial frequency is $f_0$ everywhere, and the Fourier spectrum contains a single component at $f_0$. In this special case, the local and Fourier descriptions are directly equivalent. That equivalence does not generally hold for a nonlinear phase function.

As described by Goodman previously, the connection can be seen by expanding the phase about some point $(x_0,y_0)$,

$$\phi(x,y)\approx \phi(x_0,y_0) + (x-x_0) \left.\frac{\partial\phi}{\partial x}\right|_{x_0,y_0} + (y-y_0) \left.\frac{\partial\phi}{\partial y}\right|_{x_0,y_0} +\cdots.$$

If the higher-order terms can be neglected over a sufficiently small region, the phase is locally linear and the first derivatives describe a local phase ramp. For a lens, however, the phase gradient changes continuously across the aperture. The complete aperture-limited transmittance therefore contains a distribution of Fourier spatial frequencies, and there is no general one-to-one localization between a position in the optic and a Fourier component.

This matters when applying sampling arguments. The maximum local phase gradient provides a useful engineering estimate for how finely a lens should be sampled, but it is not the same as considering the Fourier spectrum of the complete optical function.

Sampling an f/1 Metalens

To illustrate the effect, we simulated an ideal phase-only hyperboloid lens at 632 nm with a 100 µm diameter and 100 µm focal length, corresponding to f/1 and a maximum numerical aperture of approximately 0.45. The same continuous phase function was represented using square sampling periods of 300 nm, 600 nm, and 1.2 µm.

To isolate phase-profile sampling from numerical propagation sampling, all three cases were represented and propagated on the same 150 nm computational grid. The phase was held constant over progressively larger regions of that grid, while the aperture, incident power, wavelength, and propagation method were otherwise unchanged.

As shown in Figure 1, the 300 nm case closely follows the continuous phase profile, while the discretization becomes progressively more apparent at 600 nm and especially at 1.2 µm near the edge of the lens.

Figure 1. Hyperboloid metalens phase sampled at 300 nm, 600 nm, and 1.2 µm. The lens operates at 632 nm with a 100 µm diameter and 100 µm focal length. All cases use the same 150 nm propagation grid, with each phase value held constant over $2\times2$, $4\times4$, and $8\times8$ numerical samples, respectively.

Coarse Sampling Redistributes Light

Figure 2 shows the propagated x-z intensity on a shared dB scale. The 300 nm case exhibits the expected converging field with relatively little power outside the desired focusing behavior. At 600 nm, additional off-axis propagation becomes visible. At 1.2 µm, several additional branches are clearly present and a much larger fraction of the field has been redirected away from the intended focus.

All three masks have the same incident power and, because they are phase only, the same integrated field intensity immediately following the mask. The lower focal intensity is therefore due to redistribution of the transmitted field rather than reduced input power. The peak focal intensity falls to approximately 80% of the 300 nm result for the 600 nm case and 31% for the 1.2 µm case.

An undersampled phase mask therefore does not necessarily fail in an obvious way. Such a lens may still produce a recognizable focal spot while also generating substantial unwanted diffraction and stray light.

Figure 2. Simulated x-z intensity for the three sampled phase masks, shown using a common dB reference. The shared scale makes both the reduction in desired focal intensity and weaker off-axis diffraction orders visible.

The Angular Spectrum Shows the Sampling Orders

The angular spectrum provides a more direct view of where the power goes. Sampling a continuous spatial function on a periodic lattice produces shifted copies of its spatial frequency spectrum. As the sampling period increases, these copies move closer together. With sufficiently fine sampling, the additional orders remain outside the angular range of interest corresponding to propagating wavevectors. With coarser sampling, these copies can enter the propagating region and eventually overlap the spectrum associated with the desired field.

This behavior is visible in Figure 3. With 300 nm sampling, the additional spectral orders remain well separated from the desired propagating spectrum. At 600 nm, additional orders begin to enter the propagating region and carry power away from the focus. At 1.2 µm, the replicated spectra substantially overlap the angular range occupied by the desired lens field. After propagation, these additional angular components interfere in real space, producing the multiple branches visible in the x-z intensity distribution shown above in Figure 2.

Figure 3. Angular-spectrum intensity for the three sampled phase masks in normalized transverse direction-cosine coordinates. The outer circle represents the propagating region, while the dashed inner circle corresponds to the nominal lens NA of 0.45. As the sampling period increases, additional spectral orders move into the propagating region and eventually overlap the angular range occupied by the desired field.

A Local Nyquist Criterion Is Useful but Incomplete

For a lens with maximum local direction cosine equal to its NA, the maximum local spatial frequency can be approximated as

$$f_{\mathrm{local,max}}=\frac{\mathrm{NA}}{\lambda}.$$

A corresponding local Nyquist estimate of the required minimum period thus gives

$$p < \frac{\lambda}{2\mathrm{NA}}.$$

For this lens, the resulting sampling period is approximately 707 nm. The 300 nm and 600 nm cases therefore satisfy this simple criterion, while the 1.2 µm case does not. This agrees reasonably well with the simulations, but it also shows why the local frequency distinction matters. The 600 nm design retains a strong focus and satisfies the local Nyquist estimate, but additional propagating orders are already visible and its peak focal intensity is reduced. There is no contradiction with the sampling theorem: the local-gradient criterion treats a small portion of the nonlinear phase profile as a linear phase ramp, while the complete aperture-limited transmittance has a broader Fourier spectrum. The local phase gradient is therefore a useful design metric, but it is not a complete description of the spatial-frequency content of the optic.

Why the Sampling Period Matters for Metasurfaces

This sampling argument helps explain one potential advantage of metasurfaces over coarser diffractive optics. As the required diffraction angle or NA increases, the phase profile varies more rapidly. A smaller lateral period provides more samples per local phase cycle and can delay the onset of unwanted sampling orders and aliasing.

That does not mean a metasurface will necessarily outperform a conventional diffractive optic. Material choice, available phase control, fabrication constraints, polarization, and electromagnetic behavior all remain important. But when two platforms are asked to implement the same rapidly varying phase profile, the ability to use a substantially smaller period can be a real design advantage.

Summary

Metasurface phase masks are spatially sampled optical functions, and the physical period of the structure limits how rapidly the target wavefront can be represented. For locally linear phase profiles, the phase gradient provides a direct spatial-frequency description. For nonlinear functions such as lenses, it remains a useful local approximation, but is not generally equivalent to the Fourier spectrum of the complete optical transmittance.

When sampling becomes coarse, a metalens does not abruptly stop focusing. Instead, additional spectral orders enter the propagating region and optical power is redirected away from the intended field. The fine periods available with metasurfaces can therefore provide an important advantage for steep phase gradients, provided the electromagnetic response and fabrication of the underlying structures can support the intended design.

References

Goodman, Joseph W. Introduction to Fourier optics. Roberts and Company publishers, 2005.

Kamali, Seyedeh Mahsa, et al. “Highly tunable elastic dielectric metasurface lenses.” Laser & Photonics Reviews 10.6 (2016): 1002-1008.

Fröch, Johannes E., et al. “Computational imaging with meta-optics.” Optica 12.6 (2025): 774-788.

Matsushima, Kyoji, and Tomoyoshi Shimobaba. “Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields.” Optics express 17.22 (2009): 19662-19673.