Multiple-Focal-Length Achromatic Systems

• Multiple-focal-length achromatic systems are advanced optical architectures that deliver multiple, tunable focal points while eliminating chromatic aberration across broad spectral bands.
• They employ innovative phase engineering methods—such as metasurface modal control, rotational Moiré doublets, and variational N-achromat frameworks—to achieve precise, achromatic focusing.
• These systems are applicable in compact zoom imaging, AR/VR, ophthalmic lenses, and spectroscopy, offering high efficiency, stable focal lengths, and scalable fabrication.

Multiple-focal-length achromatic systems are advanced optical architectures engineered to deliver multiple, tunable, or simultaneous focal points while maintaining achromatic (wavelength-independent) focusing properties across broad spectral bands. These systems span refractive, diffractive, and metasurface-based implementations and address longstanding constraints in applications such as miniaturized zoom imaging, augmented/virtual reality, ophthalmic lenses, and spectrally invariant microscopy.

1. Theoretical Principles and Design Equations

At their core, multiple-focal-length achromatic systems seek to overcome the intrinsic wavelength dependence of optical path differences that causes chromatic aberration in conventional lenses. The fundamental phase profile required by a lens of focal length $f(\lambda)$ for wavelength $\lambda$ at radius $r$ is

$$\varphi(r, \lambda) = \frac{2\pi}{\lambda}\left( \sqrt{r^2 + f^2(\lambda)} - f(\lambda) \right)$$

This law underpins both refractive and metasurface approaches (Aiello et al., 2019, Hou et al., 27 Jul 2024, Hu et al., 2022, Laville et al., 13 Nov 2025).

Varifocality and/or multifocality are introduced in several distinct architectures:

• Metasurface Modal Control: Utilizing polarization channels, multiple focal lengths can be engineered within a single metasurface, exploiting orthogonal phase responses to produce distinct, achromatic foci (Aiello et al., 2019, Hou et al., 27 Jul 2024).
• Layered or Conjugate Elements: Hybrid constructs, such as conjugate quartic phase plates (the Alvarez concept), employ the controlled superposition of engineered phase profiles to realize tunable zoom or extended depth-of-field with chromatic invariance (Colburn et al., 2019).
• Rotational Moiré Doublets: Achromatic zoom is accomplished by phase quantization and mutual rotation of complementary metasurface layers, yielding a tunable, nearly spherical phase profile that is optimized for minimal chromatic error (Hu et al., 2022).
• Variational Refractive Combinations: The $N$-achromat formalism and its variational extensions provide a systematic method to assign geometries and glass types to a lens stack such that either discrete or continuous bands are mapped to prescribed focal strengths—enabling multiple-focal-length achromatic operation (Laville et al., 13 Nov 2025).

2. Refractive/Hybrid Achromat Design: The Variational $N$-Achromat Framework

The classical doublet achromat corrects chromatic aberration at two wavelengths via balancing lens powers and dispersions. The generalized $N$-achromat, for $N$ lenses and $N$ wavelengths, solves the system

$$\sum_{i=1}^N (n_i(\lambda_j) -1) K_i = \phi_0 \quad \forall j=1\ldots N$$

where $K_i$ are lens curvatures and $n_i(\lambda)$ the dispersions. The analytical “pentachromat” ($N=5$) provides closed-form coefficients for five-wavelength achromatization. For continuous spectral correction or multiple-focal-length bands, a variational approach is used:

• Define $M$ spectral intervals (windows) $\Delta\lambda_j$ and target powers $\phi_j$.
• Impose affine constraints $$\int_{\Delta\lambda_j}\phi(\lambda)\,d\lambda = \phi_j\,\Delta\lambda_j$$ for $j=1..M$.
• Minimize the integrated squared residual chromatic error across the union of all $\Delta\lambda_j$ while satisfying physical manufacturability criteria. This methodology enables both tailored spectral bandwidths and customized focal power assignment, extending beyond the superachromat paradigm and supporting multi-focal or multi-window designs (Laville et al., 13 Nov 2025).

3. Diffractive and Multifocal Metalens Architectures

Diffractive and metasurface-based systems leverage advanced phase engineering and nanofabrication to address miniaturization and chromatic aberration simultaneously:

• Spectral Multifocal Diffractive Lenses: The phase relief is optimized to encode multiple fixed foci at several specific wavelengths via a joint amplitude-matching merit function:

$$E[h] = \sum_{j=1}^N \int_0^R \left| T_\mathrm{smf}(h(\rho);\lambda_j) - U_{\mathrm{target},j}(\rho;\lambda_j) \right|^2 d\rho$$

This approach, demonstrated experimentally with zone plates featuring three foci fixed across blue, green, and yellow-red, yields ≈30% focusing efficiency per focus and ≤0.05 mm chromatic focal shift (Doskolovich et al., 2018).

• Metalens Doublets with Moiré Principle: Rotating two metasurface phase plates with quantized, opposing spherical profiles creates a continuously tunable (1–10×) achromatic zoom, optimized globally across three design wavelengths (440, 540, 640 nm) for efficiency ≈86% and sub-7.3% focal-length coefficient of variation (Hu et al., 2022).
• Nanofin-Based Bi-focal Metalenses: Utilizing polarization-split phase channels—co-polarized and cross-polarized responses of anisotropic nanofins—enables simultaneous off-axis foci with individual phase correction, maintaining <5% focal-length drift and >75% channel efficiency (average 80.5%) across 450–650 nm. Particle-swarm optimization is used for library selection and phase error minimization (Hou et al., 27 Jul 2024).

Platform	Achromatic Bandwidth	Number of Focal Lengths	Efficiency
Moiré metalens doublet	440–640 nm (3λ)	Tunable (1–10× zoom)	up to 86.5%
Bi-focal nanofin metalens	450–650 nm (3λ + cont.)	2 simultaneous	76–85%
Spectral diffractive	3 design λ (can be extended)	3 fixed	~90% total (3 foci)
Conjugate metasurface	400–700 nm (broadband)	Tunable (5× zoom, continuous)	~37%
Variational $N$-achromat	User-set intervals	User-defined	Glass-limited

4. Achromatic Varifocal Metalens and Hybrid Systems

The varifocal metalens paradigm achieves tunability and achromatism by building phase dispersion into the unit-cell response or introducing external control:

• Polarization-Modulated Metalens: A metasurface of asymmetric TiO₂ nanofins produces distinct phase delays for horizontal and vertical input states. By rotating the input polarization angle and employing wavelength-dependent pre-rotation using polarization optics (AQWP–LC–AQWP stack), continuous and electronically switchable zoom is achieved with achromatic focusing over 483–620 nm. Focal length is tunable from 220–550 μm at nearly diffraction-limited quality with lateral FWHM ≈λ/(2NA), and only ~13 μm focal shift after correction (Aiello et al., 2019).
• Conjugate Quartic Metasurfaces (Alvarez-like design): Two laterally shifted, oppositely signed quartic phase plates produce a tunable quadratic term (controls focal length) and a third-order term (invariance of point spread function over $\lambda$ and $d$). The system delivers a 5× zoom (4.8 mm tunable range), spectrally-invariant PSF over 400–700 nm, and full-color images after a single, wavelength-independent deconvolution. The efficiency is ~37% and PSF invariance has a correlation ≥0.8 across the visible (Colburn et al., 2019).

These approaches enable artifact-free, white-light imaging and rapid focal switching without mechanical translation, albeit often with trade-offs in maximum numerical aperture and transmission efficiency.

5. Practical Considerations and Fabrication

Fabrication protocols are tuned for high precision and efficiency:

• Metasurface Unit Cells: TiO₂ nanopillars/nanofins or silicon-nitride posts on transparent substrates, with finely controlled geometric parameters to yield the desired phase shifts and maintain robustness against polarization or wavelength variations (Hou et al., 27 Jul 2024, Hu et al., 2022, Aiello et al., 2019).
  • Minimum lateral features ≥50 nm.
  • Aspect ratios for height : width typically ≤12:1.
  • Transmission per unit cell often >0.9 at design λ.
• Large-Area Scaling: Circular nanoposts or ring-type meta-atoms are chosen for polarization insensitivity and large-area manufacturability, avoiding high-Q resonators (Colburn et al., 2019).
• System Integration: Multi-element metalenses may be cascaded with conventional refractive elements or integrated with elastic or electro-optic actuators for extended tunability (Hu et al., 2022, Colburn et al., 2019).
• Optimization Tools: Particle swarm and genetic algorithms are standard for multi-dimensional phase-error minimization given the discrete meta-atom library and multiple constraints.

6. Optical Performance and Applications

Optical characterization reports the following:

• Focal Stability: Across all designs, the focal length drift with wavelength remains within either 4–7% (metalens and hybrid metasurface systems) or within the depth of field for refractive multi-window designs. Lateral resolution at each focus matches the diffraction limit within 5–10%.
• Efficiency: Metasurface designs reach up to 86% focusing efficiency at maximum zoom, with total throughput set by meta-atom coverage and polarization usage. Bi-focal and multifocal designs allocate efficiency between channels but maintain cross-talk <5% (Hou et al., 27 Jul 2024, Doskolovich et al., 2018, Hu et al., 2022).
• Applications: Imaging (varifocal, zoom, or multi-channel), AR/VR, ophthalmic correction (multifocal IOLs), ultra-compact spectroscopy, and machine vision sensors are prominent targets for these architectures (Colburn et al., 2019, Aiello et al., 2019, Doskolovich et al., 2018).

7. Outlook, Limitations, and Future Directions

Recent advances demonstrate that multiple-focal-length achromatic systems can decisively challenge the traditional trade-offs among bandwidth, tunability, and miniaturization. However, they confront clear boundaries in aperture scaling (phase coverage and meta-atom aspect ratio), NA and off-axis aberrations, and efficiency. The emerging variational refractive design, together with multi-channel metasurface engineering and computational post-correction, define a pathway toward highly specialized, spectrally agile, and compact optics for next-generation imaging devices. Further bandwidth, field-of-view, and multi-focal capabilities are anticipated via advanced meta-atom dispersion engineering, adaptive elements, and integration with digital computation (Laville et al., 13 Nov 2025, Hu et al., 2022, Colburn et al., 2019).