Unidirectional Light Focusing via Diffractive Surfaces

• The paper presents a deep learning–optimized approach using cascaded phase-only diffractive layers that achieve forward focusing efficiency over 98% with backward suppression below 1%.
• Unidirectional focusing is defined as the selective enhancement of light propagation in one direction while suppressing reverse transmission using structured diffractive surfaces.
• The approach leverages angular-spectrum propagation and multi-angle training to ensure robustness, scalability, and effective performance across varied wavelengths and incident angles.

Unidirectional Focusing of Light Using Structured Diffractive Surfaces

Unidirectional optical systems pursue the selective transmission and focusing of light in one direction, while strongly suppressing throughput or focusing in the opposite direction. This paradigm enables functionalities such as optical isolation, back-reflection suppression, and directionally sensitive free-space interconnects—critical for applications in communication, defense, and security. The architecture reported in "Unidirectional focusing of light using structured diffractive surfaces" (Li et al., 2024) achieves strong unidirectionality using cascaded, deep-learned, phase-only diffractive layers structured at the scale of the illumination wavelength.

1. Physical Principles and Mathematical Formalism

The system operates within the constraints of reciprocity in linear, isotropic optics—ruling out approaches based on magneto-optics or nonlinearity. Unidirectional behavior is induced by asymmetric spatial structuring across cascaded diffractive layers. Each phase-only diffractive surface locally modulates the phase via position-dependent thickness, $t_k(x,y) = \exp[j\,\phi_k(x,y)]$, $\phi_k\in[0,2\pi)$, but the global stack is engineered such that the forward-propagating wavefronts experience lens-like constructive focusing (typically in the center aperture), while backward-propagating waves suffer diffraction and destructive interference, often through edge scattering reminiscent of engineered gratings.

Wave propagation between planes is modeled using the angular-spectrum formalism: $$u(x,y,z+d) = \mathcal{F}^{-1}\left\{\mathcal{F}\{u(x,y,z)\} \; H(f_x,f_y;d) \right\}$$ with transfer function

$$H(f_x,f_y;d) = \exp\left[ j\,k\,d\sqrt{1-(\lambda f_x)^2 - (\lambda f_y)^2} \right]$$

where $u$ is the optical field, $k$ the wavenumber, and $\lambda$ the wavelength. Key performance metrics are:

• Forward focusing efficiency: $$\eta_{\rm F} = \frac{\int_{\rm FOV_B} |u_{\rm out}|^2\,dx\,dy}{\int_{\rm FOV_A} |u_{\rm in}|^2\,dx\,dy}$$
• Backward suppression ratio: $$S_{\rm B} = \frac{\int_{\rm FOV_A} |u_{\rm back}|^2\,dx\,dy}{\int_{\rm FOV_B} |u_{\rm in}|^2\,dx\,dy}$$
• Unidirectional gain: $G_{\rm UD} = \frac{\eta_{\rm F}}{S_{\rm B}}$ which rigorously quantify directionality and isolation (Li et al., 2024).

2. Cascaded Diffractive-Layer Device Architecture

Unidirectionality is realized via a stack of $\phi_k\in[0,2\pi)$0 diffractive layers ($\phi_k\in[0,2\pi)$1 in simulation; $\phi_k\in[0,2\pi)$2 in experiment), each comprising an array of phase modulating pixels at approximately $\phi_k\in[0,2\pi)$3 pitch. Numerically, $\phi_k\in[0,2\pi)$4 features per layer are used ($\phi_k\in[0,2\pi)$5 in experimental demonstration at $\phi_k\in[0,2\pi)$6 mm), with lateral spans $\phi_k\in[0,2\pi)$7–$\phi_k\in[0,2\pi)$8 mm and axial interlayer spacings of $\phi_k\in[0,2\pi)$9 (simulation) or $$u(x,y,z+d) = \mathcal{F}^{-1}\left\{\mathcal{F}\{u(x,y,z)\} \; H(f_x,f_y;d) \right\}$$0 (experiment). The physical substrate is a 3D-printed photoresist (refractive index $$u(x,y,z+d) = \mathcal{F}^{-1}\left\{\mathcal{F}\{u(x,y,z)\} \; H(f_x,f_y;d) \right\}$$1 at $$u(x,y,z+d) = \mathcal{F}^{-1}\left\{\mathcal{F}\{u(x,y,z)\} \; H(f_x,f_y;d) \right\}$$2 mm), with total system axial length $$u(x,y,z+d) = \mathcal{F}^{-1}\left\{\mathcal{F}\{u(x,y,z)\} \; H(f_x,f_y;d) \right\}$$3 cm (Li et al., 2024).

The spatial arrangement—dense central lens-like phase for forward focus, grating-like edge pattern for backward suppression—breaks the forward/backward symmetry in energy transmission without violating reciprocity.

3. Deep Learning-Based Optical Design Optimization

Optimization proceeds by treating the device as a differentiable optical network: each diffractive layer maps to a computational “layer” with learnable local phases. The entire stack is trained end-to-end under the angular-spectrum propagation model.

The trainable parameters are the pixel-wise phases $$u(x,y,z+d) = \mathcal{F}^{-1}\left\{\mathcal{F}\{u(x,y,z)\} \; H(f_x,f_y;d) \right\}$$4. The core loss function is

$$u(x,y,z+d) = \mathcal{F}^{-1}\left\{\mathcal{F}\{u(x,y,z)\} \; H(f_x,f_y;d) \right\}$$5

for single angle/frequency, with $$u(x,y,z+d) = \mathcal{F}^{-1}\left\{\mathcal{F}\{u(x,y,z)\} \; H(f_x,f_y;d) \right\}$$6 to prioritize backward suppression. For robust designs under variable illumination,

$$u(x,y,z+d) = \mathcal{F}^{-1}\left\{\mathcal{F}\{u(x,y,z)\} \; H(f_x,f_y;d) \right\}$$7

across $$u(x,y,z+d) = \mathcal{F}^{-1}\left\{\mathcal{F}\{u(x,y,z)\} \; H(f_x,f_y;d) \right\}$$8 forward and $$u(x,y,z+d) = \mathcal{F}^{-1}\left\{\mathcal{F}\{u(x,y,z)\} \; H(f_x,f_y;d) \right\}$$9 backward angles/wavelengths. Phase values are clipped to $$H(f_x,f_y;d) = \exp\left[ j\,k\,d\sqrt{1-(\lambda f_x)^2 - (\lambda f_y)^2} \right]$$0; stochastic gradient descent (Adam/SGD) is used with learning rate $$H(f_x,f_y;d) = \exp\left[ j\,k\,d\sqrt{1-(\lambda f_x)^2 - (\lambda f_y)^2} \right]$$1 and early stopping for regularization (Li et al., 2024).

4. Performance Analysis: Efficiency, Robustness, and Spectral Response

The optimized design demonstrates:

• Monochromatic, normal-incidence focusing efficiency $$H(f_x,f_y;d) = \exp\left[ j\,k\,d\sqrt{1-(\lambda f_x)^2 - (\lambda f_y)^2} \right]$$2
• Backward leakage $$H(f_x,f_y;d) = \exp\left[ j\,k\,d\sqrt{1-(\lambda f_x)^2 - (\lambda f_y)^2} \right]$$3
• Under $$H(f_x,f_y;d) = \exp\left[ j\,k\,d\sqrt{1-(\lambda f_x)^2 - (\lambda f_y)^2} \right]$$4 oblique incidence, multi-angle–trained devices retain $$H(f_x,f_y;d) = \exp\left[ j\,k\,d\sqrt{1-(\lambda f_x)^2 - (\lambda f_y)^2} \right]$$5, $$H(f_x,f_y;d) = \exp\left[ j\,k\,d\sqrt{1-(\lambda f_x)^2 - (\lambda f_y)^2} \right]$$6.
• Adversarial wavefront attacks degrade backward suppression in single-angle–trained devices ($$H(f_x,f_y;d) = \exp\left[ j\,k\,d\sqrt{1-(\lambda f_x)^2 - (\lambda f_y)^2} \right]$$7 after $$H(f_x,f_y;d) = \exp\left[ j\,k\,d\sqrt{1-(\lambda f_x)^2 - (\lambda f_y)^2} \right]$$8 attack iterations), but multi-angle–trained stacks maintain $$H(f_x,f_y;d) = \exp\left[ j\,k\,d\sqrt{1-(\lambda f_x)^2 - (\lambda f_y)^2} \right]$$9 under sustained attack.
• Multi-wavelength operation ($u$0 mm): $u$1, $u$2. Multi-wavelength–trained devices achieve $u$3 efficiency and $u$4 leakage across $u$5 mm span.
• Polarization insensitivity: both scalar simulations and terahertz experiments reveal negligible polarization dependence (Li et al., 2024).

5. Experimental Validation and Physical Realization

A two-layer, $u$6 feature system fabricated in photoresist was tested at $u$7 GHz (terahertz regime). Experimental setup used a CW source, horn antenna, and lock-in detection. The measured forward-spot diameter was $u$8 mm, with efficiency $u$9–$k$0. Backward recorded fields were diffuse with $k$1, matching numerical predictions within $k$2 absolute error for both key metrics. The observed results confirm practical viability and theoretical fidelity at the terahertz scale (Li et al., 2024).

6. Universality, Scalability, and Application Prospects

Scalability to other spectral domains is attained by proportional scaling of the feature pitch and axial spacing to the design wavelength: operation in visible, NIR, and mid-IR becomes feasible at sub–$k$3m fabrication resolution. Identical deep learning–optimized design rules apply. Applications span:

• Optical security for tamper-resistant links and prevention of back-reflections
• Laser defense systems for suppression of scattered/return signals
• Unidirectional free-space/photonic couplers and isolators
• Directional, polarization-independent sensors in integrated photonics and remote sensing (Li et al., 2024)

7. Technical Significance and Future Directions

The device realizes record-level forward efficiency ($k$4) with exceptional backward suppression ($k$5) in linear, isotropic, and scalable platforms. Robustness to spectral, angular, and adversarial perturbations is attained via joint multi-condition training. The architecture establishes a universal, physically transparent, and fabrication-accessible approach to programmable unidirectionality in free-space optics, unencumbered by exotic materials or lossy nonreciprocal elements. Pursuing higher-layer count, finer pixelization, and further generalizations of the optimization loss (e.g., for arbitrary spatiotemporal transfer functions) are projected avenues for research and application (Li et al., 2024).