Optimal Diffractive Focusing of Vortex Beams

The gist The optimal pupil for diffractive ring focusing of paraxial beams carrying orbital angular momentum (OAM) is determined by an integral-kernel eigenvalue problem that yields an exact analytical solution combining Fresnel phase structure with an OAM-dependent Bessel radial modulation, establishing a fundamental performance limit for diffractive focusing of aperture-limited vortex beams.

Problem Formulation and Theoretical Framework

The research formulates the diffractive ring focusing of paraxial beams carrying orbital angular momentum (OAM) through a finite circular aperture as a variational optimization problem <ref:2610.10842#pg8>. For a fixed topological charge l, the goal is to determine the real-valued radial pupil that maximizes the intensity at a prescribed ring radius and propagation distance under a normalization constraint on the input field <ref:2610.10842#pg4>. The resulting problem admits an exact analytical solution where the optimal pupil combines the familiar Fresnel phase structure with an OAM-dependent Bessel radial modulation <ref:2610.10842#pg4>. This analysis provides a rigorous upper bound on the achievable ring intensity <ref:2610.10842#pg4>.

Vortex Beam Propagation and Optimization

The propagation of a field with OAM is governed by the paraxial wave equation <ref:2610.10842#pg4>. The input field is written as a helical phase structure Ψ0(ρ, φ) = ψl(ρ)eilφ <ref:2610.10842#pg4>. The objective is to maximize the functional J[ψl] = Ψ(ρ∗, τf)2 subject to a normalization constraint <ref:2610.10842#pg4>. This leads to a quadratic form in the pupil function, J[ψl] = ∫01 ρ dρ ∫01 ρ' dρ' Kl(ρ, ρ') ψl(ρ) ψl(ρ') <ref:2610.10842#pg4>. The propagation dynamics and target focusing radius are encoded in the real symmetric kernel Kl(ρ, ρ') <ref:2610.10842#pg4>.

Eigenvalue Problem and Optimal Solution

Variation of the functional leads to an eigenvalue problem 1/2π ∫01 ρ' dρ' Kl(ρ, ρ') ψl(ρ') = Λ ψl(ρ) <ref:2610.10842#pg4>. The largest eigenvalue Λ determines the maximum achievable ring intensity <ref:2610.10842#pg4>. The eigenvalues are given by Λ(l) ± (ρ∗, τf) = 1/2τ2f (Il ± Rl) <ref:2610.10842#pg4>. The corresponding normalized eigenfunction ψ+l(ρ) is given by ψ+l(ρ) = 1/qτ2f Λ(l) + Jl ρ∗ρ τf! cos ρ2 / (2τf − θl!) <ref:2610.10842#pg4>.

Performance Analysis and Practical Design Rules

The focusing efficiency of an arbitrary normalized pupil function ψ is defined as ηl(ψ) = J[ψ] Λ(l) + ⟨ψ, Klψ⟩ Λ(l) + <ref:2610.10842#pg4>. The decomposition of the pupil into optimal eigenmodes shows that ηl(ψ) = a2l,+ + Λ(l − Λ(+)) a2l− <ref:2610.10842#pg4>. For the Fresnel-Bessel zone plate, η(ψPFZ) = 8/π2, which is in exact agreement with the results shown in Ref. [16] <ref:2610.10842#pg4>. For nonzero OAM, two natural generalizations arise: (i) Binary Fresnel–Bessel zone mask ψPFBZ, and (ii) Amplitude-weighted Fresnel–Bessel mask ψFBZ <ref:2610.10842#pg4>. The amplitude-weighted mask yields a systematically higher ring intensity than the fully binary Fresnel–Bessel mask <ref:2610.10842#pg4>.

Experimental Validation and Conclusion

Experimental results confirm that the amplitude-weighted Fresnel–Bessel mask ψFBZ yields a systematically higher ring intensity than the fully binary Fresnel–Bessel mask ψPFBZ <ref:2610.10842#pg4>. The experimental implementation using spatial light modulation demonstrates that optimal diffractive pupils can be realized and directly compared with practical mask families <ref:2610.10842#pg4>. The results establish the fundamental limits of diffractive focusing for vortex beams <ref:2610.10842#pg4>. This framework provides a general route for analyzing and engineering optimal diffractive focusing in systems where conventional refractive optics is unavailable or impractical <ref:2610.10842#pg4>.

How it works

The core mechanism involves treating the focusing as a variational optimization problem to find the real-valued radial pupil function that maximizes intensity at a target ring radius and propagation distance <ref:2610.10842#pg4>. This optimization is solved by finding the principal eigenfunction of an integral-kernel eigenvalue problem derived from the paraxial wave equation <ref:2610.10842#pg4>.

The optimal pupil profile combines the familiar Fresnel phase structure with an OAM-dependent Bessel radial modulation <ref:2610.10842#pg4>. This structure is dictated by the natural focusing geometry imposed by the phase singularity of OAM beams, which requires concentrating energy onto an annular ring rather than a point <ref:2610.10842#pg4>.

Key Findings and Implications

The analysis reveals that for l ≠ 0, the optimal pupil is no longer determined solely by Fresnel phase matching, which is responsible for the target focal point, but also exhibits a radial Bessel structure dictated a <ref:2610.10842#pg4>. This OAM-dependent radial weighting imposes an additional penalty on masks that fail to reproduce it <ref:2610.10842#pg4>. The experimental results validate the central theoretical prediction that optimal diffractive focusing of OAM beams is governed not only by Fresnel phase matching, but also by the Bessel-type radial structure imposed by the vortex geometry <ref:2610.10842#pg4>.

Experimental Implementation and Results

The experiment was designed to compare the propagation and ring-focusing performance of different pupil functions, including the binary Fresnel–Bessel mask ψPFBZ and the amplitude-weighted Fresnel–Bessel mask ψFBZ <ref:2610.10842#pg4>. The measured intensity patterns closely follow theoretical expectations, confirming that the encoded holograms reproduce the intended diffractive pupils with high fidelity <ref:2610.10842#pg4>. The measurements demonstrate that the amplitude-weighted Fresnel–Bessel mask ψFBZ yields a systematically higher ring intensity than the fully binary Fresnel–Bessel mask ψPFBZ, in agreement with the theoretical efficiency analysis presented above <ref:2610.10842#pg4>.

Conclusion

In this work, vortex beams possess a phase singularity that imposes vanishing on-axis intensity, so the natural focusing objective is to concentrate energy onto a ring rather than a point <ref:2610.10842#pg4>. As a result, the optimal pupil is no longer determined solely by Fresnel phase matching, which is responsible for the target focal point, but also exhibits a radial Bessel structure dictated a" <ref:2610.10842#pg4>. The results establish the fundamental limits of diffractive focusing for vortex beams and provide a systematic framework for designing high-efficiency ring-focusing elements for structured waves <ref:2610.10842#pg4>.

Additional Derivations

The derivation of the OAM-projected Fresnel propagator involves transforming the paraxial Schrödinger equation into an ordinary differential equation in the Fourier domain <ref:2610.10842#pg4>. This leads to a propagation integral Ψ(ρ, τ) given by Eq.(3) <ref:2610.10842#pg4>. The kernel Kl(ρ, ρ') separates into two components fc and fs in the inner product space <ref:2610.10842#pg4>. The eigenvalue problem reduces to a 2 × 2 matrix equation which determines the eigenvalues and corresponding eigenfunctions <ref:2610.10842#pg4>.

References

[1] M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Elsevier, 2013) <ref:2610.10842#pg4>

[5] M. Sussman, American Journal of Physics 28, 394 (1960) <ref:2610.10842#pg4>

[7] L. B. Lesem, P. M. Hirsch, and J. A. Jordan, IBM Journal of Research and Development 13, 150 (1969) <ref:2610.

1. The improved AI system can perform optimal diffractive focusing of vortex beams by determining the real-valued radial pupil that maximizes the intensity at a prescribed ring radius and propagation distance under a normalization constraint on the input field. This allows for designing high-efficiency ring-focusing elements for structured waves, which has implications for nonlinear optics and strong-field physics.

2. The system can compare diffractive focusing performance with refractive techniques by deriving a simple bound that allows direct comparison between diffractive and refractive vortex focusing. This provides a rigorous benchmark for evaluating different focusing strategies in optics and other wave systems.

3. The system can design practical mask structures using the derived analytical solutions, specifically identifying the pupil structures that maximize energy concentration on the characteristic annular intensity distribution. This enables the creation of high-resolution optical elements with predictable performance limits based on OAM charge and focusing geometry.

4. The system can quantify focusing performance for arbitrary pupils by calculating focusing efficiency ηl(ψ) = J[ψ] / Λ(l)+ = ⟨ψ, Klψ⟩ / Λ(l)+, allowing it to assess the performance of any pupil function against the theoretical maximum.

5. The system can analyze OAM effects on focusing efficiency by decomposing an arbitrary pupil as ψ = al,+ψ+l + al,-ψ−l + a⊥ψ⊥ and using Eq. (14) to find that ηl(ψ) = a2l,+ + Λ(l)−Λ(l)+a2l−. This helps in understanding how the interplay between Fresnel diffraction and the radial structure imposed by OAM determines the ultimate limits of intensity concentration.

We formulate diffractive ring focusing of paraxial beams carrying orbital angular momentum (OAM) through a finite circular aperture as a variational optimization problem. For a fixed topological charge, we determine the real-valued radial pupil that maximizes the intensity at a prescribed ring radius and propagation distance under a normalization constraint on the input field. The resulting problem admits an exact analytical solution: the optimal pupil combines the familiar Fresnel phase structure with an OAM-dependent Bessel radial modulation, and the associated eigenvalue yields a rigorous upper bound on the achievable ring intensity. We further analyze the lens-assisted configuration and derive a simple bound that allows direct comparison between diffractive and refractive vortex focusing. The theory also provides a quantitative framework for evaluating the performance of practical mask designs, including binary Fresnel-Bessel zone plates and amplitude-weighted pupils. We have experimentally tested our predictions in the optical domain by implementing the corresponding vortex beam pupils and comparing their propagation and focusing performance with the optimal solution. Our results establish the vortex-beam analogue of optimal amplitude-only focusing and identify the fundamental limits governing diffractive concentration of structured waves