We introduce a compact operator-based technique that solves the paraxial wave equation for a broad class of structured light fields. Using the spatial evolution operator to propagate two families of physically apodized inputs, Gaussian apodized Whittaker integrals and Gaussian apodized Helmholtz fields, we derive closed form expressions that retain the Gaussian width and therefore describe finite energy beams. The method unifies and extends the Helmholtz Gauss families and readily generalizes to nonseparable nondiffracting architectures. Experiments on superposed Bessel Gauss beams confirm the predicted transverse rotations, demonstrating that this operator approach is a fast, transparent, and practical alternative to standard diffraction ntegral treatments
The discovery of diffraction-free Bessel beams by Durnin et al. [1,2] opened new avenues for understanding light propagation. Subsequently, other nondiffracting families emerged, Mathieu [3] and parabolic beams [4], each associated with distinct coordinate systems (cylindrical circular, elliptic, and parabolic, respectively). These separable solutions carry orbital angular momentum transferable to particles [4][5][6], enabling optical manipulation applications. Despite originating from different coordinate systems, all can be unified through the Whittaker integral [7], which describes nondiffracting beams as plane-wave superpositions weighted by an angular spectrum. This framework has recently allowed the creation of complex nonseparable beam structures [8][9][10][11][12].
Under paraxial propagation, nondiffracting beams acquire a Gaussian envelope, ensuring finite energy. Gori et al. [13] introduced the first such beam, the Bessel-Gauss, by multiplying a Bessel function with a Gaussian profile. Extensions to Mathieu-Gauss and parabolic-Gauss beams [14][15][16] followed, preserving their nonparaxial geometries while becoming experimentally accessible via spatial light modulators [17]. Chen et al. [18] further applied Fresnel diffraction and stationary phase methods to generate paraxial versions of Zannotti’s structured caustic beams [19] using phase plates and axicons.
The operator-technique method has recently been proven to be effective in solving the paraxial wave equation with various initial field distributions [20][21][22]. Here, we apply this approach to two Gaussian-apodized initial conditions at z = 0: (i) a Gaussian-modulated Whittaker integral and (ii) a Gaussian-modulated Helmholtz solution. We demonstrate that both formulations yield consistent results and provide an alternative derivation of the Helmholtz-Gauss beam family (Bessel-Gauss, Mathieu-Gauss, and parabolic-Gauss).
Whittaker-Gauss beams. In the paraxial approximation, the free-space light propagation obeys
where τ = z/k is the reduced axial coordinate, r ⊥ = (x, y) the transverse position, ∇ 2 ⊥ the transverse Laplacian, and k = 2π/λ the wavenumber. Exploiting the structural analogy with the Schrödinger equation, the solution admits the operator representation [23]:
, where the exponential acts as the spatial evolution operator in the initial field ψ(r ⊥ , 0). We consider a Gaussian-apodized Whittaker integral:
with real width parameter g and k ⊥ = k ⊥ (cos φ x + sin φ ŷ). The integral represents a coherent superposition of plane waves with wavevectors in a circle, weighted by the angular spectrum A(φ), while the Gaussian factor ensures finite energy. Applying exp iτ 2 ∇ 2 ⊥ to (2) and using the canonical commutation relations [∂ x , x] = [∂ y , y] = 1, we perform a Lie-algebraic similarity transformation [20,24] on the Gaussian envelope, yielding
with α(τ ) = -g/ω(τ ), γ(τ ) = -gτ /ω(τ ), β(τ ) = -π/4i ln(iω(τ ))/2, and ω(τ ) = 1 + 2igτ . The term β(τ ) is the squeeze operator that induces coordinate rescaling. Crucially, the plane-wave components in (2) are eigenfunctions of the transverse Laplacian with eigenvalue -k 2 ⊥ :
This property allows the operator exp[γ(τ )∇
This expression reveals the dual diffraction mechanism: the Gaussian envelope controls beam expansion and wavefront curvature through ω(τ ), while the angular spectrum undergoes coordinate renormalization r ⊥ → r ⊥ /ω(τ ). Although equivalent to Ref. [14], our operator formalism is more general and naturally incorporates the Gaussian factor, avoiding the nonphysical infinite-energy limit (g = 0) common in prior treatments [8][9][10][11][12]. Table I lists the functions A(φ) that yield separable Helmholtz solutions in Cartesian, polar, elliptic, and parabolic coordinates.
Substituting these into (5) recovers the Bessel-Gauss, Mathieu-Gauss, and parabolic-Gauss beams. Table II presents A(φ) for nondiffracting nonseparable beams-Bessel-lattice, Durnin-Whitney, Archimedes and elliptic structures-obtained by prescribing specific caustic geometries or wavevector curves.
A(φ)
Mathieu Beam cem(φ; q) + isem(φ; q)
Beam Type A(φ)
Durnin-Whitney e i(mφ+bφ 2 )
Simple plane curves e im tan -1 ky (φ)
Archimedes structure e
Elliptic structure e ik ⊥ (a+b)φ-
To demonstrate the analytical results, we generated several fields experimentally. The experimental setup consists of a 4-f optical system, where a linearly polarized He-Ne laser (λ = 632.8 nm) impinges on a spatial light modulator (SLM). A synthetic phase hologram (SPH) [25] is displayed on the SLM. A lens (f 1 = 40 cm) performs the SPH Fourier transforms, allowing a spatial filter in the Fourier plane to transmit the desired field. Afterward, a second lens (f 2 = 40 cm) performs the inverse Fourier transform to recover the encoded field; finally, a CCD records the intensity distribution. In Figs. 123, we present the analytical and experimental intensity distributions at several representative planes along the pr
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