Non-orthogonal Transformations of Structured Light Using Ellipticity-Dependent Ince-Gaussian Modes

Technical Summary: Non-orthogonal Transformations of Structured Light Using Ellipticity-Dependent Ince–Gaussian Modes

Problem Statement
Structured light fields are typically described using complete sets of solutions to the paraxial wave equation, such as Hermite–Gaussian (HG), Laguerre–Gaussian (LG), and Ince–Gaussian (IG) modes. IG modes are particularly notable as the most general separable family, offering a continuous interpolation between Cartesian (HG) and circular (LG) symmetries via an ellipticity parameter $\varepsilon$. While IG modes form a complete orthogonal basis for any fixed $\varepsilon$, modes associated with different ellipticities are not mutually orthogonal. Despite the theoretical existence of IG modes, the relationship between bases of different ellipticities remains largely unexplored. Existing modal decomposition techniques operate almost exclusively within a single orthogonal basis, and no explicit, finite analytical expression has been reported to directly transform between IG bases of arbitrary ellipticity without relying on intermediate numerical decompositions or basis truncations. Consequently, ellipticity has not been fully exploited as a functional degree of freedom for structured light engineering.

Methodology
The authors derive a compact analytic expression to connect IG modes of arbitrary ellipticity through finite superpositions. The derivation employs LG modes as an intermediate basis:

1. Theoretical Derivation: IG modes are expanded into the LG basis, and LG modes are expressed as an inverse relation back to IG modes. By combining these expressions, the authors eliminate the intermediate LG basis to establish a direct transformation between an IG basis of ellipticity $\varepsilon_1$ and another of ellipticity $\varepsilon_2$.
2. Analytical Formulation: The transformation is defined by complex weighting coefficients, $c_{p,m_2}(\varepsilon_1, \varepsilon_2)$, which are calculated either as a finite sum of products of expansion coefficients or, more directly, as overlap integrals between the two IG modes (Equation 5). These coefficients depend on the Fourier coefficients of the Ince polynomials, which encode the ellipticity dependence.
3. Experimental Implementation: To validate the theory, the authors implement an experimental setup using Spatial Light Modulators (SLMs).
   • Generation: A target IG mode ($IG^{\varepsilon_1}$) is generated using complex-amplitude modulation on the first SLM.
   • Decomposition: The generated field is projected onto a series of basis modes ($IG^{\varepsilon_2}$) using a second SLM displaying computer-generated holograms (CGH).
   • Measurement: The overlap coefficients are determined by measuring the on-axis intensity in the far field (Fourier plane) using a high-sensitivity CMOS camera. Both the amplitude (from intensity) and the intermodal phase (via interference with reference and auxiliary superposition modes) are retrieved to reconstruct the complex coefficients.

Key Contributions

• First Explicit Analytical Mapping: The paper provides the first explicit, finite analytical expression (Equation 5) that directly relates IG modes of arbitrary ellipticity. This formulation avoids the need for intermediate numerical decompositions.
• Characterization of Non-Orthogonality: The work demonstrates that IG modes with different ellipticities form a family of mutually non-orthogonal but complete bases. The transformation preserves orthogonality only in the modal order ($p$) and parity ($\sigma$), while the modal index ($m$) becomes non-orthogonal when $\varepsilon_1 \neq \varepsilon_2$.
• Ellipticity as a Tunable Parameter: The study establishes ellipticity as a continuous transformation parameter that deforms the modal basis, allowing for the controlled redistribution of modal weight across non-orthogonal representations.

Results

• Theoretical Validation: The derived coefficients were shown to reduce to the standard orthogonality relation when $\varepsilon_1 = \varepsilon_2$. When $\varepsilon_1 \neq \varepsilon_2$, the coefficients for modes with different indices become non-zero, quantifying the "distance" between bases in Hilbert space.
• Numerical and Experimental Agreement: Numerical simulations of the modal weight redistribution as a function of $\varepsilon_2$ matched the analytical predictions. Experimentally, the authors successfully decomposed a target mode ($IG^{4}_{5,3}$) into a basis of fixed ellipticity ($\varepsilon_2 = 2$). The experimentally retrieved complex coefficients ($c_{5,1}, c_{5,3}, c_{5,5}$) showed close agreement with theoretical predictions, successfully reconstructing the amplitude and phase of the target mode.
• Observation of Basis Deformation: The results confirmed that as the ellipticity mismatch increases, the modal weight redistributes from the mode with identical indices to other modes within the same order $p$, evidencing the non-orthogonal nature of the transformation.

Significance
The paper claims to introduce a practical framework for exploiting ellipticity as a tunable degree of freedom in wavefront design and structured-light engineering. By enabling explicit and measurable transformations between ellipticity-dependent IG bases, this work expands the toolbox for optical information processing beyond conventional orthogonal modal frameworks. The authors suggest this capability allows for adaptive basis selection in modal encoding schemes, where ellipticity can be tuned to optimize information capacity, minimize modal cross-talk, or match specific propagation conditions. The work confirms that these modal relationships are not only analytically exact but also experimentally accessible, offering new strategies for mode conversion and high-dimensional optical information processing.