Basis-neutral Hilbert-space analyzers

Abstract

Interferometry is one of the central organizing principles of optics. Key to interferometry is the concept of optical delay, which facilitates spectral analysis in terms of time-harmonics. In contrast, when analyzing a beam in a Hilbert space spanned by spatial modes - a critical task for spatial-mode multiplexing and quantum communication - basis-specific principles are invoked that are altogether distinct from that of 'delay'. Here, we extend the traditional concept of temporal delay to the spatial domain, thereby enabling the analysis of a beam in an arbitrary spatial-mode basis - exemplified using Hermite-Gaussian and radial Laguerre-Gaussian modes. Such generalized delays correspond to optical implementations of fractional transforms; for example, the fractional Hankel transform is the generalized delay associated with the space of Laguerre-Gaussian modes, and an interferometer incorporating such a 'delay' obtains modal weights in the associated Hilbert space. By implementing an inherently stable, reconfigurable spatial-light-modulator-based polarization-interferometer, we have constructed a 'Hilbert-space analyzer' capable of projecting optical beams onto any modal basis.

Figure 1. Concept of a generalized optical delay.

(a) Traditional temporal optical delay. The impact of a temporal delay τ on a pulse E(t) can be viewed in two ways. In the time domain (first row), the pulse is delayed, E(t − τ). In the spectral domain (second row), the pulse is a superposition of temporal harmonics (angular frequencies ω) each with a spectral amplitude c_n. The delayed pulse E(t − τ) is the result of inserting phase factors for each harmonic ω. (b) Generalized delay (GD) α in a Hilbert space spanned by a discrete modal basis . The impact of the GD on an optical beam can also be viewed in two domains. In the spatial domain (first row), the GD is not simply a shift but instead it transforms the transverse field profile E(x) → E(x; α). However, in the modal space (second row) where the field is viewed as a superposition of the modes with weights , the impact of the GD is identical to that of the temporal delay on the spectral harmonics in (a). The GD adds a phase factor to the mode amplitude, which ‘delays’ the beam by α in the Hilbert space spanned by .

Figure 2. Generalized optical interferometry for modal analysis in an arbitrary basis.

(a) Operation of a generalized interferometer in real space. Two copies of the beam E(x) are created at beam splitter 1 and subsequently combined at beam splitter 2 after one copy traverses the GD and is ‘delayed’ in the associated Hilbert space by α, E(x; α). The beam emerging from the interferometer – a superposition of the delayed beam and a reference – is collected by a bucket detector and an interferogram is recorded with α, , whose Fourier transform reveals the modal weights . (b) Operation of the generalized interferometer in the Hilbert space spanned by the modal basis on the beam (Fig. 1b). The underlying modes of the ‘delayed’ copy acquire phase shifts of the form after passing through the GD to yield a new beam . The original and ‘delayed’ beams are combined to produce an interferogram . Because the modes are orthogonal to each other, each interferes only with its phase-shifted counterpart to yield an interferogram of the form 1 + cos nα with weights – independently of the underlying basis that is traced out at the bucket detector. The sought-after weights are then revealed through harmonic analysis.

Figure 3. Inherently stable implementation of a generalized interferometer.

(a) Implementation of a 1D fFT using three generalized (variable-power) lenses L₁, L₂, and L₃ with symmetric strengths p₁, p₂, and p₁, respectively, that are selected to produce a fractional transform of prescribed order (Methods). Because the lenses are implemented by polarization-selective SLMs (affecting only the H-component), the system is in fact equivalent to the two-path interferometer in Fig. 2a, with the H- and V-components corresponding to the delay and reference arms, respectively, while the half-wave plate (HWP) and the polarizer correspond to beam splitters 1 and 2, respectively. This common-path interferometer is inherently stable. However, the V-component undergoes unwanted diffraction over the distance 2d. (b) Same as (a), except that polarization-insensitive fixed lenses (focal lengths f) are inserted in a 4 f configuration to eliminate the diffraction of the V-component. The strengths s₁, s₂, and s₁ of the generalized lenses are modified to compensate for the added lenses. (c) Folded implementation of (b). The beam is reflected onto itself from L₂, such that L₁ and L₃ are the same generalized lens and only one fixed lens is required.

Figure 4. Modal analysis in the Hilbert space spanned by 1D Hermite-Gaussian modes using generalized optical interferometry.

(a) The measured ‘delayed’ beam resulting from the input beam E(x) (which is to be analyzed into the contributions from HG modes) traversing the order-α GD (here the fFT), . Each vertical line plot represents the magnitude-squared of a 1D fFT associated with a different order α. (b) The measured interferogram resulting from superposing the delayed beam from (a) with a reference, . Each vertical line plot thus represents the magnitude-squared of the 1D spatial interferogram associated with a different order α. (c) The integrated interferogram . This interferogram is now basis-neutral. (d) The modal weights |c_n|² revealed by taking the Fourier transform of the interferogram in (c). The columns are for different input beams corresponding to modes HG₀ through HG₃. The implemented beams only approximate the pure HG modes (except for HG₀ which is exact), as shown in the insets in (d). The black mode profile in the inset is an exact HG mode while the orange plot is the approximate beam used in the experiment. The theory plots in (c) and (d) are those for the implemented approximate beams. See Supplementary Information for theory.

Figure 5. Modal analysis in the Hilbert space spanned by radial Laguerre-Gaussian modes using generalized optical interferometry.

(a–d) Same as (a–d) in Fig. 4 except that the GD operates in the space of radial LG modes. Note that in (a) and (b), the delayed beam and the interferogram are plotted with r and not x (0 ≤ r ≤ ∞). Insets show the radial intensity distribution of the beams. The columns are for different input beams corresponding to modes LG₀ through LG₂. The implemented beams only approximate the pure radial LG modes (except for LG₀ which is exact), as shown in the insets in (d). The mode profile on the left in the inset is an exact LG mode while the plot on the right is the approximate beam used in the experiment. The theory plots in (c) and (d) are those for the implemented approximate beams. See Supplementary Information for theory.

Figure 6. Modal analysis of beams comprising superimposed modes.

(a–d) Same as (a–d) in Figs 4 and 5. The Input beams are the superpositions (left column) and (right column). (e) Modal analysis of the beam , while varying θ from 0 to π/2. Plotted are the coefficients |c₀|² (blue squares) and |c₁|² (red circles), corresponding to the contributions of the modes HG₀ and HG₁. Dashed curves are theoretical predictions, for |c₀|² and |c₁|² predicated on the generated approximate modes.