Basis-neutral Hilbert-space analyzers

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.

generalized delay (GD): an optical transformation characterized by a continuous, real order-parameter that can be tuned to produce -once placed in one arm of an interferometer -an interferogram that reveals the modal weights in a prescribed functional basis via harmonic analysis. We find that GDs correspond to optical implementations of fractional transforms in the case of discrete modal bases 33,34 . For example, it can be shown 34 that the GD associated with Hermite Gaussian (HG) modes is the fractional Fourier transform 35,36 , whereas that associated with radial LG modes 37 is the fractional Hankel transform 38,39 . Sweeping the order of a fractional transform corresponds to varying a temporal delay in traditional interferometry -each in its own Hilbert space.
In the implementation presented here, we exploit electrically addressable spatial light modulators (SLMs) to realize tunable-strength cylindrical and spherical lenses that are building blocks of fractional transforms 40 . We make use of the polarization discrimination of SLMs 41,42 to construct a polarization interferometer -in lieu of a two-path interferometer -to accomplish generalized interferometry in an inherently stable configuration. Switching between Hilbert spaces -that is, examining a beam in different bases -is readily achieved in the same setup with no moving parts, simply by changing the phases imparted by the SLMs. We thus establish a versatile, basis-neutral Hilbert-space analyzer based on a generalized conception of optical interferometry.

Concept of a generalized optical delay
An optical delay τ is typically implemented by inserting an additional propagation length in a beam's path. In the time domain, a delay shifts the temporal origin E(t) → E(t − τ), whereas in the spectral domain it adds to each harmonic frequency component ω a phase e iωτ that is linear in both the delay and the frequency (Fig. 1a). In other T r a n s v e r s e p o s it io n x 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 ψ x { ( )} n with weights c n , 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 α e i n to the n th mode amplitude, which 'delays' the beam by α in the Hilbert space words, spectral harmonics {e −iωt } are eigenstates of the delay operation with eigenvalues e iωτ . Guided by this observation, we introduce a generalized delay (GD) that operates in the Hilbert space spanned by a modal basis {ψ n (x)}, such that the GD's effect on a beam described in this space is completely analogous to that of a delay τ for a pulse. A GD operates between an input plane ′ x and output plane x, and implements a unitary transformation ′ α Λ x x ( , ; ) characterized by a real, continuous order-parameter α, . Upon passage through the GD, the field is transformed according to Each mode thus acquires a phase α e in that depends linearly on its index n (Fig. 1b) -in analogy to the impact of a traditional delay with respect to spectral harmonics. For a discrete modal basis indexed by n (Equation 1), ′ α Λ x x ( , ; ) is periodic in α with period 2π. Furthermore, Λ can be generalized to two transverse coordinates and is applicable to a continuous basis 33,34 .
As an example, consider the set of one-dimensional (1D) HG modes, is the n th -order Hermite polynomial and A n is a normalization constant. This modal set is well-established as a useful basis for laser beams and arises naturally in many contexts 43 . The corresponding GD is the 1D fractional Fourier transform (fFT) 34 of angular-order α (scaled heretofore by convention from 0 to 4). Indeed, HG modes are eigenstates of the fFT 36 with eigenvalues π α e i n /2 . A beam traversing this GD is not shifted in physical space, as an optical delay shifts a pulse in time. Nevertheless, because each underlying HG mode acquires the requisite phase after the GD, the fFT 'delays' the beam in the Hilbert space of optical beams spanned by HG modes, which thus facilitates analyzing the beam in the HG basis. Alternatively the set of radial LG modes associated with zero-OAM states constitutes a modal basis for radial functions having azimuthal symmetry; here ⋅ L ( ) n is the n th -order Laguerre polynomial, B n is a normalization constant, and r is a radial coordinate. The GD here corresponds to the fractional Hankel transform (fHT); i.e., an optical implementation of the fHT 'delays' the beam in the Hilbert space spanned by radial LG modes 33 . Techniques for beam analysis into radial LG modes are lacking, leading the radial coordinate to be recently dubbed the 'forgotten' degree of freedom 29 .

Generalized optical interferometry
A GD can be exploited for the modal decomposition of an optical beam in its associated Hilbert space. The overall scheme for 'generalized optical interferometry' is a balanced two-path interferometer, in which the usual temporal delay is replaced by a GD (Fig. 2a)  such that harmonic analysis of P(α) identifies the weights c n 2 ; Fig. 2b. Each mode thus produces individually a sinusoidal interferogram ∝ 1 + cos nα. Mode-orthogonality dictates that each mode interferes only with itself. Crucially, the form of the interferogram in Eq. 3 is independent of the particular modal basis. A superposition of two HG modes of order n and m, for example, yields an interferogram that is identical to the same superposition of LG modes of order n and m -if the appropriate GD associated with each Hilbert space is implemented. This generalized interferometer is thus 'basis-neutral' . Furthermore, since the GD associated with a discrete modal basis is periodic in its order α, the resulting interferogram is in turn periodic, such that its Fourier transform yields a discrete spectrum. The number of modes that may be distinguished in this manner is determined by the sampling rate of the interferogram (the number of settings of α measured) and is ultimately Nyquist-limited.

Experimental implementation
A fFT or fHT can be implemented via combinations of cylindrical or spherical lenses, respectively, and the transform orders are varied by changing either the lens strengths or their separation (or both) 36,44 . The former approach does not require moving parts and can be realized with electrically addressable phase-only SLMs that implement generalized lenses of variable power -which is the strategy we follow here. A minimum of three generalized lenses can implement a 1D fFT 40 , where the first and last lenses have the same power and the distances separating the SLMs are equal (Methods). The fFT order can thus be varied without overall scaling or additional phases imparted to the field 40 , which is critical since we will interfere the beam with its own fFT.
The two-path interferometer in Fig. 2a requires a high degree of stability since several large components (SLMs) are introduced into one path, the overall path lengths may be large (~1 m here), and a fractional-transform-order-dependent relative phase must be included (Methods). These difficulties are obviated by introducing a novel configuration that exploits the polarization-selectivity of liquid-crystal-based SLMs 41 to construct the single-path polarization interferometer (Fig. 3a). The three SLMs impact the horizontal polarization component H, whereas the vertical component V is unaffected. After rotating the input polarization to 45°, only the H-component is transformed by the SLMs whereas the V-component is unchanged, thus serving as a reference. Projecting the output polarization at 45° allows the H and V components to interfere. However, the V-component undergoes diffraction during propagation and at the output it no longer corresponds to the original field E(x) needed as a reference. We therefore introduce lenses between the SLMs arranged in a 4-f configuration to image the V-component and reproduce E(x), and modify the strength of the lenses implemented by the SLMs accordingly ( Fig. 3b; Methods). Since the symmetry of the configuration is maintained, reflective SLMs allow folding the system such that only two SLMs and one lens are required (Fig. 3c). This stable polarization interferometer is thus in one-to-one correspondence with the two-path interferometer in Fig. 2a.
In implementing the fHT, we require that the SLMs produce simultaneously equal-order1D fFTs along x and y. Each SLM thus corresponds to equal-power crossed cylindrical lenses, or a spherical lens.

Results
We first realize modal analysis via generalized interferometry in the basis of 1D HG modes, where the associated GD is the 1D fFT. We examine beams having the separable form and focus on the x-dependence alone. The input beams are prepared by a single SLM (SLM 0 ) that imprints a phase-only pattern on a Gaussian-mode laser beam, which is then imaged to SLM 1 that constitutes the input plane to the generalized interferometer. A second SLM (SLM 2 ) reflects the beam back to SLM 1 , and the phases imparted by SLM 1 and SLM 2 are varied to cycle the fFT order α.
We report in Fig. 4 measurements carried out on 1D beams approximating the four lowest-order HG modes. For each beam, we provide: (1) the intensity of the 'delayed' beam after the fFT ( ; ) 2 while varying the 'delay' α; (2) the intensity after interfering the delayed beam with the original, ; and (4) the Fourier transform of P(α) that reveals the modal weights c n 2 . In Fig. 4a, each vertical line plot corresponds to the magnitude squared of a 1D fFT α E x ( ; ) 2 associated with a different order α, whereas each vertical plot in Fig. 4b is the correspond- These data enable us to diagnose the system and evaluate its performance, but only the interferogram P(α) is required for modal analysis, which corresponds to the temporal interferogram obtained in traditional two-path interferometers incorporating an optical delay. This interferogram is basis-neutral now that the spatial degree of freedom has been integrated over.
Whenever E(x) is a pure n th -order HG mode, the interferogram α πα ∝ + P n ( ) 1 cos /2 is a sinusoid whose Fourier transform produces a delta function at n. We verify this with modes H x ( ) is an eigenstate of the fFT, we do not observe modulation in , and the interferogram P(α) is thus a constant whose Fourier transform has a single contribution at n = 0. Next, the 1 st -order HG mode produces an interferogram having a full sinusoidal period P(α) ∝ 1 + cos α whose Fourier transform reveals the strongest contribution at n = 1. We approximate H x ( ) 1 by imparting a π-phase step (via SLM 0 ) to a Gaussian beam (Supplementary Information Sec. S3), so contributions from other modes appear in the modal analysis, and simulations provide a computed modal content that is in excellent  (x; α). The beam emerging from the interferometer -a superposition of the delayed beam and a reference α -is collected by a bucket detector and an interferogram is recorded with α, and H x ( ) 3 produce shorter period sinusoids and reveal the strongest contributions at n = 2 and n = 3, respectively. We note a discrepancy at the fFT order α = 2, whereupon the rapid variation imposed on the SLM phases results in a sudden drop in diffraction efficiency (Supplementary Information Sec. S5).
We next analyze beams into radial LG modes by implementing the fHT as the GD. The results for L r ( ) 0 2 through L r ( ) 2 2 are presented in Fig. 5. Since these beams are azimuthally invariant, we first integrate the recorded 2D intensity I(r, θ) in polar coordinates over θ to obtain a 1D radial distribution , where I(r) is the power in a thin annulus of radius r centered on the beam axis. Figure 5a depicts the 'delayed' beam I(r; α) as we vary the fHT-order α. Integrating over r after interfering the delayed beam with the reference produces the interferogram P(α). The basis-neutrality is clear when comparing the interferograms associated with H x ( ) 0 in Fig. 4 to L r ( ) 0 2 in Fig. 5; similarly for H x ( ) 1 and L r ( ) 1 2 , and for H x ( ) 2 and L r ( ) 2 2 . To highlight the versatility of this approach, we examine beams formed of various superpositions of HG modes in Fig. 6. First, we analyze the beam which we approximate by blocking half the cross section of a Gaussian beam (Fig. 6a-d). Next, we examine the field which we approximate by only varying the phase of a Gaussian beam to maximize the overlap with the desired beam ( Fig. 6a-d) (Fig. 6e).

Discussion and Conclusion
We have demonstrated that optical interferometry can be generalized to apply for any modal basis by replacing the traditional temporal delay with a generalized delay (GD): an optical transformation that 'delays' the beam in a Hilbert space spanned by the modal basis of interest. This basis-neutral strategy provides a unifying framework for modal analysis in an arbitrary basis -whether discrete, continuous, or combinations thereof for different degrees of freedom 34 . The fFT performs a rotation of the Wigner distribution associated with the field 45 , which has been exploited in tomographically reconstructing the Wigner distribution of non-classical states of light 46 . We have implemented this strategy here in the spatial domain of a scalar field using monochromatic light, but the approach is readily extended to multiple degrees of freedom of the optical field by simply cascading the associated GDs 34 . This methodology is also applicable to quantum states of light, such as one-photon or even entangled two-photon states 47 by replacing the dual delays in a phase-unlocked HOM interferometer 48 with the appropriate GDs. Our approach can thus further increase the accessible dimensionality of the Hilbert space of single photons by at least an order of magnitude 49,50 . The accessible dimension of the beam's Hilbert space is ultimately limited by the spatial resolution of the SLM pixels and the phase-step resolution for each pixel, which limit the sampling resolution of the fractional-transform order. Improvements in SLM technology may allow for real-time modal analysis over large-dimensional Hilbert spaces. One can use instead amplitude-based spatial modulators which are considerably faster, resulting in real-time modal analysis, albeit at the price of reduction in signal throughput 51 . We have found however that the physical extent of the SLM (or the number of pixels) is the main factor that limits the fidelity of modal analysis (see Supplementary Information for a detailed study).
Many new questions are now open: What is the optimal implementation of a GD when only a closed subspace of the modal basis is of interest? What is the minimum number of SLMs required to implement a GD in an arbitrary modal basis? Moreover, it is usually the case that only a few modes are activated (such as in communications protocols) or contribute significant energy -so-called modal 'sparsity' 52 . In these scenarios, uniformly sampling the GD order is not efficient. We have recently shown theoretically that optical interferometry can be modeled as a linear measurement problem and is hence subject to compressive sensing techniques that exploit the sparsity of the signal in some modal basis 52 . These findings can considerably reduce the number of measurements in the methodology presented here.
We have implemented here the GDs for the Hilbert spaces associated with HG and radial LG modes, the fFT and fHT, respectively. More generally, our approach indicates the potential utility of yet-to-be-discovered optical fractional transforms and provides a roadmap for their discovery. Given any modal set of interest, a fractional transform may be constructed out of the outer product of these functions in the diagonal representation given in Eq. 1 -and this fractional transform 'delays' the beam in its associated Hilbert space. For example, one may form a fractional transform from a basis of OAM and Bessel functions for the analysis of beams emerging from optical fibers or circular waveguides.

Properties of a generalized delay. Consider a functional basis ψ x
Using this set as a basis for a 1D finite-energy beam E(x) (in the space of square-integrable functions L 2 ), we have . Consider a linear, unitary transformation between input and output planes identified by coordinates x′ and x, respectively. The transformation has a real, continuous order-parameter α that uniquely identifies the transformation ′ α Λ x x ( , ; ). A field E(x) traversing this system is transformed according to Eq. (2), E(x) → E(x; α).
2 2 for all α and arbitrary E(x), which implies that One can thus obtain the form of the GD transformation ′ α Λ x x ( , ; ) in Eq. 1, which further entails that the set of transformations ′ α Λ x x ( , ; ) forms over α a one-parameter group. Defining the group composition operation as the cascade of two transformations, ) ( , ; ) ( , ; ), which is closed on this  , while varying θ from 0 to π/2. Plotted are the coefficients |c 0 | 2 (blue squares) and |c 1 | 2 (red circles), corresponding to the contributions of the modes HG 0 and HG 1 . Dashed curves are theoretical predictions, for |c 0 | 2 and |c 1 | 2 predicated on the generated approximate modes.
Scientific RepoRts | 7:44995 | DOI: 10.1038/srep44995 set, we have the requisite properties for a group: (I) the set has an identity ′ δ ′ Λ = − x x x x ( , ; 0) ( ); (II) the group composition operator is associative; and (III) there exists a unique inverse for any transformation ). The group is also obviously commutative. Finally, the property of the inverse and the unitarity of Λ together imply that ) ( , ; ) (5) Implementation of the 1D fFT using SLMs. The 1D fFT is defined by Eq. 1 after substituting the 1D HG functions for ψ x ( ) n . Explicitly, the 1D fFT is given by the canonical transformation fFT 2 2 where x and ′ x are normalized and unitless. Several specific angular orders of the fFT are readily recognizable. At α = 0, the system is , which is an imaging system without inversion or the identity operator; is a Fourier transform system; and at α = π, ′ π δ , which is an imaging system with inversion. The system in Fig. 3a consists of three cylindrical lenses (implemented by SLMs) of powers p 1 , p 2 , and p 1 (inverse focal lengths) separated by equal distances d, and can perform the 1D fFT of arbitrary order, without scaling or additional spatially varying phase, while using the minimal number of optical components 40 . By introducing a characteristic length scale σ (to be set shortly) to normalize x and ′ x , the impulse response function of this system at a wavelength λ is i i kd 1 2 is a unitless parameter that combines all the length scales in the system. Comparing Eq. 7 to Eq. 6, we identify the lens strengths p 1 and p 1 that are necessary to implement the fFT of angular order α: 1 2 In the case of a polarization-selective SLM, the impulse response function for the H-component is Eq. 7 whereas that for the V-component corresponds to free-space propagation for a distance 2d.
The modified system in Fig. 3b includes two identical lenses with focal lengths f in addition to the three SLMs implementing cylindrical lenses with strengths s 1 , s 2 , and s 1 , and all the separating distances are equal to f. The impulse response function for the V-component is δ ) , corresponding to imaging with inversion (a 4f imaging system). For the H-component, the impulse response function is a result of all five optical components (three when the system is folded back on itself) is given by: where η = σ λf 2 and we have introduced the transverse length scale σ as above. We identify s 1 and s 2 that implement the 1D fFT of order α This polarization interferometer thus achieves both goals: the H-component undergoes a 1D fFT whereas the V-component reference is imaged via a 4f system, both without introducing extra spatial phases or scaling. Implementation of the radial fHT using SLMs. The 2D fFT between input plane ′ ′ x y ( , ) and output plane (x, y) is separable along the two Cartesian coordinates, such that ′ ′ α α ′ α ′ α Λ = Λ Λ x y x y x x y y ( , ; , ; , ) ( , ; ) ( , ; ), x and ′ α Λ y y ( , ; ) y y are 1D fFTs of order α x (along x) and α y (along y), respectively. These fFTs may be controllably accessed independently by adding the phase patterns for the two required crossed generalized cylindrical lenses to be implemented by the SLMs. The fHT corresponds to a symmetric 2D fFT 34 Experimental setup. The optical beam is derived from a laser diode at a wavelength of 808 nm that is spatially filtered by coupling into a single-mode fiber at the operating wavelength (Thorlabs, FS-SN-4224) and collimated using a fiber-integrated collimation package. This produces an approximate Gaussian beam whose size is controlled by a variable beam expander (Thorlabs, BE02-05-B) moving along with the collimation package along a rail mount to yield a Gaussian beam with a FWHM of 0.6 mm located at SLM 0 . The beam is polarized along H and is modulated by SLM 0 to produce the desired beam. The field at SLM 0 is imaged to SLM 1 -through a beam splitter -via a 4f imaging system comprised of equal-focal-length lenses (f = 300 mm) and the polarization is rotated from H to 45° by a half-wave plate. All the SLMs are reflection-mode, polarization-sensitive Hamamatsu LCOS-SLM (X10468-02) that modulate H but not V. The angle of incidence on SLM 1 is less than 10°, the reflected beam passes through a lens L 1 (f = 500 mm) and is normally incident on SLM 2 reflecting back through L 1 to SLM 1 again. The plane of SLM 1 is then imaged to the detector plane through the beam splitter and analyzed at + 45° polarization. The image of the modified interference beam is recorded by a CCD camera (The Imaging Source, DFK 72BUC02). The SLMs are computer-controlled to synchronize the display of the phases required to implement the fFT of desired order.