Measuring the Complete Transverse Spatial Mode Spectrum of a Wave Field

We put forward a method that allows the experimental determination of the entire spatial mode spectrum of any arbitrary monochromatic wave field in a plane normal to its propagation direction. For coherent optical fields, our spatial spectrum analyzer can be implemented with a small number of benchmark refractive elements embedded in a single Mach-Zehnder interferometer. We detail an efficient setup for measuring in the Hermite-Gaussian mode basis. Our scheme should also be feasible in the context of atom optics for analyzing the spatial profiles of macroscopic matter waves.

We put forward a method that allows the experimental determination of the entire spatial mode spectrum of any arbitrary monochromatic wave field in a plane normal to its propagation direction. For coherent optical fields, our spatial spectrum analyzer can be implemented with a small number of benchmark refractive elements embedded in a single Mach-Zehnder interferometer. We detail an efficient setup for measuring in the Hermite-Gaussian mode basis. Our scheme should also be feasible in the context of atom optics for analyzing the spatial profiles of macroscopic matter waves. The temporal [1], vectorial [2] and spatial [3] degrees of freedom of electromagnetic fields with increasing complexity are exploited in applications ranging from communications to medicine. The detailed knowledge of the spatial mode spectrum of these beams, generally emitted by laser devices, is fundamental in optimizing applications as well as in research, both in classical and quantum optics. Many imaging and tailoring techniques, for instance, are based on the possibility to devise transformations induced by optical elements by determining the effects on the spatial components of generic signals [4]. Recently, an intense research activity on multimode light has burgeoned in quantum information, communication and imaging [5], with successful demonstrations spanning from nanodisplacement measurements [6], to parallel information [7], high-dimensional entanglement [8], coherent transfer of suitable prepared superpositions of vortices onto Bose-Einstein condensates [9], and hot Rubidium vapors [10]. A key ingredient in all these scenarios is spatial encoding, which conveys several independent channels of information by exploiting the access to a large number of optical transverse modes. The use of multimode beams thus demands for the development of techniques allowing to characterize their spatial spectrum and to retrieve the information encoded in different components. Fourier modes are of immediate access through a basic lens setup [4], while rotating elements lead to 'helical' spectra decomposition [11]. To the best of our knowledge, however, no direct method is known to measure the Hermite-Gaussian (HG) modes spectrum. Building on the symplectic formalism to describe firstorder optical transformations, we present in this Letter a general strategy to find an arrangement of refractive elements that enables the quantitative measurement of the transverse spatial spectrum of light beams by using a single Mach-Zehnder interferometer. Due to the generality of our approach, different experimental setups can be implemented to extract the full spatial spectrum of multimode transverse beams in many bases, including Laguerre-Gaussian (LG) modes. Here we focus on the spectrum analyzer for HG modes, as these are the most common in laser physics and appear naturally in devices where astigmatism, strain or slight misalignment drive the system toward rectangular symmetry [12]. Furthermore, the presented framework is rather suggestive of analog measurements for matter waves spatial spectra.
Spatial modes are ubiquitous. Sets of modes u satisfying the wave equation (i∂ η + ∂ 2 x + ∂ 2 y )u = 0 describe broad classes of physical systems. When the evolution variable is η = 2kz, the sets comprise the optical paraxial modes [12] with wave number k propagating along the z direction. For η = 2mt/ , the sets model instead the quantum dynamics of free particles of mass m in the xy plane. Since there are two transverse spatial variables, each entire set of mode solutions u m,n is labeled by two integers m, n. Relevant examples are the HG and LG mode bases. Any scalar field ψ obeying the above wave equation can thus be decomposed in terms of these modes as ψ = m,n C m,n u m,n . A fundamental question then arises: How can one measure the mode spectrum (weights) P m,n = |C m,n | 2 of any given scalar field?
To answer the above question we apply a symplectic invariant approach [13,14,15]. Symplectic methods have been used in theories of elementary particles, condensed matter, accelerator and plasma physics, oceanographic and atmospheric sciences, and optics [16]. Central to our work is the recognition that any linear passive symplectic transformation S acting on the canonical Hermitian operatorsx,ŷ,p x , andp y (whose only nonvanishing commutators are [x,p x ] = [ŷ,p y ] = iλ), is associated with a unitary operatorÛ (S) generated by the group Here, λ = 1/k and w 0 is the characteristic width of the spatial modes u m,n in which the analyzed wave field is to be decomposed. The set (1) satisfies the usual SU(2) algebra [L a ,L b ] = iε abcLc (a, b, c = x, y, z), withN being the only commuting generator in the group andL z describing real spatial rotations on the transverse xy plane (it is proportional to the orbital angular momentum operator along the wave propagation direction [14]). We can therefore represent the most general passive unitary operatorÛ (S) associated with S by a single exponential of linear combinations of any of the above generators [15] , with real scalar φ + and vector Φ − parameters. We show below howÛ (S) can be implemented using simple optical elements.
The method proposed here relies on exploitingN , together with specific combinations of the generatorsL x andL y , to construct two commuting unitary operators from which the associated symplectic matrices and their experimental implementation can be found relatively easily. To this end, let |m, n denote the (pure) mode states of order m + n ≥ 0 (in position-representation x, y|m, n = u m,n ). We first impose |m, n to be eigenstates of bothN and the operatorL θ,ϕ ≡ u r ·L, where u r = (cos ϕ sin θ, sin ϕ sin θ, cos θ) may be conceived as a radially oriented unit vector in the orbital Poincaré sphere [17,18]. The eigenstates |m, n depend on the choice of θ and ϕ and fulfillN |m, n = [(m + n)/2]|m, n andL θ,ϕ |m, n = [(m − n)/2]|m, n . For instance, the eigenvectors of operatorsL x andL z are the HG and LG modes, respectively [14,18]. Measurements of |ψ = m,n C m,n |m, n will involve the action of the unitarieŝ U N = e −iφ+N andÛ θ,ϕ = e −iφ−L θ,ϕ , upon variation of parameters φ + and φ − , which can be externally controlled. Notice thatÛ N is connected with the Gouy phase [19], whereasÛ θ,ϕ describes φ − -angle rotations about u r , thereby changing the mode superpositions.
In order to extract the complete spectrum P m,n of a coherent electromagnetic scalar field, an optical scheme is further developed. We remark that for quantum mechanical matter waves (e.g. Bose-Einstein condensates), a conceptually similar approach to analyze their spatial structure should currently be feasible by exploiting atom optics: interferometers [20], beam splitters in atom chips [21], focusing and storage in resonators [22], and conical lenses [23]. We use here a Mach-Zehnder interferometer with built-in refractive components performinĝ U N andÛ θ,ϕ : sets of spherical and cylindrical thin lenses. For ease of operation, it is desirable to vary φ ± in a way that minimizes the displacements of the optical elements. We explicitly show that all the required transformations in the interferometer can be achieved solely by rotation and variation of the focal lengths of the lenses. Translations of the lenses are not necessary, although it may turn out to be more convenient in certain instances to carry finite displacements in some of them. In any case, the arms of the interferometer always remain fixed.
Detection of the light intensity difference ∆I ≡ I B −I A at the two output ports (A and B) of the interferometer provides the data for reconstructing the weights P m,n . One has ∆I ∝ ψ|(Û † NÛ † θ,ϕÛ C +Û † CÛ NÛθ,ϕ )|ψ , wherê U C represents the unitary operator for the compensating lens system in the complementary arm of the interferometer. For clarity, let us assume thatÛ C comprises a similar lens set as the one forÛ NÛθ,ϕ (we will show later that this assumption can be removed), with equivalent parameters φ + and φ − . The mode spectrum and the output intensity difference are directly connected by a double Fourier-like transform where the proportionality constant equals (16π 2 ) −1 if m = n = 0, and (8π 2 ) −1 otherwise. Note that ∆I possess the symmetry properties ∆I(φ + ± 2π, φ − ± 2π) = ∆I(φ + ± 2π, φ − ∓ 2π) = ∆I(φ + , φ − ). Our scheme nontrivially generalizes that of Ref. [11], aimed to reveal the quasi-intrinsic nature of the orbital angular momentum degree of freedom for scalar waves. There, by means of the measurementÛ = e −iφLz , implemented with Dove prisms, the azimuthal index spectrum of spiral harmonic modes could readily be accessed. Here, we explore the entire transverse mode space expanded by the indices m and n. Hence, the combined effect of the two nonequivalent measurementsÛ N andÛ θ,ϕ is crucial, and would allow us to measure, for instance, either the full spectrum of HG or LG modes.
To show an explicit application of the above approach, we proceed with the characterization and design of an optical system that enables the reconstruction of the HG mode spectrum P nx,ny of a light beam (n x , n y ≥ 0). In this scenario, one needs to consider the unitary operatorŝ U N andÛ θ=π/2,φ=0 = e −iφ−Lx . The path from these unitaries to the pursued setups resorts to the Stone-von Neumann theorem [13,14,15]; it yields the symplectic matrices S + and S − which contain all systems information where c ± = cos(φ ± /2), s ± = sin(φ ± /2), and z 0 = w 2 0 /(2λ) is the Rayleigh range. From matrices S ± one can then obtain the corresponding integral transforms that govern the propagation (along the z-direction) of any input paraxial wave ψ(x, y) traversing each system. These transforms lead, in our case, to the integral kernels The output wave profiles after each S ± result from ψ ± (x, y) = dx dy K ± (x, y; x , y )ψ(x , y ). Kernels (4) exhibit the structure of those for the fractional Fourier transform [24]. Integral transforms associated to unitary operators generated byL y (rather thanL x ) have been formulated [25]. They make possible the full meridianrotation on the orbital Poincaré sphere and, in particular, the reciprocal conversion between HG and LG modes. The two proposed optical systems for S ± are symmetric and illustrated in Fig. 1(a). For S + , three fixed spherical lenses with varying radii of curvature R j (linked to the focal lengths f j by R j = (ñ j − 1)f j ,ñ j being the refractive indexes), and placed at equal distances z 0 , are sufficient. They fulfill R 1 = R 3 = z 0 /[1 − cot(φ + /4)] and R 2 = z 0 /(2 − s + ). Tunable-focus liquid crystal spherical lenses controlled by externally applied voltages have been demonstrated, displaying a wide range of focal lengths [26]. Figure 1(c) shows the operation curve describing the dependence between R 1 , R 3 and R 2 needed to cover the interval π ≤ φ + < 3π. These values can be attained with the lenses of Ref. [26]. For S − , three pairs of cylindrical lenses are required [see Fig. 1(a)]. Each pair of assembled cylindrical lenses is rotated in a scissor fashion with α = −β [ Fig. 1(b)]. The corresponding angles satisfy α 1 = α 3 = (π − Ω)/4 and α 2 = (3π − φ − )/4. The operation curve in Fig. 1(d) represents the variation of the rotation angles in accordance with the constraint imposed by cot(φ − /4) = −2 sin(Ω/2). This equation is satisfied when π ≤ φ − ≤ 3π and −(π/3) ≤ Ω ≤ (π/3), which lead to 0 ≤ α 2 ≤ (π/2) and (π/6) ≤ α 1 ≤ (π/3). The radii of curvature of the cylindrical lenses are R 1 = R 2 = R 5 = R 6 = z 0 /2 and R 3 = R 4 = z 0 /4. The distance between consecutive pairs is z 0 /2. With this scheme, the covered values for φ − ∈ [π, 3π]. The fact that both φ + and φ − are restricted to the interval [π, 3π], rather than to [0, 4π], does not constitute a fundamental limitation. It can be circumvented by employing the compensating system [see Fig. 1(a)] and the symmetry properties of ∆I. If two sequences of measurements are made, each having a different compensating system performing transformations with φ + = φ − = 0 (identity matrix) and φ + = 0, φ − = 2π (minus identity matrix), respectively, then Eq. (2) can be cast (m → n x and n → n y ) as The proportionality constant is (8π 2 ) −1 if n x = n y = 0, and (4π 2 ) −1 otherwise. The integration intervals in (5) now display the accessible ranges for φ ± . The first and second sequences of measurements can be carried out with a compensating system made of four and two (all identical) spherical lenses, respectively. By properly choosing their focal lengths, it is not necessary to displace the S ± systems nor the interferometer arms. Moreover, measuring the LG spectra would only require embedding the system S − between two fixed, mutually orthogonal, cylindrical lenses, with S + remaining invariant.
To demonstrate that our data analysis scheme is feasible and does not demand a large number of measurements for each φ ± , we have numerically simulated the transformation and processing of several input coherent waves. Figure 2 depicts the light profiles entering into the interferometer: strongly astigmatic Gaussian [ Fig. 2(a)], hexapole necklace [ Fig. 2(d)] and anisotropic multiring [ Fig. 2(g)] beams. Their exact HG distributions together with those retrieved from Eq. (5) are also displayed. To simulate the evolution of the various beams, the kernels (4) were used to calculate ∆I in Eq. (5) for 10 different values (in equal increments) per each φ ± ∈ [π, 3π]. The reconstructed and exact weights agree well (compare right and central columns in Fig. 2). In analogy with the Whittaker-Shannon sampling theorem in Fourier analysis [4], input beams that are mode-band-limited can be exactly reconstructed via our scheme [compare Figs. 2(e) and 2(f)]. That is, if the sampling increments for φ ± are smaller than the inverse of the highest contributing mode numbers, reconstruction will be exact. Misalignment of