Measuring the Complete Transverse Spatial Mode Spectrum of a Wave Field

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Gabriel F. Calvo,1 Antonio Picoń,1 and Roberta Zambrini 2 1Grup de F´ısica Teòrica, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain and 2Institute for Cross-Disciplinary Physics and Complex Systems, IFISC (CSIC-UIB), Palma de Mallorca, 07122, Spain (Dated: December 12, 2007) 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-Zender 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.

PACS numbers: 42.25.-p, 42.30.-d, 42.15.Eq, 39.20.+q

The temporal [1], vectorial [2] and spatial [3] degrees Spatial modes are ubiquitous. Sets of modes u satis- 2 2 of freedom of electromagnetic fields with increasing com- fying the wave equation (i∂η + ∂x + ∂y )u = 0, describe plexity are exploited in applications ranging from com- broad classes of physical systems. When the evolution munications to medicine. The detailed knowledge of the variable is η = 2kz, the sets comprise the optical parax- spatial mode spectrum of these beams, generally emitted ial modes [10] with wave number k propagating along by laser devices, is fundamental in optimizing applica- the z direction. For η = 2mt/!, the sets model instead tions as well as in research, both in classical and quantum the quantum dynamics of free particles of mass m in the optics. Many imaging techniques, for instance, are based xy plane. Since there are two transverse spatial vari- on the possibility to determine transformations induced ables, each entire set of mode solutions um,n is labeled by optical elements by knowing the effects on the spatial by two integers m, n. Relevant examples are the HG and spectral component of generic signals [4]. Recently, an in- LG mode bases. Any scalar field ψ obeying the above tense research activity on multimode light has burgeoned wave equation can thus be decomposed in terms of these

in quantum information, communication and imaging [5], modes as ψ = m,n Cm,num,n. A fundamental question with successful demonstrations spanning from nanodis- then arises: How can one measure the mode spectrum ! 2 placement measurements [6], to parallel information [7] (probabilities) Pm,n = |Cm,n| of any given scalar field? and high-dimensional entanglement [8]. The use of trans- To answer the above question we apply a symplectic inverse multimode beams demands, indeed, the develop- variant approach [11–13]. Symplectic methods have been ment of techniques allowing to characterize their spatial used in theories of elementary particles, condensed mat- spectrum and to access the information encoded in differ- ter, accelerator and plasma physics, oceanographic and ent components. Fourier modes are of immediate access atmospheric sciences and in optics [14]. Central to our through a basic lens set-up [4], while rotating elements work is the recognition that any linear passive symplec- allow to determine ’helical’ spectra [9]. To the best of tic transformation S acting on the canonical Hermitian our knowledge, however, no direct method is known to operators xˆ, yˆ, pˆx, and pˆy (whose only nonvanishing com- measure the Hermite-Gaussian (HG) modes spectrum. mutators are [xˆ, pˆx] = [yˆ, pˆy] = iλ), is associated with a Building on the symplectic formalism to describe first- unitary operator Uˆ(S) generated by the following group order optical transformations, we present in this Letter 2 2 (pˆ2 + pˆ2)w2 a general strategy to find an arrangement of refractive ˆ xˆ + yˆ x y 0 N = 2 + 2 , (1a) elements that enables the quantitative measurement of 2w0 8λ the transverse spatial spectrum of light beams by using 2 2 2 xˆ2 − yˆ2 (pˆ − pˆ )w a single Mach-Zender interferometer. Due to the gener- Lˆ = + x y 0 , (1b) x 2w2 2 ality of our approach, different experimental set-ups can 0 8λ 2 be implemented to determine the full spatial spectrum ˆ xˆyˆ pˆxpˆyw0 Ly = 2 + 2 , (1c) of multimode transverse beams in different basis, includ- w0 4λ ing Laguerre-Gaussian (LG) modes. Here we focus on the xˆpˆ − yˆpˆ Lˆ = y x . (1d) spatial spectrum analyzer for HG modes, as these are the z 2λ most common in laser physics and appear naturally in devices where astigmatism, strain or slight misalignment Here, w0 is the width of the spatial modes um,n and drive the system toward rectangular symmetry [10]. Fur- λ = 1/k. The set (1) satisfies the usual SU(2) alge- ˆ ˆ ˆ ˆ thermore, the presented framework is rather suggestive bra [La, Lb] = iεabcLc (a, b, c = x, y, z), with N being ˆ of analog measurements for matter waves spatial spectra. the only commuting generator in the group and Lz de- scribing real spatial rotations on the transverse xy plane 2

(it is proportional to the orbital angular momentum op- interferometer. For clarity, let us assume that UˆC com- erator along the wave propagation direction [12]). We prises a similar lens set as the one for UˆN Uˆθ,ϕ (we will can thus represent the most general passive unitary op- show later that this assumption can be removed), with ˆ " " erator U(S) associated with S by a single exponential of equivalent parameters φ+ and φ−. The mode spectrum linear combinations of any of the above generators [13] and the output intensity difference are directly connected Uˆ(S) = exp[−i(φ+ Nˆ + Φ− · Lˆ )], with real scalar φ+ and by a double Fourier-like transform Φ ˆ vector − parameters. We show below how U(S) can be 4π 4π " " implemented using simple optical elements. Pm,n ∝ dφ+ dφ− ∆I(φ+ − φ+, φ− − φ−) ˆ 0 0 The method proposed here relies on exploiting N , to- " " " " − − −− gether with specific combinations of the generators (1b)- × ei(m n)(φ+ φ+)/2 ei(m+n)(φ φ−)/2, (2) (1d), to construct two commuting unitary operators from where the proportionality constant equals (16π2)−1 if which the associated symplectic matrices and their ex- m = n = 0, and (8π2)−1 otherwise. Note that ∆I pos- perimental implementation can be found relatively easy. sess the symmetry properties ∆I(φ ± 2π, φ ± 2π) = To this end, let |m, n" denote the (pure) mode states of + − ∆I(φ ± 2π, φ ∓ 2π) = ∆I(φ , φ ). Our proposed order m + n ≥ 0. We first impose that |m, n" be eigen- + − + − scheme nontrivially generalizes that of Ref. [9], aimed states of both Nˆ and the operator Lˆ ≡ u · Lˆ , where θ,ϕ r to reveal the quasi-intrinsic nature of the orbital angular u = (cos ϕ sin θ, sin ϕ sin θ, cos θ) may be conceived as r momentum degree of freedom for scalar waves. There, a radially oriented unit vector in the orbital Poincaré ˆ by means of the measurement Uˆ = e−iφLz , implemented sphere [15, 16]. The eigenstates |m, n" depend on the with Dove prisms, the azimuthal index spectrum of spiral choice of θ and ϕ and fulfill Nˆ |m, n" = [(m + n)/2]|m, n" harmonic modes could readily be accessed. In our case, and Lˆ |m, n" = [(m − n)/2]|m, n". For instance, the θ,ϕ we explore the entire transverse mode space labeled by eigenvectors of operators Lˆ and Lˆ are the HG and x z the indices m and n. Hence, the combined effect of the LG modes, respectively [12, 16]. Measurements of |ψ" = two nonequivalent measurements UˆN and Uˆ is crucial, C |m, n" will involve the action of the unitaries θ,ϕ m,n m,n and would allow us to measure, for instance, either the ˆ ˆ ˆ −iφ+ N ˆ −iφ−Lθ,ϕ U!N = e and Uθ,ϕ = e , upon variation full spectrum of HG or LG modes. of parameters φ+ and φ−, which can be externally con- To show an explicit application of the above approach, trolled. Notice that UˆN is connected with the Gouy we proceed with the characterization and design of an ˆ phase [17], whereas Uθ,ϕ describes φ−-angle rotations optical system that enables the reconstruction of the HG u about r, thereby changing the mode superpositions. mode spectrum Pnx,ny of a light beam (nx, ny ≥ 0). In In order to extract the complete spectrum Pm,n of a this scenario, one needs to consider the unitary opera- ˆ ˆ ˆ −iφ−Lx coherent electromagnetic scalar field, an optical scheme tors UN and Uθ=π/2,φ=0 = e . By resorting to the is further developed. We remark that for quantum me- Stone-von Neumann theorem [11, 12], the symplectic ma- chanical matter waves (e.g. Bose-Einstein condensates), trices S+ and S− associated to these unitaries are a conceptually similar approach to analyze their spatial structure should currently be feasible by exploiting c± 0 z0s± 0 0 c± 0 ±z0s± atom optics: interferometers [18], beam splitters in atom S± =  , (3) chips [19], focusing and storage in resonators [20], and −s±/z0 0 c± 0  0 ∓s±/z0 0 c±  conical lenses [21]. We use here a Mach-Zender interfer-   ometer with built-in refractive components performing   where c± = cos(φ±/2), s± = sin(φ±/2), and z0 = UˆN and Uˆθ,ϕ: sets of spherical and cylindrical thin lenses. 2 w0/(2λ) is the Rayleigh range. Here, w0 provides the For ease of operation, it is desirable to vary φ± in a way characteristic width of the HG modes in which the input that minimizes the displacements of the optical elements. beam is to be decomposed. From matrices S± it is then We explicitly show below that all the required transfor- possible to obtain the corresponding integral transforms mations in the interferometer can be achieved solely by that govern the propagation (along the z-direction) of rotation and variation of the focal lengths of the lenses. any input paraxial wave ψ(x, y) traversing each system. Translations of the lenses are not necessary, although it These transforms follow from the general Collins integral may turn out to be more convenient in certain instances representation [11] and lead, in our case, to the kernels to carry finite displacements in some of them. In any " " case, the arms of the interferometer always remain fixed. " " 1 −2i(xx ± yy ) K±(x, y; x , y ) = 2 exp 2 Detection of the light intensity difference ∆I ≡ IB −IA πi|s±|w0 w0s± ) " " * at the two output ports (A and B) of the interferome- 2 2 2 2 i(x ± y + x ± y )c± ter provides the data that enables the reconstruction of × exp 2 . (4) ˆ † ˆ † ˆ ˆ † ˆ ˆ + w0s± , Pm,n. One has ∆I ∝ &ψ|(UN Uθ,ϕUC + UC UN Uθ,ϕ)|ψ", where UˆC represents the unitary operator for the com- The output wave functions after S± thus result from " " " " " " pensating lens system in the complementary arm of the ψ±(x, y) = dx dy K±(x, y; x , y )ψ(x , y ). Kernels (4) - 3

s The radii of curvature of the cylindrical lenses are R1 = (a) + R2 = R5 = R6 = z0/2 and R3 = R4 = z0/4. The Input s- Beam distance between consecutive pairs is z0/2. With this IA scheme, the covered values for φ− ∈ [π, 3π]. The fact that both φ+ and φ− are restricted to the interval [π, 3π], Output Compens Beams rather than to [0, 4π], does not constitute a fundamental IB (b) Syst ating limitation. It can be circumvented by employing the com- α em β pensating system [see Fig. 1(a)] and the symmetry prop- 3.5 /3 π erties of ∆I. If two sequences of measurements are made, 3.0 (d) each having a diﬀerent compensating system performing

0 (c) z

/ 2.5 " " 3 3

3 transformations with φ+ = φ− = 0 (identity matrix) and R

2.0 α " "

0 , , 0 1, 1,

z φ = 0, φ = 2π (minus identity matrix), respectively, /

α + −

1 1 1.5

R then Eq. (2) can be cast (m → nx and n → ny) as 1.0

0.5 π /6 3π 3π 0.4 0.5 0.6 0.7 0.8 0.9 0 π /2 nx+ny α2 Pnx,ny ∝ dφ+ dφ− (−1) ∆I(φ+, φ− − 2π) R2 / z0 "π "π .(n − n )φ + (n + n )φ FIG. 1: (Color online) (a) Schematic of the optical imple- + ∆I(φ , φ )]cos x y + x y − . mentation to measure the Hermite-Gaussian mode spectrum. + − 2 S S ) * Transformations + and − are performed using symmetric (5) sets of three fixed spherical lenses with varying focal lengths and three pairs of rotating cylindrical lenses, respectively. 2 −1 The proportionality constant equals (8π ) if nx = The compensating system involves two settings with four and 2 −1 two spherical lenses, respectively. (b) Detail of one of the as- ny = 0, and (4π ) otherwise. The integration inter- sembled pairs of cylindrical lenses required for S−. (c) and vals in (5) now display the accessible ranges for φ±. The (d) Operation curves for S+ and S−, respectively (see text). first and second sequences of measurements can be car- ried out with a compensating system made of four identical and two identical spherical lenses, respectively. By exhibit the structure of those for the fractional Fourier properly choosing their focal lengths, it is not necessary transform [22]. The singular cases, arising when s± = 0, to displace the S± systems nor the interferometer arms. can easily be handled too. Integral transforms associated To demonstrate that our data analysis scheme is fea- to the unitary operator generated by Lˆy (rather than Lˆx) sible and does not require a large number of measure- have been formulated and lead to the, so-called, gyrator ments for each φ±, we have numerically simulated the transform [23]. They allow for the full meridian-rotation transformation and processing of several input coherent on the orbital Poincaré sphere and, in particular, for the waves. Figure 2 depicts the light profiles entering into the reciprocal conversion between HG and LG modes. interferometer: an strongly astigmatic Gaussian beam The two proposed optical systems for S± are symmet- [Fig. 2(a)], an hexapole necklace beam [Fig. 2(d)] and an ric and illustrated in Fig. 1(a). For S+, three fixed spher- anisotropic multiring beam [Fig. 2(g)]. Their exact HG ical lenses with varying radii of curvature Rj (linked to mode weight distributions are displayed in the central the focal lengths fj by Rj = (n˜j − 1)fj, n˜j being the column of histograms [Fig. 2(b), (e), (h)]. The right col- refractive indexes), and placed at equal distances z0, are umn in Fig. 2 represents the weights Pnx,ny from Eq. (5) sufficient. They fulfill R1 = R3 = z0/[1 − cot(φ+/4)] retrieved after the traversing of the input beams through and R2 = z0/(2 − s+). Tunable-focus liquid crystal the system. To simulate the evolution of the various spherical lenses controlled by externally applied voltages beams, we have used the transformation kernels (4) to have been demonstrated, displaying a wide range of focal calculate the ∆I in Eq. (5) for 10 different values (in lengths [24]. Figure 1(c) shows the operation curve de- equal increments) per each φ± ∈ [π, 3π]. The recon- scribing the dependence between R1, R3 and R2 needed structed weights (right column) agree well with the exact to cover the interval π ≤ φ+ < 3π. These values can be weights (central column). In analogy with the Whittaker- attained with the lenses of Ref. [24]. For S−, three pairs Shannon sampling theorem in Fourier analysis [4], input of cylindrical lenses are required [see Fig. 1(a)]. Each beams that are mode-band-limited can be exactly recon- pair of assembled cylindrical lenses is rotated in a scissor structed via our scheme [compare Figs. 2(e) and 2(f)]. fashion with α = −β [Fig. 1(b)]. The corresponding an- That is, if the sampling increments for φ± are smaller gles satisfy α1 = α3 = (π − Ω)/4 and α2 = (3π − φ−)/4. than the inverse of the highest contributing mode num- The operation curve in Fig. 1(d) represents the variation bers, reconstruction will be exact. Misalignment of the of the rotation angles in accordance with the constraint optical elements is expected to be the main source of imposed by cot(φ−/4) = −2 sin(Ω/2). This equation is errors. The effect on the weights Pnx,ny , due to lens dis- satisfied when π ≤ φ− ≤ 3π and −(π/3) ≤ Ω ≤ (π/3), placements (tolerances) δ with respect to the beam axis, 2 which lead to 0 ≤ α2 ≤ (π/2) and (π/6) ≤ α1 ≤ (π/3). introduces a correction term ∼ (δ/w0) , which is smaller 4

3 " # " # "a# b c

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