Geometry and Dynamics in the Fractional Discrete Fourier Transform

K. B. Wolf and G. Krötzsch Vol. 24, No. 3/March 2007/J. Opt. Soc. Am. A 651

Geometry and dynamics in the fractional discrete Fourier transform

Kurt Bernardo Wolf and Guillermo Krötzsch Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Cuernavaca, Morelos 62251, México

Received August 3, 2006; accepted September 8, 2006; posted September 25, 2006 (Doc. ID 73731); published February 14, 2007 The NϫN Fourier matrix is one distinguished element within the group U͑N͒ of all NϫN unitary matrices. It has the geometric property of being a fourth root of unity and is close to the dynamics of harmonic oscillators. The dynamical correspondence is exact only in the N→ϱ contraction limit for the integral Fourier transform and its fractional powers. In the ﬁnite-N case, several options have been considered in the literature. We compare their ﬁdelity in reproducing the classical harmonic motion of discrete coherent states. © 2007 Optical Society of America OCIS codes: 070.2590, 070.6020, 030.1670.

1. INTRODUCTION that pass through F and its integer powers. We also propose a simple analog definition of coherent states. The The integral Fourier transform (IFT) participates in the time evolution of these states is explored computationally foundations of quantum mechanics and is ubiquitous in in Section 5 for the “Taipei” basis of Pei and Yeh10 and Pei signal processing. Geometrically, the IFT is a fourth root et al.,11 the eigenbasis of Mehta,12 and the “Ankara” basis of unity; its fractionalization is not unique, but one distin- of Candan, Ozaktas, and coauthors13,14 (see also Ref. 5, guished fractional IFT (FrIFT)1 is the one-parameter cy- Chap. 4). As expected, coherent states of low energy have clic subgroup U͑1͒ of the Lie group Sp͑2,R͒ of linear ca- Gaussianlike shapes and oscillate harmonically; differ- nonical transformations in one-dimensional quantum ences between the models arise only at high energies. In mechanics,2 describing the time evolution of the quantum Section 6 we briefly recall the “Cuernavaca” Fourier– harmonic oscillator (see, e.g., Ref. 3, Part IV). Its applica- Kravchuk transform and its coherent states.6,15,16 A suc- tions to optics include the paraxial wave model,4 and in cinct concluding Section 7 ends the discussion. signal processing these have been widely documented by Ozaktas et al.5 Here we examine analog finite structures that satisfy both the geometry of the FrIFT as fourth 2. FrIFT AND THE HARMONIC OSCILLATOR roots of unity and the dynamics of discrete coherent F states as finite counterparts of the classical and quantum- There exists a close relation between the IFT operator acting on the Hilbert space L2͑R͒ of quantum mechanics mechanical harmonic motion. and the harmonic oscillator evolution at one-quarter pe- The discrete Fourier transform (DFT) is approximately riod [Ref. 3, Eq. (7.197)], realized in planar multimodal optical or acoustical 6 paraxial waveguides and in symmetric one-lens F = ei␲/4 exp͓−i1 ␲͑P2 + Q2͔͒, ͑1͒ setups,7–9 where the input signal is produced by a linear 4 array of N-emitting points (with controlled phases) and where we have the Schrödinger operators of position ͑Q ͒͑ ͒ ͑ ͒ ͑P ͒͑ ͒ ͑ ͒ measured by a similar array of sensors. A corresponding :f x ªxf x and momentum :f x ª−idf x /dx, and fractional DFT (FrDFT) occurs for any length along the where e͑i␲/4͒ is the metaplectic phase (Ref. 17, Appendix guide or for spaces around the lens setup. In the litera- C). This leads to the definition of the fractional IFT op- ture we find several approaches to define a FrDFT (five of erators F␣ through a number operator N, which we review in this paper), presenting various com- F␣ ͑ 1 ␲␣N͒ N 1 ͑P2 Q2͒ 1 ͑ ͒ putational and mathematical advantages. Although we ª exp −i2 , ª 2 + − 2 1. 2 refrain from selecting a “best” FrDFT, we compare differ- The spectrum of N in L2͑R͒ is k෈͕0,1,2,...͖, nondegen- ent versions in terms of their rendering of the motion of erate, and F4 =1 is the unit operator. The normalized coherent states. eigenfunctions of N are the well-known quantum har- In Section 2 we recall some of the remarkable proper- monic oscillator functions, ties of the FrIFT, its integral kernel, and its relation with the harmonic oscillator wave functions and coherent ⌿ ͑ ͒ 1 2 ͑ ͒ ͱ k ͱ ␲ ͑ ͒ k x = exp͑− x ͒Hn x / 2 k ! . 3 states. The DFT matrix F and a standard modulo-4 frac- 2 tionalization of any fourth root of unity are presented in The ground state is ⌿0͑x͒, and there is no “top” state. Section 3. In Section 4 we fractionalize the powers ␣ of F␣ Acting on functions f෈L2͑R͒, F␣ is a unitary integral determined by Fourier eigenbases V; these are all one- transform that one can write, using either powers or ␣ ෈ ͑ ͒ ʚ ͑ ͒ ␾ 1 ␲␣ parameter cyclic groups of matrices FV U 1 V U N angles ª 2 ,as

Dimension Multiplicities N␸ ͑F␣:f͒͑x͒ = ͵ dxЈF͑␣͒͑x,xЈ͒f͑xЈ͒, ͑4͒ R N ␸ =1 −i −1 i detF 4JJ+1 JJJ−1 −i͑−1͒J i␾/2 2 2 e ͑x + xЈ ͒cos ␾ −2xxЈ J ͑␣͒ 4J +1 J +1 JJJ͑−1͒ F ͑x,xЈ͒ = exp i ͑5͒ , ␾ ͱ2␲i sin ␾ 2 sin 4J +2 J +1 JJ+1 J − ͑−1͒J 4J +3 J +1 J +1 J +1 J i͑−1͒J ϱ ⌿ ͑ ͒ ͑ ␾͒⌿ ͑ ͒* ͑ ͒ ͑11͒ =͚ k x exp −ik k xЈ . 6 k=0 Ͼ Ϸ 1 ͚ for J 0, i.e., roughly N␸ 4 N; of course ␸N␸ =N. These In the closed-form Eq. (5), the square root is understood to Fourier subspaces are mutually orthogonal; their projec- ͱ ͑ 1 ␲ ͒ͱ͉ ͉ be isªexp i 2 sign s s , and there are Dirac singulari- tor matrices are ties at F͑0͒͑x,xЈ͒=␦͑x−xЈ͒=F͑4͒͑x,xЈ͒ and F͑2͒͑x,xЈ͒=␦͑x +xЈ͒. The bilinear generating function form of Eq. (6) (Ref. 3 1 ␸−n n n ␸n ͑ ͒ 18) exposes the role of the orthonormal and complete os- P␸ = 4 ͚ F , F = ͚ P␸, 12 n=0 ␸ cillator basis ⌿k͑x͒ in Eq. (3), which are eigenfunctions of F with the four inﬁnitely degenerate eigenvalues, ͑−i͒k ␦ ෈͕1,−i,−1,i͖. satisfying P␸P␸Ј= ␸,␸ЈP␸. The probability density of the oscillator functions, 2 B. Standard Fractionalization ͉⌿k͑x͉͒ , are the invariants under ␣-evolution. Functions Fourth roots of unity, such as F, have “standard” frac- that are covariant should have one parameter undergoing ␣ harmonic motion; these are the coherent states, which tional powers FS that we consider to be purely geometri- can be deﬁned equivalently as ground states displaced by cal. They are given by a complex parameter c, or as linear generating functions 3 of the oscillator functions, also with powers of c, sin ␲͑n − ␣͒ ␣ ͑ 3 ␲͑ ␣͒͒ n ͑ ͒ FS ª ͚ exp i 4 n − 1 F 13 2 4 sin ␲͑n − ␣͒ ⌼ ͑ ͒ c /2⌿ ͑ ͱ ͒͑͒ n=0 4 c x ª e 0 x − 2c 7

3 ϱ k c 1 ␲␣ ͑ ͒ ⌿ ͑ ͒ ͑ ͒ =exp͑−i NS͒, NS ͚ nP␸͑n͒. 14 =͚ k x . 8 2 ª n=0 k=0 ͱk!

␣ ␣ ␤ ␣+␤ The evolution cycle of ⌼c͑x͒ under F is evinced in the They satisfy composition, FSFS =FS , and Eq. (13) en- ␣ n n ͑ ͒ harmonic motion of the parameter c͑␣͒, sures that for integer =n, FS =F . This U 1 S Lie group is displayed in Eq. (14) as generated by a “number ma- F␣⌼ ͑ ͒ ⌼ ͑ ͒ ͑␣͒ ͑ ͒ ͑ 1 ␲␣͒ ͑ ͒ c͑0͒ x = c͑␣͒ x , c = c 0 exp −i2 . 9 trix” NS, in the same form as in Eq. (2). The Fourier eigenspaces with eigenvalues ␸͑0͒=1, ␸͑1͒=−i, ␸͑2͒=−1, and ␸͑3͒=i are also eigenspaces of the number matrix with eigenvalues n=0, 1, 2, 3, respectively. 3. DISCRETE FOURIER TRANSFORM However simple and universal the standard FrDFT ϫ ʈ ʈ The standard N N DFT matrix is F= Fm,mЈ , with ele- [Eq. (13)] appears to be, it is not the fractionalization we ments that involve powers of the Nth roots of unity, want to consider, because while for N→ϱ the DFT matrix (10) contracts to the IFT kernel e−ixxЈ/ ͱ2␲, the standard 1 2␲mmЈ F expͩ−i ͪ. ͑10͒ FrDFT matrix (13) does not contract to the canonical in- m,mЈ ª ͱN N tegral transform kernel (5), belonging to the continuous FrIFT group on L2͑R͒. It is periodic (Fm,mЈ=Fm+kN,mЈ=Fm,mЈ+kЈN for k,kЈ inte- ͑ ͒ ͑ † gers), symmetric Fm,mЈ=FmЈ,m , and unitary F F=1 =FF†͒,so͉det F͉ =1. Its square is the inversion matrix ͑ 2͒ ␦ 4 4. FRACTIONALIZATION OF DFT IN A F m,mЈ= m,−mЈ, and F =1 is a fourth root of the unit matrix. BASIS V The last paragraph justiﬁes considering a number matrix A. Eigenvalues and Projectors different from the standard NS in Eq. (14). Its spectrum The eigenvalues of the NϫN standard DFT matrix are should be k෈͕0,1,...,N−1͖ and increase only the inter- ␸͑ ͒ ͑ ͒n ͓ ͔ →ϱ the fourth roots of unity, to be denoted by n ª −i val 0,N under contraction N . ͑ 1 ␲ ͒ ෈͕ ͖ ෈͕ ͖ =exp −i2 n 1,−i,−1,i , for n 0,1,2,3 . This divides the space of N-point complex signals into four Fourier- A. Fourier Eigenbases and Their FrDFTs invariant subspaces whose dimensions N␸ are the multi- Within each ␸-subspace one can ﬁnd orthonormal bases V ␸ ͑␸,j͒ ʈ ͑␸,j͒ʈ plicities of the eigenvalues , which have a peculiar of N-column vectors, which we indicate by v = vm , modulo-4 recurrence in the dimension N=:4J+n, given labeled by j෈͕0,1,...,N␸ −1͖, and with rows m by ෈͕1,2,...,N͖. They will satisfy K. B. Wolf and G. Krötzsch Vol. 24, No. 3/March 2007/J. Opt. Soc. Am. A 653

N␸−1 F␣ = exp͑−i1 ␲␣N ͒, ͑␸,j͒† ͑␸Ј,jЈ͒ ␦ ␦ ͑␸,j͒ ͑␸,j͒† ͑ ͒ V 2 v v v = ␾,␸Ј j,jЈ, ͚ v v = P␸ 15 j=0 2 d N ͯi F␣ ͯ , ͑19͒ V ª ␲ d␣ V ␣=0 and can be arranged in four NϫN␸ rectangular matrices ͑␸͒ ʈ ͑␸,j͒ʈ V = vm . There are slight and nonessential differences N␸−1 ␸ in the expressions of the four -cases in Eq. (11); we shall = ͚ ͚ v͑␸,j͒ diag͑4j + n͒v͑␸,j͒†. ͑20͒ not need their explicit forms. ␸͑n͒ j=0 Next we build the bilinear generating function analo- gous to Eq. (6) for the finite bases V characterized above. The eigenvectors of the number matrix are v͑␸͑n͒,j͒ with For noninteger ␣ we must further specify that ␸␣ nondegenerate eigenvalues 4j+n෈͕0,1,2,...,N−1͖. Fi- ϵ͑␸͑ ͒͒␣ ͑ 1 ␲ ␣͒ ෈͕ ͖ n ªexp −i2 n with n 0,1,2,3 . We define the nally, note that the number operator in Eqs. (19) and (20) ϫ ␣ V-FrDFT by the N N matrices FV, whose elements are can serve equally well to define the V-FrDFT group of matrices proposed in Eqs. (16)–(18).

N␸−1 C. Discrete Coherent States ͑ ␣ ͒ ͑␸,j͒ ͓ 1 ␲͑ ͒␣͔ ͑␸,j͒* ͑ ͒ The requirement that the bilinear generating function FV m,mЈ ª ͚ ͚ vm exp −i2 4j + n vmЈ 16 ␸͑n͒ j=0 form of the V-FrDFT proposed in Eq. (16) contract in the N→ϱ limit to the FrIFT in Eq. (6) suggests that we should propose discrete or discretized oscillatorlike functions, numbered by k=4j+n as above, to have “good” ␸␣͑ ͑␸͒⌽͑␸͒͑␣͒ ͑␸͒†͒ ͑ ͒ V-bases for the fractionalization of the DFT matrix F. The =͚ V V m,mЈ, 17 ␸ three models of the next section comprise essentially identical low-lying eigenvectors, and they differ most sharply in the highest ones. The FrDFTs of sampled functions, such as centered rectangles,10,11 do not provide a ⌽͑␸͒͑␣͒ ͑ ͑ ␲ ␣͒͒ ϫ ͑ ͒ ª diag exp −2i j is N␸ N␸. 18 sufficient and reliable impression of their overall fidelity to the FrIFT. Our proposal here is to examine the exis- tence and behavior of associated coherent states. Associated to the V-FrDFT in Eq. (16), we search for The last line defines the matrix ⌽͑␸͒͑␣͒, which is diagonal states given by N-vectors ⌼V͑m͒ that contract to the con- and independent of ␸—except for its dimension. The con- c tinuous coherent states ⌼ ͑x͒ in Eqs. (7) and (8). This can ditions of Eq. (15) ensure the multiplication property c ␣ ␤ ␣ ␤ ␣ ␣ be made by truncating the infinite sum to the N available F F =F + modulo 4, and the unitarity ͑F ͒† =F− ,of V V V V V functions or by displacing the ground state. The latter can each set of V-FrDFT matrices. When ␣ is integer, be done when the system is periodic and has a well- ⌽͑␸͒͑␣͒ =1, and the matrix (17) becomes the sum of the ͑ ͒ ͑ ͒ n n identified ground state j,n = 0,0 . For generic, nonperi- projectors in Eq. (12), so FV =F . The V-FrDFT matrices odic V-bases, we opt for the former choice, Eq. (8), defin- ͑ ͒ ʚ ͑ ͒ thus belong to subgroups U 1 V U N , all of which pass ing discrete coherent states as linear generating functions through the standard DFT matrix and its integer powers. of the V-basis vectors, namely, as N-point column vectors of components

B. Number Matrix of an Eigenbasis N␸−1 4j+n c ͑␸ ͒ Note that in each ␸-subspace, the numbering of the N␸ ⌼V͑ ͒ ,j ͑ ͒ c m = ͚ ͚ vm . 21 ͑␸,j͒ ͱ͑ ͒ basis vectors v by j෈͕0,1,...,N␸ −1͖ is arbitrary; our ␸͑n͒ j=0 4j + n !

construction is still purely geometric, since no dynamic ␣ “energy spectrum” is used to suggest any prefered order. When multiplied on the left by FV, each summand is mul- (Three bases will be examined in the next section.) Start- tiplied by a phase that is collected by the complex param- ͑␣͒ ͑ ͒ ͑ 1 ␲␣͒ ing with one V basis, represented by the NϫN matrix V, eter c =c 0 exp −i2 , exactly as in Eq. (9), so that the we can subject each ␸-subspace to an N␸ ϫN␸ unitary V-FrDFT is a uniform rotation of the complex c-plane. For ͑␸͒ transformation U ෈U͑N␸͒, so that through right multi- “arbitray” bases V, the vectors deﬁned by Eq. (21) may not plication we obtain a new W-basis associated to the prod- mean much, but for “good” bases (closely related to the os- ⌼V uct of matrices W=VU as well as a new one-parameter cy- cillator wave functions), the discrete coherent state c clic group of W-FrDFTs that will also pass through the should initially resemble a displaced Gaussian and standard DFT matrix F and its integer powers. The mani- should oscillate as its continuous counterparts without undue distorsion. fold of these U͑1͒V FrDFT subgroups within the 2 2 N -dimensional manifold of U͑N͒ has dimension ͚␸N␸ and can be characterized by its tangent at the origin, namely, its number matrix. 5. APPROACHES TO DISCRETE The V-FrDFT matrix in Eqs. (16)–(18) has the form of OSCILLATOR DYNAMICS Eq. (2) in continuous systems; they are Lie exponentials Since the FrIFT has close connection with the dynamics of an NϫN number matrix associated to the basis V, of the quantum harmonic oscillator, so should their 654 J. Opt. Soc. Am. A/Vol. 24, No. 3/March 2007 K. B. Wolf and G. Krötzsch

V-FrDFT analogs. In this section we describe three ap- highest values of energy k. The coherent states built out proaches: two of them based on the oscillator wavefunc- of the Mehta basis are shown in Fig. 4; since they are pe- tions ⌿k͑x͒ in Eq. (3), and one based on importing sym- riodic, for large c they “spill over” the ends of the interval 1 ͑ ͒ metry to a discrete and concrete physical model. at ± 2 N−1 . Generalizations of these Mehta functions to other simi- A. Sampled Harmonic Oscillator: Taipei Bases larly summed Fourier eigensets of special functions have It is suggestive to build bases for V-FrDFTs using been shown by Atakishiyev and others19,20 to extend into sampled oscillator functions ⌿k͑xm͒ in Eq. (3). Pei and the much wider field of discrete q-special functions. Yeh 10 and Pei et al.,11 working in Taipei, have constructed FrDFTs defined through the bilinear generating function (16). They first define the vectors ␾͑k͒, k෈͓0,N−1͔, C. Ankara Lattice and FrDFT A group of researchers and students based in Ankara in- 2␲ troduced, by analogy with continuous systems, the Hamil- ␾͑k͒ ⌿ ͩͱ ͪ ͑ ͒ 13,14 m ª k m , where 22 tonian for a periodic vibrating lattice model. This sys- N tem consists of N points of equal mass, numbered cyclically by m෈͕1,2,...,Nϵ0͖, that are on a circle, ͓− 1 N, 1 N −1͔ N even joined to their equilibrium positions and to one another m ෈ ͭ 2 2 . ͑23͒ 1 1 by springs. The states of the system are given by the N ͓ ͑ ͒ ͑ ͔͒ N odd − 2 N −1 , 2 N −1 complex quantities ͕fm͑␶͖͒ subject to time-␶ Schrödinger ʈ A ʈ These N-vectors are not quite orthogonal, except for par- evolution by a real Hamiltonian matrix HA = Hm,mЈ (A for ity ͑−1͒k (we call this the uncorrected Taipei basis); they Ankara); solving a difference equation, the solution is a are not naturally periodic in k nor m, unless so defined Green matrix GA͑␶͒, beyond their natural range; and they are not guaranteed to have k changes of sign in m as the continuous functions do. However, they do provide an approximate FrDFT d H f͑␶͒ =i f͑␶͒ ⇔ f͑␶͒ = G ͑␶͒f͑0͒, ͑25͒ when replaced directly in Eq. (16), which produces cred- A d␶ A ible harmonic motion on its coherent states. ␾͑k͒ We display the uncorrected Taipei basis m by the gray-tone matrix ͑m,k͒ in Fig. 1; there we can see the relative signs and the regions where the components are near zero. The errors are small in this uncorrected basis for the lower-lying states. For example, when N=33asin our figures, the maximum of the overlaps ͑␾͑k͒ ,␾͑kЈ͒͒ for ЈՅ 2 k k 22= 3 N is 0.01. The coherent states constructed according to Eq. (21), with the uncorrected Taipei basis (23) are shown in Fig. 2 for three values of c͑0͒, over one- quarter of the Fourier cycle. The center of the Gaussian- ͱ like bumps is at mmax͑␣͒Ϸ N/␲Re c͑␣͒; in the figure, for N=33, the peak of a c=5, coherent state is at mmax =16.2, just outside the interval m෈͓−16,16͔, yet it oscil- lates quite harmonically and does not unduly disperse. The refinement of this construction constitutes the 10 11 ͑ ͒ ␸͑k͒ work of Pei and Yeh and Pei et al., who first projected Fig. 1. Matrix m,k of the uncorrected Taipei basis m of the basis functions (22) by Eq. (12) to separate them into sampled oscillator functions in Eq. (22) for N=33. Light and dark the four ␸-eigenspaces and then applied a modified elements indicate positive and negative values, respectively. Gramm–Schmidt computer orthonormalization process within each eigenspace, with the numbering provided naturally by the energy quantum label k.

B. Mehta Eigenbasis We consider next a (previous and) very elegant basis for FrDFTs given by Mehta in 1987,12 who found that the sums of periodically displaced oscillator functions (3),

ϱ 2␲ ␮͑k͒ ␮͑k͒ ⌿ ͩͱ ͑ ͒ͪ ͑ ͒ m = m+N ª ͚ k m + Ns , 24 s=−ϱ N are N eigenvectors of the DFT matrix, F␮͑k͒ =͑−i͒k␮͑k͒, Fig. 2. Motion of coherent states built out of the uncorrected which (as the Taipei basis) is not quite orthogonal. Except Taipei basis of sampled oscillator functions in Eq. (22) for N =33. In the three columns c͑0͒=1, 3, 5. The rows show their evo- for orthogonality, this Mehta basis also fulﬁlls most ex- lution over a quarter-period for ␣=0, 0.2,…1.0. Dashed curve, pectations of Eq. (16) with k=4j+n; appears in Fig. 3. real parts; dotted curve, imaginary parts; solid curve, absolute Comparison with Fig. 1 shows that they differ only at the values. K. B. Wolf and G. Krötzsch Vol. 24, No. 3/March 2007/J. Opt. Soc. Am. A 655

͑k͒ ͑k͒ ͑k͒ ␸͑ ͒ ͑k͒ ͑ ͒ HAh = Ekh , Fh = n h , 31

where the numeration follows that of Eq. (16) with k=4j +n. The energy eigenvalues ͕Ek͖ are the roots of an Nth degree polynomial; they are real and nondegenerate (except for one double degeneracy when N is a multiple of 413), and they are naturally ordered in k by the number of sign alternations in m. Yet they are not equally spaced nor rational nor algebraic (for NϾ4), so they have to be ͑k͒ computed numerically. The hm ’s have been named Harper functions;14 they are bona ﬁde orthonormal and periodic N-point signals; their matrix is shown in Fig. 5 (cf. Figs. 1 and 3).

͑ ͒ ␮͑k͒ Fig. 3. Matrix m,k of the Mehta of oscillator functions m in D. Importation of Symmetry Eq. (24). It differs signiﬁcantly from that in Fig. 1 only in the The time evolution of the Ankara lattice is produced by highest k-columns. the Green matrix in Eq. (26), whose matrix elements are the bilinear generating functions

N−1 A ͑␶͒ ͑k͒ ͑ ␶ ͒ ͑k͒ ͑ ͒ Gm,mЈ = ͚ hm exp −i Ek hmЈ. 32 k=0

The Green matrix is unitary for all ␶෈R; but since the ͕Ek͖ are not commensurable, the ␶-line of Green matrices does not close into a U͑1͒ circle but is a Lissajous-type curve in the N2-dimensional manifold of U͑N͒. To produce a FrDFT with a modulo-4 cycle, the Ankara

Fig. 4. Motion of Mehta coherent states built out of the basis ␮͑k͒ m in Eq. (24). The rows and columns have the same parameters as those in Fig. 2.

͑␶͒ ͑ ␶ ͒ ͑ ͒ GA ª exp −i HA . 26 The ␶-evolution of the lattice deﬁnes a group of “A-transforms” that is expected to lie approximately on an A-FrDFT closed cycle (still to be deﬁned), i.e., ␣͑␶͒ Ϸ ͓ ␶͑ ͔͒ ͑ ͒ FA exp −i HA − h01 . 27

Here h0 is the lowest (vacuum) energy; this is an analog of the metaplectic phase correction in Eqs. (1) and (2) in the continuous case. 5,13,14 Fig. 5. Matrix ͑m,k͒ of the Ankara oscillator with the Harper The group in Ankara considered the natural dif- ͑k͒ Ϸ functions hm . Note the “jump” at k 1/2N between states where ference analog of Eq. (2), the soft springs contain most of the energy and those where the lattice vibration is largest in the stiff zone. 1 ͑⌬ ⌬˜ ͒ ͑ ͒ HA ª − 2 + , 28

⌬ ͑ ͒ ͑ ͒ ª circ − 2,1,0, ... ,0,1 , 29

⌬˜ ⌬ −1 ͑ 2͑␲ ͒͒ ͑ ͒ ª F F = diag − 4 sin m/N . 30 This Hamiltonian contains the kinetic energy of the circulating second-difference matrix ⌬, which is due to equal springs between first neighbors along the circle. The po- tential energy is contained in the springs that tie each mass to its equilibrium position by the diagonal ⌬˜ , soft Ϸ ϵ Ϸ 1 around m N 0 and stiff around m 2 N. As in the continuous case, the Ankara oscillator system Fig. 6. Motion of coherent states built out of the Harper func- in Eq. (28) is invariant under the DFT; F and HA can be ͑k͒ tions hm of the Ankara oscillator [Eq. (31)] with imported sym- simultaneously diagonalized with a common set of eigen- metry. The parameters are the same as in Figs. 2 and 4. In the vectors, last column we see the effect of the stiff-mode dominance. 656 J. Opt. Soc. Am. A/Vol. 24, No. 3/March 2007 K. B. Wolf and G. Krötzsch group imported the symmetry of the continuous system comparable to the models of Section 5. by replacing the physical spectrum ͕Ek͖ of the vibrating lattice with the linear number spectrum ͕k͖ between 0 A. Position, Momentum, and Number and N−1. The action of this A-FrDFT on the Harper co- Discretely quantized, the classical observables of position herent states, defined as in Eq. (21) is shown in Fig. 6. For q and momentum p are assigned not to the eigenvalues of large c, the stiff modes at the interval ends overwhelm the Schrödinger operators Q and P seen above, but to the the soft modes, so most of the energy lingers there eigenvalues of the operators Li of the Lie algebra of angu- throughout the cycle. lar momentum and spin su͑2͒, which is characterized by ជ ជ ជ Symmetry importation has maintained dialogue with LϫL=iL. The new asignments (with overbars) and their many of the works on finite models of quantum spectra are mechanics21–24 and discrete models with number-phase 25–28 Q¯ ↔L ෈ ͕ ͖ ͑ ͒ uncertainty relations. Here, the importation of sym- position: x, m − l,− l +1,...,l , 33 metry both evinces and bridges the gap between the dynamics of the physical system on one hand and the geo- momentum: P¯ ↔L , p ෈ ͕− l,− l +1,...,l͖, ͑34͒ metric requirement of a fractional DFT generated by a y number matrix. N¯ ↔L ෈ ͕ ͖ ͑ ͒ number: z + l1, k 0,1, ... ,2l , 35 E. Discrete Toroidal and Plane Phase Spaces within the spin-l representation of su͑2͒, for fixed l In order to associate a phase space to discrete systems ෈͕0, 1 ,1,...͖. This representation has dimension N=2l along the lines of continuous classical or quantum Hamil- 2 +1, as determined by the quadratic Casimir operator, tonian systems, we must put in place proper definitions of position and momentum; such may also be based on the Q¯ 2 + P¯ 2 + ͑N¯ − l1͒2 = l͑l +1͒1. ͑36͒ dynamics of coherent states, although we should be pre- pared for mismatches between the two. The simplest analogy consists in assigning to the row B. Eigenvectors of Number numbers of the state N-vectors f=ʈfmʈ the meaning of The signals sensed along the waveguide evolve through a discrete position, and to the row numbers of its DFT discrete version of the FrIFT, the fractional Fourier– ʈ˜ ʈ Kravchuk transform (FrFKT), which is built exactly as in Ff= fmЈ the meaning of discrete momentum. Their inter- twining by F implies that these discrete sets of points Eq. (2), but with the number operator N¯ in Eq. (35). It is a should be considered periodic with the period N, so phase rotation around the z axis that carries the position x axis ͑ Ј͒ ␺͑l͒ space is a discrete torus of points m,m . Integer trans- onto the momentum y axis. The basis k of the FrFKT lations along the position or momentum circles are well are the eigenvectors of the number operator Lz, measured defined, as well as a discrete Heisenberg–Weyl group with in the position basis of Lx, and are known as the Wigner a phase composition, and a discrete Wigner function can little-d functions [Ref. 32, Eq. (3.65)], also be defined29–31 with the most important properties of ␺͑l͒͑m͒ = dl ͑− 1 ␲͒ = dl ͑ 1 ␲͒ ͑37͒ its continuous counterpart. On this construction, how- k m,k−l 2 k−l,m 2 ever, understanding the role of the fractional DFT is dif- ficult, because classically it is supposed to rotate the po- ͑−1͒k 2l 2l = ͱͩ ͪͩ ͪK ͑l + m; 1 ,2l͒. ͑38͒ sition circle continuously onto the momentum circle—in 2l k l + m k 2 spite of the topological obstruction presented by the hole of the torus. The last form shows that the m-dependence is contained Discrete coherent states ⌼V as defined in Eq. (21) pro- 2l c in the square root of the binomial distribution, , vide for another picture of phase space, namely, the com- ͑l+m ͒ plex c-plane, whose real and imaginary parts take the times the symmetric Kravchuk polynomial of degree k in role of continuous position and momentum variables. The l+m෈͓0,1,...,2l͔.33 Kravchuks34 are discrete orthogonal action of the FrDFT is then in complete accordance with counterparts to the Hermite polynomials [Ref. 35, Eqs. classical expectations but collides with the notion that the (5.2.13) and (5.2.14)] and binomials to Gaussians. phase space of discrete systems should be compact. In Fig. 7 we show the matrix of the Cuernavaca basis. Note that the top states ͑kϷ2l͒ mirror the bottom states ͑kϷ0͒ with alternating signs between consecutive points, ͑ ͒ ͑ ͒ ␺ l ͑m͒=͑−1͒l−k+m␺ l ͑−m͒. This figure should be com- 6. su 2 FINITE OSCILLATOR MODEL 2l−k k „ … pared with Figs. 1, 3, and 5 of the previous bases. A discrete, finite system with the geometry and dynamics of the harmonic oscillator was proposed in Ref. 6 (see also C. Fourier–Kravchuk Transform Matrix Ref. 16), realized as a planar paraxial, shallow multimo- The FrFKT of power ␣ is a rotation around the z axis by dal waveguide that processes N-point complex signals in an angle ␾= 1 ␲␣, generated by N¯ , which carries the Q¯ x parallel, and produced and received at linear arrays of 2 axis onto the P¯ y axis, with a phase [cf. Eq. (2)], point emitters and sensors. This model, developed in Cu- ernavaca, follows with a “discrete quantization” of the K␣ exp͑−i1 ␲␣N¯ ͒ = e−i␲l␣/2 exp͑−i1 ␲␣L ͒. ͑39͒ classical harmonic oscillator. It has natural coherent ª 2 2 z states whose behavior was analyzed in Ref. 15. Here we The FrFKT matrix elements are built as bilinear generat- highlight only the main postulates and results in terms ing functions of the wave functions in Eq. (38) for N=2l K. B. Wolf and G. Krötzsch Vol. 24, No. 3/March 2007/J. Opt. Soc. Am. A 657

+1, as in Eq. (16); using standard results from angular momentum theory,32 they can be summed to a closed form,

2l ͑l,␣͒ ␺͑l͒͑ ͒ −i␲k␣␺͑l͒͑ Ј͒ mЈ−m l ͑ 1 ␲␣͒ Km,mЈ = ͚ k m e k m =i dm,mЈ 2 . k=0 ͑40͒ The set of FrFKT matrices form a group U͑1͒K ʚSU͑2͒ʚU͑2l+1͒. However, note that this group does not contain the Fourier matrix F of Eq. (10); instead, for ␣=1, K͑l,1͒ is the real matrix shown in Fig. 7, multiplied by a circulating phase, as can be seen by comparing Eqs. (40) and (37). Also, the Cuernavaca model is not periodic: In signals of N=2l+1 points, m=±l are not first Fig. 7. Matrix ͑m,k͒ of the Cuernavaca discrete wave functions k neighbors, but the two ends of the signal. of the finite oscillator, ␺ ͑m͒ in Eq. (37), for l=16 ͑N=33͒. See the m↔m and k↔2l−k parity covariances with the signs ͑−1͒k and ͑−1͒l−k+m, respectively. D. Coherent States in su„2… The ground state of the su͑2͒ finite oscillator is

2l ␺͑l͒͑m͒ = dl ͑ 1 ␲͒ = 1 ͱͩ ͪ. ͑41͒ 0 −l,m 2 2l l + m

It is even and centered in m෈͓−l,l͔; this we can imagine as a density bump on the south pole of a sphere, which is projected on the position x axis as a discrete Gaussianlike function. Rotation of this sphere around the y axis of momentum through an angle −␪, will translate the projection of its density bump in Eq. (41) along the position axis by sin ␪ [cf. Eq. (7)] without exceeding the interval. The 1 center of the bump reaches the equator at ␪= ␲ and ex- ͑l͒ 2 Fig. 8. Motion of the coherent states ␬␪ in Eqs. (42)–(45) under tends up to the north pole of the sphere for ␪=␲. Simi- fractional Fourier–Kravchuk transformations. In the three col- larly, rotation of the sphere around the x axis with umns, ␪=30°, 60°, and 90°. The rows have the same parameters exp͑i␩Q¯ ͒ will multiply the signal values by linear phases, as those in Figs. 2, 4, and 6. and this translates the momentum of the ground state in Eq. (41). exp͑−i 1 ␲␣͒c͑0,␪͒, which is the exact analog of Eqs. (9) We deﬁned the coherent states of the Cuernavaca ª 2 ͓Q¯ P¯ ͔ ͑N¯ ͒ model for 0Յ␪Յ␲ by and (21). The basic commutator here is , =i −l1 instead of the standard quantum mechanical ͓Q,P͔=i1. ␬͑l͒ ͑ ␪P¯ ͒␺͑l͒ ͑ ͒ In Fig. 8 we show the motion of Cuernavaca coherent ␪ ª exp i 0 , 42 states over one-quarter Fourier cycles. ␬͑l͒͑ ͒ l ͑ 1 ␲ ␪͒ ͑ ͒ ␪ m = dm,−l − 2 − 43

2l E. Phase Space of Discrete Systems The Cuernavaca model is based on the manifold of the Lie ͑ ͒k l ͑␪͒␺͑l͒͑ ͒ ͑ ͒ =͚ −1 dl,l−k k m , 44 algebra su͑2͒ with the operators of Eqs. (33)–(35). A k=0 proper covariant Wigner distribution is a function of ͑q,p,n͒෈R3 that we called meta-phase space.36,37 An 2l 1 l 1 k N-point discrete system belongs to the spin-l representa- dl ͑␪͒ = ͑cos ␪͒ ͱͩ ͪ͑sin ␪͒ . ͑45͒ l,l−k 2 k 2 tion ͑N=2l+1͒, whose Casimir [Eq. (36)] classically re- Ϸ 32 duces the manifold to a sphere of radius l, which is an Equation (43) follows from angular momentum theory, easily visualized symplectic manifold. Tangent to the and Eq. (44) displays coherent states as the linear gener- south pole is ordinary phase space ͑q,p͒෈R2. Evolution ␺͑l͒͑ ͒ ating function of the number eigenvector set k m [cf. along the harmonic guide (modulo a “metaplectic” phase) Eqs. (8) and (21)]; however, the coefﬁcients of the trun- is rotation of the sphere around its vertical axis; prisms k cated exponential series c / ͱk! are here replaced by Eq. rotate the sphere around its y axis; inclined slabs trans- ϳ͑ 1 ␪͒k ͱ ͑ ͒ (45) as sin 2 / k! 2l−k !. late signals in x, across the guide and with “reﬂection” at The FrFKT in Eq. (39) will rotate the sphere around the endpoints. Consideration of transformations gener- ␾ 1 ␲␣ the z axis by = 2 , so the Gaussian bump projection on ated by elements in the universal enveloping algebra of ͑l͒ the x axis of ␬␪ will oscillate harmonically with ␣ modulo su͑2͒ allow for the introduction of aberrations on the ͑ ␪͒ 1 ␪ 4, while the coherent state parameter c 0, ªsin 2 phase space of complex signals, such as those due to the in Eq. (45) is multiplied by a phase, c͑␾,␪͒ Kerr effect.38 658 J. Opt. Soc. Am. A/Vol. 24, No. 3/March 2007 K. B. Wolf and G. Krötzsch

7. CONCLUSION 16. N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Finite models of the oscillator,” Phys. Part. Nucl. 36, 521–555 It should be interesting to compare the different versions (2005). of the FrDFT with the actual output of a planar optical 17. K. B. Wolf, Geometric Optics on Phase Space (Springer- Fourier signal processor, particularly a waveguide, at fi- Verlag, 2004). nite linear arrays of sensors. We expect that in such 18. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, guides, well-collimated coherent beams will evolve with 241–265 (1980). close-to-harmonic motion. Then we could select a “best” 19. N. M. Atakishiyev, D. Galetti, and J. P. Rueda, “On V-basis so that its V-FrDFT approximates the close-to- relations between certain q-polynomial families, generated Fourier action of that particular signal processor. by the finite Fourier transform,” Int. J. Pure Appl. Math. 26, 275–284 (2006). 20. N. M. Atakishiyev, “On q-extensions of Mehta’s eigenvectors of the finite Fourier transform,” Int. J. Mod. ACKNOWLEDGMENTS Phys. A 21, 4993–5006 (2006). We are indebted to N. M. Atakishiyev (Instituto de 21. T. S. Santhanam and A. R. Tekumalla, “Quantum Matemáticas, UNAM–Cuernavaca), F. Leyvraz, W. L. Mo- mechanics in finite dimensions,” Found. Phys. 6, 583–589 chán, and L. E. Vicent (Centro de Ciencias Físicas, (1976). 22. C. Belingeri and P. E. Ricci, “A generalization of the UNAM) for discussions on the subject of this paper. Sup- discrete Fourier transform,” Integral Transforms Spec. port from the SEP–CONACYT project 44845 and Funct. 3, 165–174 (1995). DGAPA–UNAM project IN102603, Óptica Matemática, is 23. M. S. Richman, T. W. Parks, and R. G. Shenoy, “Discrete- gratefully acknowledged. time, discrete-frequency analysis,” IEEE Trans. Signal Corresponding author K. B. Wolf can be reached by Process. 46, 1517–1527 (1998). 24. M. Shiri-Garakani and D. R. 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