Optical waveguide Hamiltonians leading to step-2 difference equations

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1 Journal of Physics: Conference Series Optical waveguide Hailtonians leading to step-2 difference equations To cite this article: Juvenal Rueda-Paz and Kurt Bernardo Wolf 2011 J. Phys.: Conf. Ser View the article online for updates and enhanceents. Related content - Magnon Bose-Einstein condensation at inhoogeneous conditions Yury Bunkov and Murat Tagirov - Superfluid hydrodynaic in fractal diension space D A Tayurskii and Yu V Lysogorskiy - Cooperative Resonance Interaction Between One-and Two-Photon Superfluorescences Trough the Vacuu Field Nicolae A Enaki This content was downloaded fro IP address on 14/06/2018 at 13:47

2 GROUP 28: Physical and Matheatical Aspects of Syetry Optical waveguide Hailtonians leading to step-2 difference equations Juvenal Rueda-Paz 1 and Kurt Bernardo Wolf 1 1 Instituto de Ciencias Físicas, Universidad Nacional Autónoa de México, Av. Universidad s/n, Cuernavaca, Morelos 62251, Mexico E-ail: bwolf@fis.una.x Abstract. We exaine the evolution of an N-point signal produced and sensed at finite arrays of points transverse to a planar waveguide, within the fraework of the finite quantization of geoetric optics. In contradistinction to the coon echanical Hailtonians (kinetic plus potential energy ters), the classical waveguide Hailtonian is the square root of a difference of squares of the refractive index profile inus the optical oentu. The finitely quantized odel requires the solution of the square eigenvalue and eigenfunction proble, which leads to a step-two difference equation that contains two solutions and two signs of energy. We find the proper linear cobinations to fit the Kravchuk functions of the finite oscillator odel. 1. Introduction Most studies in syetry, supersyetry, and separation of variables, pertain Hailtonian systes of echanical nature, i.e., whose Hailtonian functions separate kinetic fro potential energy ters, H ec ( x, p) = p 2 /2M + V ( x), (1) where M is the ass. Schrödinger quantization of this classical Hailtonian provides the wellknown Schrödinger second-order differential equation, whose eigenvalues are the energies and the stationary states are the eigenvectors of such systes. Waveguides with finite sensor arrays provide Hailtonian functions which are of a different structure, and whose quantization is proposed to be distinct fro the Schrödinger one, where now the geoetric optical syste is associated to a odel where the observables of position, oentu and energy have a discrete, finite set of eigenvalues. This we call so(3)- or finite quantization. This is taylored for parallel analysis of finite signals and the processing of pixellated iages. We set up the Hailton equations of geoetric optics starting fro the two siilar right triangles in Figure 1. Let light rays be abstracted as lines in 3-space x(s), s R; and let their tangent vectors be p(s). In [1, Chap. 1] we prove succintly that, postulating the change in ray direction to be in linear response to the gradient of the refractive index n( x), as d p = n( x) ds, leads to the restriction p = n( x). Then, writing p = ( ) ( ) p x, x =, (2) pz z Published under licence by Ltd 1

3 GROUP 28: Physical and Matheatical Aspects of Syetry Figure 1. Left: The ray direction vector p decoposed into its coponent p along the screen z = 0, and p z noral to it; the length of p is the refractive index p = n( x). Right: the differential length d s along the ray, decoposed into the differential dx on the screen, and dz noral to it; dz is the differential evolution paraeter. The two triangles are siilar. The angle θ opens between the optical z-axis and the ray. the siilarity of the triangles in Figure 1 iplies that dx/dz = p/p z, dp/dz = dp/ds dz/ds = n n/p z, dp 2 z /dz = n2 / z. (3) These equations fit into the Hailtonian for dx dz = h p, dp dz = h x, dp z dz = h z, (4) with the Hailtonian function h(x, p, z) := σ n(x, z) 2 p 2 = p z = n( x) cos θ, (5) where σ = + applies to forward rays (in the +z direction), σ = to backward rays (in the z direction), and θ is the angle at the screen z = 0 between the ray and the optical axis +z. (The sign assignent can be seen easily for free propagation in a ediu n = constant.) The square-root for of the Hailtonian (5) is characteristic of optics, and we shall particularly devote our attention to the case when n is independent of z naely, when the optical ediu is a waveguide, so its refractive index is n(x). We point to the reseblance of (5) with the total relativistic energy of a freely oving ass point, E = c M 2 c 2 + p 2 = Mc M p 2 1 8M 3 c 2 p 4 +. (6) The difference between both is of course the sign of the oentu ter p 2 in the square root: in (6) the evolution paraeter is tie t, and the etric of (x, t) and (p, E) is Lorentzian, while in (5) the etric of (x, z) and (p, p z ) is Euclidean. The Schrödinger quantization of the square of (6) leads to the Klein-Gordon equation as a relativistic quantu echanical odel; it is the Dirac quantization however, which posits a linear atrix equation that is ost productive to describe Nature. Siilarly, while the wavization of the square of (5) produces the Helholtz wave equation, we feel that its atrix square-root for can yield interesting results provided with a proper interpretation as a odel of finite optics. In Section 2 we introduce the specific faily of waveguides to be treated here, whose refractive index profile is elliptic, arriving at their syetry expressed in algebraic ters through Poisson brackets. Then in Section 3 we quantize the classical syste to a finite syste given by a Hailtonian atrix defined by the square root of a self-adjoint atrix. The signs of the 2

4 GROUP 28: Physical and Matheatical Aspects of Syetry eigenvalues and eigenvector proble that stes fro a step-two difference equation are analyzed and resolved in Section 4. The concluding Section 5 refers to results on the evolution of finite Kravchuk coherent states in such waveguides, and the possible use of aberration expansions in odeling nonlinear optical edia. 2. Waveguide Hailtonians In a two-diensional (planar) geoetric optical syste x = (x, z) where the index of refraction is n(x, z), the x-coordinate easures the screen z = 0, and its z-coordinate is the independent evolution paraeter along the optical axis of the device. In this paper we shall consider edia that are invariant under z-translations, i.e. waveguides, where n n(x); and oreover, for reasons of paraetric proxiity, we propose waveguides whose refractive index profile be elliptic, naely n ν,µ (x) := + ν 2 µ 2 x 2, ν 1, µ 0, x ν/µ. (7) At the center of the guide n(0) = ν, while 1/µ estiates the width of the guide. Although physically n 1 this is not necessary for the analysis, so we consider the guide to extend between ±ν/µ. The Hailtonian (5) of the waveguide ediu (7) is h ν,µ (x, p) = σ ν 2 (p 2 +µ 2 x 2 ) (8) = σν + σ p2 +µ 2 x 2 2ν + σ (p2 +µ 2 x 2 ) 2 8ν 3 +, (9) where we have expanded the square root into a series of powers of the oscillator Hailtonian k µ := 1 2 (p2 + µ 2 x 2 ). In the waveguide (8), the evolution of phase space along z will be that of an incopressible fluid, flowing along the ellipses p 2 + µ 2 x 2 = constant < ν 2. The trajectory of lowest energy h = ν is a ray along the +z axis at the center of phase space, and that of highest energy ν is along the z axis. Two particlar cases of waveguide with ν = 1 will serve as reference: circular profile µ = 1, k 1 = 1 2 (p2 +x 2 ), (10) free particle µ = 0, k 0 = 1 2 p2. (11) Written with Poisson brackets, the Hailton equations are {k µ, x} = p, {k µ, p} = µ 2 x, {x, p} = 1. (12) The four quantities {k µ, x, p, 1} for, for µ = 1, the oscillator Lie algebra osc, while for µ = 0 they for the nilpotent algebra free. 3. Finite so(3)-quantization Finite quantization involves a deforation (or pre-contraction ) of the Lie algebra that contains the classical syste [2, 3, 4]. Poisson brackets between the classical phase space observables of position and oentu (x, p) are replaced by i ties the coutator Lie brackets between operators (X, P), closing with a pseudo-hailtonian K µ for which we assue no specific for that obey the sae geoetry and dynaics. Corresponding to (12), the two Hailton equations are: [K µ, X ] = i P, [K µ, P] = i µ 2 X. (13) The deforation is contained in the third non-standard coutator [X, P] = i K µ. (14) 3

5 GROUP 28: Physical and Matheatical Aspects of Syetry For µ = 1, the three generators {X, P, K µ } close into the rotation Lie algebra so(3), which we shall use as reference for several considerations below. For µ = 0 these generators close into the Euclidean algebra iso(2). Aong the realizations of so(3) we choose the self-adjoint irreducible representation atrices in the eigenspaces of the Casiir operator [5, 6], C := X 2 + P 2 + K 2 = j(j+1) 1, (15) whose eigenvalues (nubered by j) deterine the vector space to be of diension N = 2j+1. Thus we represent the classical quantities by the self-adjoint atrices that belong to the representation j of so(3): x X = X, will be diagonal, p P = P, skew-syetric, and h + const. K = K, syetric, with eleents [6] X, = δ,,, { j, j+1,..., j}, (16) P, = i 1 2 (j )(j + + 1) δ+1, + i 1 2 (j + )(j + 1) δ 1,, (17) K, = 1 2 (j )(j + + 1) δ+1, (j + )(j + 1) δ 1,, (18) and their su of squares (15) is C := X 2 + P 2 + K 2 = j(j+1) 1. The spectru of the position operator X is thus x { j, j+1,..., j}; the eigenvalues {ϖ r } j r= j of the oentu P, and {κ n } j n= j of the pseudo-energy operator K, take values in the sae range. Since we consider iportant to have sensor arrays that include one point x 0 = 0 at the center, we consider only integer j s, to the exclusion of the su(2) spin representations. The elliptic-profile waveguide Hailtonian h ν,µ (x, p) in (8) will be thus finitely-quantized to an N N heritian atrix which, taking (15) into account, provides the equation for its eigenvalues and eigenvectors: H ν,µ := [(ν 2 1)C (µ 2 1)X 2 ] + K 2, (19) H ν,µ Ψ ν,µ η () = η ν,µ Ψ ν,µ η (), { j, j+1,..., j}, (20) where the atrix between the square brackets in (19) is diagonal. We shall call η ν,µ the energy of the state Ψ ν,µ η (), corresponding to the value of the classical Hailtonian (8). In the special case ν = 1 = µ we have H 1,1 = K 2, and we refer to this as a Kravchuk guide, since then the eigenvalues are ηn 1,1 = ±κ n { j, j+1,..., j}, and the eigenfunctions Ψ 1,1 η () are the Kravchuk functions studied in Refs. [4, 7, 8, 9], which are finite deforations of the haronic oscillator (Herite-Gauss) wavefunctions. When µ = 0, the free Hailtonian (19) has eigenvalues ηr ν,0 = [ν 2 j(j+1) ϖr 2], with ϖ r = r having the sae range as κ n. We restrict the waveguide paraeters to 0 µ ν 1 to keep the eigenvalues of the radicand positive. Since the square root of a atrix is not uniquely defined, we propose to use its square and solve (20) through (H ν,µ ) 2 Ψ η () = λ ν,µ Ψ η (), λ ν,µ := (η ν,µ ) 2, (21) knowing that the eigenvectors of H and of its square are the sae. We thus diagonalize (21) by its eigenvector atrix, Ψ (H ν,µ ) 2 Ψ = Λ ν,µ, Ψ = Ψ ηn () Λ ν,µ = diag (λ ν,µ ax, λ ν,µ ax 1,... λν,µ in+1, λν,µ in ), (22) where the N = 2j+1 eigenvalues {λ ν,µ n } are ordered fro axiu to iniu, observing that away fro µ = 0, 1, they are non-degenerate, although j pairs are very close in various ranges of µ. In Figure 2 (left) we show the coputed eigenvalues {λ ν,µ n } for ν = 1 and ν = 2, in guides with µ = ν, and for free µ = 0. 4

6 GROUP 28: Physical and Matheatical Aspects of Syetry ( ) 2 ( ) Figure 2. Left colun: The squared eigenvalues {η 2 } of (H ν,µ ) 2. Right colun: eigenvalues {η} of the root H ν,µ with the sign asignents detailed in the text. Top row: The guide ν = 1. Botto row: A guide with ν = 2. Heavy lines ark the ground state eigenvalues η 2 0 and η 0. Throughout, j = 10 (N = 21 points) and 0 µ ν. 4. Eigenvalue signs and eigenvectors We now proceed to analyze the sign of the square roots (η ν,µ ) 2. Notice that the ground state of a waveguide has the lowest energy eigenvalue ηgr ν,µ < 0 of H ν,µ, and this appears as the highest λ ν,µ ax = (ην,µ gr )2 > 0 of (H ν,µ ) 2. In the Kravchuk case this eigenvalue is η 1,1 j = j, whose square j 2 is degenerate with that of the highest energy that the waveguide can carry, η 1,1 j = j. In the other extree case µ = 0, the ground state has zero transverse oentu ϖ 0 = 0, so ηgr ν,0 = ν j(j+1); this state also has the highest eigenvalue λ ν,µ ax = (η ν,µ gr ) 2 > 0 under (H ν,µ ) 2, and is nondegenerate. Since eigenvalues do not cross, we conclude that for all 0 µ < 1 the highest eigenvalue of (H ν,µ ) 2 corresponds to the ground state. We thus take the square root of this eigenvalue with a negative sign: η gr := λ ax. On the other hand, the state of highest energy that the waveguide can carry, η ν,µ top, is positive; and this corresponds with the next-highest square eigenvalue, so we should opt for the positive sign: η ν,µ top := + λ ν,µ Again, since eigenvalues do not cross, λ ν,µ ax. Continuing in this way with the 2j+1 ax 1 < λν,µ ax 1. eigenvalues {λ ν,µ n } of (H ν,µ ) 2 fro top to botto, we conclude that ηgr+n ν,µ = λ ν,µ ax 2n and η ν,µ top n = + λ ν,µ ax 2n 1, for n {0, 1,..., j}. In Figure 2 (right) we show the η s with this alternating sign selection. To copute the eigenvectors Ψ η Ψ ν,µ η of (H ν,µ ) 2 in (21) we ust solve a difference equation [10], which is of step-two: 1 4 (j 1)(j )(j++1)(j++2) Ψ η (+2) + [j(j+1)(ν ) 2 (µ ) η2 ] Ψ η () (23) (j+ 1)(j+)(j +1)(j +2) Ψ η ( 2) = 0, where we notice that both Ψ η () and Ψ η () are solutions to the sae equation, as well as any linear cobination of the two. Replacing in (23) returns the sae equation for Ψ ±η ( ), so the solutions will have definite parity. Equation (23) stands in effect for a pair of 5

7 GROUP 28: Physical and Matheatical Aspects of Syetry difference equations. Assue first that j is odd; then, the values of one solution, Ψ e η(), can be deterined at all the even points x by setting Ψ e η(±[j+1]) = 0, using the recurrence (23) to deterine its values Ψ e η() for all other even j 1, while its values at all odd s are zero. A second solution, Ψ e η(), nonzero at all odd points stes fro setting Ψ o η(±[j+2]) = 0 and then using the sae (23) to deterine its values at all odd points j, and zero at the even s. (When j is even, a corresponding separation into two independent solutions is ade.) Since both Ψ e η () and Ψo η () have zeros at alternating points, they appear as porcupine signals, such as shown in Figure 3 (top row). A question thus arises on the relation between these two orthogonal solutions to (23) and the two solutions Ψ ±η () entioned earlier. What we expect is to reproduce ground states with a Gaussian shape in particular the ground Kravchuk function in a Kravchuk guide [2]; and for µ = 0 a state resebling a free wave. In all cases, definite parity is guaranteed by (23). For general ν, µ it does not see possible to find a known discrete special function that will solve (23) exactly, so we ust resort to nuerical coputation to plot results. In Figure 3 (rows 2 4 ) the noralized su and difference of the first three lowest energy (η < 0) states (24), and the three highest (η > 0) states (25), in the range 0 µ ν = 1. The resulting states Ψ η () = 1 2 [Ψ e η() + Ψ o η()], (24) Ψ η () = 1 2 [Ψ e η() Ψ o η()], (25) have recognizable shapes and definite energy eigenvalues. The µ 1 liit to the Kravchuk spectru { j, j+1,..., j} thus defines the proper square root of the diagonal atrix Λ ν,µ in (22), H ν,µ := diag (η gr, η gr+1,..., η top 1, η top ). (26) The Hailtonian atrix in (19) is finally obtained through de-diagonalizing (26) by eans of the transpose of the eigenvector atrix, H ν,µ = Ψ H ν,µ Ψ. (27) 5. Concluding rearks The z-evolution along the waveguide is generated by the proper Hailtonian atrix (27), exponentiated a 1-paraeter group of unitary N N atrices U ν,µ (z) := exp(iz H ν,µ ) U(N), (28) for z R. This is a line within the N 2 -diensional copact anifold of U(N) that will in general not close. In Ref. [10] we coputed the z-evolution of the finite Kravchuk coherent states in a Kravchuk guide, and plotted the result using the so(3) Wigner function [11] to ascertain that one reproduces the usual forward z > 0 circulation of phase space. The oveent of backward rays is obtained siply by letting z < 0, which is equivalent to exchanging the signs of all energies in (26). Also, we have let the finite coherent states evolve in a non-kravchuk guide with ν = 2 and µ = to observe the nonlinear aberration of the Wigner function that results fro classical phase space points oving on concentric ellipses p x2 = constant < 2 at different velocities. Presently we are investigating the behavior of the coherent states in waveguides whose refractive index is different fro the elliptic one in (7), such as a rectangular index profile with zeros at the endpoints, and an interediate faily of sooth profiles. 6

8 5 GROUP 28: Physical and Matheatical Aspects of Syetry Figure 3. Solutions Ψ η () to the step-two difference equation (23), for N = 21 points ( j = 10); lines connect their values at the integer s. Top row: The ground porcupine states, Ψ e gr () and Ψo gr (), in a Kravchuk guide ν = 1 = µ, corresponding to the largest eigenvalue η 2 in Fig. 2, and yielding both the ground and top energy states. Second row: The su and difference (24) (25) of the previous porcupines produces the ground, and top states of the Kravchuk guide indicated by a heavy line; with light lines we show the ground and top states of guides with µ = 0.95, 0.9,..., 0.8, 0.7,..., 0.0. Third row: The sae as the previous row, but for the next-to-largest eigenvalue η 2 of Fig. 2, corresponding to the next-to-lowest ground and next-to-top states. Fourth row: Siilarly, for the following lowest and highest states. In this paper our interest was to exaine optical Hailtonian operators that are defined by the square root of a positive, self-adjoint and otherwise well known operator, which is a proble seldo addressed in echanics. The resulting step-two difference equation separates solutions into two porcupine coponents that see to be inseparable fro the sign abiguity of the square root operation. Since several group-theoretic developents exist which (rightly or wrongly) include the square root of a positive operator for purposes of noralization, usually, this operation ay shown to have soe hidden difficulties. As suggested by the aberration expansion (9) for the geoetric optical Hailtonian, we also considered the perturbative description of waveguide evolution as a su of powers of haronic oscillator Hailtonians. The ain difficulty in using this expansion for finite waveguide odels is that the atrix series does not converge: in the Kravchuk so(3) case the axiu eleent 7

9 GROUP 28: Physical and Matheatical Aspects of Syetry of H osc := P 2 + X 2 = C K 2 is j(j+1) j 2 = j, and hence a bound for the axiu of (H osc ) n is j n (2j+1) n 1 ; eanwhile, the coefficient of this ter in the (1 x) series is (2n 3)!!/(2n)!!. Yet, the first two powers can be used to odel paraxial and nonlinear Kerr edia [12, 13, 14], as done in Ref. [15]. The purpose of our exploration of discrete and finite systes based on representations of copact groups such as so(3) and so(4), and also of discrete infinite systes [4], is to understand finite signal analysis and pixellated iage processing in ters of the syetries behind their geoetric optical realizations. Acknowledgents We acknowledge the support of the Óptica Mateática projects, DGAPA-UNAM IN and SEP-CONACYT 79899, and we thank Guillero Krötzsch (ICF-UNAM) for his careful help with the graphics. References [1] K.B. Wolf, Geoetric Optics on Phase Space (Springer-Verlag, Heidelberg, 2004). [2] N. M. Atakishiyev, G. S. Pogosyan and K. B. Wolf, Finite odels of the oscillator, Phys. Part. Nuclei 36, (2005). [3] C.A. Muñoz, J. Rueda-Paz and K.B. Wolf, Discrete repulsive oscillator wavefunctions, J. Phys. A 42, art (12 p.) (2009). [4] K.B. Wolf, Discrete systes and signals on phase space, Appl. Math. & Inforation Science 4, (2010). [5] R. Gilore, Lie Groups, Lie Algebras, and Soe of their Applications (Wiley-Interscience, New York, 1978). [6] L.C. Biedenharn and J.D. Louck, Angular Moentu in Quantu Physics, Encyclopedia of Matheatics and Its Applications, Vol. 8 (Addison-Wesley Publ. Co., Reading, Mass., 1981). [7] M. Krawtchouk, Sur une généralization des polinôes d Herite, C.R. Acad. Sci. Paris 189, (1929). [8] N. M. Atakishiyev and K. B. Wolf, Fractional Fourier-Kravchuk transfor, J. Opt. Soc. A. A 14, (1997). [9] N.M. Atakishiyev, G.S. Pogosyan and K.B. Wolf, Contraction of the finite one-diensional oscillator, Int. J. Mod. Phys. A 18, (2003). [10] J. Rueda-Paz and K.B. Wolf, Finite signals in planar waveguides J. Opt. Soc. A. A subitted, (). [11] N. M. Atakishiyev, S. M. Chuakov and K. B. Wolf, Wigner distribution function for finite systes, J. Math. Phys. 39, (1998). [12] R. Tanaś, In: Coherence and Quantu Optics V, L. Mandel and E. Wolf Eds. (New York: Plenu, 1984), p [13] G.J. Milburn, Quantu and classical Liouville dynaics of the anharonic oscillator, Phys. Rev. A 33, (1986). [14] G.J. Milburn and C.A. Holes, Quantu coherence and classical chaos in a pulsed paraetric oscillator with Kerr nonlinearity, Phys. Rev. A 44, (1991). [15] S.M. Chuakov, A. Frank and K.B. Wolf, Finite Kerr ediu: Macroscopic quantu superposition states and Wigner functions on the sphere, Phys. Rev. A 60, (1999). 8