Group approach to the paraxial propagation of Hermite–Gaussian modes in a parabolic medium

Annals of Physics 383 (2017) 257–277 Contents lists available at ScienceDirect Annals of Physics journal homepage: www.elsevier.com/locate/aop Group approach to the paraxial propagation of Hermite–Gaussian modes in a parabolic medium S. Cruz y Cruz *, Z. Gress Instituto Politécnico Nacional, UPIITA, Av. Instituto Politécnico Nacional 2580, Col. La Laguna Ticomán, C.P. 07360, Ciudad de México, Mexico h i g h l i g h t s •Ladder and shift operators for the Hermite–Gaussian modes are constructed. •These operators are used to obtain the generators of the dynamical algebras. •The Barut–Girardello and Perelomov coherent states are determined. •The uncertainty relations are considered in connection to the beam quality factor. article info Article history: Received 3 March 2017 Accepted 30 May 2017 Available online 13 June 2017 Keywords: Hermite–Gaussian modes Dynamical algebras Coherent states abstract A group-theoretical approach to the paraxial propagation of Hermite–Gaussian modes based on the factorization method is presented. It is shown that the su(1,1) and the su(2) algebras generate the spectrum of propagation constants at any fixed transversal plane. The complete set of HG modes is decomposed into hierarchies that are used to establish the representation spaces of SU(1,1) and SU(2). The corresponding families of generalized coherent states are constructed and the variances of the quadratures and canonical variables are determined. ©2017 Elsevier Inc. All rights reserved. 1. Introduction The set of Hermite–Gaussian (HG) modes has been one of the most studied families of paraxial beams. They play an important role in the study of laser resonators and optical waveguides [1] and have a number of applications in particle trapping [2], communications and signal processing [3], microand nano-manipulation of matter [4,5] and high resolution imaging [6] among other areas *Corresponding author. E-mail address: [email protected] (S. Cruz y Cruz). http://dx.doi.org/10.1016/j.aop.2017.05.020 0003-4916/©2017 Elsevier Inc. All rights reserved. 258 S. Cruz y Cruz, Z. Gress / Annals of Physics 383 (2017) 257–277 (see also [7] and references quoted therein). Yet, the study of HG modes is far from being completely exhausted. New developments about their properties, generation and applications are currently addressed [8–11]. In the paraxial approximation, where the directions of the normals to the wavefronts are close to the optical axis, the exact wave equation can be reduced to a parabolic-type one, the paraxial wave equation, by neglecting the second derivative of the field amplitude with respect to the longitudinal coordinate. The HG modes form a complete, orthogonal set of transverse modes with rectangular symmetry that span the solution space of this equation in a homogeneous medium [1]. The corresponding solution in generic weakly inhomogeneous media has been considered in [12] by using curvilinear coordinates. In that work the authors are led to the paraxial wave equation for either a homogeneous or a parabolic medium in the particular case of optical systems admitting separation of the transversal variables. Specifically, the transverse quadratic refractive index characteristic of the parabolic medium has deserved a lot of interest because of its self-focusing and exact collimation properties in the paraxial regime. The propagation and diffraction of Gaussian beams in parabolic media have been considered as well [13–15]. Indeed, it can be shown that the HG modes form an orthogonal set of exact solutions to the paraxial wave equation in optical media with quadratic index profiles. On the other hand, it is well known that the propagation of paraxial light can be mapped to the evolution of a two-dimensional harmonic oscillator. This is due to the formal equivalence between the parabolic wave equation, describing wave propagation in the paraxial regime, and the Schrödinger equation, governing the time-evolution of a quantum particle in an interaction potential [16–18]. In this context, some operator methods, originally addressed to the study of quantum mechanical systems, have been adapted for the description of light propagation in free space as well as in the presence of optical systems. Such techniques include, e.g., group-theoretical approaches [19–22], the density matrix formalism [23] and the coherent state method [23–26]. In particular, the factorization method is an algebraic technique used to solve the eigenvalue problem and to describe the spectral properties of quantum mechanical systems associated with exactly solvable potentials [27–30]. In this method the second order differential Hamiltonian is factorized, up to an additive constant, as the product of two first order differential operators. These factors can be used to construct the basis elements of the dynamical algebra of the system leading to the group approach of the factorization method (see, [31,32]). In this work we take advantage of the formal equivalence between the Schrödinger and the paraxial wave equations in order to establish a group-theoretical approach, based on the factorization method, to the paraxial propagation of HG beams in a parabolic medium. We find that the Lie algebras su(1,1) and su(2) are the generating algebras of the system at each fixed transversal plane. In this way, the complete set of HG modes can be decomposed into a denumerable set of hierarchies, labeled by a discrete parameter, in two different forms. This means that the whole set of transversal modes is covered either by a set of infinite-dimensional representations of SU(1,1) or by a set of finite-dimensional representations of SU(2). Considering these representations we construct different families of generalized coherent states and determine the corresponding variances for the su(1,1) and su(2) quadratures as well as for the canonical variables of position and propagation direction, these last in connection to the propagation factors of each family of coherent states. The material in this article is organized as follows: In Section 2we summarize the construction of the fundamental Gaussian-type solution to the paraxial wave equation in a parabolic medium. Next, higher order HG modes are determined by means of the factorization method. In Section 3we define ladder operators for the HG modes and show that they are propagation invariants. In Section 4 we present the group-approach in terms of the generating algebras and construct the corresponding families of coherent states. In Section 5we summarize our results and present some conclusions. 2. The complete set of Hermite–Gaussian modes 2.1. The fundamental Gaussian mode Consider a weakly inhomogeneous medium with a quadratic refractive index profile n2(r)=n2 0(1−Ω2 2r2),Ω2r2≪1,(1) S. Cruz y Cruz, Z. Gress / Annals of Physics 383 (2017) 257–277 259 where ris the radial transversal coordinate, Ωis a constant parameter characterizing the focusing properties of the medium, and n0is the refractive index at the optical axis. The homogeneous case is recovered by properly taking the limit Ω→0. The corresponding paraxial wave equation is given by i k0 ∂U ∂z={−1 2k2 0n0(∂2 ∂x2+∂2 ∂y2)+n0 Ω2 2r2}U=HU,(2) with U=U(r,z) a representative component of the electromagnetic field, r=(x,y) the transversal position vector and k0the wave number in free space. As the medium is homogeneous in the zdirection, we can find solutions to Eq. (2) as (stationary) eigenstates of H(the guided HG modes), where the corresponding eigenvalues constitute the spectrum of propagation constants. However, we are interested in non stationary Gaussian wavepacket-type solutions of the form [12,13] U(r,z)=N(z)eiS(z)r2,(3) where the normalization factor Nand the coefficient Sin the exponent are complex-valued functions of the longitudinal coordinate zfulfilling the set of equations d dz (2 k0n0 S)+(2 k0n0 S)2 +Ω2=0,(4) d dz ln N(z)= − 2 k0n0 S.(5) Eq. (4) is a complex Riccati equation that can be reduced to the real Ermakov one d2w dz2+Ω2w=4 k2 0n2 0w3(6) by writing 2 k0n0 S(z)=1 R(z)+2i k0n0w2(z),1 R(z)=d dz ln w(z).(7) In this way, the function w(z) comprises all the information of the evolution of the wavepacket and is determinant in the description of the propagation processes of the light beam. Actually, the functions w(z) and R(z) turn out to be, respectively, the width of the beam and the radius of curvature of the wavefront. By substituting (7) into (5) we get the normalization factor N(z)=N0 w(z)e−iχ(z),d dz χ(z)=2 k0n0w2(z),(8) where N0is a constant to be fixed and the function χ, known as the Guoy phase, is a cumulative on-axis phase shift experienced by the wavefront in passing through a focal plane [1] which can be interpreted as a geometric phase [22,33]. It is well known that the solution of (6) can be constructed in terms of the solutions of the corresponding linear equation [34,35] d2w dz2+Ω2w=0.(9) Hence, by setting w(0) =w0we have w(z)=w0[cos2(Ωz)+1 (ΩzR)2sin2(Ωz)]1/2 ,zR=k0n0w2 0 2.(10) The parameter zRis known as the Rayleigh range or collimation distance. The expression for the Gaussian mode is then U(r,z)=N0 w(z)e−iχ(z)e ik0n0r2 2R(z)e−r2 w2(z),(11) 260 S. Cruz y Cruz, Z. Gress / Annals of Physics 383 (2017) 257–277 with R(z)=tan(Ωz) Ω(1−Ω2z2 R)[1+(ΩzR tan(Ωz))2], χ(z)=2 k0n0 ∫zdt w2(t).(12) Observe the oscillating nature of these functions due to the focalization properties of the medium. The beam experiences, at the same time, the wave propagation (spreading) effects and the medium confinement, resulting in a periodically self-reproducing wavepacket of period π Ωfor Ω>0. The Guoy phase shift between two maxima of the beam width is given by χ(z)=arctan (tan(Ωz)/ΩzR). In the limit Ω→0 we obtain the well known width, radius of curvature and Guoy phase shift for a Gaussian beam supported by a homogeneous medium [1] w(z)=w0√1+(z zR)2 ,R(z)=z[1+(zR z)2],(13) and χ(z)=arctan (z/zR). In this case w0is the minimum width and is known as the waist radius of the beam. This means that z=0 is the focal plane of the system. The beam doubles its cross section at a distance z=zRfrom the focus (w(zR)=√2w0). Within this distance we say that the beam is well collimated since the width remains nearly constant. As the beam propagates out of this region the width increases as a linear function of z. On the other hand, in the particular case Ω=1 zR, the beam propagates with constant width w(z)=w0all along the optical axis, the radius of curvature tends to infinity and the Guoy phase becomes proportional to z. In any case, the initial width w0and the Rayleigh range zRcan be considered as the transversal and longitudinal characteristic distances of the beam respectively. 2.2. Higher order Hermite–Gaussian modes The existence of more general localized wavepacket-type solutions of the paraxial wave equation with parabolic refractive index is well known [12]. Indeed, one can construct solutions of the form U(r,z)=1 w(z)e ik0n0r2 2R(z)e−iβχ(z)F(u, v),u=√2x w(z), v =√2y w(z),(14) where the function F(u, v) fulfills the stationary Schrödinger-type equation for a two dimensional oscillator −(∂2 ∂u2+∂2 ∂v2)F+ρ2F=2βF, ρ2=u2+v2.(15) The square integrability condition in the (u, v) plane must be imposed if localized beams, carrying finite transverse optical power, are to be constructed. In rectangular coordinates, a complete set of solutions of (15) can be determined from the fundamental solution F00(u, v), by the subsequent application of ladder operators (see, e.g., [22,36]). Thus Fnm(u, v)=1 √n!m!(a+)n(b+)mF00(u, v),F00(u, v)=1 √πe−ρ2 2,(16) where a±=a±(z)=1 √2(∓∂ ∂u+u),b±=b±(z)=1 √2(∓∂ ∂v +v)(17) are operators that fulfill the single mode boson algebra [a−,a+]=[b−,b+]=I,[a±,b±]=0.(18) The application of these ladder operators on the functions Fnm yields a+Fnm =√n+1Fn+1,m,a−Fnm =√nFn−1,m,(19) b+Fnm =√m+1Fn,m+1,b−Fnm =√mFn,m−1.(20) S. Cruz y Cruz, Z. Gress / Annals of Physics 383 (2017) 257–277 261 (a) Ω=0. (b) Ω=0.5/zR. (c) Ω=0.9/zR. Fig. 1. Longitudinal plane y=0 of the optical intensity distribution for the HG mode U12(r,z) as it propagates along the z direction, with Ω=0 (a), ΩzR=0.5 (b) and ΩzR=0.9 (c). The transversal and longitudinal coordinates are measured in units of w0and zRrespectively. The solution of the eigenvalue problem (15) takes the form Fnm(u, v)=1 √π2n+mn!m!e−1 2ρ2Hn(u)Hm(v), βnm =n+m+1,(21) leading to the well known field amplitude of the HG mode Unm(r,z)=1 √π2n+mn!m! 1 w(z)e−i(n+m+1)χ(z)e ik0n0r2 2R(z)e−r2 w2(z)Hn(u)Hm(v),(22) which is normalized in such a way that the total transverse optical power satisfies P0=∫R2|Unm(r,z)|2 da =1, with da =dxdy the differential element of area in the transversal plane. Notice that the lowest order HG mode U00(r,z) coincides with the fundamental Gaussian mode (11). In Fig. 1 is shown the longitudinal plane y=0 of the optical intensity distribution for the HG mode U12(r,z) as it propagates in the zdirection. For Ω=0 (a), the wavepacket spreads as zincreases due to the propagation effects of the beam. On the other hand, for Ω=0.5 zR(b), the beam focuses itself with a period 2πzR. The beam becomes well collimated as Ω→1 zR(c). 3. Ladder operators of Hermite–Gaussian modes It turns out that the HG modes can be generated from the fundamental one U00(r,z) by the subsequent application of ladder operators [25]. Indeed, using (16) the HG mode (22) can be rewritten in the form Unm(r,z)=1 √n!m!e−i(n+m)χ(z)e ik0n0r2 2R(z)(a+)n(b+)me−ik0n0r2 2R(z)U00(r,z).(23) This suggests the introduction of the operators A+=e−iχe ik0n0x2 2R(z)a+e−ik0n0x2 2R(z),A−=(A+)†(24) B+=e−iχe ik0n0y2 2R(z)b+e−ik0n0y2 2R(z),B−=(B+)†(25) which are such that Unm(r,z)=1 √n!m!(A+)n(B+)mU00(r,z).(26) 262 S. Cruz y Cruz, Z. Gress / Annals of Physics 383 (2017) 257–277 Fig. 2. Action of the boson operators A±,B±on the Hermite-Gaussian modes. The algebraic structure of A±and B±is inherited from the boson operators a±,b±. We have [A−,A+]=I,[B−,B+]=I,[A±,B±]=0,(27) [Nx,A±]= ±A±,[Ny,B±]= ±B±,(28) with Nx=A+A−and Ny=B+B−. Additionally, according to (19)–(20), the action of the operators A±, B±,Nx,Nyon the HG modes (see Fig. 2) produces A+Unm(r,z)=√n+1Un+1,m(r,z),A−Unm(r,z)=√nUn−1,m(r,z),(29) B+Unm(r,z)=√m+1Un,m+1(r,z),B−Unm(r,z)=√mUn,m−1(r,z),(30) NxUnm(r,z)=nUnm(r,z),NyUnm(r,z)=mUnm(r,z).(31) It is not difficult to verify that A±and B±are invariant operators under propagation along the optical axis. In the formal operator description of the paraxial wave optics, the canonical variables of transverse position rand propagation direction pare considered as Hermitian operators r=(x,y),p=(px,py)= − i k0(∂ ∂x,∂ ∂y),(32) acting on the Hilbert space L2(R2) of square integrable transversal modes of the electromagnetic field [16,18]. It is convenient to introduce the vectors |U(z)⟩to denote the normalized modes of the electromagnetic field. The field amplitude as a function of the position can be expressed as U(r,z)= ⟨r|U(z)⟩. In this way, the paraxial wave equation i k0 ∂ ∂z|U(z)⟩ = H|U(z)⟩,H=p2 2n0+n0 2Ω2r2,(33) reveals that the dynamics of the beam is encoded in the family of unitary operators W(z,z0) describing the propagation process from an initial point z0at the optical axis, to an arbitrary point zin the form |U(z)⟩ = W(z,z0)|U(z0)⟩.(34) As the Hamiltonian does not depend explicitly on the longitudinal coordinate, we have W(z,z0)= e−ik0H(z−z0). This means that His the generator of the propagation processes of the light beam along S. Cruz y Cruz, Z. Gress / Annals of Physics 383 (2017) 257–277 263 the optical axis. Thus, by choosing z0=0, the HG mode |Unm(z)⟩at an arbitrary point zin the optical axis can be constructed as the unitary transformation |Unm(z)⟩ = W(z,0)|Unm(0)⟩.(35) Moreover, the boson operators A±,B±are the evolved versions of the harmonic oscillator ladder operators a±(0) and b±(0) since A±=e−ik0Hza±(0)eik0Hz,B±=e−ik0Hzb±(0)eik0Hz.(36) The above expressions are useful to show that A±,B±and H=Nx+Ny+Iare constants of motion as they satisfy the condition dO dz = −ik0[O,H]+∂O ∂z=0.(37) In terms of the canonical variables (r,p) these operators are given by A+= −ie−iχw[k0px 2−¯ Sx],A−=ieiχw[k0px 2−Sx],(38) B+= −ie−iχw[k0py 2−¯ Sy],B−=ieiχw[k0py 2−Sy],(39) H=k0zR w2 w2 0[p2 2n0−1 k0n0(Sp·r+¯ Sr·p)+2 k2 0n0|S|2r2]+I,(40) where the bar stands for complex conjugation. In the case Ω=0, the expressions (38) coincide with the creation and annihilation operators reported by Nienhuis and Allen for the HG modes in free space [25]. Those operators have also appeared as the conserved creation and annihilation operators of the harmonic states for the free particle [37], and as generalized (invariant) ladder operators of the parametric harmonic oscillator [38]. 4. Mode field hierarchies and generalized coherent states In the (n,m)-plane of the parameters that define the whole set of HG modes (depicted in Fig. 2), the operators A±and B±induce the first order mappings A±:(n,m)→(n±1,m),B±:(n,m)→(n,m±1). We say that A±(B±) intertwine the HG modes of m(n) fixed. From Fig. 2 one can note that second order compositions of ladder operators allow to intertwine some other sets of modes. In fact, in the context of the multiphoton representation of SU(n,m) and SU(n) [39–42], the generators of the su(1,1) and su(2) algebras can be realized as compositions of two single mode boson operators. In the non-degenerate case, these realizations correspond to the Schwinger representations of those algebras [43] (see also [44]). Accordingly, we can construct irreducible representations of su(1,1) and su(2) in terms of the ladder operators A±,B±. This fact enables the classification of the complete set of HG modes into hierarchies of two different forms. Namely, those associated to the sets of modes with fixed parameter |m−n|, and those corresponding to the sets of definite total mode index n+m. Next, we are interested in constructing ladder operators connecting the elements of a given hierarchy. Such operators turn out to be the generators of the corresponding Schwinger representations. 4.1. Mode fields with fixed parameter |m−n| Consider the second order compositions K+=B+A+,K−=B−A−,K0=1 2(Nx+Ny+I)=1 2H.(41) 264 S. Cruz y Cruz, Z. Gress / Annals of Physics 383 (2017) 257–277 Fig. 3. Action of the operators K±on the Hermite-Gaussian modes. They shift both labels nand min one unit leaving invariant the quantity |m−n|and thus connecting members of the same k-hierarchy. These operators satisfy the su(1,1) commutation relations [K−,K+]=2K0,[K0,K±]= ±K±.(42) For the sake of simplicity let us denote the HG mode by the ket |n,m⟩. The operator K+(K−) raises (low) both labels nand min one unit: K±:(n,m)→(n±1,m±1) (see Fig. 3). The HG modes are thus transformed in such a way that the quantity |m−n|is left invariant. Let us define the parameter k=1 2(|m−n|+1). For each fixed k, the sets of modes for which m−n<0 and m−n>0 correspond to two different k-hierarchies leading to two realizations of SU(1,1). We say that K±intertwine modes of the same k-hierarchy (Fig. 3). Additionally, the Cassimir operator C2=1 4(N2 x+N2 y−2NxNy−I)is proportional to the identity: C2=k(k−1)I, with the Bargman index k=1 2,1,3 2,2, . . .. Therefore, the complete mode space decomposes into the direct sum of the set of subspaces {Hk +,k=1 2,1,3 2,...,m>n}⋃{Hk −,k=1 2,1,3 2,...,m<n}, each one spanned by a particular k-hierarchy of HG modes and corresponding to a particular representation of SU(1,1). If we introduce the parameter µ=1 2(m+n+1),µ=k+n,n=0,1,2, . . . for m>n, and µ=k+m,m=0,1,2, . . . for n>m. Assuming that m>n(the case m<nis easily constructed by interchanging the roles of nand m), k=1 2(m−n+1)and n=µ−k,m=µ+k−1. It is now convenient to define the ket |k, µ⟩Ksuch that |n,m⟩=|µ−k, µ +k−1⟩=|k, µ⟩K. In this way the action of K±,K0on the HG modes can be stated as K+|k, µ⟩K=√(µ+k)(µ−k+1)|k, µ +1⟩K,(43) K−|k, µ⟩K=√(µ−k)(µ+k−1)|k, µ −1⟩K,K−|k,k⟩K=0,(44) K0|k, µ⟩K=µ|k, µ⟩K.(45) 4.2. Barut–Girardello coherent states We can construct the generalized Barut–Girardello coherent states |ξ⟩k BG, for each k-hierarchy, as eigenmodes of the SU(1,1) annihilation operator K− K−|ξ⟩k BG =ξ|ξ⟩k BG,(46) with ξ= |ξ|e−iϕ. By using (44) we get |ξ⟩k BG =eikϕ √|ξ|I2k−1(2|ξ|) ∞ ∑ µ=k ξµ √Γ(µ+k)Γ(µ−k+1)|k, µ⟩K,(47) S. Cruz y Cruz, Z. Gress / Annals of Physics 383 (2017) 257–277 265 (a) ϕ=0. (b) ϕ=π 4. (c) ϕ=π 2. Fig. 4. Intensity distributions of the Barut–Girardello coherent states, in the focal plane z=0, for k=1 2,|ξ| = 5 and different values of the phase ϕ. The variables x,yare measured in units of w0. (a) ϕ=0, k=1 2. (b) ϕ=0, k=5 2. (c) ϕ=0, k=9 2. (d) ϕ=π 2,k=2. (e) ϕ=π 2,k=5. (f) ϕ=π 2,k=8. Fig. 5. Intensity distributions of the Barut–Girardello coherent states Uk BG(r,z), as functions of |ξ|at y=0, for different values of kand ϕ. The transversal variable xis measured in units of w0. where Iν(z) is the modified Bessel function of the first kind [45]. The corresponding position representation Uk BG(r,z)= ⟨r|ξ⟩k BG is given by Uk BG(r,z)=eikϕ √|ξ|I2k−1(2|ξ|) ∞ ∑ µ=k ξµ √Γ(µ+k)Γ(µ−k+1)Uµ−k,µ+k−1(r,z).(48) In Fig. 4 it is shown the transversal plane z=0 of the optical intensity pattern of the coherent states (48) for k=1 2,|ξ| = 5 and different values of ϕ. Around the points ϕ=sπ,s=0,1,2, . . ., these are narrow distributions presenting a series of local maxima according to the value of |ξ|. As ϕapproaches integer multiples of π 2, the wavepackets turn into rectangular patterns centered at the optical axis. Fig. 5, on the other hand, shows the intensity at the longitudinal plane y=0, as a function of |ξ|, for different values of kand ϕ. Note that for ϕ=0 the pattern becomes narrower and exhibits more than 272 S. Cruz y Cruz, Z. Gress / Annals of Physics 383 (2017) 257–277 Fig. 13. Action of the operators J±on the Hermite-Gaussian modes. correspond to different irreducible representations of SU(2). Also n=j+ν,m=j−ν=2j−n≥0, n=0,1,...,2j, and hence ν=n−j= −j,−j+1,...,j−1,j. Now, by fixing the vector |j, ν⟩Jin such a way that |n,m⟩ = |j+ν, j−ν⟩ = |j, ν⟩J, the action of the operators J±,J0reads J+|j, ν⟩J=√(j−ν)(j+ν+1)|j, ν +1⟩J,J+|j,j⟩J=0,(81) J−|j, ν⟩J=√(j+ν)(j−ν+1)|j, ν −1⟩J,J−|j,−j⟩J=0,(82) J0|j, ν⟩J=ν|j, ν⟩J.(83) 4.5. SU(2) Perelomov coherent states The Perelomov approach for constructing generalized coherent states may now be applied to these hierarchies. By using the mode |j,−j⟩Jas the extremal state one has |ξ⟩j P=D(ξ)|j,−j⟩J,with D(ξ)=eξJ+−¯ ξJ−.(84) Iterating the action of J+on the extremal state we get (J+)s|j,−j⟩J=√s!(2j)! (2j−s)!|j,−j+s⟩J,for s≤2j.(85) Then, by using the disentangling formula D(ξ)=eζJ+eln(1+|ζ|2)J0e−¯ ζJ−, ζ =ξ |ξ|tan|ξ|,(86) we arrive at |ξ⟩j P=ζj√(2j)! (1+|ζ|2)j j ∑ ν=−j ζν √(j+ν)!(j−ν)!|j, ν⟩J.(87) In the position representation the corresponding mode amplitude reads Uj P(r,z)= ⟨r|ξ⟩j P=ζj√(2j)! (1+|ζ|2)j j ∑ ν=−j ζν √(j+ν)!(j−ν)!Uj+ν,j−ν(r).(88) S. Cruz y Cruz, Z. Gress / Annals of Physics 383 (2017) 257–277 273 (a) |ζ| = 0. (b) |ζ| = 1. (c) |ζ| = 100. Fig. 14. Intensity distributions of the Perelomov coherent states at the transversal plane z=0, for j=1, ϕ=π 2and the indicated values of |ζ|. In the case |ζ| = 1 the pattern exhibits axial symmetry due to the balanced contribution of the HG modes to the coherent state. The variables x,yare measured in units of w0. In Fig. 14 we present the optical intensity carried by these modes at the transversal plane z=0, for j=1, ϕ=π 2and different values of |ζ|. The pattern consists of a set of 2j+1 maxima rotating around the origin as |ζ|increases. For the special value ζ=i(|ζ| = 1, ϕ=π 2) the beam exhibits axial symmetry due to the balanced contribution of the HG modes in the superposition (87).Fig. 15, on the other hand, shows the behavior of the optical intensity pattern as a function of |ζ|for y=0, ϕ=π 2, and different values of j. For this value of yand integer values of jthe pattern presents only one peak in the region |ζ|<1, consistently with Fig. 14(a). For |ζ| = 1, in all cases, the distribution shows two peaks (compare to Fig. 14(b)) that increase their distance for larger values of j. In the regime |ζ|>1 the pattern exhibits the whole 2j+1 maxima due to its rotation around the optical axis. Finally, the evolution of these coherent states along the optical axis is presented in Fig. 16. In this figure is depicted the longitudinal plane y=0 for the choice Ω=0.5 zR,j=3, ϕ=π 2, and |ζ| = 0,1,100. Note the correspondence between these plots and that of Fig. 15(c) for different values of |ζ|. Variances and squeezing The commutation relations (80) imply the inequality ∆J1∆J2≥1 2|⟨J0⟩|,(89) with the quadratures J1,2defined by J1=1 2(J++J−),J2=1 2i(J+−J−).(90) For the SU(2) coherent modes we get (∆J1)2= −1 2[1−(2Re(ζ) 1+|ζ|2)2]⟨J0⟩0,(91) (∆J2)2= −1 2[1−(2Im(ζ) 1+|ζ|2)2]⟨J0⟩0,(92) ⟨J0⟩j P=1−|ζ|2 1+|ζ|2⟨J0⟩0,(93) 274 S. Cruz y Cruz, Z. Gress / Annals of Physics 383 (2017) 257–277 (a) j=1. (b) j=3 2. (c) j=3. Fig. 15. Optical intensity of the SU(2) Perelomov coherent states as functions of |ζ|, for y=0, ϕ=π 2and j=1,3 2,3. The variable xis measured in units of w0. (a) |ζ| = 0. (b) |ζ| = 1. (c) |ζ| = 100. Fig. 16. Longitudinal plane y=0 of the optical intensity ⏐⏐⏐Uj P(r,z)⏐⏐⏐ 2 with Ω=0.5/zR,j=3 and ϕ=π 2, for the values of |ζ| indicated in each case. These patterns are consistent with those shown in Fig. 15(c). The coordinates xand zare measured in units of w0and zRrespectively. with ⟨J0⟩0=J⟨j,−j|J0|j,−j⟩J= −j. As ∆J1∆J2=1 2⎡ ⎣(1−|ζ|2 1+|ζ|2)2 −(4Re(ζ)Im(ζ) (1+|ζ|2)2)2⎤ ⎦ 1 2 ⟨J0⟩0,(94) the equality sign in (89) is valid whenever Re(ζ)=0 or Im(ζ)=0, i.e., either for ζ=0 or for ϕ=sπ 2, s=0,1,2, . . ., and any value of |ζ|. The squeezing of the quadratures J1or J2arises for the values of ζthat make the corresponding variance smaller than the average uncertainty 1 2⏐⏐⏐⟨J0⟩j P⏐⏐⏐. As it is shown in Fig. 17, this occurs around the points ϕ=sπand ϕ=(2s+1)π 2,s=0,1,2, . . ., respectively. There are, however, some zones of the complex plane, in the vicinity of the points ϕ=π 4,3π 4, . . ., for which the variances of both quadratures are larger than the corresponding average uncertainty. The propagation factor In order to determine the variances of the canonical variables r,pit is necessary to establish the action of the ladder operators A±and B±on |ξ⟩j P. After some calculations we get A−|ξ⟩j P=√2j 1+|ζ|2ζ|ξ⟩j−1 2 P,B−|ξ⟩j P=√2j 1+|ζ|2|ξ⟩j−1 2 P.(95) S. Cruz y Cruz, Z. Gress / Annals of Physics 383 (2017) 257–277 275 Fig. 17. (Color online) Regions of the complex plane where squeezing is expected, the gray zone corresponds to squeezing in J1and the blue one to J2. The white zones correspond to points where the variances of both quadratures are larger than the average uncertainty. The propagation factor (64) in this case turns out to be independent of ζ, as it is reduced to M2=2j+1=n+m+1=βnm.(96) This last result coincides with the quality factor M2 HG of the HG mode |n,m⟩. 4.6. Comparison with the Glauber coherent states In the Glauber approach, it is possible to construct coherent states of a finite number of modes as the product of single mode ones [52] (see also [44]). In the case of two independent modes the coherent state is given by |αx, αy⟩G=exp [−1 2(|αx|2+|αy|2)]∞ ∑ n,m=0 αn xαm y √n!m!|n,m⟩, αx, αy∈C.(97) In our case, it is straightforward to show that ∆x=∆y=w 2,∆px=∆py=w k0|S|,∆xpx=∆ypy=w2 2k0 Re(S),(98) so that these states also minimize the Schrödinger–Robertson uncertainty relation on each degree of freedom: (∆x)2(∆px)2−(∆xpx)2=(∆y)2(∆py)2−(∆ypy)2=1 4k2 0 .(99) This fact leads to M2=1 which indicates that the modes (97) are not affected by the propagation effects. Remark that, as in the Glauber case, the generalized coherent states here constructed in the Barut– Girardello approach, as eigenstates of the su(1,1) annihilator, also minimize the inequality involving the variances of the corresponding quadratures. This is no longer true for the Perelomov case, for which the inequality associated to the variances of the su(1,1) or su(2) generators is saturated only for some particular values of the parameters. It should be also noted that the Schrödinger–Robertson uncertainty relation involving the variances of the canonical variables, rand p, is not minimized for any of the generalized coherent states but for the fundamental Gaussian mode. This means that diffraction effects are always expected to some extent. Yet, the generalized coherent states here discussed encode information about the algebraic structure of the mode space. The decomposition of this space into hierarchies unveils the symmetries of the system which are usually connected with dynamical variables that are of physical interest and allow the possibility of constructing different families of localized modes with a number of interesting properties. 276 S. Cruz y Cruz, Z. Gress / Annals of Physics 383 (2017) 257–277 5. Conclusions We have constructed the complete set of HG modes in a parabolic medium through the factorization method, by mapping the parabolic wave equation at a fixed transversal plane to a Schrödingertype equation for a two dimensional harmonic oscillator. The oscillating behavior of these modes as they propagate along the optical axis is encoded in the parameter w(z) representing the beam width as a function of the longitudinal coordinate. We have also constructed z-dependent ladder operators A±, B±for the HG modes that generalize creation and annihilation operators for the harmonic oscillator, and new second order operators K±and J±that act as ladder operators in each infinite or finite dimensional hierarchy of the set of HG modes. Thus, we have shown that either the su(1,1) or the su(2) algebras generate the spectrum of propagation constants at any fixed transversal plane. A relevant point is that the index labeling the SU(2) hierarchies is connected to the propagation constant βnm. Hence, the corresponding subspaces are composed of modes with the same Gouy phase shift. We have applied the Barut–Girardello and Perelomov approaches to construct some families of coherent states, and determine the corresponding propagation factors. In the case of the coherent states associated to each k-hierarchy it was shown that the propagation factor M2behaves as 2kfor small values of the parameter |ξ|labeling the coherent states. Therefore, a nearly Gaussian behavior is expected for moderate values of k. In the case of the j-hierarchies the propagation factor M2turns out to be the same as the propagation factor M2 HG of each of its HG components. This group approach can be extended to paraxial mode space of different geometries. The generating algebras in each case lead to the classification of the mode basis into hierarchies associated to the symmetries of the system. 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