AN OPTICAL HAMILTONIAN EXPERIMENT AND THEBEAM DYNAMICS

A. Bazzani∗, P.Freguglia†, L.Fronzoni∗∗ and G. Turchetti‡

∗Dept. of Physics and CIG, University of Bologna, INFN sezione di Bologna, Italy†Department of Pure and Applied Mathematics, University of L’Aquila, Italy

∗∗Department of Physics and CISC, University of Pisa, Italy‡Department of Physics and CIG, University of Bologna, INFN sezione di Bologna, Italy

Abstract. The analogy between geometric optics and Hamiltonian mechanics is used to propose an experiment that simulatesthe beam propagation in a focusing magnetic lattice of a particle accelerator. A laser beam is reflected several times by aparabolic mirror and the resulting pattern is registered bya photo-camera. This experiment allows to illustrate some aspectsof nonlinear beam transport in presence of nonlinearities and stochastic perturbations. The experimental results arediscussedand compared with computer simulations.

INTRODUCTION

The geometric optics studies the light propagation inthe limit of a small wave lengths and it is based on theFermat’s variational principle[1]

δ∫ b

an(x) ds = 0 (1)

wheren(x) is the refraction index,s is the arc-length ofthe optical path anda,b two fixed points in the space. TheFermat’s principle is analogous to the Maupertuis’ vari-ational principle for the particle dynamics in a potentialV (x)[2]

δ∫ b

a

√

2(E −V (x)) ds = 0 (2)

where the constantE is the mechanical energy. In thecase of discontinuous potentials or discontinuous refrac-tion indexes, both the variational principles (1) and (2)imply the Snell’s diffraction law and the usual reflectionlaw at the border surfaces. According to the Hamilto-nian formulation of Classical Mechanics, we can writethe principle (2) in the form[3, 4]

δ∫ b

a~p ·d~x = 0 E(~p,~x) = E (3)

where~p are the conjugated momenta of the coordinates~x. In beam dynamics a charge relativistic particle is con-fined by using a magnetic field associated to the vectorpotential~A = Az(x,y,z)z where ˆz is the direction of thereference orbit[5, 6]. The vector potential~A has a dis-continuous dependence on the longitudinal coordinatez

due to the structure of the magnetic lattice, but it is an-alytic in the transverse coordinatesx,y. The normalizedenergyE/p0c is

Ep0c

=

√

(

pz −eAz

p0c

)2

+ p2x + p2

z +

(

mcp0

)2

(4)

wherep0 is the momentum of a particle along the refer-ence orbit ande the electric charge[6]. According to theMaupertuis’ principle (2), the transverse dynamics (beta-tronic motion) of a particle is described by the Hamilto-nian

H(px, py,x,y,z) = −pz = −

√

1− (p2x + p2

y)−eAz

p0c(5)

wherez is the orbits parameter. The vector potential~Acan be written in the form

Az

p0c= Re∑

n≥1

kn(z)n!

(x + iy)n (6)

wherekn(z) are the multipolar magnetic coefficients andare discontinuous functions ofz[5]. The key magneticstructure that provides the linear transverse stability ofthe reference orbit is the FODO cell. The FODO cellis defined by one focusingk2 < 0 and one defocusingk2 > 0 quadrupole separated by two dipoles or two emptyspaces[7]. In the real accelerators the dynamics in aFODO may be nonlinear due to multipolar errors inthe dipoles or to multipolar magnets (i.e. sextupoles oroctupoles) inserted in the lattice. In beam dynamics onecan study the single particle dynamics by integrating the

orbits of the Hamiltonian system (5) or one can considerthe evolution of the phase space densityρ(px, py,x,y,z)that satisfies the Liouville equation

∂ρ∂ z

+[ρ ,H] = 0 (7)

where[, ] is the Poisson bracket operator[2]. The singleparticle dynamics allows to study the long term dynam-ics aperture in hadron circular accelerators[8] whereasthe Liuoville equation is useful in the nonlinear transportproblem where collective effects are relevant In partic-ular in high intensity beams we have a self-consistentHamiltonian which depends on the distributionρ itself(Poisson-Vlasov problem[9]).Due to the periodic structure of the magnetic lattice (i.e.the Hamiltonian (5) is a periodicz-function of periodL),it is usually convenient to introduce the Poincarè map at afixed sectionz = z0 of the phase space. The Poincarè mapM is a 4-dimensional symplectic map with an ellipticfixed point corresponding to the reference orbit, whoselinear eigenvalues define the phase advances[6]. The mapM is used in the tracking simulations to compute thedynamics aperture and it allows to solve the Liouvilleequation according to the relation

ρ(px, py,x,y,zn) = ρ0(M−n(px, py,x,y)) (8)

whereρ0(px, py,x,y) is the initial phase space distribu-tion andzn = z0 + nL.In this paper we take advantage of the analogy betweenthe beam dynamics and geometric optics[10, 6] to studythe possibility of planning an optical experiment whichperforms analogical simulations. We show that using aparabolic mirror it is possible to construct periodic opti-cal device which simulates a nonlinear dynamics analo-gous to the dynamics of a FODO lattice with multipo-lar effects. A laser beam is injected in the device andthe luminosity at a transverse section is registered by aphoto camera. In our experimental set up only a smallnumber of periods can be detected so that our results areonly related to the nonlinear transport problem. The un-avoidable presence of dust particles and the roughness ofmirror surfaces produce an emittance growth, a beam in-tensity loss and a halo formation that could simulate theeffects of random perturbations or Coulomb collisionsin beam dynamics[11, 12]. On the contrary we expectthat the diffraction effects due to the finite wavelengthare negligible. Finally we have compared the experimen-tal results with the numerical solution of the Liouvilleequation (7).

AN OPTICAL FOCUSING CELL

In the thin lens approximation the effect of a focusingquadrupole is similar to a optical lens that focuses in

thex-plane and defocuses in they-plane. In a defocusingquadrupole the role of thex andy planes is exchanged.The composed effect of a focusing and a defocusingquadrupole separated by an empty space is a focusingdynamics in both planes[7]. An optical model of a FODOcell can be built by using two saddle parabolic mirrorsseparated by an empty space and rotated of 90◦ onerespect to the other. An optical beam injected in thesystem is reflected several times by the mirrors and eachmirror defines a Poincarè section. The Poincarè mapturns out to be a symplectic map with an elliptic fixedpoint and the nonlinear terms are given by a polynomialexpansion around the fixed point. A Poincarè map withthe same linear part can be obtained by using a singleparabolic mirror in front of a flat mirror, if one choosesappropriately the mirror focus. This experimental set upis much less expensive than the previous one and itcan equally describe the dynamics of a FODO cell inpresence of nonlinear effects. Let

z−a(x2+ y2) = 0 (9)

the equation of the parabolic mirror surface, a light raythat hits the mirror at the point(x,y) with a direction~p,will be reflected at a new direction~p′ according to

~p′ = ~p−2(~p · n)n (10)

wheren is the normal versor to the surface (9)

n =(−2ax,−2ay,1)

√

1+4a2(x2 + y2)(11)

When (x2 + y2) ≪ 1 and(p2x + p2

y) ≪ 1 we apply thelinear approximation and the map (10) reduces

p′x = px −4ax

p′y = py −4ay

p′z = −pz

(12)

so that the mirror focus is at a distancef = 1/4a alongthe optical axis. The map (10) introduces a nonlin-ear coupling between thex andy planes. Choosing theparabolic mirror as Poincarè section, the dynamics ofthe reflections sequence is described by a 2-degrees offreedom symplectic map which is the composition of themap (10) with the free propagation map

~x′ = ~x + l~p

~p′ = ~p

(13)

wherel is the optical path length between two succes-sive reflections.l depends both from the distanced be-tween the mirrors and on the transverse coordinatesx,y

Laser

Parabolic Mirror Mirror

FIGURE 1. Scheme of the experimental device; the laserbeam is injected by using a small hole of diameter 1.5mm inthe flat mirror and the spots due to successive reflections on theparabolic mirror are registered by a photo camera.

of the incident point. Due to the rotation symmetry, thePoincarè map has a first integral of motion[2], but ingeneric cases the dynamics turns out to be not integrable.In the linear approximation (12)l ≃ d and we can com-puted the phase advanceα of the optical cell accordingto the equation cosα = 1−d/ f = 1−4ad.

THE EXPERIMENT

The experimental device is built up by using a flat mir-ror in front of a concave parabolic mirror with the opticalaxis perpendicular to the plan of the first mirror. A smallhole of diameter 1.5mm in the flat mirror allows to injecta beam laser into the device, as showed in the apparatusscheme 1. We have used a He-Ne laser with wavelength632.8nm and a power of 15 mW. The beam section is 1mm so that we do not expect relevant diffraction effects.The beam is injected in the experimental device on a hor-izontal plane aty = −4 mm. In the figure 2 we show apicture of the experimental set up. The beam is reflectedseveral times by the mirrors and the light spots on theparabolic mirror are observed by a photo-camera(see fig.2) using the specular images on the flat mirror, so thatrotation versus is inverted in the experimental observa-tions. The small dust particles, that are always presenton the mirror surfaces, are able to scatter a part of thebeam that is registered by the photo camera. Of coursethis phenomenon is at the origin of the beam energylosses that limit drastically the number of observable pe-riods. In order to compensate this limitation one couldincrease the laser power or the sensitivity of the photocamera. In our experiment we have used a commercial(not sophisticated) photo-camera with a long expositiontime that allows to detect≃ 17 light spots in the experi-ment. We have used a parabolic mirror with a focusf ata distance≃ 25 mm on the optical axis, that produces a

phase advance> π/2 in the beam dynamics. This valueis certainly unrealistic in accelerator physics where thephase advance are< π/2, but a smaller phase advancewould have not allowed to observe nonlinear phenom-ena due to the limited number of experimental periods.However we remark that, in any case, the Poincarè mapof the optical system is a symplectic map that can beviewed as the Poincarè map of a periodic magnetic lat-tice and we expect to observe analogous phenomena inboth cases. The distanced between the mirrors is chosenin the interval 31mm ≤ d ≤ 37 mm and the laser beamis injected in the experimental device with an inclinationpx =−tg15◦ py = 0. Therefore we completely determinethe initial spot at the parabolic mirror from the distanced. The mirrors distanced is defined on the axis of theparabolic mirror and turns out to be a crucial parameterfor the beam dynamics since both the optical path lengthl in eq. (13) and position of the incident point(x,y) at theparabolic mirror in eq. (10) depend ond. If the inclina-tion of the injected beam is kept constant, by increasingthe distanced between the mirrors we increase automat-ically both the optical path lengthl (see eq. (13)) and thedistance

√

x2 + y2. Changing the beam inclination at afixed mirrors distance is more difficult from an experi-mental point of view, since it requires new alignments ofthe laser beam with the hole in the flat mirror. At a dis-tanced ≃ 40mm the observed light spots were very nearto the border of the parabolic mirror whose diameter is76mm.In the experiment the transverse momentum is not negli-gible with respect to its longitudinal component and thelinear approximation (12) cannot be applied. Indeed ac-cording to the linear approximation, we expect a phaseadvance≃ 5/9π but the measured phase advance atd = 31 mm is 4/5π that corresponds to a 5-order res-onance in the phase space. This value is in agreementwith the phase advance measured in the simulations byusing the Poincarè map of the system. Due to the ra-dial symmetry, when the dynamics is linear we expectto see elliptic spots that lie on an ellipse. But the nonlin-ear terms deform both the elliptic pattern and the initialbeam section. The presence of dust and the roughness atthe mirror surfaces introduce a beam intensity loss and abeam diffusion whereas the diffraction due to the finitewavelength is negligible. The beam energy dissipationmay simulate the beam losses in a particle acceleratorand the beam diffusion may have an analogy with theemittance growth due to stochastic perturbation or intra-beam scattering due to Coulomb interactions[11, 12]. Atthe present state of the experiment it is difficult to quan-tify these effects.

FIGURE 2. Photo of the experimental set up: on the rightwe observe the He-Ne laser that injects the light beam througha small hole in the flat mirror. The beam is reflected severaltime by the parabolic mirror and the light spots are registeredby a photo camera using the specular images on the flat mirror.

EXPERIMENTAL RESULTS ANDSIMULATIONS

Our aim is to study the nonlinear beam transport and theproblem of the comparison between experimental resultsand theoretical simulations in order to point out anal-ogous problems in beam physics. The experimental re-sults turn out to be very sensitive to the initial conditionthat has to be measured with high accuracy. This sensi-tivity is due to the strong focusing effect of the paraboliclens. The first experiment is performed using a distanced = 31 mm between the mirrors that corresponds to a5-th order resonance in the phase space and the resultsare reported in the fig. 3 left. The initial spot appears asthe most shining spot in the right right part of the pic-ture due to the reflection effect. In fig. 3 right, we alsoreport the simulation results obtained by solving the Li-ouville equation (7) for a initial uniform circular distribu-tion with a diameter of 1mm without momentum disper-sion. The periods number is limited to 20. The solutionis computed by iterating a uniform grid according to theequation (cfr. eq. (8))

ρ(M n(px, py,x,y),zn) = ρ0(px, py,x,y) (14)

wherezn is the longitudinal coordinate of the beam at theparabolic mirror aftern reflections. In this way we canrelate the points density in the simulations with the lightintensity in the experimental measures. In order the com-pensate the effects of the photo-camera inclination withrespect to the flat mirror plane (the inclination is≃ 30◦,see fig. 2), a linear transformation has been applied to thesimulations results for a better comparison with the ex-periment. This operation does not affect the spots relativeposition and the spots deformation and we have a very

good agreement between theory and experiments. Theexperiment has been repeated by increasing the distanced of the mirrors keeping constant the initial beam incli-nation. In this way we both increase the phase advanceand the nonlinear effects as discussed in the previous sec-tion. In the figures 4,5and 6 we compare the experimen-tal results and the simulations atd = 33,35,37mm. Thespot corresponding to the initial condition is in the bot-tom right part of the figures. The change of the spots pat-tern is due to the increasing of the phase advance from4π/5 towardsπ . We remark that even when we haveonly a qualitative agreement among the experiments andthe simulations, it is possible to observe the same kindof deformation due to the nonlinearity of the dynamicsin the corresponding spots. This is particularly clear inthe last figure (fig. 6) where the nonlinear effects arethe very strong and the beam is near the border of theparabolic mirror. However the experimental results con-tain much more dynamical information than the simula-tions due to the presence of perturbations not includedin the geometric optics. In particular a beam halo is ob-served around the spots that could remind analogous ef-fects due to space charge force in high intensity beams.Further studies are required to reproduce the details ofthe spots pattern.

CONCLUSIONS

We have performed an Hamiltonian dynamics experi-ment using a laser beam in the geometric optics approx-imation that has some analogies with the betatronic mo-tion in particle accelerators. Only a limited numbers ofperiods (≃ 17) can be detected and, on our opinion, thisnumber cannot be increased without using a more sophis-ticated and expensive experimental device. However theoptical experiment performs the solution of the Liouvilleequation for a nonlinear Hamiltonian system for a giveninitial distribution. The light scattering at the mirror sur-faces and the multiple reflections may have analogouseffects of the beam losses and the intra-beam scatteringdue to Coulomb collisions. The comparison with simu-lations allows to study the relevance of the theoreticalapproximations and the effects of the perturbations onthe experimental data. For these reasons we believe thatthis experiment has an methodological physical interestin simulating nonlinear phenomena.

REFERENCES

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-20 -10 0 10 20X (mm)

-20

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mm

)

1

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45

FIGURE 3. Left: experimental spots in thex,y observed by the photo-camera on the parabolic mirror; the distance between themirror was 31mm. The spots in the top-right part of the picture are due to a spurious reflection. Right: geometric optics simulationof the experiment; the axis unities aremm. The numbers identify the appearance order of the spots.

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FIGURE 4. Left: experimental spots in the configuration spacex,y of the parabolic mirror observed by the photo-camera; thedistance between the mirror was 33mm. Right: geometric optics simulation of the experiment; theaxis unities aremm.

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FIGURE 5. Left: experimental spots in the configuration spacex,y observed by the photo-camera on the parabolic mirror; thedistance between the mirror was 35mm. Right: geometric optics simulation of the experiment; theaxis unities aremm.

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)

FIGURE 6. Left: experimental spots in the configuration spacex,y observed by the photo-camera on the parabolic mirror; thedistance between the mirror was 37mm. The mirror border is visible on the right of the photo. Right: geometric optics simulationof the experiment; the axis unities aremm.

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