Delft University of Technology

Helmholtz Natural Modes: The universal and discrete spatial fabric of electromagneticwavefields

El Gawhary, Omar

DOI10.1088/1367-2630/aa57c3Publication date2017Document VersionFinal published versionPublished inNew Journal of Physics

Citation (APA)El Gawhary, O. (2017). Helmholtz Natural Modes: The universal and discrete spatial fabric ofelectromagnetic wavefields. New Journal of Physics, 19(1), [013021]. https://doi.org/10.1088/1367-2630/aa57c3

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New J. Phys. 19 (2017) 013021 doi:10.1088/1367-2630/aa57c3

PAPER

Helmholtz Natural Modes: the universal and discrete spatial fabric ofelectromagnetic wavefields

Omar ElGawhary1,2

1 VSLDutchMetrology Institute, Thijsseweg 11 2629 JADelft, TheNetherlands2 Optics ResearchGroup,Delft University of Technology, Lorentzweg 1 2628CJDelft, TheNetherlands

E-mail: oelgawhary@vsl.nl

Keywords: propagation, invariant opticalfields, free-space optical communications, radial and azimuthalmodes, spatial spectroscopy

AbstractThe interaction of electromagnetic waves withmatter is at the foundation of thewaywe perceive andexplore theworld around us. In fact, when a field interacts with an object, signatures on the object’sgeometry and physical properties are recorded in the resulting scatteredfield and are transported awayfrom the object, where they can eventually be detected and processed. An optical field can transportinformation through its spectral content, its polarization state, and its spatial distribution. Generallyspeaking, the field’s spatial structure is typically subjected to changes under free-space propagationand any information therein encoded gets reshuffled by the propagation process.Wemust ascribe tothis fundamental reason the fact that spectroscopywas known to the ancient civilizations already, andfounded asmodern science in themiddle of seventeenth century, while to date we do not have anestablished scientific offield of ‘spatial spectroscopy’ yet. In this workwe tackle this issue andwe showhow any field, whose evolution is dictated byHelmholtz equation, contains a universal and invariantspatial structure.When expressed in the framework of this spatial fabric, the spatial informationcontent carried by any field reveals its invariant nature. This opens theway to novel paradigms inoptical digital communications, inverse scattering,materials inspection, nanometrology andquantumoptics.

1. Introduction

Our understanding of themany physical phenomena taking place inNature often requires extractinginformation fromdifferent types of wavefields. Electromagnetic fields play a special role in this sense, because:

(a) It deals with infinite-range fields. Hence, they tell us on events that happened remotely in space and time.

(b) Scattered light gives us insights on the structure of matter, especially in the ultraviolet, visible and infraredpart of the electromagnetic spectrum, due to the fact thatmost of the electronics andmolecular transitionsfall in that spectral range [1, 2].

Of equal importance are also applications of wavefields in communications. The growing demand for aninterconnected society drives the need for an increased capacity of free-space communications byfinding newways of encoding large amount of information in an electromagnetic wave [3–6]. Formonochromatic fieldspropagating in free space we know that the spectral content consists of a single wavelengthλ, which remainsinvariant in propagation. Also the polarization state of afield remains essentially unaffected by propagation,except for the special cases of strongly focusedfields which can shownontrivial polarization changes in the focalregion. The situation is quite different for the spatial degree of freedoms, though. In fact, except for the verylimited classes of the so-called diffraction-free and shape-invariant fields [7–11], thefirst class offields being notphysically realizable and the second being only strictly definedwithin the paraxial regime, until now it seemed tobe not possible to represent the spatial information present in anyfield in away that remainsmanifestly invariant

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under propagation. In some sense this is not to be expected because, on an purely intuitive point of view, a freefield does not interact with anythingwhich could destroy the information content originally imprinted in it and,after all, thefields equations are deterministic by nature.With the risk of oversimplifying the description, onecan picture the situation as that of an objectmade of amass of a restless fluid constrainedwithin a closed plasticbag: if the objectmoves, the fluid totalmass remains constant while the bag continuouslymutates its shape.While one observer would clearly see a shape changingwith time, itmight exist a particular reference framewhere thefluidwould be at rest and the shape of the plastic bagwould stay frozen. The current work has beenmotivated by this simple reasoning andwewill show that, indeed, it is possible tofind such a privilegedrepresentation of afield. Once described in this way, any effect of propagation is removed and the field simplyreveals its spatial content, which is, as expected, invariant for anyfield. This result opens new excitingpossibilities inmany differentfields, as wewill discuss later on in themanuscript.More importantly, Helmholtzequation is not bound to only electromagnetism as all. In fact, all the things we are going to present here can beapplied to acoustic waves as well and to any otherfield theory which satisfies a similar equation ofmotion. Themanuscript is organized as follows: in section 2we introduce themain formalism and the concept ofHelmholtznaturalmodes (HNMs) andwe discuss theirmain properties. In section 3we show an explicit example ofapplicationwhich helps visualize the essence of the invariance. Finally, in section 4we summarize themainachievements of thework.

2.Helmholtz equation, non paraxialfields andfields representations

Without any loss of generality, let us consider a scalarmonochromatic optical beam that propagates in a regionof spacewhere aCartesian reference frame, ( )O x y z, , , has been defined, being z themain direction ofpropagation for thefield.We know that the fieldmust satisfyHelmholtz equation, which reads

+ =( ) ( ) ( )U x y z k U x y z, , , , 0 1202

with p l=k 20 . ( )U x y z, , denotes thefield distribution of the (scalar)field. If ( )U x y, , 0 represents its valueon a reference input plane z=0, the following Fourier integral representation holds

Dò ò p= +

Î( ) ( ) [ ( )] ( )

( )( )U x y A p q px qy p q, , 0 , exp i2 d d , 2

p q,

0

where ( )( )A p q,0 is the angular spectrumof the field, and (p, q) the spatial frequencies in Fourier space. Theintegration domainD is a subset of the homogeneous domain only as a consequence of the fact that evanescentwaves are not present in the integral representation in equation (2). This is becausewe are considering fields thatpropagate on distances larger than thewavelengthλ such that any contribution to thefield coming fromevanescent waves can be safely neglected.When dealingwith freely-propagating fields, one usually refers to thisfact as the low-pass filtering effect of propagation.Once the angular spectrum ( )( )A p q,0 is known,Helmholtzequation tells us how to compute it to another plane z. In fact, by denoting such a spectrum at plane z as

( )( )A p q,z , we have

pl

r j pl

r

= - -

= -

( ) ( )

( ) ( )

( ) ( )

( )

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

A p q A p q z p q

A z

, , exp i21

, exp i21

. 3

z 02

2 2

02

2

For convenience later on, in equation (3)wehave introduced a circular coordinate system in Fourier domain,r j( ), such that r j=p cos and r j=q sin . Given that the angular spectrum r j( )( )A ,z is defined on afinitesupport (the disk on Fourier space of radius l1 ) itmust be possible to represent it in terms of a discrete, bi-dimensional,modal decomposition. This is a remarkable property, if we consider that we are dealingwithclassical free-fields. In puremathematical terms, the problemoffinding an orthogonal base throughwhich onecan decompose a field distribution on a circular domain can be solved in different ways, by introducing differenttypes offield expansions. In optics, themost used decomposition is that introduced by Zernike andNijboeraround the 40s of the last century [12–16], nowadays better known asNijboer–Zernike’s unit circle polynomialsexpansion.Nijboer–Zernike expansion owns its popularity in optics to its applications to the characterization ofaberrations of optical systems. Another base,mostly used in patterns recognition and nowadays also in adaptiveoptics, is represented by the so-called disk-harmonics [17, 18]. These are eigenfunctions of the Laplacianoperator andwhat they sharewith theNijboer–Zernike’s decomposition is the fact that it deals with anorthogonal base on the unit disk aswell. However, all these bases also share a common limitation: they are all notcompatible withHelmholtz equation, which leads themviolate propagation invariance. The lack of propagationinvariance gives rise to cross-talks among themodes andmakes afield decomposition not unique. This inpracticemeans that the samefield gives rise to different decompositions if analyzed at different reference planes.

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New J. Phys. 19 (2017) 013021 OElGawhary

Amodal decomposition, orthogonal but also propagation invariant,must necessarily account for the naturalfabric of electromagnetism, in other words, itmust contain the natural spatialmodes of the physical system.Recently, it has been shownhow such a base can be definedwithin the paraxial approximation [19]. If amodaldecomposition for the full non paraxial case exists, itmust reduce to that paraxial base decompositionwithin theparaxial limit. On the other hand, the transition from a paraxial to fully non paraxial regime is not trivial norguaranteed. In order to appreciate this, we can for instance recall the difficulty offinding a close-form expressionfor aGaussian beamunder non paraxial regime or, again, the large amount of works describing the physics ofvortex beams in analytical terms, in themajority of the cases limited to the known existing paraxial solutions(like Laguerre–Gauss beams or Ince–Gauss beams) and very few non paraxial cases (like Bessel beams orMathieu beams [7, 20, 21]). Interestingly enough, howwe are going to show inmoment, it is possible tofind theanalytical expression for amodal decomposition, orthogonal but also propagation invariant, which iscompatible withHelmholtz equation. The relevance of this finding stems from the fact thatHelmholtz equationapplies tomany field theories, not only electromagnetism. In the next sectionwewill introduce such a base.

2.1.Helmholtz naturalmodesIf one aims atfinding a description of afield in terms of orthogonalmodes that is however also compatible withthefield equations, there are nomany possibilities to choose from. From equation (3)we know that free space

propagation alters the field angular spectrumby the factor p r-l( )zexp i2 1 2

2 and one should try tofind a

basewhich shares a similar functional dependency. Inspired by this principle, wewrite ( )( )A p q,0 in thefollowingway

år jp l r

rj=

-

-

l

l

( )( )

( ) ( ) ( )( )

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥A c

mn,

exp i2exp i , 4

m nm n

0

,,

1 2

1 21 4

2

2

wherem and n are integers, with = ¼m n, 0, 1, 2, . Aswewill show, equation (4) is the core result of thiswork andwewill refer to themodes of the base as theHelmholtzNaturalModes (HNMs). If we define

r rl= ( )5

then the expansion in equation (4) is defined on the unit disk and takes the form

år jp r

rj=

-

-( )

( )( )

( ) ( )( )A cm

n,exp i2 1

1exp i . 6

m nm n

0

,,

2

2 1 4

In equation (6) the coefficients cm n, have been rescaled to include a constant factor l . In terms of the newnormalized variable, equation (3) becomes

r j r j pl

r= -( ) ( ) ( )( ) ( ) ⎜ ⎟⎛⎝

⎞⎠A A z, , exp i2

11 . 7z 0 2

The orthogonality of themodes in equation (6) is consequence of the relations (denoting again r as ρ)

ò p r p rr

rr d- - -

-=( ) ( ˜ ) ( )˜

⎡⎣⎢⎢

⎤⎦⎥⎥m mexp i2 1 exp i2 1

1d 8m m

0

12 2

2,

and

ò j j j pd- =p

( ) ( ˜ ) ( )˜n nexp i exp i d 2 . 9n n0

2

,

The coefficients cm n, in equation (4), which represent in fact theHNMs spectrum for the field ( )U x y z, , , can becomputed as follows

ò òpr j

p rr

j r r j=- -

--

p( )

( )( )

( ) ( )( )⎡⎣⎢⎢

⎤⎦⎥⎥c A

mn

1

2,

exp i2 1

1exp i d d . 10m n,

0

1

0

20

2

2 1 4

Infigure 1, the phase profiles, and the amplitude common to all themodes, are shown for themodes withindices =m n, 0, 1, 2, 3, 4. It is important to point out that each of the fundamentalmodes appearing inequation (6) carries the samefinite amount of energy, equal to p4 , whichmakes it physically realizable. Thepropagation invariance of this expansion can be appreciated if one tries to compare the expression for thecoefficients cm n, for the angular spectrum r j( )( )A ,0 in equation (6)with those of the propagated spectrum

( )( )A p q,z in equation (7). TheHNMspectrum for ( )( )A p q,z reads

3

New J. Phys. 19 (2017) 013021 OElGawhary

ò òpr j j

p r

rr r j= -

- - -

-

p l( )˜ ( ) ( )

( )( )( )

⎧⎨⎪⎩⎪

⎡⎣ ⎤⎦ ⎫⎬⎪⎭⎪

c A nm1

2, exp i

exp i2 1

1d d . 11m n

z

,0

1

0

20

2

2 1 4

By comparing equations (10) and (11)we see that the following trivialmap exists

= a-˜ ( )c c , 12m n m n, ,

where a l= z . Thismap simply corresponds to a lateral shift of thewholeHNMs spectrum along the axis ofradial indexm. In particular, at an infinite set of distances l=z ll , with = ¼l 0, 1, 2, , the spectrumof cm n,

shifts by an integer amount of units along the radial axis. Hence, we have found a representation of any solutionofHelmholtz equation in orthogonalmodeswhich is also invariant.Wewould like to recall that by invariance isintended that, except for some trivial scaling, shifting, or rotation, the relation =c̃ cm n m n, , holds. Onemightwonder what kind offield distribution corresponds to the decomposition in Fourier space in equation (4). Thatcan be easily obtained by just Fourier inverting theHNMexpansion of ( )( )A p q,z with coefficients c̃m n, , whichgives

Figure 1.Amplitude and phase profiles of someHNMs in Fourier space.While all theHNMs share the same amplitude profile, theirphase distribution depends on the indices m n, of themode.

4

New J. Phys. 19 (2017) 013021 OElGawhary

ò

å q

p l r

rpr r r

=

´+ -

-

l l

l( )

( ) ( )

( )( ) ( )

⎧⎨⎪

⎩⎪

⎡⎣⎢

⎤⎦⎥

⎫⎬⎪

⎭⎪

U x y z c n

m zJ r

, , exp i

iexp i2

2 d , 13

m nm n

nn

,,

0

11 2

1 21 4

2

2

where q=x r cos , q=y r sin . The fundamentalmodes in real space consist of vortices in the azimuthalvariable θ and integrals of Bessel functions in the radial variable r. This form for thefield decompositionmakesalso evident that the only contribution to thefield on the optical axis is (r = 0), comes from the coefficients cm,0,as expected. In fact, all the other termswould lead to a undefined phase at the points ( )z0, 0, , due to thepresence of the vortex q( )nexp i in the phase profile, whichwould be not physical. Also, it is important to stressthat the decomposition in equation (13) lends itself to a convenient quantization of a non paraxial field. In fact, itis discrete by nature and it consists of orthogonalmodes already, which can easily be turned into field operators.We can say that themodes in equation (13) parallel the role thatHermite–Gaussmodes have in quantumoptics,which are however, as well known, limited to the paraxial regime only.

3.One example of application

In order to help the reader visualize what are themain features of theHNMs, wewill nowdiscuss a concreteexample. Infigure 2 (panel a1), we show themeasuredwavefront of a collimated optical beam, asmeasured by

using awavefront sensor, and the same beam after a phase shift p r-[ ]exp i2 1 2 has been applied to thefield(figure 2, panel a2). This phase shift just represents the effect of propagation under a distance l=z . In panel b1

we show the expansion of thefield in panel a1 in terms of the firstfifteen Zernike’s circle polynomials. Panel b2

shows again the Zernike’s circle polynomials decomposition for the propagatedfield shown in panel a2. As it isevident from the two pictures, Zernike’s decomposition changes completely, being affected by propagation-induced cross-talk problems,mostly among the radial part of themodes. On the other hand, panels c1 and c2

show the decomposition in terms ofHNMs, only for the coefficients cm,1, for the two situations. The complete2DHMNdecomposition is reported in panels d1 and d2 for both the collimated and defocused field,respectively. As it is clear from the picture, the description in terms ofHNMs remains the same and does notsuffer from any cross-talk. ThewholeHNMs spectrum just shifts along the radial indexm, while rigidly keepingits shape.While it would be difficult, at afirst look, to judgewhether panels b1 and b2 refer to the same field,there are no doubts about it if one looks at panels c1 and c2. This is the essence of this work.

Panels d1 and d2 exemplify what is the impact ofHNMs in the area of optical communications. Eachmode,corresponding to a specific combination of indices m n, is in fact one independent channel, which can beeventually digitized and transmitted. At the receiver place, all channels can be resolved and the informationretrieved.Wewould like to emphasize that theHNMs are defined on the full angular spectrumof radius l1which ismuch larger than that of any other decomposition definedwithin the paraxial regime. This extends thecommunication bandwidth,making available, besides towavelength and polarization, also fully non paraxialspatial channels to encode information [22, 23]. Aswewrote at the end of the previous section, quantization ofopticalfields and opticalmetrology, wavefront characterization and corrections are also areas of applications fortheHNMs.More on these subjects will be presented in upcoming separate works.

A reader familiar with signals theorymight wonderwhat is themain difference between using theHNMs aschannels to convey the information carried by thewavefield and themore commondecomposition based onShannon’s sampling theorem. A similar question has been, in fact, raised by one of the reviewers of thismanuscript as well. Because of this, it is probably worthwhile to briefly discuss on this point, too. The samplingtheorem, in its two-dimensional form, can be applied to awavefield ( )U x y z, , by taking enough samples tofulfil Nyquist’s condition. Because thefield is band-limited, it is sufficient to sample it with sampling periods

lD =x 2 and lD =y 2 in order to avoid aliasing. The originalfield can then be recovered by applying a lowpassfilter cutting the spatial spectrumof the sampled signal in the range l l-1 , 1 for both dimensions. If wedo so, then thefield can be expressed in the followingway

å pl

pl

= D D - D

´ - D

( ) ( ) ( )

( ) ( )

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

U x y z U m x n y z x m x

y n y

, , , , sinc2

sinc2

, 14

m n,

wherewe have used the commonnotation =( ) ( )x x xsinc sin [24]. In equation (14) the information channels

are now represented by the basic functions - Dpl

( )⎡⎣ ⎤⎦x m xsinc 2and - Dp

l( )⎡⎣ ⎤⎦y n ysinc 2

, while the pieces of

information encoded in each channel are represented by the values D D( )U m x n y z, , . A readerwho has

5

New J. Phys. 19 (2017) 013021 OElGawhary

followed thewhole reasoning behind this workwould immediately recognize themain problembehind thedecomposition in equation (14), namely the fact that the information D D( )U m x n y z, , in each channel is notpreserved as soon as the field propagates. This because, for a genericfield,

D D ¹ D D( ) ( )U m x n y z U m x n y z, , , ,1 2 every time ¹z z1 2. Only for the very special case of diffraction-freebeams (like a Bessel beam) the informationwould remain somehow invariant, given that in that case

D D = D D∣ ( )∣ ∣ ( )∣U m x n y z U m x n y z, , , ,1 2 , for any values of z1 and z2. However, as it was already pointed outin the introduction, Bessel beams do not exist in reality, as they carry an infinite amount of energy. Thus, thisspecial case remains an unrealizable exception. For the same reason, a decomposition of a solution of theHelmholtz equation directly in terms of Besselmodes, which could be yet another way to represent theinformation content of awavefield, is of no practical use given that each Besselmode is not physically realizable.This is amain difference withHNMs,which on the contrary all have the samefinite energy.

Before concluding this sectionwewould like to get back to the concept of spatial spectroscopy that we havementioned in the abstract.Wewill do this with the help of an informative box, shown infigure 3. In the upperpanel of thefigure, a classical spectroscopy scheme is presented. In order tominimize any chance of confusion, itis better to refer to temporal spectroscopy for this case. The adjective temporal is used to stress that the spectrumwe are talking about comes from a 1DFourier transformof the temporal response of a system. It deals with a 1D

Figure 2.Comparison between a standardmodal decomposition, in this case a Zernike’s circle polynomials decomposition, and theHelmholtz naturalmodes decomposition presented in this work. Panel a1 shows awavefront of a roughly collimated beam(experimentally obtained by collimating the light coming from a single-mode optical fiber). Panel a2 shows the same beam after a

defocus term, represented by the additional phase term p r-( )exp i2 1 2 , is added. This defocus term corresponds to the phase shiftimposed by propagation under a distance l=z . Panels b1 and b2 show the Zernike’smodal decomposition on the unit disk for thetwo situations, respectively. As expected, although it deals with the samefield, which has just propagated on afinite distance z,Zernike’s decomposition changes completely, being affected by propagation-induced cross-talk problems among the radial part of themodes. In contrast to that, panels c1 and c2 show the behavior ofHelmholtz naturalmodes of indices ( )m, 1 . As it is easy to recognize,themodal structure behaves as a solid bodywhich simply shifts along the radial axis as a whole, without changing its overall shape.Panels d1 and d2 show the structure for the full 2DHelmholtz naturalmodes decomposition for the original and propagated field,respectively. Again, the reader can appreciate the complete invariance of themodal structure under propagation.

6

New J. Phys. 19 (2017) 013021 OElGawhary

spectrumbecause time has one dimension. Because the temporal spectral content of afield can be, in large partandmost of the cases, considered invariant during propagation, any change occurred in the spectrum after thefield has interactedwith an object,must be completely ascribed to the object itself. Thismeans that changes inthefield spectral composition are a signature of the unknown object. On a similar way, in the lower panel offigure 3, a scheme for a spatial spectroscopy concept is presented. In that case, the 2D spatial spectrumof theinputfield (in terms ofHNMmodes) ismodified by the interactionwith an unknownobject. Sincewe have justproven that such a spatial spectrumwould remain invariant for a free-field, any change in the spectrumhas to beagain ascribed to the interactionwith the object. Hence, knowledge of the initial spatial spectrum and theobservation of themodified spectrumprovides a signature of the unknown object. Some of the researchperformed in the past, which aims atmapping changes of optical vortices carried by paraxial light beams, wheninteractingwithmatter, can be seen as an embryonic formof spatial spectroscopy [10, 25, 26]. However, agenuine and true spatial spectroscopy can only be realizedwhen it is not limited to a special class offields (like

Figure 3. Informative box showing analogies between a classical temporal spectroscopy scheme and a spatial spectroscopy one. Incommon spectroscopy (upper panel) the different colors in the 1Dfield spectrum represent the different temporalmodes carried bythe field. After the interactionwith a unknown object, such spectral content changes by virtue of the object internal temporalmodes.The resulting field gives, in this way, information on the object. On a similar way, in a spatial spectroscopy scheme (lower panel) thedifferent colors in the 2Dfield spatial spectrum represent the different spatialmodes carried by thefield. After the interactionwith aunknown object, such spectral content changes by virtue of the object internal spatialmodes. Once again, knowledge on the spatialspectral composition of the incident field and the detection of the emerging field gives provides information on the unknownobject.

7

New J. Phys. 19 (2017) 013021 OElGawhary

Laguerre–Gauss or Bessel–Gauss beams) and to a special regime (like the paraxial one) and can be applied to anyopticalfield relevant for the applications. This is whatwe believe the introduction of theHNMs can offer to thescientific community operating in this area of research.

4. Concluding remarks

To conclude, in this workwe have presented the natural spatial structure of classical fields solutions of theHelmholtz equation. Such a basic structure consists of fundamental, orthogonal and propagation invariantmodes that are fully compatible withHelmholtz equation. For this reasonwe have denoted these fundamentalmodes asHelmholtzNaturalModes. As such, they appear to be the preferential description thatNature haschosen to represent freely-propagating electromagnetic fields. Because theirmain properties directly originatefrombeing solutions ofHelmhotlz equation, thesemodes are not bound to electromagnetism at all but in factare the fundamentalmodes for anywave theory that share similar equations ofmotion for the fields.We havealso discussed one explicit example which helped point out the essence of representing afield in terms ofHNMs.We expect thework presented here to be relevant forfields such as quantumoptics, classical and quantumcommunications, opticalmetrology, inverse problems.

Acknowledgments

The author gratefully acknowledges interactions on the subject presented in this workwith PaulUrbach (DelftUniversity of Technology). Additionally, the authorwould like to thank Petro Sonin (VSL) for discussions onthe subject as well as for providingwavefrontmeasured data and support with the preparation of some figure.Finally, the author is thankful toNandini Bhattacharya (Delft University of Technology) for useful discussionson the subject.

This workwas partly funded through the projects ENG53 (EMRP) and 14IND09 (EMPIR). The EMRP isjointly funded by the EMRPparticipating countries within EURAMET and the EuropeanUnion. The EMPIRinitiative is co-funded by the EuropeanUnionHorizon 2020 research and innovation programme and theEMPIRparticipating States.

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