Nuclear Instruments and Methods in Physics Research A 654 (2011) 502–507

Contents lists available at ScienceDirect

Nuclear Instruments and Methods inPhysics Research A

0168-90

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/nima

FEL beam characterization from measurements of the Wignerdistribution function

Bernd Schafer a,n, Bernhard Floter a, Tobias Mey a, Pavle Juranic b, Svea Kapitzki b, Barbara Keitel b,Elke Plonjes b, Klaus Mann a, Kai Tiedtke b

a Laser-Laboratorium Gottingen, Hans-Adolf-Krebs-Weg 1, 37077 Gottingen, Germanyb Deutsches Elektronen-Synchrotron, Notkestraße 85, D-22603 Hamburg, Germany

a r t i c l e i n f o

Article history:

Received 25 March 2011

Received in revised form

1 July 2011

Accepted 16 July 2011Available online 23 July 2011

Keywords:

Wigner distribution

Spatial coherence

FEL beam

02/$ - see front matter & 2011 Elsevier B.V. A

016/j.nima.2011.07.031

esponding author.

ail address: bernd.schaefer@llg-ev.de (B. Scha

a b s t r a c t

The Free-Electron-Laser FLASH at DESY has been characterized by a quantitative determination of the

Wigner distribution function. The setup, comprising an ellipsodial mirror and a moveable extreme UV

sensitive CCD detector, enables the mapping of two-dimensional phase spaces corresponding to the

horizontal and vertical coordinate axes, respectively. For separable beams this yields the entire Wigner

distribution, offering comprehensive information about spatial coherence properties, wavefront, beam

profiles, as well as beam propagation parameters.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

A couple of FEL applications require a more complete beamcharacterization than can be gained from standard techniques as,e.g. caustic measurements [1], beam profiling [2] or wavefrontsensing [3–6]. In particular for partially coherent sources theconsideration of the 2nd order correlations, i.e. the mutualintensity [7] of the stochastic wave field, is mandatory for reliablebeam propagation or for the design of complex optical systems.Regarding the emission of SASE-type FELs (SASE¼self-amplifiedspontaneous emission), the stochastic nature of the beam gen-eration and amplification process [8] leads to inherent instabil-ities and pulse-to-pulse fluctuations and, furthermore, influencesthe spatial coherence properties as predicted by theory [9,10] andsupported by several experimental investigations [11–14]. Thetheoretical approach, however, usually makes some simplifyingassumptions regarding system alignment, stability or, e.g. theelectron beam shape, and uses extensive numerical simulationsespecially in the non-linear regime. The experiments, on the otherhand, are based on Young’s double slit experiment and investiga-tion of the corresponding two-beam interference patterns.The latter, however, are rather difficult to evaluate and it isextremely time consuming to map the four-dimensional repre-sentation space of 2nd order correlation functions that way. Incontrast, it is known [15–17] and has already been demonstrated

ll rights reserved.

fer).

for X-ray synchrotron sources [18,19], that the Wigner distribu-tion [20], being a two-dimensional Fourier transform of themutual intensity, can be fully reconstructed from two-dimen-sional intensity profiles of the beam via tomography, usingrefractive and/or reflective optics exclusively. A correspondingfully automated setup for the purpose of characterization ofexcimer laser beams has been described recently [21].

In this paper, after a brief review of the Wigner distributionapproach, we present results of its reconstruction from causticmeasurements at FLASH beam line 2 (BL2) at a mean wavelengthof 10.8 nm. The evaluation of beam parameters including globaldegree of coherence, wavefront and modal composition of the FELbeam is described.

2. Theory

The Wigner distribution h of a quasi-monochromatic paraxialbeam is defined in terms of the mutual intensity J as a two-dimensional Fourier transform of the latter according to [22]:

hðx,uÞ ¼k

2p

� �2ZZJðx�s=2,xþs=2ÞeikuUsd2s, ð1Þ

where x¼(x, y), s¼(sx, sy) are two-dimensional spatial and u¼(u, v)angular coordinates in a plane perpendicular to the directionof beam propagation and k the mean wave number of light,respectively. As J is Hermitian, h is real, although it may becomenegative in some regions. However, its marginal distributionswith respect to x and u are always non-negative and yield the

B. Schafer et al. / Nuclear Instruments and Methods in Physics Research A 654 (2011) 502–507 503

irradiance (near field) I(x) and the radiant intensity (far field)IFF(u) respectively [20].

The propagation of the Wigner distribution through static andlossless paraxial systems, signified by a 4�4 optical ray propaga-tion ABCD matrix S from an input (i) to an output (o) plane writes[15,20]:

hiðDx�Bu,�CxþAuÞ ¼ hoðx,uÞ: ð2Þ

Likewise, the four dimensional Fourier transform ~h of h obeysa similar transformation law under propagation [15]:

~hiðAT wþCT t, BT wþDT tÞ ¼ ~hoðw,tÞ, ð3Þ

where (w,t) are the Fourier-space coordinates corresponding to(x,u).

Considering a set {p} of parameters and a set of irradianceprofiles I{p}(x,y) recorded at positions which are connected to anarbitrary reference plane via the corresponding ray transforma-tion matrices S{p}, one obtains, according to the marginal propertyof h and well known Fourier relations:ZZ

hfpgðx,y,u,vÞd2uv¼ Ifpgðx,yÞ2FT ~I fpgðwx,wyÞ ¼

~hfpgðwx,wy,0,0Þ ð4Þ

and from (3) and (4) [19]:

~href ðATfpgw,BT

fpgwÞ ¼~I fpgðwÞ: ð5Þ

In particular, for a separable beam, the Wigner distribution canbe written as a product

hðx,y,u,vÞ ¼ hxðx,uÞUhyðy,vÞ ð6Þ

and, provided S{p} is of the aligned astigmatic type, i.e.,

S¼

Ax 0 Bx 0

0 Ay 0 By

Cx 0 Dx 0

0 Cy 0 Dy

0BBBB@

1CCCCA

Eq. (4) and (5) decompose likewise leading to:

~href ,xðAfpg,xwx,B pf g,xwxÞ ¼~I fpg,xðwxÞ

~href ,yðAfpg,ywy,Bfpg,ywyÞ ¼~I fpg,yðwyÞ, ð50Þ

~Ix,y being the Fourier-transformed marginal of I(x, y). For theexperimental arrangement described below, parameter p corre-sponds to the detector position z and, with Ax¼Ay¼1, Bx¼By¼z,

Fig. 1. Experimental setup for measurement of the separable Wigner distribution

of the FLASH beam at BL2 (cf. text).

Fig. 2. FEL irradiance profiles at five positions in the vicinity of the beam waist. The m

right) z¼0 mm, z¼30 mm, z¼60 mm, z¼90 mm and z¼120 mm.

Eq. (50) writes:

~href ,xðwx,zUwxÞ ¼~I fzg,xðwxÞ

~href ,yðwy,zUwyÞ ¼~I fzg,yðwyÞ: ð500Þ

3. Experimental

Fig. 1 shows the setup employed for beam caustic measure-ments at FLASH BL2 at a wavelength of 25.9 nm. This experimentis described in detail [6] elsewhere; therefore, only the featuresrelevant for evaluation of the Wigner distribution are brieflyreported. A carbon-coated ellipsoidal mirror with 2 m focal lengthfocuses the FEL beam to approximately 20 mm FWHM. Optimumalignment of the mirror was achieved by minimizing its wave-front aberration, as detected with a EUV Hartmann sensorin-stalled in the FEL beam �4820 mm behind the ellipsoidal mirror.Approximately 2.6 nm rms wavefront aberration was observedfor best focusing conditions [6]. A niobium filter with 202 nmthickness was used to attenuate the beam both for the Hartmannand the caustic measurements. Due to the chromaticity of thisfilter, the fundamental at l¼25.9 nm is efficiently blocked andonly the second and third harmonic at l¼13 and 8.6 nm aretransmitted (pulse energy fractions of 0.35% for the second and0.4% for the third harmonic [25]), leading to a mean wavelengthof l¼10.8 nm for evaluation. The caustic sensor, a phosphorousscreen imaged onto a CCD chip by a 10� magnifying microscopeon a linear translation stage, senses the FEL beam near the focalplane of the ellipsoidal mirror. Beam profile measurements weretaken at Nz¼32 z-positions within an interval of 124 mm evenlydistributed around the beam waist and covering approximately2.5–3 Rayleigh lengths (zR) in both axial directions. At eachz-position 100 single pulses were recorded.

Data evaluation starts with background correction and deter-mination of the centroids and 2nd order spatial moments ox24 ,oy24 and oxy4for each profile, followed by a rotation of thereference frame in order to minimize the normalized mixedmoments averaged over all measurement positions

s¼N�1z

Xi29/xySi9=ð/x2Siþ/y2SiÞ: ð7Þ

Provided the mixed moments oxy4 can be neglected afterthis rotation, a standard evaluation yields the 2nd order beammatrix [1] at the reference plane, from which important beamparameters as M2, beam divergence and waist can be calculated[1] for x- and y-directions, respectively (zero twist has to beassumed).

For reconstruction of the Wigner function each rotated two-dimensional record is separately integrated over x and y, deliver-ing the one-dimensional marginal distributions Ix,y. Their Fouriertransforms ~Ix,y are then mapped into two-dimensional sub-spacesaccording to (500), using the nearest neighbor method for inter-polation onto a regular two-dimensional grid of size 256�256.

easurement positions with respect to the camera reference frame are (from left to

B. Schafer et al. / Nuclear Instruments and Methods in Physics Research A 654 (2011) 502–507504

Grid points corresponding to non-measured z-positions zA(0,124)are addressed via interpolation from neighboring measurements i

and iþ1 with zA(zi, ziþ1) according to

~Ix,yðwx; zÞ ¼z�zi

ziþ1�zi

~Ix,y

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2

Rx,yþðz�z0x,yÞ2

z2Rx,yþðziþ1�z0x,yÞ

2

vuut wx; ziþ1

0@

1A

þziþ1�z

ziþ1�zi

~Ix,y

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2

Rx,yþðz�z0x,yÞ2

z2Rx,yþðzi�z0x,yÞ

2

vuut wx; zi

0@

1A

with the waist positions z0x,y in x- and y-directions, respectively.In a final step hx,y are obtained from two-dimensional inverse

Fourier transforms and the Wigner distribution h via Eq. (6). There-after, the desired beam characteristics, e.g. irradiance I(x) andwavefront w(x), are calculated according to [26] using the relation

rT wðx,zÞ ¼ l2P�1

ZuUhzðx,uÞdu ð8Þ

with the total beam power P and rT the two dimensional Nablaoperator . Coherence related quantities, as mutual intensity J, globaldegree of coherence K and the modal composition {Cij} of the beam,can be obtained via [23,24]:

Jðx,sÞ ¼

ZZhðx,uÞe�ikusd2u, K¼

l2c

P2

ZZ9hðx,uÞ92

d2xd2u ð9Þ

and

Cij,x ¼

ZZJxðx,sxÞHi xþ

sx

2

� �Hj x�

sx

2

� �dxdsx, ð10Þ

for the horizontal direction, with the Hermite–Gaussian basisfunctions H.

Table 1Beam parameters M2, Rayleigh range zR, waist diameter d0, waist position z0, rms

wavefront deformation wrms and global degree of coherence K from numerical

reconstruction of the Wigner distribution for the FEL beam. Each sequence

represents a beam profile measurement at 32 z-positions evenly distributed over

an interval of 124 mm around the beam waist.

Sequence 1 Sequence 2 Sequence 3

Mx2/My

2 4.14/2.93 4.09/2.97 3.92/2.90

zRx/zRy (mm) 27.6/19.4 26.5/19.9 27.7/19.9

d0x/d0y (mm) 0.04/0.028 0.038/0.029 0.039/0.028

z0x/z0y (mm) 56.8/68.8 56.5/68.1 58.2/68.5

wrms (nm) 0.43 0.47 0.83

Kx/Ky 0.29/0.36 0.29/0.36 0.3/0.38

4. Results

Fig. 2 shows the irradiance distribution of the FEL beam for fivepositions around the beam waist. Apart from the hyperbolicexpansion of the diameter, the beam shows a smooth intensitydistribution near the waist and obviously develops a horizontalmodulation when approaching the far field region.

Furthermore, no significant z-dependence of the profile rotationis observed; according to Eq. (7) an overall rotation angle of –51 isdetermined, yielding a value of s¼0.04351. The latter resultindicates that Eq. (6) applies and the beam is, at least in first order,separable.

Fig. 3. Wigner distribution hx(x,u) (left) and hy(y,v) (right

Fig. 3 shows the reconstructed Wigner distribution for bothcoordinate axes evaluated at the average beam waist position0.5(z0xþz0y). The double peak appearing in the horizontaldistribution clearly resolves two components, which cause thebeam modulation in the far-field region, but are spatially stillnon-separated at waist position.

Results of the beam parameter evaluation are shown in Table 1for 3 z-sequences of 32 profiles each, giving an average beampropagation factor M2 of 4.05 and 2.93 as well as a global degreeof coherence K of 0.29 and 0.36 for the horizontal and verticaldirection, respectively. Furthermore, the rms wavefront aberra-tion wrms at waist was determined to approximately 0.8 nm.

Regarding the small value of wrms, the reduced beam quality(M2�3–4) is mainly a consequence of the low spatial coherence

rather than being induced by wavefront aberrations. The latterare dominated by astigmatism, as shown in Fig. 4, where thecoefficients of a 6th order Zernike expansion and the correspond-ing wavefront aberration are displayed. We note here, that Fig. 4show a typical wavefront averaged over 3 sequences. For indivi-dual sequences the observed peak-valley irregularities variedbetween 2.8 and 3.3 nm. Likewise there was an approximately20% variation in the Zernike coefficients of defocus and astigma-tism as well as for the wrms value, as can be seen from Table 1.

In concordance with the reduced global degree of spatialcoherence the modal spectrum, shown in Fig. 5 for a 10th degreeHermite–Gaussian mode expansion, contains a considerableamount of higher order modes. The modes are uncorrelated tofirst order, with a cross correlation figure Sn,kan9Cnk9

2/Sn9Cnn92

below 0.15 for all sequences. Furthermore, data evaluation runsbased on 32 as well as 16 z-positions were performed on several

) of the FEL beam at average waist position (cf. text).

Fig. 4. Zernike coefficients Zn (left) and aberration (right) of the FEL beam wavefront evaluated at average waist position and averaged over 3 sequences (cf. text).

Fig. 5. Spectrum 9Cnn9 of the first ten Hermite–Gaussian modes of the FEL beam expansion for the horizontal (left) and vertical (right) direction, respectively.

Table 2Beam parameters, global degree of coherence and coefficients of a Hermite–Gauss

expansion at average waist position from theory as well as from numerical

reconstruction of the Wigner distribution for a synthetic Gauss-Shell beam.

Profiles at 32 z-positions evenly distributed over an interval of 124 mm around

the beam waist were used for reconstruction.

Theory Reconstruction

Beam parameter

Mx2/My

2 4.02 3.98/3.98

zRx/zRy (mm) 24.9 24.5/24.5

d0x/d0y (mm) 0.0371 0.037/0.037

z0x/z0y (mm) 73.70 �3.77/3.74

wrms (nm) 0.94 0.89

Kx/Ky 0.249 0.253/0.253

Mode coefficients

9C009 0.401 0.42/0.38

9C119 0.241 0.24/0.23

9C229 0.145 0.14/0.15

9C339 0.087 0.08/0.09

9C449 0.052 0.05/0.06

9C559 0.032 0.03/0.04

9C669 0.019 0.02/0.02

B. Schafer et al. / Nuclear Instruments and Methods in Physics Research A 654 (2011) 502–507 505

sequences for expansion degrees of 10 and 14, respectively. In allcases, the four largest coefficients for both coordinate directionschange by no more than 20% in magnitude.

As an additional test to estimate the systematic error intro-duced by the limited number of profile recordings with the finitespacing Dz¼4 mm as well as the limited z-interval of 124 mm, areconstruction was performed on a separable Gauss-Shell modelbeam with the mutual intensity at z¼0:

Jxðx,sxÞJyðy,syÞ ¼NI0e� ðx2=ð2s2ÞÞþ ðs2

x =ð2e2ÞÞð Þe� ðy2=ð2s2ÞÞþ ðs2

y=ð2e2ÞÞð Þeikðððxsx�ysyÞ=RÞÞ:

ð11Þ

Here I0 is the maximum intensity, N a normalization factor,s and e denote the beam width and the coherence parameter,respectively, whereas R accounts for astigmatism along the coor-dinate axes. In total 32 profiles at equidistant positions werecalculated within the interval [�62 mm, 62 mm] by numericalpropagation of (11) on a 512�512 grid with 0.672 mm pixel sizeand using values of s¼9.3 mm, e¼4.5 mm and R¼163 mm, yield-ing theoretical values of M2

¼4.02 and Kx,y¼0.249, a Rayleigh rangeof zR¼24.9 mm and an astigmatic waist difference of 7.3 mm.The synthetic profile data were used as input for reconstruction ofthe Wigner distribution, which was performed as described underSection 3. The results given in Table 2 and Fig. 6a and b shows avery good concordance of theoretical and reconstructed beamparameters and expansion coefficients. Likewise a good agreementbetween the reconstructed wavefront at average waist (Fig. 6b) aswell as intensity distributions at different z-positions within themeasurement interval (Fig. 6a) has been found.

5. Conclusion

The results presented in Section 4 clearly show that a valuableamount of information, especially on spatial coherence properties,can be gained from reconstruction of the Wigner distribution of

an FEL beam, even for a reduced data set originally recorded forthe purpose of standard beam parameter estimation only [6].However, the implications of the latter as well as several under-lying approximations, in particular the assumption of a separableand quasi-monochromatic beam, are worth to be briefly discussedat the end of this paper.

a.

Due to the fact that profiles beyond �2.5zR have not beensampled, the resulting gap in the reciprocal phase space(approximately 20% in the wxtx- and wyty-planes) was filledby interpolation, reducing the accuracy of reconstructed pro-files in the far field region. As a consequence, the beamcoherence might be slightly underestimated.

0.1 mm

-60 mm 0 60 mm

36µm

9·10-10m

36µm

9.4·10-10m-9·10-10m -9.4·10-10m

Fig. 6. (a) Theoretical (upper row) and reconstructed (lower row) beam profiles at three z-positions for an astigmatic Gauss-Shell model beam according to Eq. (11) with

s¼9.3 mm, e¼4.5 mm and R¼163 mm (cf. text). (b) Reconstructed (left) and theoretical wavefront distribution at average waist position for an astigmatic Gauss-Shell

model beam according to eq. (11) with s¼9.3 mm, e¼4.5 mm and R¼163 mm (cf. text).

B. Schafer et al. / Nuclear Instruments and Methods in Physics Research A 654 (2011) 502–507506

b.

Since at least two adjacent harmonics of the 25.9 nm funda-mental beam with comparable weights are monitored, it isobvious that the quasi-monochromatic beam model is only arough approximation. Strictly speaking, all beam characteris-tics then become frequency-dependent, and spectrallyresolved measurements would yield more accurate results.Only in the absence of lateral chirp and for a frequency-independent wavefront distribution, time-averaged quantitiesmay be determined with comparable accuracy from a poly-chromatic measurement. However, for an optimized operationof FLASH the 2nd order harmonic should disappear, and inmany other cases it may be permissible to narrow thespectrum to a satisfactory extent by selecting appropriatefilter combinations.

c.

Although the experimental results support that the investi-gated beam is indeed separable to a sufficient degree, this doesno longer hold in the presence of considerable wavefrontaberrations, twisted phase, or more complex mode content.In those cases a proper characterization of partially coherentbeams requires a full four-dimensional approach. In particular

when coherence properties of the beam are of major impor-tance, the comprehensive information gained from four-dimensional measurements may outweigh the higher experi-mental effort.

Taking these points into consideration, more accurate mea-surements of the Wigner distribution function of FELs are plannedfor the future.

Acknowledgement

This work was partly supported by ‘IRUVX-PP’, an EU co-funded project under FP7 (grant agreement 211285). Support ofthe FLASH user facility, in particular the funding of wavefrontsensors through the BMBF programme FSP301-FLASH, is great-fully acknowledged. We also acknowledge support from DeutscheForschungsgemeinschaft within SFB755 ‘Nanoscale Photonic Ima-ging’ as well as from the German federal ministry of economicsand technology within the InnoNet joint research project SOL.

B. Schafer et al. / Nuclear Instruments and Methods in Physics Research A 654 (2011) 502–507 507

References

[1] ISO 11146-1,2,3, Lasers and Laser-Related Equipment—Test Methods forLaser Beam Widths, Divergence Angles and Beam Propagation Ratios, Inter-national Organization for Standardization (ISO), Geneva, 2004/2005.

[2] ISO/DIS 13694, Lasers and Laser Related Equipment—Test Methods for LaserBeam Power (energy) Distribution, International Organization for Standardi-zation (ISO), Geneva, 1998.

[3] ISO 15367-1,2, Lasers and Laser Related Equipment—Test Methods forDetermination of the Shape of a Laser Beam Wavefront, InternationalOrganization for Standardization (ISO), Geneve, 2003.

[4] B. Floter, et al., Beam parameters of FLASH beamline BL1 from Hartmannwavefront measurements, Nucl. Instr. and Meth. A (2010). doi:10.1016/j.nima.2010.10.016.

[5] P. Merc�ere, et al., Opt. Lett. 28 (2003) 1534.[6] B. Floter, P. Juranic, S. Kapitzki, B. Keitel, K. Mann, E. Plonjes, B. Schafer,

K. Tiedtke, New J. Phys. 12 (2010) 083015.[7] L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed.,

Cambridge University Press, Cambridge, 1995.[8] S. Krinsky, Y. Li, Phys. Rev. E 73 (2006) 066501.[9] E.L. Saldin, E.A. Schneidmiller, M.V. Yurkov, Opt. Commun. 186 (2000) 185.

[10] E.L. Saldin, E.A. Schneidmiller, M.V. Yurkov, New J. Phys. 12 (2010) 035010.[11] R. Mitzner, et al., Opt. Exp. 16 (2008) 19909.[12] A. Singer, I.A. Vartanyants, M. Kuhlmann, S. Duesterer, R. Treusch, J. Feldhaus,

Phys. Rev. Lett. 101 (2008) 254801.

[13] R. Mitzner, S. Rolling, B. Siemer, M. Wostmann, K. Tiedtke, J. Feldhaus,H. Zacharias, Direct Evaluation of the Temporal and Spatial CoherenceProperties of FLASH Pulses, DESY Annual Report, 2009.

[14] R. Ischebeck, et al., Nucl. Instr. and Meth. A 507 (2003) 175.[15] B. Eppich, G. Mann, H. Weber, Measurement of the four dimensional Wigner

distribution of paraxial light sources, in: Proceedings of the SPIE 5962-2,D1-D11, 2005.

[16] D.F. McAlister, M. Beck, L. Clarke, A. Mayer, M.G. Rayner, Opt. Lett. 20 (1995)1181.

[17] A. Camara, T. Alieva, J.A. Rodrigo, M.L. Calvo, Tomographic Reconstruction ofthe Wigner Distribution of Non-separable Beams, in: PIERS Cambridge 2010Proceedings, ISSN:1559–9450, 2010.

[18] C.Q. Tran, A.G. Peele, A. Roberts, K.A. Nugent, D. Paterson, I. McNulty, J. Opt.Soc. Am. A—Opt. Image Sci. Vis. 22 (8) (2005) 1691.

[19] C.Q. Tran, G.J. Williams, A. Roberts, S. Flewett, A.G. Peele, D. Paterson,M.D. de Jonge, K.A. Nugent, Phys. Rev. Lett. 98 (22) (2007) 224801.

[20] M.J. Bastiaans, J. Opt. Soc. Am. A 3 (8) (1986).[21] B. Schafer, K. Mann, New J. Phys. 13 (2011) 043013.[22] M.J. Bastiaans, Opt. Acta 28 (1981) 9.[23] R. Martınez-Herrero, P.M. Mejas, Opt. Commun. 94 (1992) 197.[24] B. Eppich, Definition, meaning and measurement of coherence parameters,

in: Proceedings of SPIE 4270, 2001, pp. 71–79.[25] S. Dusterer, et al., Opt. Lett. 31 (2006) 1750.[26] A. Torre, Linear Ray and Wave Optics in Phase Space, 1st ed., Elsevier,

Amsterdam, 2005.