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arXiv:1203.6442v1 [physics.optics] 29 Mar 2012

DEUTSCHES ELEKTRONEN-SYNCHROTRON

Ein Forschungszentrum der Helmholtz-Gemeinschaft

DESY 12-051

March 2012

Pulse-front tilt caused by the use of a grating

monochromator and self-seeding of soft X-ray

FELs

Gianluca Geloni,

European XFEL GmbH, Hamburg

Vitali Kocharyan and Evgeni Saldin

Deutsches Elektronen-Synchrotron DESY, Hamburg

ISSN 0418-9833

NOTKESTRASSE 85 - 22607 HAMBURG

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Pulse-front tilt caused by the use of a grating

monochromator and self-seeding of soft X-ray

FELs

Gianluca Geloni,a,1Vitali Kocharyanband Evgeni Saldinb

aEuropean XFEL GmbH, Hamburg, Germany

bDeutsches Elektronen-Synchrotron (DESY), Hamburg, Germany

Abstract

Self-seeding is a promising approach to significantly narrow the SASE bandwidth

of XFELs to produce nearly transform-limited pulses. The development of such

schemes in the soft X-ray wavelength range necessarily involves gratings as dis-

persive elements. These introduce, in general, a pulse-front tilt, which is directly

proportional to the angular dispersion. Pulse-front tilt may easily lead to a seed

signal decrease by a factor two or more. Suggestions on how to minimize the

pulse-front tilt effect in the self-seeding setup are given.

1 Introduction

Asaconsequenceofthestart-up from shotnoise,thelongitudinalcoherence

of X-ray SASE FELs is rather poor compared to conventional optical lasers.

Self-seedingschemeshavebeenstudiedtoreducethebandwidthofSASEX-

ray FELs [1]-[19]. In general, a self-seeding setup consists of two undulators

separated by a photon monochromator and an electron bypass, normally a

four-dipole chicane. For soft X-ray self-seeding , a monochromator usually

consists of a grating [1]. Recently, a very compact soft X-ray self-seeding

scheme has appeared, based on grating monochromator [17, 18].

In[19]westudiedtheperformanceofthiscompactschemefortheEuropean

XFEL upgrade. Limitations on the performance of the self-seeding scheme

related with aberrations and spatial quality of the seed beam have been

1Corresponding Author. E-mail address: gianluca.geloni@xfel.eu

Preprint submitted to30 March 2012

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Fig. 1. Schematic representation of the electric field profile of an undistorted pulse

beam (left) and of a beam with pulse front tilt (right). The z axis is along the beam

propagation direction (adapted from [22]).

extensivelydiscussed in[17,18]andgobeyondthe scopeofthispaper.Here

we will focus our attention on the spatiotemporal distortions of the X-ray

seedpulse.Numericalresultsprovidedbyray-tracingalgorithmsappliedto

grating designprograms give accurate information onthe spatial properties

of the imaging optical system of grating monochromator. However, in the

case of self-seeding, the spatiotemporal deformation of the seeded X-ray

optical pulses is not negligible: aside from the conventional aberrations,

distortions as pulse-front tilt should also be considered [20, 21, 22]. The

propagation and distortion of X-ray pulses in grating monochromators can

be described using a wave optical theory. Most of our calculations are,

in principle, straightforward applications of conventional ultrafast pulse

optics [20]. Our paper provides physical understanding of the self-seeding

setup with a grating monochromator, and we expect that this study can be

useful in the design stage of self-seeding setups.

2Theoreticalbackgroundfortheanalysisofpulse-fronttiltphenomena

2.1 Pulse-front tilt from gratings

Ultrashort X-ray FEL pulses are usually represented as products of electric

field factors separately dependent on space and time. The assumption of

separability of the spatial (or spatial frequency) dependence of the pulse

from the temporal (or temporal frequency) dependence is usually made for

the sake of simplicity. However, when the manipulation of ultrashort X-ray

pulses requires the introduction of coupling between spatial and temporal

frequency coordinates, such assumption fails. The direction of energy flow

-usually identified as rays directions- is always orthogonal to the surface of

constant phase, that is to the wavefronts of the corresponding propagating

wave. If one deals with ultrashort X-ray pulses, one has to consider, in

addition, planes of constant intensity, that is pulse fronts. Fig. 1 shows a

schematic representation of the electric field profile of an undistorted pulse

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Fig. 2. Geometry of diffraction grating scattering.

and one with a pulse-front tilt. A distortion of the pulse front does not

affect propagation, because the phase fronts remain unaffected. However,

formostapplications,includingself-seedingapplications,itisdesirablethat

these fronts be parallel to the phase fronts, and therefore orthogonal to the

propagating direction.

A pulse-front tilt can be present in the beam dueto the propagation through

a grating monochromator. As shown in Fig. 2, the input beam is incident on

the grating at an angle θi. The diffracted angle θDis a function of frequency,

according to the well-known plane grating equation. Assuming diffraction

into the first order, one has

λ = d(cosθi− cosθD) ,

(1)

where λ = 2πc/ω, and d is the groove spacing. Eq. (1) describes the basic

working of a grating monochromator. By differentiating this equation one

obtains

dθD

dλ

=

1

θDd,

(2)

where we assume grazing incidence geometry, θi ≪ 1 and θD ≪ 1. The

physical meaning of Eq. (2) is that different spectral components of the

outcoming pulse travel in different directions. The electric field of a pulse

including angular dispersion can be expressed in the Fourier domain {kx,ω}

as E(kx−pω,ω), while the inverse Fourier transform from the {kx,ω} domain

to the space-time domain {x,t} can be expressed as E(x,t + px), which is the

electric field of a pulse with a pulse-front tilt. The tilt angle γ is given by

tanγ = cp. More specifically

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Fig.3.Reflectionoftheprimarybeamfromthelatticeplanesinthecrystalaccording

to Bragg law.

p =dkx

dω= kdθD

dω

=λ

c

dθD

dλ

=

λ

cθDd.

(3)

Therefore one concludes that the pulse-front tilt is invariably accompanied

by angular dispersion. It follows that any device like a grating monochro-

mator, that introduces an angular dispersion, also introduces significant

pulse-front tile, which is problematic for seeding.

2.2Spatiotemporal transformation of X-ray FEL pulses by crystals

The development of self-seeding schemes in the hard X-ray wavelength

rangenecessarilyinvolvescrystalmonochromators.Recently,thespatiotem-

poral coupling in the electric field relevant to self-seeding schemes with

crystal monochromators hasbeen analyzed in the frame of classical dynam-

ical theory of X-ray diffraction [23]. This analysis shows that a crystal in

Bragg reflection geometry transforms the incident electric field E(x,t) in the

{x,t}domainintoE(x−at,t),thatisthefieldofapulsewithalesswell-known

distortion, first studied in [24]. The physical meaning of this distortion is

thatthe beamspot sizeisindependentoftime,butthe beamcentral position

changes as the pulse evolves in time. One of the aims of this subsection is to

disentangle what is specific to the transformation by a crystal and what is

intrinsic tothegrating case.Ourpurpose hereisnotthatofpresentingnovel

resultsbut,rather,toattemptamoreintuitive explanationofspatiotemporal

coupling phenomena in the dynamical theory of X-ray diffraction, and to

convey the importance and simplicity of the results presented in [23].

We begin our analysis by specifying the scattering geometry under study.

The angle between the physical surface of the crystal and the reflecting

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atomic planes is an important factor. The reflection is said to be symmetric

if the surface normal is perpendicular to the reflecting planes in the case of

Bragg geometry. We shell examine only the symmetric Bragg case, Fig. 3.

Letus consider an electromagnetic planewave in the X-ray frequencyrange

incident on an infinite, perfect crystal. Within the kinematical approxima-

tion,accordingtotheBragglaw,constructive interferenceofwavesscattered

from the crystal occurs if the angle of incidence, θiand the wavelength, λ,

are related by the well-known relation

λ = 2dsinθi.

(4)

assuming reflection into the first order. This equation shows that for a given

wavelength of the X-ray beam, diffraction is possible only at certain angles

determined by the interplanar spacings d. It is important to remember the

following geometrical relationships:

1. The angle between the incident X-ray beam and normal to the reflection

plane is equal to that between the normal and the diffracted X-ray beam. In

other words, Bragg reflection is a mirror reflection, and the incident angle

is equal to the diffracted angle (θi= θD).

2. The angle between the diffracted X-ray beam and the transmitted one

is always 2θi. In other words, incident beam and forward diffracted (i.e.

transmitted) beam have the same direction.

We now turn our attention beyond the kinematical approximation to the

dynamical theory of X-ray diffraction by a crystal. An optical element in-

serted into the X-ray beam is supposed to modify some properties of the

beam as its width, its divergence, or its spectral bandwidth. It is useful to

describe the modification of the beam by means of a transfer function. The

reflectivity curve - the reflectance - in Bragg geometry can be expressed in

the frame of dynamical theory as

R(θi,ω) = R(∆ω + ωB∆θcotθB) ,

(5)

where ∆ω = (ω−ωB) and ∆θ = (θi−θB) are the deviations of frequency and

incident angle of the incoming beam from the Bragg frequency and Bragg

angle, respectively. The frequency ωBand the angle θBare given by the

Bragg law: ωBsinθB= πc/d. We follow the usual procedure of expanding ω

in a Taylor series about ωB, so that

ω = ωB+ (dω/dθ)B(θ − θB) + ... .

(6)

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Consider a perfectly collimated, white beam incidenton the crystal. In kine-

matical approximation R is a Dirac δ-function, which is simply represented

by the differential form of Bragg law:

dλ/dθi= λcotθi.

(7)

In contrast to this, in dynamical theory the reflectivity width is finite. This

means that there is a reflected beam even when incident angle and wave-

length of the incoming beam are not related exactly by Bragg equation. It is

interesting to note that the geometrical relationships 1. and 2. are still valid

in the framework of dynamical theory. In particular, reflection in dynamical

theory is alwaysa mirror reflection. We underline here thatif we havea per-

fectly collimated, white incident beam, we also have a perfectly collimated

reflected beam. Its bandwidth is related with the width of the reflectivity

curve. We will regard the beam as perfectly collimated when the angular

spread of the beam is much smaller than the angular width of the transfer

function R. It should be realized that the crystal does not introduce an an-

gular dispersion similar to a grating or a prism. However, a more detailed

analysisbased on the expression for the reflectivity, Eq. (5),shows that a less

well-known spatiotemporal coupling exists. The fact that the reflectivity is

invariant under angle and frequency transformations obeying

∆ω + ωB∆θcotθB= const (8)

is evident, and corresponds to the coupling in the Fourier domain {kx,ω}.

The origin of this relation is kinematical, it is due to Bragg diffraction. One

might be surprised that the field transformation derived in [23] for an XFEL

pulse after a crystal in the {x,t} domain is given by

Eout(x,t) = FT[R(∆ω,kx)Ein(∆ω,kx)] = E(x − ctcotθB,t) ,

(9)

where FT indicates a Fourier transform from the {kx,ω} to the {x,t} domain,

andkx= ωB∆θ/c.Ingeneral,onewould indeedexpectthe transformation to

besymmetricinboth the{kx,ω}andinthe{x,t}domainduetothesymmetry

of the transfer function2. However, it is reasonable to expect the influence

of a nonsymmetric input beam distribution. In the self-seeding case, the

incoming XFEL beam is well collimated, meaning that its angular spread

2There is a breaking of the symmetry in the diffracted beam in the {kx,ω} domain.

While the symmetry is present at the level of the transfer function, it is not present

anymore when one considers the incident beam. We point out that symmetry

breaking in [23] is a result of the approximation of temporal profile of the incident

wave to a Dirac δ-function.

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is a few times smaller than angular width of the transfer function3. Only

the bandwidth of the incoming beam is much wider than the bandwidth of

the transfer function. In this limit, we can approximate the transfer function

in the expression for the inverse temporal Fourier transform as a Dirac

δ-function. This gives

Eout(x,t) = FT[R(∆ω,kx)Ein(∆ω,kx)]

1

2π

= ξ(t)b(x − ctcotθB) ,

where we applied the Shift Theorem twice, and where

= ξ(t) ·

?

dkxexp(−ikxctcotθB)exp(ikxx)Ein(0,kx)

(10)

ξ(W) =

1

2π

?

dYexp(iYW)R(Y) (11)

is the inverse Fourier transform of the reflectivity curve.

In the opposite limit when the incoming beam has a wide angular width

and a narrow bandwidth we take the transfer function in the inverse spatial

Fourier transform as a Dirac δ-function. This gives

Eout(x,t) = ξ(xtanθB/c)a(t − (x/c)tanθB) ,

where ξ(x) is given in Eq. (11). These two limits represent the two sides of

the symmetry of the transfer function.

(12)

The last expression, Eq. (12) is the field of a pulse with a pulse front tilt.

Typically one would think that a pulse front tilt can be introduced only by

dispersive elements like gratings or prisms. Here we presented an example

inwhichnodispersiveelementsexists,andwestressthatangulardispersion

can be introduced by non dispersive element like crystals too.

3In [10] we pointed out that: ”In our case of interest (hard X-ray self-seeding

with wake monochromator) we have an angular divergence of the incident photon

beam of about a microradian, which is much smaller than the Darvin’s width of

the rocking curve (10 microradians). As a result, we assume that all frequencies

impinge on the crystal at the same angle in the vicinity of the Bragg diffraction

condition. Note that mirror reflection takes place for all frequencies and, therefore,

thereflectedbeamhasexactly thesamedivergenceas theincoming beam”. Inother

words, the description of our problem includes a small parameter, the small ratio

betweenthebeamangularwidthandthewidthofthecrystaltransferfunction.This

justifies the application in [10] (with an accuracy of about 10%) of the plane wave

approximation for the first transmission maximum. We thus avoided difficulties

related with spatiotemporal coupling.

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Fig.4. Optics forthecompact grating monochromatororiginally proposedat SLAC

[17, 18] for the soft X-ray self-seeding setup.

Although we began by considering a case of reflection transfer function

in Bragg reflection geometry, none of our arguments depends on that fact.

Eq. (5) still holds if the transfer function R is referred to the transmittance

in Bragg reflection geometry. For the transmitted beam, all derivations are

worked out in the same way we have done here and gives asymptotic

expression like Eq(10) , Eq. (12) for field of forward scattered pulses.

3Modeling of self-seeding setup with grating monochromator

A self-seeding setup should be compact enough to fit one undulator seg-

ment. In this case its installation does not perturb the undulator focusing

system and allows for the safe return to the baseline mode of operation. The

design adopted for the LCLS is the novel one by Y. Feng et al. [17, 18], and

is based on a planar VLS grating. It is equipped only with an exit slit. Such

design includes four optical elements, a cylindrical and spherical focusing

mirrors, a VLS grating and a plane mirror in front of the grating. The optical

layout of the monochromator is schematically shown in Fig. 4.

A simplified diagram for analyzing the grating monochromator is shown

in Fig. 5. We will assume that the optical system used for imaging purposes

is the well-known two-lens image formation system. With reference to Fig.

4, the VLS grating is represented by a combination of a planar grating with

fixed line spacing and a lens, with the focal length of the lens equal to the

focal length of the VLS grating. The analysis of the grating monochromator

is simplified by recognizing that the grating can be shifted from a position

immediatelybefore the lensto aposition immediatelyafterthe object plane.

Themonochromatoristreatedassumingnoaberrations.Thisapproximation

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Fig. 5. Diagram of the self-seeding grating monochromator used in theoretical

analysis. For simplicity the grating is depicted in transmission mode, and tilt is not

shown. The dashed lines refer to different planes at which the field is calculated

(adapted from [20]).

is useful, since for the design shown on Fig. 4 the aberration effects are

negligible [17, 18]. This simplifies calculations and allows analytical results

to be derived.

Theangulardispersion ofthegrating causesaseparation ofdifferentoptical

frequencies at the Fourier plane of the first focusing element (lens). There-

fore, this system becomes a tunable frequency filter if a slit is placed at the

Fourier plane. We assume that the two lenses in Fig. 5 are not identical, so

that this scheme allows for magnification by changing the focal distance of

the second lens.

It is important to analyze the output field from the grating monochromator

quantitatively. In our analysis we calculate the propagation of the input

signal to different planes of interest within the self-seeding monochroma-

tor, as indicated in Fig. 5. We start by writing the input field in plane 1,

immediately before the grating, as

E1(x,t) = Re[a(t)b(x)exp(iω0t)] ,

(13)

where ω0is the pulse carrier frequency, which is linked to k by k = ω0/c. We

assume that the input signal is Gaussian in the transverse direction, that is

b(x) = 1/

?

?√2πσ

exp(−x2/2σ2). The field in plane 2, immediately after the

grating (assuming diffraction into the first order) may be written as

E2(x,t) =

?β

2πRe

?

d∆ωA(∆ω)b(βx)exp(ip∆ωx)exp[iωt],

(14)

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where the astigmatism factor β = θi/θD, results due to the difference in

input and output angles, and p = λ/(cθDd).

OuranalysisexploitstheFouriertransformpropertiesofalens.Inparticular

we consider the propagation of a monochromatic one-dimensional field in

paraxial approximation. The field distribution in the focal plane of the first

lens, which we call plane 3, is given by

E3(x,t) =

1

2π?λfβ

Re

?

d∆ω

√

iA(∆ω)ˆb

?

k

βf(x + η∆ω)

?

exp(iωt) ,

(15)

whereˆb(kx)referstothespatialFouriertransform oftheinputspatialprofile,

η = fλ2/(2πcdθD) is a spatial dispersion parameter, which describes the

proportionality between spatial displacement and optical frequency. In the

case of a Gaussian input beam we haveˆb(kx) = exp(−σ2k2

field in the Fourier plane is written as

x/2). Therefore, the

E3(x,t) =

1

2π?λfβ

Re

?

d∆ω

√

iA(∆ω)exp

−(x + η∆ω)2

2σ2

f

exp(iωt) ,

(16)

where σf= βf/(kσ) is the rms of the focused beam at the Fourier plane for

any single frequency component.

We now add a slit at the Fourier plane, that we regard as a particular spatial

mask with a real transmission function S(x). The field in plane 4, that is

directly after the slit is simply given by

E4(x,t) = S(x)E3(x,t) .

(17)

Let us first consider the limiting case of a δ-function slit that is, physically,

a slit with much narrower opening than the spot size of a fixed individual

frequency, centered at transverse position x′. For a Gaussian input beam,

the square modulus of the transmittance of the monochromator, that is the

frequency response, is given by

|T(∆ω)|2= A0exp

?

−

?λσ

cdθi

?2

(∆ω − ∆ω′)2

?

.

(18)

The center frequency of the passband , ∆ω′= x′/η, is determined by the

transverse position oftheslit. The spectral resolution ofthe monochromator

depends on the spot size at the Fourier plane related with the individual

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frequencies, σf, and on the rate of spatial dispersion with respect to the

frequency (determined by η). The FWHM of the monochromator spectral

line is

∆ω

ω

= 1.18dθi

2πσ∼

1

πN,

(19)

whereN ∼ 2σ/(dθi)isthenumberofgroovesilluminatedbytheinputbeam.

For any single frequency the spot size at the Fourier plane, and hence the

bandwidth transmitted through a narrow slit, is inversely proportional to

theinputspotsize.Sincethetemporalspreadintheoutputpulseisinversely

proportional to the transmitted bandwidth, the output pulse duration is

proportional to the input spot size.

To get to the plane prior to the second undulator entrance, that will be

called plane 5, we simply perform a second spatial Fourier transform. The

resulting field is located at an output plane at distance f′behind the second

lens with focal distance f′, and is given by

E5(x,t) =

1

2πλ?f′fβ

dx′dx′′S(x′′)b(x′)exp

Re

?

d∆ω iA(∆ω)exp(iωt)

×

? ?

?ikx′(x′′+ η∆ω)

βf

?

exp

?ikxx′′

f′

?

(20)

Performing the integral over x′′first we obtain

E5(x,t) =

1

4π2

?

fβ

f′Re

?

d∆ω iA(∆ω)exp(iωt)

×

?

dXexp(iη∆ωX)b(βfX/k)ˆS(X + kx/f′) ,

(21)

where we define

ˆS(k) =

?

dxS(x)exp[ikx] .

(22)

We now turn to consider the limiting case when the spot size of any given

individual frequency at the Fourier plane is small compared to the spatial

scaleoverwhichthetransmission oftheslitsvaries.Thislimitingsituationis

the opposite of the previously analyzed case of δ-function slits; here the slits

do not modify the spatial profiles of the individual frequency components.

Mathematically this corresponds to the substitution of the functionˆS in the

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Fig. 6. Intensity profile in the space-time domain for α = 100

expression for E5(x,t) with a Dirac δ-function. In other words we assume

that slits are absent. One obtains

E5(x,t) = Re

?

i

2π

?

fβ

f′b

?

−βfx

f′

?

×

?

d∆ωA(∆ω)exp

?−ip∆ωfx

f′

?

exp(iωt)

?

.

(23)

In the grating monochromator, a low spectral resolution is equivalent to a

large slit size compared to the spot size of a given individual frequency at

the Fourier plane, and to a small slit size compared to the spot size of whole

spectrum. In this limit, the slit size does not modify the spatial profile of the

output beam, but modifies spectrum. The output field in this case can be

expressed as

E5(x,t) = Re

?

i

2π

?

fβ

f′b

?

−βfx

f′

?

×

?

d∆ωS(η∆ω)A(∆ω)exp

?−ip∆ωfx

f′

?

exp(iωt)

?

.

(24)

Theoptimal choiceforthetwo-lenssystem magnification iswhenβf/f′= 1.

This is the case when the field after the grating monochromator is perfectly

matched to the FEL mode in the second undulator.

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Fig. 7. Intensity profile in the space-time domain for α = 0.1

Fig. 8. Intensity profile in the space-time domain for α = 2

We fix the slit function S as

S(x) = 1 for |x| < ds

S(x) = 0 for |x| > ds

Given a slit with half size ds, we introduce a normalized notion of slit size

(25)

α =ds

σf

= dskσ

βf

(26)

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Fig. 9. Resolving power normalized to the asymptotic case for α ≪ 1 as a function

of α.

We can now plot the intensity profile in the space-time domain for different

values of the normalized slit size α. Fig. 6 and Fig. 7 qualitatively show the

two limiting situations respectively for α = 100 and α = 0.1. The spatiotem-

poral coupling is evident in Fig. 6. Fig. 8 shows the analogous plot in an

intermediate situation, for α = 2.

Itis possible to show the output characteristics of the radiation asa function

of the slit size by means of universal plots. We first consider the resolving

power R = (∆ω/ω)−1

to the inverse of the maximal bandwidth in Eq. (19), that is the bandwidth

in the limiting case for α ≪ 1, as

FWHM. We introduce the resolving power Rnnormalized

Rn= R

?1.18θid

2πσ

?

.

(27)

The behavior of Rnas a function of α is shown in Fig. 9. The resolution of

monochromator increases as the slit size decreases. The 90% of the maximal

resolution level is met for normalized slit width less than α < 1. However,

the energy of the seed pulse decreases proportionally to the decrease of the

slit width. Moreover, decreasing the slit width will also cause an increase of

the output beam size. This will lead to spatial mismatch between the seed

beam and the FEL mode in the second undulator. The relationship between

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Fig. 10. Transverse spot size of the photon beam normalized to the asymptotic case

for α ≫ 1 as a function of α. We assume that the magnification of the two-lens

optical system of the monochromator compensates the astigmatism introduced by

the grating, that is fθi/f′θD= 1.

the beam transverse size(in terms of FWHM)andslit width is shown in Fig.

10, where we plot the transverse spot size of the photon beam normalized

to the asymptotic case for α ≫ 1 as a function of α. To summarize, it is not

recommended that the normalized slit width be narrower than unity if a

reasonable seed field amplitude is required.

Finally, a useful figure of merit measuring the spatiotemporal coupling can

be found in [24]. Considering the angular dispersion this parameter can be

written as

ρ =

?

dkxd∆ωI(kx,∆ω)

kx∆ω

< (δkx)2>1/2< (δω)2>1/2,

(28)

where

< (δkx)2>=

?

?

dkxd∆ωI(kx,∆ω)k2

x,

< (δω)2>=

dkxd∆ωI(kx,∆ω)∆ω2,

I(kx,∆ω) = |E(kx,∆ω)|2

(29)

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Fig. 11. Dependence of the spatiotemporal coupling as a function of α as from Eq.

(28).

The range of ρ is in [−1,1] and readily indicate the severity of these distor-

tions.

To estimate the pulse front tilt distortion calculate the pulse front tilt param-

eter ρ as a function of the slit width α. The results are shown in Fig. 11. It is

foundtobelargerthan50%foraslitwidthα > 1.Therefore,standardtuning

of the seed monochromator will lead to significant spatiotemporal coupling

in the seed pulses. The effect of pulse front tilt distortion can be reduced if

the slit width will be narrower than α < 1. However, the reduction of the

pulse front tilt influence is accompanied by significant loss in seed signal

amplitude.

4Conclusions

To the best of our knowledge, there are no articles reporting on the impact

of pulse front tilt distortions of the seed pulse in the performance on self-

seeding soft X-ray setups. Spatiotemporal coupling is natural in grating

monochromator optics, because the monochromatization process involves

the introduction of angular dispersion, which is equivalent to pulse front

tilt distortion. In general, it is desirable that the resulting seed pulse be

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free of such distortion. This can be achieved only by decreasing the width

of the monochromator slits. On the one hand, decreasing the slit width

increases the resolving power and suppresses the pulse front tilt distortion.

On the other hand, it decreases the seed power and increases the transverse

mismatch with the FEL modein the second undulator. Asa result, a tradeoff

must be reached between achievable resolution and effective level of the

input signal.

Transverse coherence of XFEL radiation is settled without seeding. This is

due to the transverse eigenmode selection mechanism: roughly speaking,

only the ground eigenmode survives at the end of amplification process. It

follows that the spatiotemporal distortions of the seed pulse do not affect

the quality of the output radiation. They only affect the input signal value.

Therefore, the relevant value for self-seeded operation is the input coupling

factor between the seed pulsed beam and the ground eigenmode of the FEL

amplifier.

In order model the performance of a soft X-ray self-seeded FEL with a grat-

ing monochromator, one naturally starts with the grating monochromator

optical system. One aspect of optimizing the output characteristics of the

self-seeded FEL involves the specification of spectral width, peak power,

pulse-front tilt parameter and transverse size of the seed pulse as a function

oftheslitwidth.Thiscanbeachievedbypurelyanalyticalmethods.Another

aspect of the problem is the modeling of the FEL process including a seed

pulse with spatio-temporal distortions and transverse mismatching with

the ground FEL eigenmode. This study can be made only with numerical

simulation code.

Inthisarticlewerestrict ourattention tothefirstpartoftheself-seedingpro-

cess, discussing spatiotemporal coupling of the electric field in seed pulses.

The field amplitude at the exit of the self-seeding grating monochromator

can be obtained using physical optics rather than a geometrical approach.

An analysis of the behavior of an X-ray optical pulse passing through a

grating monochromator is given in terms of analytical results. In order to

make results useful for practical applications, we have included numerous

universal graphs which can be useful to find a balance between resolving

power, seed power and pulse quality during the design phase.

5 Acknowledgements

WearegratefultoMassimoAltarelli,ReinhardBrinkmann,SergueiMolodtsov

and Edgar Weckert for their support and their interest during the compila-

tion of this work.

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References

[1]

[2]

J. Feldhaus et al., Optics. Comm. 140, 341 (1997).

E. Saldin, E. Schneidmiller, Yu. Shvyd’ko and M. Yurkov, NIM A 475

357 (2001).

E. Saldin, E. Schneidmiller and M. Yurkov, NIM A 445 178 (2000).

R. Treusch, W. Brefeld, J. Feldhaus and U Hahn, Ann. report 2001 ”The

seeding project for the FEL in TTF phase II” (2001).

A. Marinelli et al., Comparison of HGHG and Self Seeded Scheme for

the Production of Narrow Bandwidth FEL Radiation, Proceedings of

FEL 2008, MOPPH009, Gyeongju (2008).

G.Geloni,V.KocharyanandE.Saldin,”Schemeforgenerationofhighly

monochromatic X-rays from a baseline XFEL undulator”, DESY 10-033

(2010).

Y. Ding, Z. Huang and R. Ruth, Phys.Rev.ST Accel.Beams, vol. 13, p.

060703 (2010).

G. Geloni, V. Kocharyan and E. Saldin, ”A simple method for control-

ling the line width of SASE X-ray FELs”, DESY 10-053 (2010).

G.Geloni,V.Kocharyan andE.Saldin,”ACascadeself-seedingscheme

with wake monochromator for narrow-bandwidth X-ray FELs”, DESY

10-080 (2010).

[10] Geloni, G., Kocharyan, V., and Saldin, E., ”Cost-effective way to en-

hance the capabilities of the LCLS baseline”, DESY 10-133 (2010).

[11] Geloni,G., KocharyanV.,and

seeding scheme for hard X-ray FELs”, Journal of Modern Optics,

DOI:10.1080/09500340.2011.586473

[12] J. Wu et al., ”Staged self-seeding scheme for narrow bandwidth , ultra-

short X-ray harmonic generation free electron laser at LCLS”, proceed-

ings of 2010 FEL conference, Malmo, Sweden, (2010).

[13] G. Geloni, V. Kocharyan and E. Saldin, ”Scheme for generation of fully

coherent, TW power level hard x-ray pulses from baseline undulators

at the European XFEL”, DESY 10-108 (2010).

[14] Geloni, G., Kocharyan, V., and Saldin, E., ”Production of transform-

limited X-ray pulses through self-seeding at the European X-ray FEL”,

DESY 11-165 (2011).

[15] W.M. Fawley et al., Toward TW-level LCLS radiation pulses, TUOA4,

to appear in the FEL 2011 Conference proceedings, Shanghai, China,

2011

[16] J. Wu et al., Simulation of the Hard X-ray Self-seeding FEL at LCLS,

MOPB09,toappearintheFEL2011Conferenceproceedings,Shanghai,

China, 2011

[17] Y. Feng et al., ”Optics for self-seeding soft x-ray FEL undulators”,

proceedings of 2010 FEL conference, Malmo, Sweden, (2010).

[18] Y. Feng, at al. ”Compact Grating Monochromator Design for LCLS-

I Soft X-ray Self-Seeding”, https://sites.google.com/a/lbl.gov/realizing-

[3]

[4]

[5]

[6]

[7]

[8]

[9]

Saldin,E.,”A novel Self-

19

Page 20

the-potential-of-seeded-fels-in-the-soft-x-ray-regime-workshop/talks

[19] G. Geloni, V. Kocharyan and E. Saldin, ”Self-seeding scheme for the

soft X-ray line at the European XFEL”, DESY 12-034 (2012).

[20] A. M. Weiner. Ultrafast Optics. Wiley, New Jersey, 2009

[21] J. Hebling, ”Derivation of the pulse front tilt caused by angular disper-

sion”, Opt. Quantum Electron. 28, 1759 (1996)

[22] G. Pretzler, et al., Appl. Phys. B 70, 1-9 (2000)

[23] R. R Lindberg and Y. V. Shvyd’ko, ”Time dependence of Bragg for-

ward scattering and self-seeding of hard x-ray free-electron lasers”,

http://arxiv.org/abs/1202.1472

[24] P. Gabolde et al., ”Describing first-order spatio-temporal distortions

in ultrashort pulses using normalized parameters”, Optics Express 15,

242 (2007)

20