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Femtosecond X-ray Production

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					                  Femtosecond X-rays via Thomson Scattering
                                  Oliver Williams
                                    Physics 199
                    U.C.L.A. Department of Physics and Astronomy
                                    Winter 2003




       Within many disciplines of science, the theory behind various nanoscale ultra-fast
dynamics has surpassed the abilities of experimental verification. There has been a
growing desire for tools capable of ultra-fast, high resolution imaging in the Angstrom
wavelength range. These femtosecond x-ray sources offer physics, chemistry, biology,
and other applied sciences a new, powerful tool for experimental research.
       Light travels the length of a virus in one femtosecond (10-15 second). Many
chemical and biochemical processes occur in a few picoseconds to less than 100
femtoseconds.    The onset of thermal excitation in atoms is approximately 100
femtoseconds [1]. Hence, the necessity of femtosecond sources is obvious.
       Ultra-fast sources which probe at x-ray wavelengths offer many advantages. The
Angstrom scale wavelengths of x-rays have the advantage compared to other light
sources (e.g. Ti:S lasers, wavelength ~ 800 nm) in that they give excellent resolution of
incident structures. X-rays are also capable of penetrating atoms and exciting deeper core
electrons, a useful feature in atomic physics. Within biology, current sources are only
capable of imaging surfaces and are unable to probe with wavelengths corresponding to
the “water window” (~22-45 Angstroms) where many biological processes occur and
water absorbs much less radiation. X-rays also offer the advantage in studying crystalline
structures, as the wavelength is approximately that of the lattice structure. The above
aspects of femtosecond x-ray sources have prompted much research, and different
methods of producing this radiation have been suggested.
       In this paper, we will focus on sources that are compatible with University-scale
research programs, but larger proposed source facilities will be briefly mentioned. The
allotted budget and space for a source is a consideration as well as the user’s desired




                                            1
source characteristics. To be useful in university research, the source should have the
following aspects:
      Low cost and size
      Easy tunability to accommodate various users’ experimental parameters
      Achievable short wavelength (< 50 Angstroms)
      Ultra-fast pulses (few picoseconds to <100 femtoseconds)
      High brightness and photon flux/pulse (brightness=1020* is “ideal” [2])


       *units for brightness (photons/s mm2 mrad2 0.1% BW)


       Much research has gone into the production of femtosecond x-rays by
laser/plasma interactions. These sources can produce high fluxes of x-rays in less than
600 femtosecond pulses. However, this process is not very tunable and as the emitted x-
rays are emitted in a large (4π) solid angle - it is naturally low brightness. Another source
has been proposed to be built at the Brookhaven Accelerator Test Facility (ATF) - a laser
synchrotron source (LSS). These sources would be capable of producing femtosecond
pulses of very high peak and average brightness but requiring a 150 MeV electron beam
and laser pulse energies in the range of 10-20 J [3]. The cost of such facilities could be as
high as several hundred million dollars , putting it clearly out of range for University use.
X-ray free electron lasers (XFEL’s) are also a future possibility and one has been
proposed utilizing the last third of the linac at Stanford Linear Accelerator Center
(SLAC). The XFEL, like the LSS, would be able to produce sub-100 fs pulses with peak
brightness ten orders of magnitude higher than third generation synchrotron sources [4].
The price, unfortunately, is also comparable to LSS’s.
   The recent development of small, affordable, high-power (1 TW and 100 mJ)
lasers based on chirped pulse amplification (CPA), capable of producing femtosecond
pulses has produced an interesting insight into tabletop size x-ray source production. The
interaction of this high intensity laser beam (commonly called table-top terawatt lasers, or
T3 lasers) with a relativistic electron beam gives rise to the production of high-energy
photons through a process called Thomson scattering (or Inverse Compton scattering) [5].
Using the laser as an effective undulator with much shorter period than the commonly


                                             2
used magnetic undulators in synchrotron light sources, a much less energetic electron
beam can be used [6], hence reducing the size and cost of the required facility. This
Thomson source also offers a wide tuning range, where source parameters such as photon
energy, brightness, bandwidth, and pulse length can be adjusted directly through the laser
and electron beam parameters as well as the interaction geometry of the beams to obtain
various x-ray characteristics [6]. The Thomson scattering source has the desired high-
brightness, short pulse, “user-friendly” tuning features of a University research tool
capable of measuring on an atomic scale with high spatial and temporal resolution.
   The next section is devoted to the theory behind Thomson scattering and includes the
relevant equations for determining source parameters. Using the theory behind Thomson
scattering in the various interaction schemes, an experimental design of the U.C.L.A.
PEGASUS Laboratory electron beam interacting with a terawatt laser pulse produced by
a compact, commercial T3 laser, is then investigated. Further sections discussed are the
necessary modifications to the beam and laser pulse parameters in determining the most
plausible method of obtaining sub-100 fs x-ray pulses. The possible applications using
this source are then discussed and a summary of user source requirements is included.


Theory
   The fundamental workings of radiation production through the interaction of a laser
and relativistic electron beam can be based on the understanding of undulator theory. For
a typical magnetic undulator in a synchrotron light source, as the electrons move down
the undulator, they are deflected transversely by the alternating magnetic field. However,
always associated with an alternating magnetic field in the relativistic electron reference
frame is an alternating electric field (i.e. a propagating electromagnetic field). Using this
analogy, and replacing the magnetic undulator with a laser beam of period 104 times
smaller, the electrons are deflected many more times and contribute to coherent energy
gain using 100 times less energetic electron beams [7] resulting in a wavelength
downshift of the emitted radiation proportional to 1/2. Following Brau [8], we begin
with the resonance condition:
          2 2  ,
         W     z   X                                                                            (1)



                                             3
where the magnetic undulator period is given by W, the emitted radiation wavelength is
X, and  is the electron energy in units of its rest energy. It is obvious then, if the laser
wavelength replaces the magnetic undulator period, a much shorter wavelength of the
emitted radiation can be obtained. This interaction can also be viewed as a collision
between an electron and photon due to light’s wave-particle duality. This treatment of
the system gives rise to the name Thomson scattering. Depending on the reference
frame, it can appear as if the electrons have been energized by the photons upon collision
(Thomson scattering) or, in the reference frame (laboratory frame) where the electrons
are moving relativisticaly, the energy is instead transferred from electrons to photons,
therefore the opposite of Compton scattering (Inverse Compton scattering) [9]. In either
case, the electron beam must be “bent” (accelerated transversely) to produce/observe the
scattered radiation.
       The frequency of the scattered photons off the relativistic electron beam is
dependent on the interaction angle (  ) at which the two beams meet (typically ranging
from 90o to 180o, although small-angle Thomson scattering (SATS) has also been a
proposed interaction configuration [10]).         Below is a diagram of the interaction
configuration.




Figure 1: Diagram of interaction orientations between the electron bunch and laser pulse
where We and Le correspond to x and z, respectively,  (stated as  earlier) is the
interaction angle with =180 indicating a head-on collision and x-rays are emitted in a


                                              4
cone of half-angle, . Note that the z-coordinate in the expression for the longitudinal
dimension of the electron bunch is also defined to be along the axis of bunch propagation
(z-axis).



         A frequency upshift of the emitted radiation occurs and is expressed as a function
of the interaction angle, given as [7&11],
                2 2L(1  cos )
          
          X                        ,                                                           (2)
                1  a L 2   2 2
                       2




where the incident laser frequency is  L  2c  . The maximum observed frequency of
                                                  L


the emitted radiation occurs at =0 with aL2<<1 typically. The energy of the emitted
radiation is therefore just [7],
         EX[keV ]  X ,                                                                      (3)

with  h/2 =6.582x10-16 eV-s and Planck’s constant is h and the energy of the emitted
radiation given in keV’s. The wavelength is then simply expressed as [5,7],
                  12 .4
         x               ,                                                                   (4)
                E X [keV ]

where the radiation wavelength is approximately given in Angstroms.

Laser Parameters
         A few parameters of the laser beam appear in expressions later in this theory
section and are covered next.
         As seen in equation (2), the effects of the laser intensity appear in the undulator
strength, aL, and can play a role in the scattered radiation frequency (hence, wavelength).
The peak laser intensity is inverse-square proportional to the laser spot size and can be
written as,
                 PL
         I               ,                                                                    (5)
              2 w
                      2




with PL the peak power of the laser beam and w the spot size. The normalized vector
potential of the laser field (wiggler strength) is then expressed in convenient laboratory
units:

                                              5
                                 1
          a L  0.85  10 9 I 2 [W / cm 2 ]L[m] .                                           (6)

        A point of interest is the effects of very high laser intensity (I>1016 W/cm2) such
that aL approaches unity and greater.            Non-linear effects in the form of generated
harmonics appear which may be used to extend the tuning range of the x-ray source.
However, ponderomotive scattering also starts to occur for aL>1, where the electrons are
deflected from the laser focus point before they are able to scatter the photons therefore
decreasing the total flux of emitted x-rays [10]. Throughout this paper it is assumed
aL2<<1.
The Rayleigh length of the laser beam is given by the general expression,
                       w 2
            ZR              .                                                                 (7)
                       L

        The Rayleigh length is important as it partly determines the interaction region of
the two beams. A longer Rayleigh range would allow more interaction cycles between
the laser field and the electrons (primarily for backscattering, discussed later) and hence
more radiation production. The relevant parameters are the laser spot size W and the
wavelength of the incident laser beam, L.
        The observed shape of the laser beam depending on the orientation of the
interaction is also of importance.          The RMS Gaussian width of the laser beam is
expressed by [12],

             1             1   1 
                      c2  2  2  ,
                                                                                            (8)
               2
                           L w 

where the rms temporal width is , the spatial laser pulse length is L and c is the speed
of light.

Electron Beam Parameters

        Basic properties of the electron beam have also already been shown to influence
the resultant source parameters. The Lorentz correction factor, first appearing in equation
(2), is just the electron beam’s energy given in units of its rest energy and is computed
by,
            Eb  mc 2                                                                         (9)

                                                   6
where Eb is the energy of the electron beam and m is the mass of an electron. A good
approximation is mc2=0.511 MeV.
           There also exists a limitation on the transverse focusing of the electron beam,
which is related to the normalized emittance of the electron bunch, n, the initial beam
size at the last focusing magnet (thin lens approximation used on quadrupole magnet
array), 0, the “focal length” (distance) after this magnet, f (s), and the Lorentz correction
factor, given by,
                     s n
            (s)           .                                                                                (10)
                      0

This is derived1 by substituting the expression for the beta-function,  into the rms
divergence angle formula, θrms, describing the resultant beam spread following the last
quad due to uncorrelated beam energies, and assuming fs. The focal spot size then
becomes σ(s) by multiplying the angle by s:

     n       0 2
0       
              n

             n
 rms 
             

 ( s)  f rms  s rms .

Configurations


           Having established the fundamentals, we turn to the analysis of specific
geometries in Thomson scattering.                 There are primarily two modes of operation.
Orthogonal Thomson scattering is the configuration where the electron bunch and laser
pulse meet at a 90o angle and the laser pulse essentially “slices out” only a portion of the
electron bunch with which it interacts. The other case is Thomson backscattering where
the two beams collide head-on at 180o. Here, all the electrons in the bunch interact with
the laser pulse but at the cost of an increased radiation pulse length compared to the


1
    Expressions not numbered as no reference is made to them later; they are only for derivation purposes.


                                                       7
orthogonal case. Because both of these interaction schemes are quite different in the
resultant characteristics of the emitted x-rays, they shall be discussed and analyzed
separately.


Orthogonal Thomson Scattering (=90o)
         In the 90 degree interaction scheme the laser pulse crosses the electron bunch
transversely and so the interaction geometry of the two must be considered more closely.
         The effective number of laser periods seen by the electrons is dependent on the
temporal rms Gaussian laser width which can be expressed as [12],
                   2  c 
         N eff                     .                                                         (11)
                         L

Each time the electron is deflected (i.e. cycles through one laser period) radiation is
emitted. The number of x-rays per electron emitted from the interaction is [6],
                 
         n xe   N eff a L ,
                         2   2
                                                                                              (12)
                3

where 1/137 is the fine-structure constant. The strong dependence on aL should be
noted.
         The number of scattered x-rays per pulse is related to eqn. (12) but considers the
geometrical orientation of the electrons forming the bunch as well as including other
parameters such as the total number of electrons in the bunch, Ne, the energy of the laser
pulse in Joules, UL, which corresponds to the number of photons in the pulse, and z, the
electron bunch length (note: all ’s below given in m). The total number of x-rays in
each pulse is NX and is expressed in a more useful “lab units” form as [7],
                                             113N eU L  L
         NX                                                                      .           (13)
                      x
                             2      2
                                        
                                 w  z  x w L
                                               2      2      2       2
                                                                         
                                                                          1
                                                                              2



         The electron beam energy and normalized emittance are strongly correlated to the
brightness of the x-ray source. The expression for brightness can take the form [6],

                          U            b 2
         B  f rep      N e L2 L x 2 L              ,                                         (14)
                     2    mc  z Z R  n  X  T 2



                                                                 8
where the repetition rate of the system, frep, is considered as well as the total beam
divergence, T, and the electron beam rms opening angle, b. The total beam divergence
is composed of three contributing factors: the electron beam rms opening angle, the
intrinsic x-ray beam opening angle (int), and the divergence contribution from the finite
bandwidth (). Equations for these beam aspects [6] are shown below:

       T 2   b 2   int 2    2                                                         (15)

               n
       b                                (r=1.2x)                                          (16)
               r

                          1
        int                                                                                 (17)
                    2
                          NL      
                                  1/ 2




              
                              1/ 2

                
                                .                                                       (18)
                      

The minimum bandwidth, /, is discussed in more detail later in Common
Characteristics.
       The x-ray pulse length has a similar form to eqn. (14) as it is determined by the
interaction geometry. The length of the x-ray pulse will be dependent on how long the
laser and electron beam interact. This indicates that tight beam focusing and fast laser
pulses will result in fast x-rays due to a minimized interaction time. The relation between
these parameters and the resultant source pulse length is expressed as [5,7],


                            2  2                     2

         X 
                              z           x       w       L
                                                                   ,                          (19)
                  c z  x  w  L
                               2              2       2        2



where X is the temporal pulse length of the scattered x-rays, x and z are the transverse
and longitudinal electron beam sizes and w and L are the transverse and longitudinal
laser beam sizes, respectively.


Thomson Backscattering (=180o)



                                                                       9
       When the electron beam and laser pulse interact in a head-on collision, the
photons scatter backwards off the electrons with increased energy. In this interaction
scheme, the transverse beam sizes of the electron and laser beam must be matched for
optimal electron/photon interaction and hence x-ray production. The effective number of
laser periods seen by an electron is directly proportional to the temporal laser pulse length
and is given by [3],
                  c L
        N eff           .                                                                      (20)
                  L

       This leads to an x-ray flux generated by each electron interaction and has the
same form as the orthogonal Thomson scattering case, given by eqn. (13). It is seen in
eqn. (20) then, that increasing the laser pulse length increases the effective number of
laser periods interacting with the electrons, which corresponds to higher x-ray
production.
       Because the beams are assumed to be matched transversely (i.e. x=W) and are
counter-propagating, the interaction is much simpler than the 90o case and the number of
x-ray photons produced for each pulse is geometrically only dependent on the Rayleigh
length of the laser. This is the effective longitudinal interaction region where a smaller
Rayleigh length indicates a tighter laser focus (7) (but not smaller than the electron beam
focus), and hence denser photon/electron interactions, seen in the expression for the total
flux per pulse [7,12],
                              N eU L
        N X  4  10 3                 ,                                                        (21)
                             Z R [ m]

which has been conveniently expressed in units of experimental parameters with the laser
energy given in Joules. The brightness of the backscattered photons is then given by an
expression [5] dependent on the transverse and temporal electron bunch sizes, beam
energy, and the number of electrons in the bunch,
             NX 2
        B                ,                                                                     (22)
           4( x ) 2  b

where the temporal bunch length is calculated as b=z/c.




                                             10
       This simplified interaction geometry, unlike the orthogonal scattering, makes the
interaction time negligibly dependent on the laser pulse length and the x-ray pulse length
is determined almost entirely by the length of the electron bunch[5,7]:
                        L
        X b              .                                                                          (23)
                        4 2

       Having described the special case of 180o scattering, it is worth mentioning
“small-angle” scattering.              Small-angle Thomson scattering has been considered as a
viable interaction scheme to produce femtosecond pulses of x-rays due to the laser pulse
slicing the electron bunch from behind as the electron bunch travels. This scheme is
discussed in [10] and the expression for the pulse duration is cited as,
                                                       1
                    x2   y2   2
           L 1  1           2  .                                                             (24)
                            2 2 
                           4 L c  
                                       

       The interaction angle here is for a small-angle (i.e. 1). The other required
beam parameters are investigated in [10] and it should be noted that eqn. (24) assumes
 a condition which cannot easily be met by most University-controlled electron beam
  1,
sources and hence will not be discussed further in this paper.


Common Characteristics
       A characteristic common among both interaction configurations is the minimum
bandwidth (25) of the source [3,5].                        This has three factors which determine it:
contribution due to the laser periods (26), the intrinsic energy spread of the electron beam
(27), and the emittance broadened spectral width (28). The bandwidth can be expressed
as,

                    ' 2   ' ' 2    ' ' ' 2
                                                      where,                                           (25)
                                

         '         1
                         ,                                                                             (26)
                   N eff

         ''       2E
                       , and                                                                           (27)
                    E


                                                           11
         '''        n 2 [mm  mrad ]
                                       .                                                      (28)
                            x 2 [ m]

       The peak brilliance of the x-ray source is then just the brightness in a spectral
width [4].
       The scattered (”wiggled”) electrons emit radiation as they interact with the laser
beam, but because the electrons are moving so fast, the radiation emitted normal to the
longitudinal (wiggler) axis is Lorentz contracted in the laboratory frame, forming a cone
of radiation in the direction of travel. More energetic electrons therefore cause less
radiation divergence off the longitudinal axis. The entire spectrum of radiation can be
observed within the collection angle of the cone, 2 (i.e. 100% BW [3]), given by the
close approximation [6],
              1
              .                                                                              (29)
             2

       In the next section, a conceptual experimental design of a commercially available
femtosecond terawatt laser interacting with the electron beam produced by the
PEGASUS laboratory at U.C.L.A. shall be discussed as well as the necessary
modifications to the laser and electron beam parameters in both configurations to obtain
sub-100 fs pulses.

The PBPL Femtosecond X-ray Facility
       The PEGASUS (Photoelectron Generated Spontaneous Radiation Source)
Laboratory at U.C.L.A., outfitted with a Ti:S T3 laser, could be used to create the Particle
Beam Physics Laboratory (PBPL) Femtosecond X-ray Facility. A brief discussion of the
electron beam produced in the laboratory follows.
       The current Plane Wave Transformer (PWT) Injector utilizes an interchangeable
cathode design, allowing for the use in either thermionic emission or photoinjector mode.
The thermionic emitter is designed to provide cost-effective, high charge (1 nC) bunches.
A LaB6 cathode is currently acting as the emitter. The RF power system has been
designed to provide 20 MW of power to the RF photoinjector, a standing wave S-band
electron source [13&14]. The electron beam parameters in photoinjection mode are
shown in Table 1.


                                            12
       The Ti:S T3 laser is an obtainable, affordable, and compact high-power
femtosecond light source. Its parameters are shown in Table 2. The PEGASUS electron
beam interacting with this laser should theoretically be capable of femtosecond x-ray
pulses with relatively high collimation and an experimentally interesting x-ray flux given
beam parameters in Tables 1 and 2. The resultant x-ray characteristics are given in
Tables 3 and 6, while improved parameters are listed in Tables 4, 5, 7, and 8. An
example diagram of an experimental, compact Thomson scattering system is given in
Figure 2, below.




Figure 2: Diagram of example experimental system [15].


       The parameters listed in Table 1 assume an electron beam size of 50 m. This is
an acceptable value, which gives an electron beam focusing distance following the last
magnet of (using eq. 10) s=0.3 m assuming the initial transverse beam size of 1 mm.
This should allow enough space after the final focusing magnet for a reasonably sized
interaction region. The reduction of the electron beam size to 25 m (Tables 4 and 7)
results in s=0.15 m, still an acceptable length for an interaction region.


Table 1: PEGASUS Electron Beam Parameters
              Parameter                              Value



                                             13
Energy (Eb)                             15 MeV (30)
Energy Spread (E/E)                    0.15 %
Normalized Emittance (n)               5 mm-mrad
Charge (electrons/bunch) (Ne)           1 nC (6x109 e-/bunch)
Electron Bunch Length (z/c=b)         5 ps
Beam Size (x)                          50 m
Peak Current (Ib)                       200 A




Discussion of 90o Orientation
       An electron beam, with the parameters given above, scattering an incident laser
beam (Table 2) in the 90o orientation will depend on a fast laser pulse and tight focusing
to decrease the interaction time and hence produce x-rays on the order of femtoseconds.
This advantage of pulse duration primarily being determined by the transverse beam sizes
and laser pulse length (eqn. 19), is appealing as a reduction in these parameters is
accomplishable by PBPL through stronger focusing, experience which PBPL has in the
design and construction of permanent magnet quadrupoles (PMQ’s) [2].


Table 2: T3 Ti:S Laser Beam Parameters for Thomson scattering
              Parameter                            Value
Wavelength (L)                         800 nm
Peak Power (PL)                         1 TW
Pulse Energy (UL)                       100 mJ
Pulse Duration (W/c=L)                100 (300) fs
Laser Spot Size (W)                    50 m


       A disadvantage and technical obstacle to overcome is that synchronization
between laser and electron pulses is essential to maximize the amount of photon/electron
interactions and therefore obtain maximal output parameters where picosecond timing


                                           14
jitters are common [1, 2, 13] but <0.5 ps jitters are desired [2]. However, here PBPL is
also experienced in advanced synchronization methods as well as streak-camera imaging
[2] where precision synchronizing electronics are required, thus orthogonal scattering
might be an option for PBPL. Below is a diagram of an example synchronization system.




       Figure 3: Diagram of example synchronization system necessary for orthogonal
       Thomson scattering. Here, as well as in the calculated x-ray parameters, a system
       repetition rate of 10 Hz is theorized.

        The parameters of the emitted x-rays are given in Table 3. Because =90o in eqn.
1, the frequency is shifted by 22 and the resultant x-ray wavelength is 4.4 Angstroms. It
is seen that this orientation is capable of producing 250 fs pulses with a total flux of
5.1x105 photons/pulse emitted into a collection angle of 33 mrad, spanning the entire
spectral width (i.e. /=100%). While this interaction orientation produces very short
pulses of x-rays, the total flux and brightness are rather uninteresting for use as a research
tool (more details are discussed in the Applications section). This can be understood by
the minimal amount of interactions between the two beams due to the transverse
interaction geometry of this orientation (eqn. 13 and 14).

Table 3: Emitted X-ray Characteristics (90o)
             Parameter                                   Value
X-ray Energy (EX)                          2.8 keV (x=4.4 Angstroms)


                                             15
Pulse Duration (X)                       250 fs
Total Photon Flux (NX)                    5.1x105 photons/pulse
Peak Brightness (B) (/=0.1%)           1.5x104*
Collection angle (2)                     33 mrad


*units for brightness (photons/s mm2 mrad2 0.1% BW)


       Reasonably obtainable modifications to the laser and electron beam parameters
and the resultant changes in the source characteristics are given below.


Table 4: Source characteristics after modifications (90o)
           Modifications                 Brightness      Total Flux        Pulse Length
Spot size decrease: x , w = 25 m
Increased Laser Power: PL=2 TW           B=1.4x105*      Nx=2.0x106         x=150 fs
Increased Laser Energy: UL=200 mJ


       Obtaining Sub-100 Femtosecond X-ray Pulses
       The desire for sub-100 fs pulses is common across many scientific disciplines. To
obtain these ultra-short pulses through orthogonal scattering the beams must be focused
down to reduce the interaction time between the electron and laser beam. Reducing the
spot sizes of the laser and electron beam in the transverse dimensions to 25 m lowers
the x-ray pulse duration to 150 fs from 250 fs (Table 4). If, in addition to reducing the
spot sizes even more to 20 m, the laser pulse is also compressed to 20 fs, we may be
able to obtain x-ray pulses around 100 fs (Table 5).
       This tighter focusing and a possible laser energy increase to 200 mJ will
correspondingly increase the total x-ray flux and brightness (equations 13 and 14,
respectively). This produces fluxes on the order of 106 photons per pulse. When only the
parameters required for 100 fs pulses are considered, the resultant flux is 1.3x106
photons/pulse and the brightness is 2.1x104*. This reduced brightness is due to the
decreased laser pulse length in eqn. 14. The effects of a reduction of the electron bunch
length are also shown in Table 5, but are considered primarily for the head-on collision.


                                            16
Table 5: Modifications necessary for sub-100 fs x-ray pulses at (90o)
           Modifications                  Brightness       Total Flux      Pulse Length
Laser pulse length decrease: L20 fs     B=2.1x104*       NX=1.3x106        X100 fs
 Spot size decrease: x , w = 20 m
   Bunch compression: b=100 fs           B=7.5x105*       NX=9.3x106         X90 fs


Discussion of 180o Orientation
       This scattering configuration has some definite advantages over the orthogonal
interaction. Matching the laser and electron beam sizes transversely and aligning them
in the counter-propagating configuration is of relative ease compared to the necessity of
synchronization in the 90o case. The electron beam parameters are given in Table 1 and
the same laser parameters in Table 2 are used except for an increased laser pulse length of
300 fs to make the effective laser periods encountered by the electrons the same in both
cases. The resultant x-ray characteristics are shown in Table 6. The frequency is up
shifted by 42 due to =180o in eqn. (2). This results in an x-ray wavelength of 2.2
Angstroms. The total x-ray flux is three orders of magnitude higher for backscattering
and is calculated (using eqn. 21) to be 2.4x108 photons/pulse, which is also emitted into a
33 mrad collection angle across the entire x-ray spectrum. The brightness is drastically
higher than for orthogonal scattering and is found to be 4.4x1014* using eqn. (22) for the
given parameters.


Table 6: Emitted X-ray Characteristics (180o)
                Parameter                                    Value
X-ray Energy (EX)                             5.6 keV (x=2.2 Angstroms)
Pulse Duration (X)                           5 ps
Total Photon Flux (NX)                        2.4x108 photons/pulse



                                            17
Peak Brightness (B)(/=0.1%)                4.4x1014*
Collection angle (2)                         33 mrad




       Because the pulse length is primarily determined by the electron bunch length,
much longer laser pulses (picosecond and even nanosecond pulses) may be used to
increase the effective interaction cycles between the electrons and undulator laser field,
therefore increasing the number of photons scattered (x-rays produced) for each electron
(eqn.’s 13&20) [5, 10].
       A disadvantage of this interaction is that in order to produce femtosecond x-rays,
femtosecond electron bunches are required (eqn. 23). For our given electron beam
parameters, a bunch length of 5 ps is available, therefore only 5 ps x-ray pulses may be
created. Unfortunately, compressing the electron bunch to sub-100 fs levels is much
more difficult for PBPL to accomplish than tighter focusing and faster laser pulses
(necessary for 90o scattering) as a compressor chicane is not currently available.
Considering the aspects of both configurations, the backscattering orientation promises
the highest brightness and twice as energetic x-rays than the orthogonal scattering as well
as not requiring beam synchronization at the interaction region, making it the most
desirable set-up if electron bunch compression can be achieved. Following is a table
showing the resultant changes in the source characteristics after reasonably obtainable
modifications are made to the laser and electron beam parameters.


Table 7: Source characteristics after modifications (180o)

            Modifications                 Brightness         Total Flux    Pulse Length
 Spot size decrease: x , w = 25 m
 Increased Laser Power: PL=2 TW          B= 5.2x1016*        NX= 7.2x109    No Change
 Increased Laser Energy: UL=200 mJ




                                            18
           Obtaining Sub-100 Femtosecond X-ray Pulses
           As stated earlier, the x-ray pulse length in the 180o orientation is just that of the
electron bunch length.          This necessitates the compression of the electron bunch if
femtosecond x-rays are desired. There are various methods through which this may be
achieved. Direct production of a femtosecond electron bunch can be done through a
somewhat “exotic” method by the use of a photocathode gun and plasma-based laser
accelerator techniques. However, a high acceleration gradient is needed in order to
minimize the space-charge effect of the bunch [17], overall not making this a feasible
method.
           Bunch selection is another method of creating smaller bunches. This technique
has been used at BNL ATF where the longitudinal dependency of the electron beam on
the transverse position (dispersion) is utilized to select a small bunch out of a larger one
[17]. This obviously reduces the number of electrons per bunch, an undesirable effect
resulting in decreased x-ray flux (eqn. 21).
           In addition to the already mentioned magnetic compression where the electron’s
path length dependence on energy is taken advantage of [17], there is also wake field
compression. In this method, it is possible to use the physical imperfections of the
interior of a waveguide to slow down faster electrons in the bunch and in effect compress
it2.
           Compressing the electron bunch will have positive effects in both scattering
configurations. Decreasing the electron bunch length from 5 ps to 100 fs increased the
peak brightness for =180o by nearly two orders of magnitude to 2.2x1016*. For =90o,
the result is somewhat less dramatic but still impressive. The brightness and x-ray flux
increased to 7.5x105* and 9.3x106 photons/pulse, respectively (Table 5). As expected for
backscattering, the x-rays are emitted in a 100 fs pulse, but for the orthogonal case, the
pulse is actually reduced to only 90 fs (eqn. 19). The adjustments to the beam parameters
and the corresponding changes in the characteristics of the scattered x-rays are
summarized in Table 8.


2
    Undocumented presentation given by Dr. Sven Reiche at UCLA on bunch compression and its application
to femtosecond x-ray production (2003).


                                                   19
Table 8: Modifications necessary for sub-100 fs x-ray pulses in 180o scattering
                                            Modifications                     Brightness                  Total Flux         Pulse Length
Bunch compression: b=100 fs                                                 B=2.2x1016*                  No Change           X=100 fs




                                           Below are graphs which show the tuning range of the x-ray wavelength in the
Thomson scattering source by adjusting the interaction angle.                                                          The simplicity of
wavelength tuning in this source makes it highly desirable by the user. These plots were
made based on varying  in equation (2). Figure 4 shows the range of coarse tuning
(=10o) from 90o to 180o. Figures 5 and 6 show fine tuning around the interaction
limits (90o to 95o and 175o to 180o with =1o).




                                                            Interaction angle vs. X-ray wavelength
                                                               (Coarse tuning 90 to 180 degrees)

                                 5


                                4.5


                                 4


                                3.5
 X-ray wavelength (angstroms)




                                 3


                                2.5


                                 2


                                1.5


                                 1


                                0.5


                                 0
                                      85       95     105   115        125           135            145   155   165    175
                                                                      Interaction angle (degrees)



Figure 4: Coarse tuning range of the x-ray wavelength (90o to 180o)




                                                                                        20
                                                        Interaction angle vs. X-ray wavelength
                                                             (Fine tuning 90 to 95 degrees)

                                4.45



                                 4.4



                                4.35
 X-ray wavelength (angstroms)




                                 4.3



                                4.25



                                 4.2



                                4.15



                                 4.1



                                4.05
                                       89   90         91             92                93             94         95   96
                                                                   Interaction angle (degrees)




Figure 5: Fine tuning range of x-ray wavelength near 90o


                                                        Interaction angle vs. X-ray wavelength
                                                           (Fine tuning 175 to 180 degrees)

                                2.223



                                2.222



                                2.221
 X-ray wavelength (angstroms)




                                 2.22



                                2.219



                                2.218



                                2.217



                                2.216
                                     174         175        176                177               178        179        180
                                                                   Interaction angle (degrees)




Figure 6: Fine tuning range of x-ray wavelength near 180o




                                                                                     21
Applications
          There are many possible applications for this highly collimated, ultra-fast x-ray
source which have a variety of different source requirements. The possibility of x-ray
microscopy arises when considering the applications. Here, there is a five to ten fold
improvement of resolution compared to that of visible light. Its use in structural biology
is based on the x-ray absorption properties of carbon and nitrogen in the “water window”
(wavelength ~ 23.2-43.6 Angstroms) [18]. Within this range, water absorbs radiation less
readily, therefore good contrast with biological samples can be achieved and less energy
must be deposited onto the sample to view it [18]. For the method of holography, the
optimal wavelength extends just outside the upper water window (44 Angstroms) due to
scattering by carbon near its K-edge. The required energy for imaging is high due to
inefficiencies in the imaging optics, yet spatially coherent x-rays are not necessary,
allowing the total flux to be used. In all cases, colloidal gold labeling can be used to
reduce the required source energy. Here, a suspension of gold atoms envelops the sample,
absorbing and scattering the radiation very efficiently [19]. This causes a background
picture of the sample to be formed based on the “shadow” of the gold labeling. As this
radiation is incident on the sample, however, thermal motion will onset and there will be
degradation of the image. This thus requires a fast pulse (<30 ps) for high doses (>2x10 6
Gy where 1 Gy1 J/kg) of x-rays so that an image can be made before the sample is
destroyed by radiation [19].      Thomson backscattering would be well-suited for this
application with higher brightness than orthogonal scattering and still an acceptable pulse
length.
          The ability to create smaller and smaller circuits has created a necessity for
advanced lithography techniques. X-ray lithography offers the higher resolution imaging
necessary to create these tiny circuits. 1 keV x-rays may be used in proximity printing
where the radiation is shined onto a mask and a pattern is transferred to the silicon chip.
High brightness and collimation are desired for this application.
          Another form of lithography is deep-etch x-ray lithography. This technique can
be used for micro-machinery (and possibly nano-machinery) where structural heights of
up to 500 microns with lateral dimensions in the micron range with sub-micron precision
can be created. This corresponds to aspect ratios (ratio of height to smallest lateral


                                             22
dimension) of ~100 while for film lithography the ratio3 is only ~2-3 [17]. This allows
for the creation of thicker microstructures and hence more mass and strength.          A
proposed nanostructure fabrication facility at LBL’s Center for X-ray Optics would use
just this technique to penetrate deeply into target materials with 5 keV (X3 Angstroms)
x-ray beams to produce highly complex structures and even MEMS (micro-
electromechanical systems) [20]. The beam requirements for deep etch x-ray lithography
are short wavelengths of only 2-3 Angstroms with high flux and collimation [18]. This is
exactly the wavelength range for the 180o interaction, which also produces substantial x-
ray flux. No mention of pulse length requirements were mentioned, therefore it is
assumed that this is not a crucial parameter, hence again Thomson backscattering offers
the most promising source parameters for this application.
           Time resolved x-ray absorption fine structure (XAFS) has been proposed for use
in imaging atomic structures during chemical reactions and phase transitions.        The
relevant parameters for this technique are photon energies of 1-20 keV (0.62-12.4
Angstroms), and an 80 fs pulse length with a high repetition rate to image the dynamics
of the structure in time [4].
           A specific application and proposed experiment at LCLS is the observation of
giant coulomb explosions in atomic clusters (GCEC). Due to the photoionization of the
core electrons of many nuclei (~1012) upon being irradiated, a coulomb ball of charge is
formed. This ball explodes and produces very fast nuclei, resulting in damage to the
irradiated material.        The desired source parameters call for a 15 angstrom nominal
wavelength due to the larger inner-shell ionization cross sections, the shortest pulse
possible as higher quality data can be obtained with shorter pulses at 15 Angstroms, and
very high photon flux is necessary to achieve maximum ionization of the sample. As an
example model of source parameters, the x-ray source at LCLS has the following
characteristics [21]:
                       Photon energy: EX = 850-1000 eV
                       Total photon flux: NX = 2 x 1013 photons/pulse
                       Pulse length: X = 233 fs


3
    Given ratio is from 1993.


                                                    23
       The necessary photon energy given here is easily obtainable by PBPL. The total
photon flux in each pulse is somewhat high compared to what PBPL can provide, but no
minimum required flux is mentioned, therefore maybe some useful data can be gathered.
This flux is best obtained by Thomson backscattering, however, the pulse length of 233 fs
produced by the LCLS source is only realistically obtainable through the orthogonal
scattering scheme if bunch compression is not an option. Following is a summary table
of the applications and their associated source parameter requirements.


Table 9: Summary of applications and required source parameters
      Application            Pulse Length         Photon Flux            X (Angstroms)
GCEC                                 233 fs             2 x 1013?                        15
Holography                            <30 ps                     ?                        44
X-ray Lithography                           ?                    ?                        12
Deep Etch X-ray Litho.                      ?                    ?                      2-3
X-ray Microscopy                      <30 ps                     ?               23.2-43.6
Time resolved XAFS                      80 fs                    ?               0.62-12.4


Summary
       The recent development of table-top terawatt lasers has allowed the production,
within a small facility, of high fluxes of collimated x-rays in ultra-fast pulses. The source
wavelength (photon energy) is dependent on the interaction angle between the electron
and laser beams and other beam parameters can be adjusted to give specific changes to
the source parameters, giving a high degree of tunability.           It has been shown that
scattering a laser pulse off a counter-propagating electron beam (backscattering) gives
high fluxes of photons with the pulse length determined by the electron bunch length and
twice as energetic photons than in the 90o orientation. In the 90o case, however, much
faster pulse lengths can be achieved due to being dependent on the transverse interaction
time between the two beams. Total flux in this orientation is significantly (a few orders
of magnitude) lower than in backscattering and the brightness is even lower.              An
experimental design using the PEGASUS electron beam at U.C.L.A. and a commercially



                                             24
available table-top terawatt laser has been investigated and the conclusion made that
Thomson backscattering is a promising radiation source.




                                          References
[1] http://www-als.lbl.gov/als/science/sci_archive/femto2.html
[2] PBPL Proposal to Stockpile Stewardship (SS) for PLEIADES source
[3] E. Esarey, P. Sprangle, and A. Ting, NIMA, 331, (1993) 545-549.
[4] Conceptual Design of a 500 GeV e+e- Linear Collider with Integrated X-Ray Laser
     Facility, Volume II.
[5] I.V. Pogorelski, et al, Physical Review STAB, vol. 3, 090702 (2000).
[6] E. Esarey and W.P. Leemans.
[7] http://accelconf.web.cern.ch/accelconf/pac97/papers/pdf/3V014.PDF
[8] C. A. Brau, Free-Electron Lasers, San Diego, 1990.
[9] http://www.astro.utu.fi/~cflynn/astroII/l7.html
[10] Y. Li, et al, Physical Review STAB, vol. 5, 044701 (2002).
[11] http://accelconf.web.cern.ch/AccelConf/p95/ARTICLES/TPG/TPG07.PDF
[12] K.J. Kim, S. Chattopadhyay, and C.V. Shank, NIMA, 341, (1994) 351-354.
[13] http://pbpl.physics.ucla.edu/pool/pbpl-0600-2002-000056.pdf
[14] S. Telfer, et al, Proceedings of the 2001 Particle Accelerator Conference, Chicago,
     2263.
[15] http://atfweb.kek.jp/icfa/2001/box/w2-5_10p.pdf
[16] J. Yang, Proceedings of EPAC 2002, Paris, France,
     http://accelconf.web.cern.ch/AccelConf/e02/PAPERS/TUPRI076.pdf
[17] X.J. Wang, Proceedings of the 1999 Particle Accelerator Conference, NY.
[18] LBL-35023, SLAC Report-430, UC-400, December 1-2, 1993.
[19] SLAC Report- 414, October 21, 1992.
[20] LBL Center for X-Ray Optics, January 10, 1992. http://www.lbl.gov/Science-
     Articles/Archive/x-ray-lithography.html
[21] LCLS: The First Experiments, September 2002.



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