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A Padè Type Analysis of the Harmonic Generation Mechanism in Free Electron Lasers
G. Dattoli, L. Giannessi, P. Ottaviani, A. Segreto
To cite this version:
G. Dattoli, L. Giannessi, P. Ottaviani, A. Segreto. A Padè Type Analysis of the Harmonic Generation Mechanism in Free Electron Lasers. Journal de Physique I, EDP Sciences, 1997, 7 (9), pp.1039-1051.
�10.1051/jp1:1997107�. �jpa-00247381�
A Padb Type Analysis of the Harmonic Generation Mechanism in Free Electron Lasers
G. Dattoli (*), L. Giannessi, P-L- Ottaviani (**) and A. Segreto (***)
ENEA, Dipartimento Innovazione, Divisione Fisica Applicata, Centro Ricerche Frascati CP 65, 00044 Frascati, Rome, Italy
(Received 17 February 1997, revised 22 May 1997, accepted 2 June 1997)
PACS.41.60.Cr Free-electron lasers
Abstract. The dynamics of the harmonic generation induced by the Free Electron Laser is reproduced analytically by using appropriate forms of approximants of the Padb type. The
analysis includes in homogeneous broadening effects and the comparison with the data from simulation is shown to be excellent.
1. Introduction
In a previous paper the theory of Padb approximant has been applied to the description of the saturation dynamics of a Free Electron Laser (FEL) [1]. The main results of the quoted analysis are summarized below,
a) the evolution of the optical field has been obtained in analytical form, even for field intensities significantly larger than the saturation intensity;
b) gain parametrization formulae yielding the dependence on the laser intensity have been
derived;
c) the dependence of the saturation intensity on parameters like the detuning and energy spread has been clarified either on physical ground either quantitatively.
AS iS well known the FEL dynamics iS a fairly complicated process. Initially the electron(e)-
beam undergoes an energy modulation, which transforms into Spatial bunching followed by
coherent emission. When the laser intensity increases, higher order bunching occurs along
with a Substantive coherent emission at higher harmonics, if it iS allowed by the undulator geometry. It Should also be added that during this interplay a degradation of the e-beam energy occurs and consequently gain and bunching efficiency are reduced.
The dynamical aspects of harmonic generation in FEL has been discussed by various au- thorS [2-4] and the relevant mechanisms have Suggested the possibility of producing very Short
(*) Author for correspondence (e-mail: dattolitlfrascati.enea.it)
(**) ENEA, DIP. Innovazione, Divisione Fisica Applicata, Centro Ricerche Bologna
(*** ENEA Fellow
© Les #ditions de Physique 1997
Table I. List of8ymbo18.
go small signal gain coefficient
a complex field amplitude
p~ inhomogeneous broadening parameter Is saturation intensity
wavelength coherent radiation in the vacuum-ultraviolet (VUV) and X-ray region, by exploit- ing the process of seedless amplification by a prebunched e-beam [5-7]. A Semianalytical model for the analysis of the intracavity harmonic generation in FEL oscillators has been developed
in reference [8]. In this last paper the method of the bunching coefficients has been exploited
to Study the evolution of the longitudinal phase-Space distribution of the e-beam and the consequent emission process.
In this paper we will complete the line of research developed in references 11, 3.8], by deriving
a Set of approximant functions, which provide an accurate description of the dynamical aspects of the harmonic generation mechanism.
The plan of the paper iS the following. In Section 2 we will briefly review the main elements of the bunching coefficient treatment of the harmonic generation in FEL and the method of
the approximants. Section 3 contains a PadA like treatment of the complex amplitudes of the fields of the coherently generated harmonics. Section 4 iS finally devoted to concluding remarks and to a comparison of the analytical results with a numerical Simulation.
2. Bunching Coefficients, FEL Dynamics and Approximant Forms The Liouville equation ruling the FEL longitudinal phase-Space distribution iS [8]
)P(v,(;T) = -v~P(v,(;T) + a(°)Sinl())P(v, (;T) (i)
where v and ( are the FEL longitudinal canonical coordinates, T iS the dimensionless, interac- tion time and a(b) iS the input complex field amplitude (See Tab. I) for further Specification).
By expanding the distribution p in Fourier Series, namely
+m
P(v,(;T)
=
~j bn(Tjem< (2j
n=-«
we obtain the following equation (2) for the coefficients bniv,T)
I)bnlv,T) = nvbnlv>T) +
~)~ ) ibn-i
Iv>T) bn+iIv>T)1 (3)
if the e-beam iS not initially bunched, we can add to the above differential-difference equation the following initial conditions
bnlv>°)
= °> n # °; bo(v>°)
= f(v, vo)
= j)~~ exP (-)~(jj)~j (4)
where the function flu, vo) is the e-beam energy distribution in the v-space (see Ref. [3] for further comments and Tab. I for the symbols). The evolution of the laser field is linked to
the bunching coefficients bi by the relation [9] (~)
~ +m
palT) = 217rgo / bi
Iv,T)dv (5)
-«
where go is the FEL small signal gain coefficient. An analogous relation holds for the field of the nth harmonic which depends on bn according to the relation [9]
~ +m
panlT) = 217rgn / bnlv,T)dv
16)
-«
with gn being the gain coefficient of the nth harmonic. In reference ill it has been shown that the solution of equation (4) can be cast in the form
m
a(T) = a(0) + 217rgo ~ ai,k(T)(a(°))~~~~ ~~)
k=0
which provides a perturbative expansion in terms of the input field amplitude a(0). The
amplitudes al,k(T) of the perturbative expansion [7] have been calculated up to k
= 2. At this
order we obtain fairly correct description of the onset of saturation. The possibility of extending the validity of the expansion to deep saturation, without calculating further perturbative terms, is offered by the theory of PadA approximants [10], which allows to approximate the r-h-s- of
equation (7) with the following rational functions
~(~) ll(°) ~ ~~~T90
~ ~~j
~~))~
~~jj2 ~ (~j~ (~j
'
(~)
~(°)~ ~~~ ~ ~(°)~ ~~~ ~~~j2 ~~~
~i,0 T ~l,0 T
It has furthermore been shown that the ai,k are reproduced by the following approximants (valid for a monochromatic e-beam)
Ai,kexp[-iai,kvTj
ai,k(Tj ~ T~+~~ ~~~~
~
2))))j2 ~~~~~~~~~~'~(~~)~~~~~~~~(~~~~'~~~)
(9j
(VT)~
~~~~ 2(~i,k)~
The numerical values of the coefficients (A, B, a, fl, 1, ~) are given in Table II. In the following
section we will extend the previous results to the higher harmonics generation.
3. High Order-Harmonic Generation and Bunching Coefficient
The last equation of the previous section provides the time evolution of the perturbative coef- ficients of the expansion (7) in the hypothesis that inhomogeneous broadening effects can be
neglected. As to the higher harmonics it is easily proved that equation (7) can be generalized
as follows (gn is the small signal gain coefficient of the nth harmonic)
m
an(T)
= 217rgn ~an,k(T)[a(0)]~~+" (10)
k=0
(~ The field of the fundamental harmonic will be denoted by a(T) and the subscript 1 will be dropped.
Table II. CoejJicient8 of equation (9).
s Ai,~ Bi,~ oi,~ fli,~ ii,s ~li,s
1 0.083 7.44X10~~ 0.5
ai,1 4.472 5.331
2 -9.425X10~~ -1.677X10~~ 0.868 0.628 3.372 11,2 3 1.19X10~~ -1.035X10~~ 1.228 1.013 2.655
11,3
Table III. CoejJicient8 of equation (11) n
= 3.
s A~,~ B~,~ a~,~ fl~,~ 13,s ~13,s
1 7X10~3 1.762X10~~ 1.798 1.973 2.364 2.963
2 -5.06X10~~ -1.143X10~~ 2.116 2.291 2.077 2.077
3 2.174X10~~ 4.157X10~~ 2.453 2.666 1.883 2.383
and the expansion amplitudes an,k(T) are specified by a relation of the type (9), namely An,kexp(-ian,kvT)
(~~j2
an,k(Tj ~ T~"+~+~~ ~~~~ (~ 2j7nkj2 ~~"'~(~~)~~~~(~~~"'~~~)
ill)
~~~
(VT)~
~ ~(lln,k)~
The explicit calculation of the amplitudes an,k has been performed by using the method out- lined in reference [1] however, being the computation procedure rather cumbersome (see Ap- pendix), the perturbative series has been truncated at the order k
= 3. The method of the PadA approximants can be used to extend the range of validity to larger a(0) values. Even
though equation (11) is valid for the generic harmonic n, we have not been able to get general formulae connecting the various coefficients A, B... etc., which, for the case n
= 3, are given in Table III.
By using the same approximant leading to equation (8), by evaluating the functions an,k at
v = 0 and T
= 1 and by recalling that
la(°)l~
=
°.8T~x, ~
= I/Is (12)
where I is the input laser power density and Is the saturation power density of the FEL, we find
~
~~~~~ ~ ~~
(1 + /~~~~~.9~2)2 ~~~~
An idea of the role played by the PadA approximation is provided by Figure 1a, where we have plotted (a3(~ ver8u8 ~ by using either equation (13) and the series (10) truncated at k
= 3.
The PadA approximant and the series exhibit substantial agreement for x < 0.1. For larger x values the truncated series diverges, while (13) behaves in a "reasonable" way. However, by confronting (13) with the numerical results (Fig. 1b) we find that the PadA approximant is not
adequate to reproduce the real behavior. We can conclude that the order of the expansion is not sufficient to obtain PadA approximant provided by a rational function. Before discussing
0.02 0.10 a)
,',
w w ,'
= =
~ ~ ,
~ 0.01 ~
J5 J5
~ ~
o
Fig. 1. a) Comparison between Padb approximant (Eq. (13) dotted line) and series expansion (Eq.
(10) continuous line) vo
= 0, n = 3. b) Comparison between Padb approximant (Eq. (13) dotted line)
and numerical results (solid points), and equation (15) (solid line).
o-lo
j~
f 0.05
~h
o o.5
~
i-o
Fig. 2. Comparison between equation (15) and the numerical data (solid points) vo # 0, n = 3.
The simulation has used the following parameters as input go
= 0.2, k
=
vi, N
= 40, a~ = 10~~
approximants of non-rational nature, we note that, compared to the analytical case, the inten- sity of the third harmonic has a maximum shifted towards smaller ~ value. This means that
if assume that a rational function of the type (13) is the approximant, its denominator must
have a degree larger than 4. A possible solution might be a slightly modified form of equation (13), namely
913.871 ~3
(14) '~3'~ ~ ~~
(1 + 5.65x + 12.9x2 + aX~ + bX~ + CX~)~
the analysis of data has shown that the numerical results are reproduced satisfactorily is summarized in Figure 2; for x < 1, equation (14) reproduces the data with an average accuracy better than 5$l.
Even though equation (14) is fairly useful for a value of the detuning around zero, its ex- tension to a larger v interval is not trivial. Since the expansion up to k
= 3 (see Eqs. (10,
11)) is not sufficient to get predictive rational approximants, we have looked for non-rational approximants which are known to accelerate the convergence of the usual PadA approxima- tion. Unfortunately, unlike the rational case [10], rigorous methods to deal with nonrational
approximants have not been developed we have therefore proceeded by successive attempts.
0.04
a)
~f
~~~ 0,04
I 0.02
b)
I
0.01
I 0,02
I
o o.5
~
i-o o o.5 1-o
0.02
c)
w
f 0.01
£
o o.5
~
i-o
Fig. 3. Comparison between equation (14) (dotted line) and numerical results (continuous line) for different values of vo, n
= 3, a) vo = 0, b) vo = 2.6, c) vo
= 3.9, p~ = 0.16. The simulation has used the following parameters as input. go = 0.2, k = vi, N
= 40, a~ = 1$lo.
Among the possible candidates the form which yields a better approximation for a larger
v-interval is given below (~)
The comparison between (15) and the numerical results is given in Figure 3. The reliability of the non rational approximant (15) is fairly good for v in the interval (-3, 3).
In the forthcoming section we will include the effect of the inhomogeneous broadening.
4. Harmonic Generation and Energy Spread Contribution
The inclusion of the effect of the energy spread in the dynamics ofharmonic generation can be achieved by following the method discussed in reference [1], I-e- by performing a convolution
(~) Even through the choice of equation (15) may sound artificial we note that it is a trivial consequence of the assumption that 1+ ~~'~~~~(ao(~ + ~~'~~~~(ao(~ corresponds to the first three terms of the
a3,0(T) a3,0(T) Taylor expansion of the function ii + c(ao(~)~.
0.010
a)
o-oio
W
I 0.005 I
~ ~
# i
~#
0 0.5
~
l.0 0 0.5
~
l.0
o.oio
c)
I
o.oos
#
o o-5 1-o
Fig. 4. Same as Figure 3. a) vo
= 0, b) vo =1.3, c) vo = 2.6, ~1~
= 0.32, a~ = 2$lo.
of (9) on the Gaussian energy distribution (4), thus obtaining (n > 1)
an,k(V, #e) " An,kH(V, p~, an,k>'fn,k) lBn,kH(V, p~, fin,k>lln,k)§~ V> #e, fin,k. ~)l'j ~ ~
~n.k
i
p~
(16a)
where
~ ~ ~
~~~'~~'~''~~
7~
(7rp~
)2
~~~ ~'~~~~~
~(~~~~ ~ ~ (16b)
4(~> Pe> CY,'f) " 'f~(~ i~(~Pe)~)l'f(~ ~~(~Pe)~)~ + 31~PE)~l'
The approximant (15) can be exploited to get an idea of the dependence of the intensity of the coherently generated third harmonic on p~ as shown in Figure 4, which also provides a
comparison between numerical and analytical results.
We will now discuss the possibility of gaining a deeper insight into the harmonic generation dynamics by exploiting a quasi-analytical model based on the results of references [11].
It has indeed been shown that the equilibrium intracavity dimensionless intensity is given by
~~ ~
~
~~ ~~~ ~~G ~~~ /~ ~ ~~~~
where G is the small signal maximum gain, eventually including inhomogeneous broadening, slippage, lethargy effects and so on. Denoting by To the initial dimensionless seed we can follow the round trip evolution of the laser signal by means of the so called logistic function
~~~~ ~
xoexP nil -11)G llini
j~~~
I + )iexPii(1 -11)G llini 11
6
~
4
= ,,
g i
i
d 1
~ i
2 '
0 100 200 300
Fig. 5. Intracavity harmonic generation versus the number of round trips. The dotted line is the
intensity of the fundamental harmonic and the continuous line is that of the coherently generated third harmonic (multiplied by 807r~) (G
= 0.2, go
= 0.2, ~
= 0.03, vu = 2.6, p~
= 0.16).
which can be shown to be the solution of the following Ginzburg-Landau equation (~)
)
= [(1 ~)G ~] (1- I)x, (19)
il Xe
with n being the number of cavity round trips and ~ the cavity losses. From equation (18) we can infer a practical formula to estimate the rise time, namely the time necessary to reach lie
of the equilibrium intensity (Lc is the optical cavity length)
~~"~~ ~ ~ ~~ ~ (1
~j~~-
~~~
(~~-1~~o~
~~~~
We can couple equations (18, 15) to get the evolution, round trip after round trip of both the laser field and of the coherently generated power of the third harmonic (see Fig. 5).
Further considerations on this last point will be presented in the forthcoming section.
5. Concluding Remarks
In the previous part of this paper we have presented a general formalism useful to analyze the
dynamics of the harmonic generation. This formalism has been tested for the third harmonic
only. We have applied the same approximant method for the case of the sth (for the relevant coefficient see Tab. IV) and the comparison with the numerical results are summarized in
Figures 6. The agreement can be considered satisfactory.
We must underline that, albeit the nonrational PadA approximant we have exploited works fairly well up to n = 7, its accuracy decreases with the increasing of the harmonic number.
More accurate approximants, either of rational type can be obtained by evaluating a larger number of perturbative terms of the series (10). This aspect of the problem is under active
consideration and will be discussed elsewhere.
Before concluding this paper we want to emphasize some final points. The maximum of the coherently generated power occurs at zero detuning. An idea the dependence on v of the
dimensionless intensity x maximizing the third harmonic is given in Figure 7.
(~) To be more precise equation (19) is only the real part of the Ginzburg-Landau equation
0.006 0.006
a) b)
~ ~
i 0.004 I
~ ~
$
,
$
0.002 " ',
o o.5
~
i-o o o-5
~
i-o
Fig. 6. Same as Figure 4, n
= 5, p~ = 0.160, a) vo
= 0, b) vo
= 1.3.
Table IV. Coejficient8 of equation (it ) n
= 5.
s A5,s 85,s a5,s fl5,s 75,s ~5,s
1 1.127X10~~ 6.912X10~~ 3.157 3.258 1.829 1.984
2 -1.725X10~~ -9.636X10~~ 3.452 3.556 1.679 1.779
3 1.408X10~~ 6.1X10~~ 3.763 3.92 1.561 1.561
In the previous section we have discussed the evolution of the intracavity signal along with the relevant coherent emission on the third harmonic. It is evident that if the system reaches
a large saturation level in a short rise-time, the harmonic generated power rises up suddently and decays very fast. If one is interested in utilizing the power generated at higher harmonics for longer time one may use an optical cavity with relatively large losses, thus reducing the
equilibrium intracavity power and getting a "stable" output at higher harmonics as shown in
Figure 8.
This might be particularly useful in FEL operating at shorter wavelength (250 nm or below)
to get a significant amount of coherent power in the region below 100 nm.
A further possibility of enhancing the intracavity higher-harmonic generated power is of- fered by the dependence of the power of the fundamental on the cavity length. This depen-
dence occurs in FELS operating with short electron pulses and is due to the so-called lethar-
gic effect [2]. Accordingly, by mismatching the cavity from the optimum cavity length [12],
o.44
( 0.42
~~
~ 0.40
0.38
0 0.5 1.0 1.5 2.0 2.5 3.0
v
Fig. 7. Dependence on v of the value of x maximizing the coherently generated power of the third harmonic.
6
4
=«
~
~
J5 h
2
0 100 200 300
Fig. 8. Same as Figure 5, but (G
= 0.2, g3
= 0.3, ~ = 0.1, vo
= 2.6, p~
= 0.16).
I-e- the length ensuring the optimum overlapping between electron and photon bunches after each round trip (~) the intracavity power of the fundamental can be reduced thus yielding a
larger and more stable power generated at higher harmonics, this aspect of the problem has been preliminarly investigated by using the analysis developed in this paper and the results,
and the expected behaviour has been confirmed.
Appendix
In the following we sketch the procedure to get the perturbative amplitudes an,k of equations (10, 11).
Equation (3) defining the evolution of the coefficients bn is more conveniently handled in the
Laplace domain, where it reads [8]
~~~ ~~~~~~~
~~~
~~ ~ ~ ~
~~~ ~ ~~~~~
(A.1) Bn(v,s)
= bn(v, T)exp(-sT)dT.
By using ao as a perturbative parameter we can look for perturbative solution of the type
and it is easily shown that Bn,k have the following general form
~"'~~~'~~
~
~~~~~~)~"'~'~'~ ~~~~~~
n
+~~ ~ven (A.3)
° otherwhise
(~ The cavity length during the EEL macropulse evolution can be controlled by using the experimental set-up of reference [12].
where Pn,k,m (s) is a polynomial function in s. The above relation can be antitransformed after conversion to partial fractions, thus finding
Examples of bn,k coefficients are provided below
bll(~>T) " )(1 ~~pl~i~~)))fl~)
~~~ ~~ ~ ~
V~ 2 ~ ~~~~)~XP(~IVT) +
~~~(~21VT)
~ l~~~~ + I (I ~xPi-ivT)
+
~Pij2ivT)
j~~~~
~~~~~ ~j
~
~ 2 ~
4 2 ~
2 ~~~~ ~~~~
2
~ ~~~~
d /~~j (A.5)
' ~~
~pi-~~~~)
~Pii~~~~~ ~~
~ ~
~ (~ ~
~~~
~~~~ ~~~~ ~
~ ~~~
~~ f(U)
~~ exp(-2ivT) ~~~~ ~~~~~ ~~~
2
The field of the nth bunching coefficient is linked to the nth harmonic dimensionless amplitude by the following identity [3]
d +m
-an(T)
= 27rign bn(v, T)dv. (A.6)
dT ~_~
The integral on the r-h-s- of (A.6) can be more conveniently evaluated in s-domain of the
Laplace transform. We remind, indeed, that
/ /~~bn(v, )dvj
=
/ ~j
~° ~/~~bnk(v,
)dvl=
~j ~° ~/~~ Bn k(v,s)dv
~
2i
~
' 2i
_~
'
k k
(A.7a)
and
/ ~ n(T)j
= sAn(s) an(0)
= sAn(s) (A.7b)
dT
~~~~~
An(s)
=
217r~" ~ ~~~~j~/~ In.k(S) ~~'~~~
s
~ 7r
-m
furthermore
~~
In,k(s)
= ~« Bn,k(v,s)dv. (A.7d)
