Wave-guide free electron laser: gain inhomogeneous broadening and saturation

I February

1996

OPTICS

COMMUNICATIONS Optics Communications

123 ( 1996) 535-542

Wave-guide free electron laser: gain inhomogeneous broadening and saturation G. Dattoli, A. Doria, L. Giannessi, A. Segreto ENEA. Dipartirnento lnnovazione.

’

Settore Fisica Applicata, C.R. Frascati, C.P. 65-00044 Frascati, Rome, Italy Received 30 May 1995

Abstract We discuss wave-guide Free Electron Laser gain formulae including inhomogeneous broadening and saturation effects. The analysis is not limited to the low gain regime. It is also proven that, after a proper redefinition, most of the “empirical” formulae exploited for the free space case can be utilised to model the wave-guide evolution. PACS: 42.5S.T.b.

1. Introduction Wave-guide Free Electron Lasers (W.G. FEL) have been successfully operated in the far infrared and millimetre region of the spectrum [ 11. The experimental results have supported the theoretical analysis which predicted new dynamic effects provided by the dispersive properties of the wave-guide [ 2-61. Systematic efforts to characterise the W.G. FEL gain and dynamics in terms of the system parameters have not been undertaken yet. It has been however proved that, owing to the dispersive properties of the waveguide. inhomogeneous broadening and slippage effects may provide less significant gain degradation than for the vacuum case [ 4-61. The problems connected to the high gain and strong signal regime have been treated only numerically. In this paper we present a comprehensive treatment of the W.G. FEL dynamics including inhomogeneous broadening. high gain and strong signal contributions,

’ ENEA Guest. 0030-40 I8 /96/ $12.00 6 I996 Elsevier Science B.V. All rights reserved SSD10030-401 X(95 100570-6

we will in particular recover the results of the previous investigations [ 4-61 and prove that most of the scaling relations developed to treat the free space FEL can be exploited to characterise the W.G. FEL. The plane of the paper is the following; in Section 2 we discuss low gain and inhomogeneous broadening effects. Section 3 is addressed to the W.G. FEL high gain problem, where we also derive approximant gain forms which can be exploited for practical purposes. The saturated regime is treated in Section 4, where we use the Pad6 point of view, developed in Ref. [ 71, to derive analytical formulae accounting for the gain dependence on the laser intensity. Section 5 is finally devoted to concluding remarks. 2. W.G. FEL gain The basic theory of a W.G. FEL has been developed in Refs. [ 2-51, according to which it can be modelled by using a pendulum like equation leading, in the hypothesis of linear regime, to a gain equation of the type 151:

536

G. Dattoli et al. /Optics Communications 123 (1996) 535-542

Table I Expauation of symbols ( 1+

F?)fF uh3N’

I A

Mode indices set

Radiation transverse section

W

Normalized frequency %

See Ref. [ 91

0”

Free-space frequency

Cut-off wavenumber

w( 0)

lnhomogeneous parameter d

P’J

& In

(Au/w)

(do/oh,

47r-N k

W=vW+plgc

where go is the small signal gain coefficient given in Table 1 and v(o) is the W.G. FEL detuning parameter specified below:

(2.4)

where Eis the relative energy shift. The gain depression due to the energy spread inhomogeneous broadening is therefore obtained convolving the gain function on a gaussian relative energy distribution with r.m.s. o,, thus getting [ 61:

(2.2) with /3: being the electron reduced longitudinal

velocity, L, the undulator length, k, = 27r/h, the undulator wave-number and A, the undulator period. Finally kR is the free-space natural wave-vector. Furthermore wave-number and frequency are related by the dispersion relation [ 81:

= -(k-t-k,,)

0 !!!

(2.3)

c

and wAis the cut-off frequency of the mode (for further details on symbols see Table 1). The effects of the inhomogeneous broadening are evaluated using a procedure analogous to that of the free-space [ 91. Considering the v(w) shift due to the off-energy electrons we find

t( 1 -t)

exp(iv(w)t)

0

2

k2=

;Re

Xexp

[

(VOJ)t)* 2

1’ dt

(2.5)

where --- 1 k

(2.6)

It is clear that PJ o) reduces to ps, i.e. to the vacuum case inhomogeneous broadening parameter, when

Table 2 High gain regime Pad6 expansion coefficients

I

2-

3

1

--I 3 25X8! 293

2 33

307607

4332X8! 30740245I

38 6997

489896 131988412234022823

G 5700 fi

5700

13029314366125200

J5zii&?

-

1

IO!x 5“ X 2h 204687

4% 38

6fi Jr;j

G. Daftoli et al. /Optics

Communications

123 (1996) 535-542

537

Specialising to the wave-guide case the approximant gain formula of Ref. [ lo], we write

(2.7) 0.2

0.4

0.6

0.6

1

1.2

1.4

1.6

w

Fig. 1. Gain as a function of the normalizedfrequencyw (see Table I)forh,=2.5cm,N=10,y=5,K=1,1=4A,a=O.5cm,6=0.385 cm. Dot line: inhomogeneous case (CL.= 0.8). Solid line: homogeneous case. Gain 40,

I

I

r

I

I

I

I

The degree of reliability of the above approximant is offered by Fig. 2 where EQ. (2.7) is compared with the exact gain formula and the agreement can be considered satisfactory. The advantage offered by EQ. (2.7) is that it allows the computation of the integral (2.5) in exact form. Replacing Eq. (2.7) in Eq. (2.5) and calculating the integral we find indeed:

a)

Xexp[i($$ -40

1 0

L 0.2

L 0.4

, 0.6

v, 0.6

I 1

I 1.2

I 1.4

40

,

I

r

I

I

I 1

I 1.2

1 1.4

’ f?’

-60 1 0

I 0.2

I 0.4

I 0.6

I 0.6

(w)12

-5)201~$$)~~]z

Y_y]

w 10 Xsin Y(W) 20+[7r/,4,(w)]2 1 Z’ [

I 1.6

w Gain

J20+[;;

$+u

S[d~), P,(W)1 =

(2.8)

bl

i 1.6

W

Fig. 2. Gain as a function of w. Solid line: exact formula. Dot line: approximantgivenby Eq. (2.7). (a) 6=0.385cm. (b) 6=0.45 cm.

& + 1 and when the mode is essentially free. The same comment holds for the function (2.1) . An idea of the effect on the p&w) factor on the gain is offered by Fig. I. It is qualitatively evident that when I*.~(w) # 0 the gain curve is broadened and the peak value is reduced. In the following we will analyse, non quantitatively, this aspect of the problem and point out the difference with the open cavity case. Crucial to this analysis will be a set of gain approximant forms, already discussed in Ref. [ 101 allowing an analytical treatment of the inhomogeneous broadening effects.

401 0

I

1

I

I

I

I

0.6

0.8

1

I.2

14

16

I

I

0.2

04

W

Gain 40ti

*0

L

I

I

I

I

I

I

1

I

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

W

Fig. 3. Inhomogeneouslybroadenedgain as a functionof w F. = 0.8. Solid line: convolution using the approximantgain function (2.8). Dot line: convolution using the exact gain formula (2.5). (a) b=0.385 cm. (b) 6=0.45 cm.

~738

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et al. /Optics

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(1996)

535-542

values we plot the gain (see Fig. 5). (iii) We compare the behaviour of Fig. 5 with the trial functions: G,..,x( w*, EL,) G MAX 0.45

1+1.7/.&(o*)’

=

J 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

uE Fig. 4. Normalized frequency position of the maximum inhomogeneous gain as a function of the parameter pCL,.

0

0.2

0.4

0.6

0.8

1.2

1

1.4

1.6

P&

Fig. 5. Value of the maximum of the gain as a function of the parameter @L,.Solid line: Eq. (2.8). Dot line: Eq. (2.9a). Dashed line: Eq. (2.9b). See text for further comments.

Error 0.05 l-----T -0.11 0

' 0.2

' 0.4

' 0.6

exp[

-O.~P,(W*)~~, (2.9a) G MAX

G MAX

tw*,

EL,)

=

(2.9b)

1+[1.7&F(O*)]’

where GM,, is the maximum homogeneous gain. Eqs. (2.9) are similar to those used to parametrize the open cavity FEL gain. The agreement between (2.9) and the exact dependence is rather good and the agreement is within few percent (see Fig. 6 where the relative error is represented). The function F( u* ) tends to unity when the W.G. FEL approaches the free-space configuration (typical values are reported in Fig. 6). However since in general F( W*) > 1, we can conclude that the W.G. FEL is slightly less sensitive to inhomogeneous broadening effects than the open cavity case. The results of this section confirm the analysis of Ref. [ 561 and in addition yield new practical formulae, which can be exploited in the design of a wave-guide FEL.

3. The high gain regime

1 0.8

1 1

1 1.2

1 1.4

I 1.6

FE

It is well known that a gain formula of the type ( 1.1) is valid for FEL operating with small signal gain coefficients, smaller than 30% [ 91. For larger values the high gain contributions should be included. The equation providing the optical field evolution in the high gain non saturated regime is given below [ 93 :

Fig. 6. Relative error between Eqs. (2.8) and (2.9) as a function of pL,.Solid line: errorbetween (2.8) and (2.9a). Dotlineerrorbetween (2.8) and (2.9b).

The level of accuracy of Eq. (2.8) is provided by the comparison between approximate and exact convolutions. An important information which can be derived from (2.8) is the behaviour of the maximum gain versus the parameter /.L,.Unlike the ordinary case the inhomogeneous broadening parameter is a function of k, we therefore proceed as follows: (i) Evaluate the position of the maximum gain as function of pE (see Fig. 4). (ii) In correspondence of these Fig. 3 showing

X

I

exp[-iv(w)+]

a(~-I’)

~‘dr’,

(3.1)

0

where a( 7) is the dimensionless optical field. The above equation is of the Volterra integro-differential type and can be integrated either exactly or using g, as expansion parameter. We have denoted by 7 a dimensionless time ranging from 0 to 1. The perturbative solution offers the possibility of evaluating the

G. Daftoli et al. /Optics

-5

I

,

0

0.2

I

I

I

0.4

0.6

0.8

,

1

I

1.2

,

1.4

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123 (1996) 535-542

The maximum gain versus g, is plotted in Fig. 8 which also shows that it can be reproduced by a function of the same type used for the free-space case (see e.g. Ref. [ 93 ) . According to the results of this section the waveguide high gain effects can be treated quite straightforwardly, further comments will be presented in the concluding remarks.

1.6

w Fig. 7. Gain behaviour versus w

in the high gain regime. Solid line:

4. The strong signal regime

go = 7. Dot line: gt, = IO.

deviation from the linear case when g, does not exceed 10. The gain is linked to the modulus of a( T) by the relation:

The previous sections have been devoted to the analysis of the W.G. FEL far from the saturation. Here we discuss how the gain is reduced by the growing optical signal. A quantity of noticeable importance within the context of saturated FEL behaviour is the saturation intensity [ 91, which in the present case is defined as:

(3.2) and keeping an expansion

up to gi we find

(4.1) As for open cavity FELs, as well as for conventional lasers, Is is equivalent to the optical intensity halving the small signal gain. According to Ref. [7] we can evaluate low gain strong signal evolution using a perturbative series in terms of the parameter (I is the optical field intensity) : x = Ills,

thus getting for the optical field at r= 1 an expression of the type (for an explicit form of the functions a,.,( V(O) ) see Appendix A while for the derivation seeRef. [7]):

where [ lo] (see also Appendix A) G,[v(w)]

= iexp[-

G[v’(w)]=

T]sin[%],

- +]coS[g],

-$exp[

Gain 15 _,

I

I=

(4.2)

10 -

12235.341504 5 -

X exp I_ -$$$I

sin[s],

(3.4) 0

The behaviour of the gain versus the frequency for different values of g, is given in Fig. 7. It is evident that when g,, increases also the high frequency component of the gain is affected by the high gain corrections and may reach significant values.

0

2

4

8

10

Fig. 8. Maximum of the gain as a function of the gain parameter go. Solid line: EZq.(3.3). Dot line: approximant given by 0.55 q0 + 0.048 (T&y

+0.003

(qoY.

s40

Gain

G. Dattoli et al. /Optics

0.004

1

I

I

,

I

I

I

Communications 123 (1996) 535-542

5. Concluding remarks

I

0.002

-0.006 t 0

, 0.2

I 0.4

I 0.6

I 0.6

I

I

I

1

1.2

1.4

1 1.6

W

Fig. 9. Gain saturation as a function of the normalized frequency w and for two different values of the intensity parameter X. Solid line: Y= I. Dot line: x = 3.

a( 1) =a(O)

Atv(o),xl

1+27rigo ~AbW,xl R

=a,.,[v(w)l

+a,.2i v(w) 1 x+al.3[4@>1x2.

(4.3)

Albeit the series is truncated at x2 its validity is limited toX < 1, the following two Padt approximants have been shown to be able to provide the behaviour of A even for large x values (x=4, 5 . . . i.e. strong saturation).

The analysis developed in this paper has shown that many relations, already used in the study of the open cavity FELs, can be exploited design a W.G. FEL. We have indeed proven that the gain degradation due to energy spread is reproduced by a simple Lorenzian and that the gain versus intensity scaling can be accounted for by a formula analogous to that providing saturation for conventional laser systems [ 121. The validity of these results has been checked with an ad hoc developed code. Regarding the saturation, we have limited the analysis to the low gain homogeneously broadened regime. Albeit these parts derive a careful treatment, a preliminar analysis has however shown that the inclusion of these effects does not provide significant variations with respect to the conclusions of the previous section. To give an example in the case of a W.G. FEL operating with non-negligible inhomogeneous broadening contributions, the gain saturation formula is still of the type (4.5), provided that GMAXis replaced by the maximum small signal gain including the inhomogeneous degradetion and Z, is corrected as follows: Zs(&o*))

=a,.,[~(~>3

In Fig. 9 we show the effect of the saturation on the W.G. FEL gain. Another important information is relevant to the maximum gain behaviour as a function of X. This information is provided by Fig. 10 and it is important to emphasise that the gain versus x trend is reproduced by an equation of the type: %+x(x)

z &Ax

1-

9

(5.1)

where a is a number depending on the cut-off frequency. The same conclusion holds for the high gain regime, namely the (4.5) is still valid, but GM,& and Z,are suitably modified to include high gain corrections. We have limited the considerations relevant to the inhomogeneous broadening without including emittantes effects. The inclusion of these terms, apart from complicating the relevant gain expression, do not pro-

4.21 dW)lX + 1-(~,.~t~(~)l~~,.2[~(~)1)~’ A[v(w),xl

=ZsJi&.S%

exp( - (WC) CtX ’

Gain

0.005

1

0.003

-

0.002

-

0.001 0

(4.5) which is clearly similar to that exploited for open cavity FELs (seeRef. [ 111).

0.5

1

1.5

2

2.5

3

3.5

X

Fig. 10. Behaviour of the maximum of the saturated gain as a function of the normalized intensity*. Solid line: Eq. (4.3). Dot line: approximated expression as indicated in Eq. (4.5).

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123 (1996) 53%S42

541

Table 3 Summarizing schemeof the exact and approximatedformulae Exact fomula Homogeneousgain

Pad* approximations

G=TrgoF(~(w)) F(“(w))=

a - k+ll_okR c av(w)

High gain

exp[ -iv(w)+]

Saturation

Not

a(~-?‘)

existing

vide any further conceptual progress. The gain reduction is still controlled by parameters of the type (2.6) and their effect on the gain is slightly less severe than the free space case. Finally in Table 3 we have summarized the main formulae used in the previous sections making a comparison between the exact FFL equations and the one obtained with the PadC expansions. Appendix

Y db

approximated) provided by:

1 3v(w)6

Gz[v(w)l=

A

1 1;3[V(@)I

Eq. ( 1.3) is a Volterra integro-differential equation, it can be solved perturbatively by means of a Neumann expansion using g, as perturbation parameter. The congruence of the solution is ensured for any g, value. For go< 10 expansion up to go3 is sufficient and the gain can be cast in the form (3.3) with the exact (non

G, [ .Y( W) ] and G3 [ v ( o) ] functions

=

6Ov( 01)~

X(11520(1-cos(v(w)])-9OOOu(w)sin[v(o)] +360v(w)2cos[v(o)2]

+480v(w)*sin[v(w)]

+20v(w)4(1+2cos[v(w)])+~(w)5sin[v(w)]], (A.1)

542

G. Dattoli et al. /Optics

The functions amplitude have c= 100/1.284):

a,,S specifying the following

the complex form (s=2,3

Communications

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123 (1996) 535-542

[21 A. Amir, 1. Boscolo and L.R. Elias, Phys. Rev. A 32 (1985) 2864. [31 SK. Ride, R.H. Pantell and J. Feinstein, Appl. Phys. Lett. 57 (1990) 1283. 141 A. Doria, G.P. Gallerano (1991) 417.

and A. Renieri, Optics Comm.

I51 R. Bartolini, A. Doria, G.P. Gallerano Instr. and Meth. A304 (1991) 417. 161 F. Ciocci,

G. Dattoli, L. Giannessi

80

and A. Renieri, Nucl. and A. Terre, IEEE J.

Quantum Electron. 30 ( 1994) 180.

- iBI.1 v4exp( -ia,,,u(w))

ex

The various parameter are given in Table 2 and derivation of the Eq. (A.2) can be found in Ref. [ 7 1.

171 G. Dattob, L. Giannessi, P.L. Ottaviani and A. Segreto, Free electron laser saturation: an analytical description, Phys. Plasma, to be published. [81 R.E. Collin, (McGraw-Hill,

Foundations for New York, 1992).

microwave

Engineering

[91 G. Dattoli, A. Renieri, A. Terre, Free Electron Laser theory and related topics (World Scientific, Singapore,

1993).

[ IO] G. Dattoli, P.L. Ottaviani, A. Segreto and G. Ahobelli, J. Appl. Phys. 77 (1995) 6162.

References 1I ) R.L. Elias

and J.C. Gallardo, Appl. Phys. B 31 (1983) 229; F. Ciocci, R. Bartolini, A. Doria, G.P. Gallerano, E. Giovenale, M.F. Kimmitt, G. Messina and A. Renieri, Phys. Rev. Lett. 70 (1993) 928; M. Asakawa et al., Nucl. Instr. and Meth. A 341 (1994) 72.

[ 11 I G. Dattoli, B. Fatz, L. Giannessi and P.L. Ottaviani, J. Appl. Phys. 76 (1994)

2598.

[ 121 W.W. Rigrod, IEEE J. Quantum Electron. QE-14 (1978) 377.