The 14th Symposium on Accelerator Science and Technology, Tsukuba, Japan, November 2003

Analysis of an FEL oscillator at zero detuning length of an optical cavity in theearly stage of the field evolution
Nobuyuki Nishimori∗ , JAERI, Ibaraki, Japan
Abstract
The evolution of the field of an FEL oscillator at zero detuning length of an optical cavity (δL = 0) is studied analytically. The field on the leading edge at the first round trip isthe same as that of a self-amplified spontaneous emission(SASE) FEL characterized by an FEL parameter ρ. The
field evolves with round trips by interaction with electrons.
The field in the early stage of the evolution is found to scalewith the FEL parameter ρ and the round-trip number, andis similar to that of SASE with high electron beam density.

INTRODUCTION
An experiment in the Japan Atomic Energy Research Institute (JAERI) has shown that an efficiency of an FEL oscillator can become maximum at the zero detuning lengthof an optical cavity (δL = 0) despite the lethargy effect[1]. The sharp increase of an efficiency near zero detuning
length has been also observed in BINP or KAERI, recently[2]. A time dependent simulation including shot-noise effects has successfully reproduced the efficiency detuning
curve obtained in the experiment in JAERI-FEL [3], but
the physics responsible for the FEL at δL = 0 has not beenclearly proposed yet. A few theoretical studies have followed the experiment and attributed the sideband instability [4] or the superradiance in short-pulse FELs [5] to thelasing at δL = 0. However, those studies are still based on
numerical simulations.
The main difference of FELs between δL = 0 andδL < 0 is whether incident electrons interact with the fieldcharacterized by the steep intensity gradient on the leading edge similar to SASE [6]. At δL < 0, an optical field
is pushed forward with round trips, and electrons interact
with a field much stronger than that on the leading edge at
δL = 0. An analysis of the interaction between the field
similar to SASE and electrons has been performed recently[7]. The analysis shows that intense few-cycle FELs are
generated as a result of intensive energy transfer from electrons to the field at the peak at δL = 0. The present study
focuses on the early stage of the field evolution at δL = 0,
and shows that the field scales with the FEL parameter ρ[8] and the round-trip number n during the evolution.

and aw are undulator period and parameter respectively. In
order to deal with few-cycle fields, I choose unity for thenumber of undulator periods, through which electrons passin a scaled time, instead of 1/(4πρ) [8] or the total number of undulator periods Nw [10]. The fundamental FEL
parameter ρ [8] is defined by(1)
ρ = [eaw F ne /(0 m)/(4ckw )]2/3 /γ0in MKSA units, where ne is the electron beam density andF is unity for a helical undulator or Bessel function [JJ]
for that of planar type [8]. I define dimensionless time byτ = ct/λw , dimensionless optical field bya(ζ, τ ) =2πeaw λw FE(ζ, τ ) exp[iφ(ζ, τ )]γ0 2 mc2(2)

with phase φ(ζ, τ ), and dimensionless beam current by[4πρ(ζ, τ )]3 as similar to Ref. [10]. Here E(ζ, τ ) is therms optical field strength. The longitudinal position, dimensionless energy and phase of the i-th electron are respectively defined by ζi (τ ) = [zi (t) − ct]/λ, µi (τ ) =4π[γi (t) − γ0 ]/γ0 and ψi (τ ) = (kw + k)zi (t) − ωt. In
the present definition, the electron dynamics is represented
by the following pendulum equations [10]:dµi (τ )= 2|a[ζi (τ ), τ ]| cos{ψi (τ ) + φ[ζi (τ ), τ ]},dτ(3)dψi (τ )/dτ = µi (τ ).

(4)

The evolutions of FEL phase and amplitude are given by[10]∂φ(ζ, τ )/∂τ = [4πρ(ζ, τ )]3 /|a(ζ, τ )|×sin{ψi (τ ) + φ[ζi (τ ), τ ]}ζi =ζ ,(5)|a(ζ, τ )|/∂τ = −[4πρ(ζ, τ )]3 ×
cos{ψi (τ ) + φ[ζi (τ ), τ ]}ζi =ζ .

(6)

The angular bracket shows an average over electrons
around ζ within λ along the propagating direction [9].

FIELD AT THE 1ST ROUND TRIP
BASIC EQUATIONS
The present study is performed under the slowly varying envelope approximation (SVEA) [9]. The initial electron energy γ0 mc2 is assumed to be resonant for radiation
wavelength λ = λw (1 + a2w )/(2γ02 ), where λw = 2π/kw∗ nisi@milford.tokai.jaeri.go.jp

A field with uniform phase over length of N λ is formed
from an initial incoherent field after passage of electrons
through N undulator periods [11]. This formation corresponds to the spectrum narrowing in the frequency domain [6, 12], and directly leads to a backward coherent
state. The average of the coherent field at time τ is approximately given by the solution of the cubic equation

The 14th Symposium on Accelerator Science and Technology, Tsukuba, Japan, November 2003

1

for an input weak field with uniform amplitude and phasea(0) = |a(0)|eiφ(0) [10]:principal peakleading edge

0.01

a(ζ)

=

0.0001

front
edge

|a|

The coherent field is ranged at ζ < −τ , and corresponds
to the steady state regime described in Ref. [8]. Since theevolution of the field at ζ = −τ stops at time τ due to theslippage of electrons, the field at ζ ≥ −τ is given by

|a|

5π

π

a(ζ, τ ) = [a(0)/3][exp(4πρτ ei 6 ) + exp(4πρτ ei 6 )π(7)+exp(4πρτ e−i 2 )].

1e-06
ζp

0

1e-08

ζp

0

incident electron

5ππ[a(0)/3][exp(−4πρζei 6 ) + exp(−4πρζei 6 )π(8)+exp(−4πρζe−i 2 )].

This field corresponds to the leading edge described in Ref.
[8]. In the high gain regime defined by −ρζ > 0.1 or2(−4πρζ)3 > 4, the field on the leading edge is approximately given byπa(ζ) = [a(0)/3][exp(−4πρζei 6 ).

(9)

The phase of a(ζ) is given byφ(ζ) = φ(0) − 2πρζ.

(10)

FIELD AT THE N TH ROUND TRIP IN
THE EARLY STAGE OF THE EVOLUTION
The output field at the first round trip in an oscillator is
identical to that of SASE given by Eq. (9), and becomesan input field for the 2nd round trip. The head of a roundtrip FEL coincides with that of an incident electron pulse
at the entrance to an undulator at δL = 0. Therefore astudy of the interaction with the field given by Eq. (9) andelectrons is indispensable to understand the FEL evolutionat δL = 0.
Figure 1 shows a semi-log plot of an FEL amplitude atsaturation as a function of longitudinal position in units ofresonant wavelength λ. This is obtained in a time dependent simulation. The right hand side shows the front edgeof the FEL field. The zero is the position of the head ofincident electrons at the entrance to an undulator. The position of the principal peak of the field is represented asζp . The region from 0 to ζp is the leading edge. The inset
shows the linear plot of the amplitude.
I assume that the output field on the leading edge at theround-trip number n is the same as that of SASE with FEL
parameter ρn . The assumption is exact at the 1st roundtrip of an oscillator where ρ1 = ρ. The validity of theassumption for the subsequent round trip will be confirmed
later. Under the assumption, the field on the leading edge
is given byan (ζ) = [an (0)/3] exp[−4πρn ζeiπ/6 ].

(11)

The phase of the ith electron remains almost unchanged
during the interaction with the field given by Eq. (11):ψi (τ ) ≈ ψi (0). This is because the field on the leading edge is weak as shown in Fig. 1 and the synchrotron

Figure 1: Semi-log plot of an FEL amplitude |a| at δL = 0
after saturation with respect to the longitudinal position ζ(solid line) together with an electron pulse at the entrance
to an undulator. The positions of the front edge and theprincipal peak are 0 and ζp , respectively. The inset is alinear plot of |a|.
oscillation of electrons occur at ζp . The energy modulation given to the ith electron at τ  is represented by∆µi (τ  ) = 2|an [ζi (τ  )]| cos{ψi (0) + φn [ζi (τ  )]}∆τ  from
Eq. (3). The phase modulation at τ due to ∆µi (τ  ) is givenby ∆µi (τ  )(τ − τ  ) from Eq. (4). The phase of the ithelectron is derived from the sum of those phase modulations during τ : τ√ψi (τ ) = ψi (0) + 2|an [ζi (0)]|e2π 3ρτ ×
0cos{ψi (0) + φn [ζi (τ  )]}(τ − τ  )dτ  .(12)Equation (12) can be used when the gain of the field is
low and the field remains almost unchanged during passthrough an undulator. Substitution of Eq. (11) into Eq.
(12) yields
ψi (τ ) = ψi (0) + {|an [ζi (τ )]|/(8π 2 ρ2 )}×
cos{ψi (0) + φn [ζi (τ )] − π/3}√

−e−2π 3ρτ cos{ψi (0) + φn [ζi (0)] − π/3}√
−4πρτ e−2π 3ρτ cos{ψi (0) + φn [ζi (0)] − π/6} . (13)The phase shift and gain of a(ζ, τ ) due to an electron micro-bunch in units of λ, whose initial positions arearound ζ + τ , are obtained by substituting Eq. (13) into
Eqs. (5) and (6) respectively as follows:∂φn (ζ, τ )/∂τ = 4π(ρ3 /ρ2n ) ×√{1/2 − e−2π 3ρn τ [cos(2πρn τ + π/3) +(14)
4πρn τ cos(2πρn τ + π/6)]},[∂|an (ζ, τ )|/∂τ ]/|an (ζ, τ )| = 4π(ρ3 /ρ2n ) ×√√{ 3/2 − e−2π 3ρn τ [sin(2πρn τ + π/3) +(15)
4πρn τ sin(2πρn τ + π/6)]},
when |an (ζ)|/(8π 2 ρn 2 )  1 is satisfied or synchrotron
oscillation does not occur.

The 14th Symposium on Accelerator Science and Technology, Tsukuba, Japan, November 2003

7

The field gain per round trip due to electrons[∂an (ζ)/∂n]/an (ζ)=[∂|an (ζ)|/∂n]/|an (ζ)| +i[∂φn (ζ)/∂n] is derived from integrations of Eqs.
(14) and (15) from 0 to −ζ as follows:iπ/6) − 2(1 − e4πρn ζe

iπ/6

)].

(16)

[∂an (ζ)/∂n]/an (ζ) = (ρ/ρn )3 (−4πρn ζeiπ/6 − 2), (17)for ρn ζ < −0.2. The differentiation of Eq. (11) by n gives
the field evolution per round trip:(18)

When the gain from electrons represented by Eq. (17) ismuch higher than α/2, the gain is equal to the field evolution given by Eq. (18). This leads to the evolution of ρn as
a function of n:ρn ≈ ρ(3n − 2)1/3 .

4Eq. (19)3

amplitude
phase

2

Equation (16) is asymptotically equal to[∂an (ζ)/∂n]/an (ζ) = −4πeiπ/6 (∂ρn /∂n)ζ.

5
ρn/ρ

[∂an (ζ)/∂n]/an (ζ) = (ρ/ρn )3 [−4πρn ζeiπ/6 ×(1 + e4πρn ζe

6

(19)

Equation (19) confirms the assumption that the field on theleading edge at the round-trip number n is the same as thatof SASE with FEL parameter ρn . As the slope of the amplitude increases with ρn , the peak intensity on the leadingedge increases and the pulse length becomes short.

COMPARISON WITH A CALCULATION
The gain of the field at the 2nd round trip is given by(−4πρζeiπ/6 − 2) from Eq. (17) together with ρ1 = ρ.
The amplitude gain and phase shift amount to 0.5 and 1.4,
respectively, for −ρζ = 0.23 or 2(−4πρζ)3 = 50. In thecase of an oscillator with higher gain, the gain and phaseshift become grater than the above values. It is therefore
unclear whether the above gain and phase shift are consistent with the assumption used for derivation of Eq. (12) thatthe field remains unchanged during the evolution. The consistency will be confirmed, if Eq. (19) agrees well with ρnobtained from a time dependent calculation using Eqs. (3),(4), (5) and (6). The calculation for an input field given by
Eq. (9) with ρ = 0.0044 is performed under an assumptionthat an optical cavity loss is 0 and the amplitude |a(0)|atζ = 0 is constant. The parameter ρn at ζ is obtained fromthe slope of the amplitude or phase of the calculated fieldat the round-trip number n. The ratios of ρn to ρ obtained
from those slopes at ζ = −0.23 with respect to n are shownin Fig. 2. They agree well with Eq. (19). The calculations
with input fields with different ρs show also similar results.

CONCLUSION
The field on the leading edge at the 1st round trip is identical with SASE and is given by the exponentially increasing term of the solution of the cubic equation. The FEL

1
10

20

30

40

50

60

70

80

90

100

round-trip number

Figure 2: The ratio of ρn to ρ as a function of the roundtrip number n. The solid line shows Eq. (19). The dashedline and dash-dotted line are derived from the slopes of theamplitude and phase of the calculated field, respectively.
interaction between the field similar to SASE and electrons can be analyzed under an assumption that the field remains almost unchanged during the evolution. The analysis
shows that the field on the leading edge scales with the FEL
parameter ρ and the round-trip number n when the gain ismuch smaller than an optical cavity loss. The validity of theassumption that the field remains almost unchanged during
pass through an undulator is also confirmed.

REFERENCES[1] N. Nishimori, R. Hajima, R. Nagai, and E.J. Minehara,
Phys. Rev. Lett. 86 (2001) 5707; Nucl. Instr. and Meth. A
483 (2002) 134.
[2] N. Vinokurov et al., in Proceedings of the 25th International
Free-Electron Laser Conference, Tsukuba, 2003; Y. Jeong etal., ibid.
[3] R. Hajima, N. Nishimori, R. Nagai, and E.J. Minehara,
Nucl. Instr. and Meth. A 475 (2001) 270.
[4] Z.-W. Dong et al., Nucl. Instr. and Meth. A 483 (2002) 553.
[5] Ryoichi Hajima and Ryoji Nagai, Phys. Rev. Lett. 91 (2003)
024801.
[6] Kwang-Je Kim, Phys. Rev. Lett. 57 (1986) 1871; Nucl. Instr.
and Meth. A 250 (1986) 396.
[7] Nobuyuki Nishimori, in Proceedings of the 25th International Free-Electron Laser Conference, Tsukuba, 2003.
[8] R. Bonifacio et al., Riv. Nuovo Cimento 13 (1990) 9.
[9] W.B. Colson and S.K. Ride, Phys. Lett. 76A (1980) 379.
[10] W. B. Colson, in Laser Handbook, edited by W. B. Colson,
C. Pellegrini, and A. Renieri (North Holland, Amsterdam,
1990), Vol.6, pp. 115 – 193.
[11] N. Nishimori, R. Hajima, R. Nagai and E.J. Minehara, Nucl.
Instr. and Meth. A 507 (2003) 79.
[12] C.A. Brau, Free-Electron Lasers (Academic, San Diego,
1990).