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Radio-frequency-injector-driven Cherenkov free-electron laser
NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH
Nuclear Instruments and Methods in Physics Research A318 (1992) 772-774 North-Holland
Section A
Radio-frequency-injector-driven Cherenkov free-electron laser John E. Walsh and Emily E. Fisch
Department of Physics and Astronorm: Dartmouth College, 6127 Wilder Laboratory, Hanover, NH 03755-3528, USA The match between the electron beam parameters typical of radio-frequency injectors and the requirements for Cherenkov free-electron laser operation are explored. Conditions for high-peak-gain, high-peak-power and short-pulse operation at far-infrared wavelengths are defined. i. Introduction
The parameters of typical radio-frequency-driven injectors [11 are ideally matched to Cherenkov freeelectron-laser (C-FED requirements [2]. Beam energies may range from 0.5 to 5 MeV, the current is a series of pulses a few ps long, each containing a charge in the nC range, and the beam quality can be outstanding. Taken together, these characteristics will produce high gain at far-infrared (FIR) wavelengths in a C-FEL resonator. In comparison with other sources, the C-FEL has a number of advantages in the FIR region . It is difficult to extend operation of high-energy (25-50 MeV), beans-driven, wiggler-coupled FELs to this regime, and other approaches such as optically-pumped lasers lack the simultaneous capabilities needed for producing tunable, coherent, high-peak-power, short pulses of radiation. The properties of an injector-driven C-FEL, which does have this potential will be explored in more detail in the following sections of the paper.
The relative overlap of the beam and the field is illustrated on the upper portion of the drawing. On the lower portion, the match of the beam and the transverse profile of the mode is shown. Overlap of the electron beam and the optical mode must be maintained in order to efficiently use the electron beam's kinetic energy. The conditions required may be quantified with the following argument. The scale of the evanescence is set by the transverse wavenumber, q = %,k"
_ w, lc"
[cm
_~
l
where, as usual, k represents the axial wavenumber and m its angular frequency (c is the speed of light). E~ x.0)
evanescent field
2. Cherenkov free-electron laser theory A brief review of C-FEL theory will be used to highlight issues which become important when RF-injector-produced electron beams are used to drive CFEL resonators. The five central concerns are: coupling, tuning, gain, saturation, and the role of beam quality. These will be addressed in order. 2.1 . Coupling geometry
The basic geometry of the C-FEL coupling is illustrated in fig. 1. On the central portion, a section of the electron beam moving over a dielectric film waveguide is displayed. As the beam moves over the dielectric, it interacts with a surface wave which is evanescent along the direction normal to the film surface (the x-axis).
Fig. 1. Schematic view of the C-FEL interaction .
0168-9002/92/$05 .00 © 1992 - Elsevier Science Publishers B.V . All rights reserved
J. E. Wulsft, E. E. Fiscft / RF-injector-driven
The coupling between the beam and the wave peaks when the modal phase velocity and the beam velocity arc approximately equal. Hence the condition may be invoked (ß is the relative velocity of a beam electron and y = 1/ 11 - ß2 the relative energy of a beam electron). Eqs. (1) and (2) may be combined to relate the evanescence scale and the operating frequency : 9 = w/cßy [em -1 ]. It is then illuminating to introduce a dimensionless coupling factor j.,,.,, defined by
29o ,h., , where Qt,_, is the beam thickness. Invoking the usual relation between w and the free-space wavelength A, the parameters which appear in the coupling factor may be rearranged in the form : 4a tL h .t A
tLh, -_
In general, in order to make full use of the beam cross section, tL1, < 4, although at sufficiently high beam current density, operation with lit,_, > 4 has been demonstrated . The relationship defined by eq . (5) defines the beam thickness required in order to operate efficiently with a given beam energy at a chosen wavelength . 2.2. Tuning The tuning characteristics of the C-FEL are determined by the dispersion relation for the guided mode and the beam velocity. Detailed discussions of this point have appeared in earlier publications, and only brief comments will be included here . Expressing the dispersion relation in symbolic form by wd (6) D , kd =0 c (where d is the film thickness), a tuning relation cod (ad DT =0 c cß ) is defined with the aid of eq. (3). It is clear from the form of eq. (7) that for a given value of dielectric constant, the ratio d/A will be a universal function of the relative kinetic energy of the electron beam (y - 1). Thus the operating wavelength will scale with film thickness [2].
Cherenkot - FLL
773
lead to the same expression for dimensionless gain . The equations of motion may be written in a dimensionless form [3), which is the same for any free-electron laser. They are :
d`6 = -E sin ~, (g) dT` de j,,(Sln (9) dT where e =kz -wt (10) is the relative phase, T =ß ctlL is dimensionless time, and E L
ntc 2/e
(ar L/c)
(11)
(ß"Y y ) ;
is the dimensionless electric field (E is the electric field strength and L is the interaction length). When the beam is matched to the optical mode the drive current, in dimensionless form, becomes hnL, _ J ~hv L (12) gofc 1e rnC ; le Um (f where Jt,o = the beam current density, L = the interaction length, Q h ,.= the beam width, Qm = the width of the optical mode, g = the universal gain function, f~ = the coupling function. The universal gain function, g is given by 8.10 2- yT) (kd) kd E ' ß5y7
y 'Y` Ji where yT = E/(E - 1) is the square of the Cherenkov threshold energy. The expression for go is universal in that it may be expressed solely as a function of kd and thus it has the same properties with respect to scaling and self-similarity displayed by the dispersion relation for the surface waves. (Note that ß = ß(kd) and y = y(kd ) when eqs. (3) and (8) are invoked). At moderately relativistic beam energies the coupling factor, f., may be held near unity (for FIR and longer wavelengths). In order to estimate the parameters needed for a practical experiment, it is useful to estimate je. The demarcation je < 1 defines the boundary between the low and high gain regions.
2.3. Gain
2.4. Saturation and output power estimate
The gain of the C-FEL may be computed beginning from a variety of formally-different points of view. All
General considerations based on an estimate of the scale of beam velocity modulation, occurring at the IX . UNCONVENTIONAL SCHEMES
J. E. Walsh, E.E. Fisch / RF-injector-drit en Cherenkoi- FEL
774
onset of saturation gives for energy change the approximate expression :
); ay 13 A (ßy . -= ~ L (y-1)
(14)
This qualitative estimate is supported by more detailed examination of numerical integration of the equations of motion .
5 Beant quality In order to have the beam energy distribution fall within the gain line width, the beam energy spread must satisfy the condition aT T-1
AT(y+ 1 ) ' L
(15)
This is easily met with the projected operating parameters of a radio-frequency injector. Table I Projected experimental parameters for a RF-injector Cherenkov FEL y-1
Baa l~n~ A at ~. _ 4
d [Lm] A [lim] at ul, = 0.2 mm j` [cm-4 at JI,=100Afory-1=1 JI,=10kAfory-1=10
The general expressions given above may now be used to make specific projections for injector-driven C-FEL performance . These are summarized in table 1 .
1
7-3 1 .80
60 360
9.8
10
1.2
0.30 2 60 7.5
3. Conclusions It is clear from the estimated gain and output power levels listed on the table that an injector-driven C-FEL would be an interesting source of radiaton . In view of its potential for high-peak power, short-pulse operation at FIR wavelengths is would have many applications in transient spectroscopy of condensed matter and biophysical systems. Acknowledgements Vermont Photonics Inc., Westminster, Vt. and ARO Contract DAAL03-91-G-0189 are gratefully acknowledged for their support . References [1] K. Batchelor et al., these Proceedings (13th Int . Free Electron Laser ConE, Santa Fe, USA, 1991) Nucl. Instr. and Meth. A318 (1992) 372 . [2] J. Walsh, B. Johnson . G. Dattoli and A. Renieri, Phys. Rev. Lett. 53(8) (1980 779 . [3] C. Brau, Free Electron Lasers (Academic Press. San Diego, CA. 1990). and references therein .
Nuclear Instruments and Methods in Physics Research A318 (1992) 772-774 North-Holland
Section A
Radio-frequency-injector-driven Cherenkov free-electron laser John E. Walsh and Emily E. Fisch
Department of Physics and Astronorm: Dartmouth College, 6127 Wilder Laboratory, Hanover, NH 03755-3528, USA The match between the electron beam parameters typical of radio-frequency injectors and the requirements for Cherenkov free-electron laser operation are explored. Conditions for high-peak-gain, high-peak-power and short-pulse operation at far-infrared wavelengths are defined. i. Introduction
The parameters of typical radio-frequency-driven injectors [11 are ideally matched to Cherenkov freeelectron-laser (C-FED requirements [2]. Beam energies may range from 0.5 to 5 MeV, the current is a series of pulses a few ps long, each containing a charge in the nC range, and the beam quality can be outstanding. Taken together, these characteristics will produce high gain at far-infrared (FIR) wavelengths in a C-FEL resonator. In comparison with other sources, the C-FEL has a number of advantages in the FIR region . It is difficult to extend operation of high-energy (25-50 MeV), beans-driven, wiggler-coupled FELs to this regime, and other approaches such as optically-pumped lasers lack the simultaneous capabilities needed for producing tunable, coherent, high-peak-power, short pulses of radiation. The properties of an injector-driven C-FEL, which does have this potential will be explored in more detail in the following sections of the paper.
The relative overlap of the beam and the field is illustrated on the upper portion of the drawing. On the lower portion, the match of the beam and the transverse profile of the mode is shown. Overlap of the electron beam and the optical mode must be maintained in order to efficiently use the electron beam's kinetic energy. The conditions required may be quantified with the following argument. The scale of the evanescence is set by the transverse wavenumber, q = %,k"
_ w, lc"
[cm
_~
l
where, as usual, k represents the axial wavenumber and m its angular frequency (c is the speed of light). E~ x.0)
evanescent field
2. Cherenkov free-electron laser theory A brief review of C-FEL theory will be used to highlight issues which become important when RF-injector-produced electron beams are used to drive CFEL resonators. The five central concerns are: coupling, tuning, gain, saturation, and the role of beam quality. These will be addressed in order. 2.1 . Coupling geometry
The basic geometry of the C-FEL coupling is illustrated in fig. 1. On the central portion, a section of the electron beam moving over a dielectric film waveguide is displayed. As the beam moves over the dielectric, it interacts with a surface wave which is evanescent along the direction normal to the film surface (the x-axis).
Fig. 1. Schematic view of the C-FEL interaction .
0168-9002/92/$05 .00 © 1992 - Elsevier Science Publishers B.V . All rights reserved
J. E. Wulsft, E. E. Fiscft / RF-injector-driven
The coupling between the beam and the wave peaks when the modal phase velocity and the beam velocity arc approximately equal. Hence the condition may be invoked (ß is the relative velocity of a beam electron and y = 1/ 11 - ß2 the relative energy of a beam electron). Eqs. (1) and (2) may be combined to relate the evanescence scale and the operating frequency : 9 = w/cßy [em -1 ]. It is then illuminating to introduce a dimensionless coupling factor j.,,.,, defined by
29o ,h., , where Qt,_, is the beam thickness. Invoking the usual relation between w and the free-space wavelength A, the parameters which appear in the coupling factor may be rearranged in the form : 4a tL h .t A
tLh, -_
In general, in order to make full use of the beam cross section, tL1, < 4, although at sufficiently high beam current density, operation with lit,_, > 4 has been demonstrated . The relationship defined by eq . (5) defines the beam thickness required in order to operate efficiently with a given beam energy at a chosen wavelength . 2.2. Tuning The tuning characteristics of the C-FEL are determined by the dispersion relation for the guided mode and the beam velocity. Detailed discussions of this point have appeared in earlier publications, and only brief comments will be included here . Expressing the dispersion relation in symbolic form by wd (6) D , kd =0 c (where d is the film thickness), a tuning relation cod (ad DT =0 c cß ) is defined with the aid of eq. (3). It is clear from the form of eq. (7) that for a given value of dielectric constant, the ratio d/A will be a universal function of the relative kinetic energy of the electron beam (y - 1). Thus the operating wavelength will scale with film thickness [2].
Cherenkot - FLL
773
lead to the same expression for dimensionless gain . The equations of motion may be written in a dimensionless form [3), which is the same for any free-electron laser. They are :
d`6 = -E sin ~, (g) dT` de j,,(Sln (9) dT where e =kz -wt (10) is the relative phase, T =ß ctlL is dimensionless time, and E L
ntc 2/e
(ar L/c)
(11)
(ß"Y y ) ;
is the dimensionless electric field (E is the electric field strength and L is the interaction length). When the beam is matched to the optical mode the drive current, in dimensionless form, becomes hnL, _ J ~hv L (12) gofc 1e rnC ; le Um (f where Jt,o = the beam current density, L = the interaction length, Q h ,.= the beam width, Qm = the width of the optical mode, g = the universal gain function, f~ = the coupling function. The universal gain function, g is given by 8.10 2- yT) (kd) kd E ' ß5y7
y 'Y` Ji where yT = E/(E - 1) is the square of the Cherenkov threshold energy. The expression for go is universal in that it may be expressed solely as a function of kd and thus it has the same properties with respect to scaling and self-similarity displayed by the dispersion relation for the surface waves. (Note that ß = ß(kd) and y = y(kd ) when eqs. (3) and (8) are invoked). At moderately relativistic beam energies the coupling factor, f., may be held near unity (for FIR and longer wavelengths). In order to estimate the parameters needed for a practical experiment, it is useful to estimate je. The demarcation je < 1 defines the boundary between the low and high gain regions.
2.3. Gain
2.4. Saturation and output power estimate
The gain of the C-FEL may be computed beginning from a variety of formally-different points of view. All
General considerations based on an estimate of the scale of beam velocity modulation, occurring at the IX . UNCONVENTIONAL SCHEMES
J. E. Walsh, E.E. Fisch / RF-injector-drit en Cherenkoi- FEL
774
onset of saturation gives for energy change the approximate expression :
); ay 13 A (ßy . -= ~ L (y-1)
(14)
This qualitative estimate is supported by more detailed examination of numerical integration of the equations of motion .
5 Beant quality In order to have the beam energy distribution fall within the gain line width, the beam energy spread must satisfy the condition aT T-1
AT(y+ 1 ) ' L
(15)
This is easily met with the projected operating parameters of a radio-frequency injector. Table I Projected experimental parameters for a RF-injector Cherenkov FEL y-1
Baa l~n~ A at ~. _ 4
d [Lm] A [lim] at ul, = 0.2 mm j` [cm-4 at JI,=100Afory-1=1 JI,=10kAfory-1=10
The general expressions given above may now be used to make specific projections for injector-driven C-FEL performance . These are summarized in table 1 .
1
7-3 1 .80
60 360
9.8
10
1.2
0.30 2 60 7.5
3. Conclusions It is clear from the estimated gain and output power levels listed on the table that an injector-driven C-FEL would be an interesting source of radiaton . In view of its potential for high-peak power, short-pulse operation at FIR wavelengths is would have many applications in transient spectroscopy of condensed matter and biophysical systems. Acknowledgements Vermont Photonics Inc., Westminster, Vt. and ARO Contract DAAL03-91-G-0189 are gratefully acknowledged for their support . References [1] K. Batchelor et al., these Proceedings (13th Int . Free Electron Laser ConE, Santa Fe, USA, 1991) Nucl. Instr. and Meth. A318 (1992) 372 . [2] J. Walsh, B. Johnson . G. Dattoli and A. Renieri, Phys. Rev. Lett. 53(8) (1980 779 . [3] C. Brau, Free Electron Lasers (Academic Press. San Diego, CA. 1990). and references therein .