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Comparison of grating and wiggler based free-electron lasers
Nuclear
2s . __ _.
Instruments
and Methods in Physics Research
NUCLEAR INSTRUMENTS 8 METHODS IN PHVSICS RESEARCH
A 375 (1996) 257-259
-_
ll!B
SectIon A
ELSEVIER
Comparison
of grating and wiggler based free-electron
lasers
Kenneth J. Woods*, John E. Walsh Department
of Physics
and Astronomy.
There has been considerable theoretical interest in a grating based free-electron laser (FEL) over the last decade 11.21. The motivations have been the ease of construction and a significantly smaller cost than the magnetic field wiggler counterpart. This paper will compare the spontaneous emission rates from the two devices and discuss the feasibility of a grating based far-infrared (FIR) FEL. Since the spontaneous emission from an electron beam passing over a grating was first observed in 1953 by Smith and Purcell [3], a device incorporating feedback can be referred to as a Smith-Purcell FEL. A calculation of the spontaneous radiation based on a grating structure comprised of a series of metallic strips has been reported previously [4]. In this model the surface charge on the grating structure is approximated by modulating the surface charge induced by an electron on an infinitely conducting sheet with a square wave. The spontaneous emission is calculated from the expression for the radiated power from an arbitrary current density [5]
Dartmouth
College.
Hanover.
NH 03755. USA
from the top of the grating surface to the electron. coupling length A, is defined by
h=PYA
(3)
47F ’
c
where p is the velocity of the 1 r n in units of c, y is the relativistic factor y = Y=+ l/(1 - @-). and A is the operating wavelength. The expression for the radiated power can be maximized in the relativistic limit for the angle corresponding to cos(0) = emission j/(y - I)l(y + I ) [7]. In this limit the emission goes as dl dw
eZ em%‘A, -
eOc
-~Rn~2+.
(4)
2
A similar calculation can be performed for the radiation from a long wiggler. The electron’s velocity in a wiggler can be computed and when the cross products with the normal vectors are performed, keeping terms of order 1 ly yields (Ref. [8], p. 69) n^ X fi X u = i d5 f uw sin(k,z) ,
X J exp[io(t
- ri *r/c)] 1
(1)
and is found to depend on the square of the Fourier coefficients of the square wave expansion. A more rigorous numerical calculation based on the method of van den Berg [6] produces the same result with the Fourier coefficients being replaced by the radiation coefficient for the grating. For a long grating, the expression for the radiated power from a single electron in the plane which contains the electron and is perpendicular to the grating surface is given by
(2) where 0 is the emission angle measured from the forward direction, w is the radiated frequency, E,, is the permittivity of free space, c is the speed of light, and x0 is the distance * Corresponding
author.
0168-9002/96/$15.00 Copyright SSDI 0168.9002(95)01234-6
The
where k, is related to the wiggler period A, = 2rlk_. wiggler parameter a, is given by
(5) The
eBh, a,=-
27rmc
’
where B is the strength of the wiggler’s magnetic field and is the electron rest mass. Since the current density for an electron passing through the wiggler is m
J = e&(r), the integration in Eq. (1) only involves the time integration and results in dl we’ _=_--a2. dw 167r1& The resonance 0=
27FL /3y’
(6)
w
condition
for a wiggler
4ay2c A,( 1 + u; + y’e’)
can be invoked and Eq. (6) becomes 0 1996 Elsevier Science B.V. All rights reserved IV. LONG WAVELENGTH
FELs
2.58
K.J. Woods. J.E. Walsh I Nucl. Instr. and Mrth. in Phw. Res. A 375 (1996) X7-X9
dl
er
dw
E”C 1 +a:
a:.
1
L
+ y’@ 27rp A,.
(7)
Inspection of Eqs. (4) and (7) reveal many similarities, and the grating factor can be seen to correspond to the wiggler parameter
(8) when the electron beam is within the evanescent field length for the grating. The similarities in the spontaneous emission rates strongly suggests that the stimulated emission rates could be made comparable with the two devices. There have been numerous Smith-Purcell FEL resonator configurations proposed in the literature which incorporate a feedback mechanism, however to our knowledge none of these resonators have ever been constructed to measure the cavity quality. A possible new configuration illustrated in Fig. 1 is one which has been in use for years in tunable dye lasers, known as the Littman-Metcalf resonator [9]. A dye laser is similar to a Smith-Purcell source in that they both have a large spectral range of emission. The resonator is constructed such that the cavity length, an integral multiple of half wavelengths, and the grating equation are consistent. Incorporating the Smith-Purcell relation results in the relationship 1lp = 2 sin(i9, ) + sin(&) ,
so a solution only exists for /? >f. In the relativistic region, p doesn’t change appreciably as a function of energy so the angle of the tuning mirror will determine the operating wavelength. If the Smith-Purcell emission angle is small, then the wavelength is a rapidly varying function of angle and a large frequency range can be covered with small changes in tuning angle. In order to make estimates of stimulated emission rates the equations for the gain length and the Pierce parameter for wiggler based FELs can be used with IR,l* replacing a’,. The gain length Lr and the Pierce parameter p are related by
zeroth
with p defined as (9) where 1, = 17 kA is the Alfven current and 2 is the mode area. In a practical device operating in the far-infrared, the mode area will most likely be diffraction limited by the optical mode. Using Gaussian optics allows Eq. (9) to be rewritten in favor of the operating wavelength and overall cavity length Lc as
(10) For the operating parameters of I = IO A. 1= 1 mm, y = 3 (I MeV), IR,I’= 1.0, A=500 km, and L, = 10 cm the gain length is calculated to be L, = 35.3 mm. Since the cavity length is about three gain lengths, a substantial amount of gain is expected. Typical wigglers have an a W- 1 with a wiggler period A, - 1 cm, resulting in a gain length which is more than two times longer than the gain length for the grating based device. In the grating based device described above the electron beam focus would need to be on the order of the coupling length A, = hpyl 47~ = 0.11 mm. Operation at a shorter wavelength could be achieved by increasing the beam energy. A 5 MeV device operating at a wavelength of 100 Pm and an IRnl” = 0.2 would have a gain length on the order of the 10 cm cavity length. These numbers indicate that investigation of a Smith-Purcell free-electron laser is warranted and that the possibility of a grating based coherent tunable source of far-infrared radiation exists in the future. There are currently two research programs, one at Stanford [lo] and one at CREOL [ 111, which are trying to develop tunable far-infrared source based on permanent magnet wigglers. The Stanford project is using an RF injector with an energy of 3.2 MeV to develop a picosecond source while the CREOL project uses a tandem Van de Graaff for developing a continuous wave source. Development of a grating based device could offer similar performance at a fraction of the cost for a wiggler based device.
References
Order Reflection
Ill J.M. Wachtel, J. Appl. Phys. 50 (1979) 49. I21 L. Schachter and A. Ron. Phys. Rev. A 40 (1989) 876. 131 S.J. Smith and E.M. Purcell, Phys. Rev. 92 (1953) 1069. r41 J. Walsh. K. Woods and S. Yeager. Nucl. Instr. Meth. A 341 (1994) 217.
Fig. 1. The Smith-Purcell calf design.
resonator
based on the Littman-Met-
ISI J.D. Jackson, 1975).
Classical
Electrodynamics,
2nd Ed. (Wiley,
K.J. Woods, J.E. Walsh
I Nucl. Instr.
[6] P.M. van den Berg, J. Opt. Sot. Am. 63 (1973)
and Meth.
1588. [7] J.E. Walsh, private communication, 1995. [8] CA. Brau. Free Electron Lasers (Academic Press, 1990). [9] M.G. Littman and H.H. Metcalf, Appl. Opt. 17 (1978) 2224.
in Phys. Res. A 375 (1996)
257-259
259
[lo]
Y.C. Huang, J.F. Schmerge. J. Harris. G.P. Gallerano. R.H. Pantell and J. Feinstein, Nucl. Instr. and Meth. A 318 (1992) 765. [I I] L.R. Elias, I. Kimel, D. Larson and D. Anderson, M. Tecimer and Z. Zhefu. Nucl. Instr. and Meth. A 304 (1991) 219.
IV LONG WAVELENGTH
FELs
2s . __ _.
Instruments
and Methods in Physics Research
NUCLEAR INSTRUMENTS 8 METHODS IN PHVSICS RESEARCH
A 375 (1996) 257-259
-_
ll!B
SectIon A
ELSEVIER
Comparison
of grating and wiggler based free-electron
lasers
Kenneth J. Woods*, John E. Walsh Department
of Physics
and Astronomy.
There has been considerable theoretical interest in a grating based free-electron laser (FEL) over the last decade 11.21. The motivations have been the ease of construction and a significantly smaller cost than the magnetic field wiggler counterpart. This paper will compare the spontaneous emission rates from the two devices and discuss the feasibility of a grating based far-infrared (FIR) FEL. Since the spontaneous emission from an electron beam passing over a grating was first observed in 1953 by Smith and Purcell [3], a device incorporating feedback can be referred to as a Smith-Purcell FEL. A calculation of the spontaneous radiation based on a grating structure comprised of a series of metallic strips has been reported previously [4]. In this model the surface charge on the grating structure is approximated by modulating the surface charge induced by an electron on an infinitely conducting sheet with a square wave. The spontaneous emission is calculated from the expression for the radiated power from an arbitrary current density [5]
Dartmouth
College.
Hanover.
NH 03755. USA
from the top of the grating surface to the electron. coupling length A, is defined by
h=PYA
(3)
47F ’
c
where p is the velocity of the 1 r n in units of c, y is the relativistic factor y = Y=+ l/(1 - @-). and A is the operating wavelength. The expression for the radiated power can be maximized in the relativistic limit for the angle corresponding to cos(0) = emission j/(y - I)l(y + I ) [7]. In this limit the emission goes as dl dw
eZ em%‘A, -
eOc
-~Rn~2+.
(4)
2
A similar calculation can be performed for the radiation from a long wiggler. The electron’s velocity in a wiggler can be computed and when the cross products with the normal vectors are performed, keeping terms of order 1 ly yields (Ref. [8], p. 69) n^ X fi X u = i d5 f uw sin(k,z) ,
X J exp[io(t
- ri *r/c)] 1
(1)
and is found to depend on the square of the Fourier coefficients of the square wave expansion. A more rigorous numerical calculation based on the method of van den Berg [6] produces the same result with the Fourier coefficients being replaced by the radiation coefficient for the grating. For a long grating, the expression for the radiated power from a single electron in the plane which contains the electron and is perpendicular to the grating surface is given by
(2) where 0 is the emission angle measured from the forward direction, w is the radiated frequency, E,, is the permittivity of free space, c is the speed of light, and x0 is the distance * Corresponding
author.
0168-9002/96/$15.00 Copyright SSDI 0168.9002(95)01234-6
The
where k, is related to the wiggler period A, = 2rlk_. wiggler parameter a, is given by
(5) The
eBh, a,=-
27rmc
’
where B is the strength of the wiggler’s magnetic field and is the electron rest mass. Since the current density for an electron passing through the wiggler is m
J = e&(r), the integration in Eq. (1) only involves the time integration and results in dl we’ _=_--a2. dw 167r1& The resonance 0=
27FL /3y’
(6)
w
condition
for a wiggler
4ay2c A,( 1 + u; + y’e’)
can be invoked and Eq. (6) becomes 0 1996 Elsevier Science B.V. All rights reserved IV. LONG WAVELENGTH
FELs
2.58
K.J. Woods. J.E. Walsh I Nucl. Instr. and Mrth. in Phw. Res. A 375 (1996) X7-X9
dl
er
dw
E”C 1 +a:
a:.
1
L
+ y’@ 27rp A,.
(7)
Inspection of Eqs. (4) and (7) reveal many similarities, and the grating factor can be seen to correspond to the wiggler parameter
(8) when the electron beam is within the evanescent field length for the grating. The similarities in the spontaneous emission rates strongly suggests that the stimulated emission rates could be made comparable with the two devices. There have been numerous Smith-Purcell FEL resonator configurations proposed in the literature which incorporate a feedback mechanism, however to our knowledge none of these resonators have ever been constructed to measure the cavity quality. A possible new configuration illustrated in Fig. 1 is one which has been in use for years in tunable dye lasers, known as the Littman-Metcalf resonator [9]. A dye laser is similar to a Smith-Purcell source in that they both have a large spectral range of emission. The resonator is constructed such that the cavity length, an integral multiple of half wavelengths, and the grating equation are consistent. Incorporating the Smith-Purcell relation results in the relationship 1lp = 2 sin(i9, ) + sin(&) ,
so a solution only exists for /? >f. In the relativistic region, p doesn’t change appreciably as a function of energy so the angle of the tuning mirror will determine the operating wavelength. If the Smith-Purcell emission angle is small, then the wavelength is a rapidly varying function of angle and a large frequency range can be covered with small changes in tuning angle. In order to make estimates of stimulated emission rates the equations for the gain length and the Pierce parameter for wiggler based FELs can be used with IR,l* replacing a’,. The gain length Lr and the Pierce parameter p are related by
zeroth
with p defined as (9) where 1, = 17 kA is the Alfven current and 2 is the mode area. In a practical device operating in the far-infrared, the mode area will most likely be diffraction limited by the optical mode. Using Gaussian optics allows Eq. (9) to be rewritten in favor of the operating wavelength and overall cavity length Lc as
(10) For the operating parameters of I = IO A. 1= 1 mm, y = 3 (I MeV), IR,I’= 1.0, A=500 km, and L, = 10 cm the gain length is calculated to be L, = 35.3 mm. Since the cavity length is about three gain lengths, a substantial amount of gain is expected. Typical wigglers have an a W- 1 with a wiggler period A, - 1 cm, resulting in a gain length which is more than two times longer than the gain length for the grating based device. In the grating based device described above the electron beam focus would need to be on the order of the coupling length A, = hpyl 47~ = 0.11 mm. Operation at a shorter wavelength could be achieved by increasing the beam energy. A 5 MeV device operating at a wavelength of 100 Pm and an IRnl” = 0.2 would have a gain length on the order of the 10 cm cavity length. These numbers indicate that investigation of a Smith-Purcell free-electron laser is warranted and that the possibility of a grating based coherent tunable source of far-infrared radiation exists in the future. There are currently two research programs, one at Stanford [lo] and one at CREOL [ 111, which are trying to develop tunable far-infrared source based on permanent magnet wigglers. The Stanford project is using an RF injector with an energy of 3.2 MeV to develop a picosecond source while the CREOL project uses a tandem Van de Graaff for developing a continuous wave source. Development of a grating based device could offer similar performance at a fraction of the cost for a wiggler based device.
References
Order Reflection
Ill J.M. Wachtel, J. Appl. Phys. 50 (1979) 49. I21 L. Schachter and A. Ron. Phys. Rev. A 40 (1989) 876. 131 S.J. Smith and E.M. Purcell, Phys. Rev. 92 (1953) 1069. r41 J. Walsh. K. Woods and S. Yeager. Nucl. Instr. Meth. A 341 (1994) 217.
Fig. 1. The Smith-Purcell calf design.
resonator
based on the Littman-Met-
ISI J.D. Jackson, 1975).
Classical
Electrodynamics,
2nd Ed. (Wiley,
K.J. Woods, J.E. Walsh
I Nucl. Instr.
[6] P.M. van den Berg, J. Opt. Sot. Am. 63 (1973)
and Meth.
1588. [7] J.E. Walsh, private communication, 1995. [8] CA. Brau. Free Electron Lasers (Academic Press, 1990). [9] M.G. Littman and H.H. Metcalf, Appl. Opt. 17 (1978) 2224.
in Phys. Res. A 375 (1996)
257-259
259
[lo]
Y.C. Huang, J.F. Schmerge. J. Harris. G.P. Gallerano. R.H. Pantell and J. Feinstein, Nucl. Instr. and Meth. A 318 (1992) 765. [I I] L.R. Elias, I. Kimel, D. Larson and D. Anderson, M. Tecimer and Z. Zhefu. Nucl. Instr. and Meth. A 304 (1991) 219.
IV LONG WAVELENGTH
FELs