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The waveguide transmission of a free electron laser
NUCLEAR INSTRUMENTS &METHODS IN PHYSICS RESEARCH
Nuclear Instruments and Methods in Physics Research A318 (1992) 870-873 North-Holland
Section A
The waveguide transmission of a free electron laser Ming Chang Wang
Shanghai Institute of Optics and Fine Jtfechanics. Academia Sinica, PO Bar 8211, Shanghai, Peoples Republic of China
An approach using waveguide laser theory to treat the radiation transmission of free electron lasers is proposed . The investigation shows that there is a low loss mode in the cavity. The low loss mode is EH for a dielectric waveguide and TE , for a metallic waveguide. 1-he mode coupling loss is also discussed.
L int
coon
A free electron laser usually has a long drift tube with a small diameter. The radiation modes transmitted inside the tube can be neither treated by laser theory which discusses open cavities nor by microwave theory which considers only cavities with size matching the wavelength. Consider the cavity of a free electron laser. For a wavelength A = I Wm. a tube radius a = 5.0 mm, and a length L = 30 m. the factor a 2/AL is calculated to be 0.83. This is different from the design of a conventional cavity of a FEL [1]. A new approach using waveguide laser theory to treat the radiation transmission is proposed and analyzed in this paper. In 1964, E.A. Marcatili discussed hollow metallic and dielectric waveguides for long distance optical transmission and lasers [2]. New devices, e.g. the CO, waveguide laser, have been developed based on this theory [3]. Using the waveguide laser theory, we were able to treat the mode transmission of a free electron laser and also calculate the transmission loss. It is found that EH, i is the lowest loss mode in a dielectric waveguide with a refractive index v = 1 .50. The loss is about 1 .48 x 10-5 dB/m . TE , is the lowest loss mode in a gold guide with an attenuation of 5.37 x 10-4 dB/m. The guides act as single-mode filters for EH and TE ,. The mode coupling loss of a free electron laser cavity is discussed . This study is an extension of the earlier work on the cavity of a free electron laser [1]. 2. Theoretical model For simplicity, consider a waveguide consisting of a circular cylinder of radius a and free-space dielectric constant E as shown in fig. 1 .
The waveguide wall is either dielectric or metallic with a complex dielectric constant E. The magnetic permeability p. is assumed to be that of free space for both media. Although the transmission characteristics of metallic waveguides are well kr. jwn for microwave frequencies, the theory is invalidated for operation at optical wavelengths because the metal no longer acts as a good conductor but rather as a dielectric having a large E. Both the dielectric and metallic waveguides are considered as special cases of a hollow circular waveguide. The following conditions of radiation in a waveguide are assumed: Ka>>IvIU,, ,,,, ly/K- 1I < 1,
(1) (2) where K = ao E A = 2-rr/A is the free-space propagation constant, U , is the rnth root of the equation J _ , (U ,) = 0 with n and m integers characterizing the propagation mode, v = E/E is the complex refractive index of the external medium, and y is the axial propagation constant of the mode under consideration. The first inequality states that the radius a is much larger than the free-space wavelength A. If the external medium is a metal, the I v I value is large but finite at an optical frequency. The second inequality restricts our analysis to low-loss modes with their propagation constant y nearly equal to that of free space. The E")'0 --o
Z
2a
501-
Fig. 1 . Schematic of a cylindrical dielectric waveguide.
0168-9002/92/$05 .00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
r
of a FEL
Mingchang Wang / Wareguide transmission
radiation in these modes propagates within the tube, bouncing at grazing angles against the wall. The field components of hybrid modes EH , inside a waveguide can be written as [4] 0
E,r ,
x
v1
Ka
H.rnr =
E() All
J K-r sin(n0) ex 1/2
1
z
(Ki/K )` = I - (y/K )`.
-toi»,
From the condition (2) we know y/K = 1 and, therefore, K;/K = 0. The right-hand side of eq. (6) is close to zero. Using a perturbation technique and approximation, we have
.rnr
Et) )1/2 r = _ Hnm E,;,,, ctn t?0 ILO
The propagation constant y can be obtained from eq. (4) (3)
where the complex propagation constant y satisfies the relationship , K ; =K - y - , (4) n and m are the numbers of periods of each field component in the 0 and r directions, respectively . In a hollow dielectric waveguide, the EH hybrid mode has the lowest transmission loss. The six components of this mode are Eil =Jl,(Ul1a
Eil = -1 HdI1
Hr
__
cos 0 exp(i(yz-£ut)),
r
E = Jl,(U1 ,
sin 0 exp(i(yz -cot)),
U1 I
(Ka ) 1/2
EO
JI U11
(
r) a
sin 0 exp(i(yz -- oit »,
Er
11+
2
El) _ _ EB 11, 11 -
Ao El)
Hi, = - (110
) 1/2
1)1/,
Ki a = .(1 ., - iv /Ka). U
(-1£I)
r
-
2(v-'
Divide both sides of eq. (4) by K2 to get
r
Eu ) 1,/2
Hnrnr -
Under the conditions (1) and (2), the characteristic equation for the propagation constant is simplified to be
vu =
__I Jj Kir)
x sin(n0) exp(i(yz -cot )), E~
3. The propagation constant of waveguide modes
J,r -1(Kia) = iv r(Ki1K)J,(Kia ), (6) where for EH mode, cos(n0) cxp(i(yz - wt)),
i unnr J,r- 1(Kir) + /Ar
r Ernt
1371
Ej, ctn 0,
where 0 and r refer to the axial and radial components in circular cylindrical coordinates, respectively .
By neglecting the second order term, we have
=K
1-
1 -i 2 y., Ka
Ur,~A 2:ra
y=K 1 -
1 ( U mA 2
( 1 _, ,,,,A
2 7ra
ra
The phase and attenuation constants of each mode are the real and imaginary parts of y, respectively. ( )2[, 1 - 1 U n,A rA + Im y A 2 2.ra r~ a ) 2 A2 U"n, ~T Rc(v ) . amn = 2r
'
For guides made of glass, where v is usually real and independent of A, the phase and attenuation constants become 27r ßnm = A [ 1 "I ) 2 _ U" anm 2r
1 A..
U _ A `' ( 2ra
(10)
a 3 vn'
Using A=10.6Wm,a=5 mm, v=1.5andU =2.405 in eq. (10), we obtained all = 1.91 x 10 -4/m = 1 .66 x 10 - ' dB/m. (Note : 20 log(E/E (,) = dB = 20 log exp(a L) = a L x 8.686.) The attenuation of this mode is negligible compared with the gain of a free XI. FEL OPTICS
Mingchang Wang / Wareguide transmission of a FEL
2
Fig. 2. Attenuation of EH mode a" versus A (Lm) for a dielectric waveguide.
electron laser even in the low gain regime G = 0.37/m [5] . Eq. (10) shows that the attenuation constant is propotional to A~/a3. Consequently, the losses can be made arbitrarily small by choosing the radius of the tube, a, sufficiently large relative to the wavelength A. The attenuation of EH mode has been plotted in fig. 2 as a function of A for v =1.5, and a = 1, 5 and 10 mm. 4. The coupling loss of waveguide modes A laser resonator consists of two mirrors. The resonator mode must result in a reproduction of the original amplitude and phase distribution after a round trip propagation in the cavity . Consider the end of a guide as a source of radiation. If the phase front of the wave arriving at one of the mirrors matches the mirror surface profile, the coupling loss of the mode back to the waveguide is the smallest . At any position z from the end of the guide we can position a mirror of radius R. It might be expected that optimum coupling at position z would occur when the radius R of a quasi-Gaussian bean. is equal to the wavefront radius R' given by b
z '
where b = -rrwo/A is a confocal parameter of the quasi-Gaussian beam. For the case when R = R', each Gaussian mode is imaged back on the guide end. When the radius at the mode waist w() = 0.643a . the coupling loss is the lowest [6].
J.J. Degnan had discussed the coupling loss of finite-aperture waveguide-laser resonators [7] . There are three types of resonators with low coupling loss : (1) large radius of curvature mirrors close to the guide, (2) large radius of curvature mirrors whose center of curvature lies approximately at the guide entrance, (3) small radius of curvature mirrors whose focus is positioned close to the guide entrance. Clearly, 100% efficiency coupling is achieved by placing a flat mirror directly against the guide. This efficiency is degraded by replacing the flat mirror with a smaller radius of curvature mirror or by increasing its separation from the guide. The coupling loss of EH mode is given to a good approximation by the formula ( d ) 3/2 (12) a; = 6.05 Ka, where d is the separation of the flat mirror from the guide. In order to reduce this loss, we need to place the mirror close to the guide. However, the cavity of a free electron laser requires a space d for the electron beam to enter and exit from the waveguide . Type 1 resonator is not suitable . In type 2, the matching is more efficient for mirrors with large radius of curvature (small a) since the phase front becomes approximately spherical as shown in fig . 3. This is similar to a concentric cavity of a conventional laser where z = R. Type 3 is like a confocal cavity where z = R/2. Degnan derived the coupling loss of a EH waveguide mode: a P a 1 2QJ,(Uo, ) x X
I0tt -dw, exp i
pw ,
J (~ dwn exp i J t ` pwo 2~ l )(U04010) 2 2 XjO(O1~_ù_j()£01)~ 1
1
(13)
where p = ka2/R, Q = dIR and h --- c/a . R is the
2a
T
IMMEEZZZE, d Fig. 3. Geometry for calculating the coupling loss .
Mingchang Wang / Wareguide transmission ofa FEL mirror radius of curvature and c is the radius of mirrors . When the value of the product ph = Kac/R is equal to the second root of the Bessel function J (U) the mirror captures only the large central lobe of the EH far-field distribution (ph = 5.520). If the value of ph is equal to the third root of the Bessel function, the mirror captures the large central lobe and the first outer ring (ph = 8.853). The minimum coupling loss is 0.06 dB. 5. The mode loss in metallic waveguide
E°, =J,(Ki r) exp[i(yz-cut)], Hôl
_ -
Hö1 =
-i
Eo
) 1/2
luo Eo (_110)
Jt(Ki r) exp[i(yz -art )]
1/2
(14)
ut) i
exp[i(yz-cvt)] . Ka J,(Kir)
The attenuation constant for TEO, mode is Uo~ Re(v ), (15) ( 2,ir ) ~~ a where Re(v )=(v-'- 1) -1 / 2 , UO = 3.832, v=0.19i6.1 for gold at A = 1.0 W m, v = 1 .49 -122.2 at A = 4.0 pm and v = 7.41 -153.4 at A = 10.0 Wm. The circular electric mode has the lowest loss in the metallic waveguide, while the circular magnetic and hybrid modes are rapidly attenuated. The attenuation constant a O for the lowest loss TEO, mode is plotted in fig. 4 for wavelengths in the range 1 .0 gm < A < 10.0 pm for a = 1 .0, 5.0 and 10.0 mm. Those data show a considerably lower loss compared with the lowest loss mode EH ,, for the dielectric guide . The attenuation of TEO , mode for the gold guide with a radius a = 1 .0 mm and wavelength A = 1 .0 pm is only 5.37 x 10 -4 dB/m. For a radius of 5 mm, the minimum loss of TEO, mode is ao, = 4.29 x 10 -6 dB/m. The metallic waveguide is superior to the hollow dielectric waveguide for long distance optical transmisao, =
ar (dB/M ) 10-' r10-2
a=1mm
10-3
5mm
10-4 10-5
10 mm
10-6
The approximate field components of the TE , mode in a metallic waveguide are (
873
1
2
3
4
5
6
7
8
9 a(Ji)
10
Fig. 4. Attenuation of TE , mode a , versus A (wm) for a gold waveguide . sion. Because of the relatively large dielectric constant exhibited by gold at an optical frequency, the attenuation constant for the lowest loss mode TEo, is very small and less sensitive to the curvature of the guide axis. The conclusion is that the mode transmission of a free electron laser can be treated by waveguide laser theory. The transmission and coupling losses can be calculated easily. References [1] Mingchang Wang et al., Chinese J. Lasers 17(7) (1990) 386. [2] E.A.J. Marcatili et al., Bell Syst. Tech. J. 43 (1964) 1783. [3] Mingchang Wang et al., Chinese J. Lasers 6(6) (1979) 55; Mingchang Wang et al., Chinese J. Lasers 7(4) (1980) 19. [4] J.A. Stratton, Electromagnetic Theory (McGraw-Hill, New York/London, 1941) p. 524. [5] Mingchang Wang et al., Acta Optica Sinica 3(9) (1983) 779. [6] R.L. Abrams, IEEE J. Quantum Electron. QE-8 (1972) 838. [71 J.J. Degnan, IEEE J. Quantum. Electron. QE-9 (1973) 901 .
XI. FEL OPTICS
Nuclear Instruments and Methods in Physics Research A318 (1992) 870-873 North-Holland
Section A
The waveguide transmission of a free electron laser Ming Chang Wang
Shanghai Institute of Optics and Fine Jtfechanics. Academia Sinica, PO Bar 8211, Shanghai, Peoples Republic of China
An approach using waveguide laser theory to treat the radiation transmission of free electron lasers is proposed . The investigation shows that there is a low loss mode in the cavity. The low loss mode is EH for a dielectric waveguide and TE , for a metallic waveguide. 1-he mode coupling loss is also discussed.
L int
coon
A free electron laser usually has a long drift tube with a small diameter. The radiation modes transmitted inside the tube can be neither treated by laser theory which discusses open cavities nor by microwave theory which considers only cavities with size matching the wavelength. Consider the cavity of a free electron laser. For a wavelength A = I Wm. a tube radius a = 5.0 mm, and a length L = 30 m. the factor a 2/AL is calculated to be 0.83. This is different from the design of a conventional cavity of a FEL [1]. A new approach using waveguide laser theory to treat the radiation transmission is proposed and analyzed in this paper. In 1964, E.A. Marcatili discussed hollow metallic and dielectric waveguides for long distance optical transmission and lasers [2]. New devices, e.g. the CO, waveguide laser, have been developed based on this theory [3]. Using the waveguide laser theory, we were able to treat the mode transmission of a free electron laser and also calculate the transmission loss. It is found that EH, i is the lowest loss mode in a dielectric waveguide with a refractive index v = 1 .50. The loss is about 1 .48 x 10-5 dB/m . TE , is the lowest loss mode in a gold guide with an attenuation of 5.37 x 10-4 dB/m. The guides act as single-mode filters for EH and TE ,. The mode coupling loss of a free electron laser cavity is discussed . This study is an extension of the earlier work on the cavity of a free electron laser [1]. 2. Theoretical model For simplicity, consider a waveguide consisting of a circular cylinder of radius a and free-space dielectric constant E as shown in fig. 1 .
The waveguide wall is either dielectric or metallic with a complex dielectric constant E. The magnetic permeability p. is assumed to be that of free space for both media. Although the transmission characteristics of metallic waveguides are well kr. jwn for microwave frequencies, the theory is invalidated for operation at optical wavelengths because the metal no longer acts as a good conductor but rather as a dielectric having a large E. Both the dielectric and metallic waveguides are considered as special cases of a hollow circular waveguide. The following conditions of radiation in a waveguide are assumed: Ka>>IvIU,, ,,,, ly/K- 1I < 1,
(1) (2) where K = ao E A = 2-rr/A is the free-space propagation constant, U , is the rnth root of the equation J _ , (U ,) = 0 with n and m integers characterizing the propagation mode, v = E/E is the complex refractive index of the external medium, and y is the axial propagation constant of the mode under consideration. The first inequality states that the radius a is much larger than the free-space wavelength A. If the external medium is a metal, the I v I value is large but finite at an optical frequency. The second inequality restricts our analysis to low-loss modes with their propagation constant y nearly equal to that of free space. The E")'0 --o
Z
2a
501-
Fig. 1 . Schematic of a cylindrical dielectric waveguide.
0168-9002/92/$05 .00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
r
of a FEL
Mingchang Wang / Wareguide transmission
radiation in these modes propagates within the tube, bouncing at grazing angles against the wall. The field components of hybrid modes EH , inside a waveguide can be written as [4] 0
E,r ,
x
v1
Ka
H.rnr =
E() All
J K-r sin(n0) ex 1/2
1
z
(Ki/K )` = I - (y/K )`.
-toi»,
From the condition (2) we know y/K = 1 and, therefore, K;/K = 0. The right-hand side of eq. (6) is close to zero. Using a perturbation technique and approximation, we have
.rnr
Et) )1/2 r = _ Hnm E,;,,, ctn t?0 ILO
The propagation constant y can be obtained from eq. (4) (3)
where the complex propagation constant y satisfies the relationship , K ; =K - y - , (4) n and m are the numbers of periods of each field component in the 0 and r directions, respectively . In a hollow dielectric waveguide, the EH hybrid mode has the lowest transmission loss. The six components of this mode are Eil =Jl,(Ul1a
Eil = -1 HdI1
Hr
__
cos 0 exp(i(yz-£ut)),
r
E = Jl,(U1 ,
sin 0 exp(i(yz -cot)),
U1 I
(Ka ) 1/2
EO
JI U11
(
r) a
sin 0 exp(i(yz -- oit »,
Er
11+
2
El) _ _ EB 11, 11 -
Ao El)
Hi, = - (110
) 1/2
1)1/,
Ki a = .(1 ., - iv /Ka). U
(-1£I)
r
-
2(v-'
Divide both sides of eq. (4) by K2 to get
r
Eu ) 1,/2
Hnrnr -
Under the conditions (1) and (2), the characteristic equation for the propagation constant is simplified to be
vu =
__I Jj Kir)
x sin(n0) exp(i(yz -cot )), E~
3. The propagation constant of waveguide modes
J,r -1(Kia) = iv r(Ki1K)J,(Kia ), (6) where for EH mode, cos(n0) cxp(i(yz - wt)),
i unnr J,r- 1(Kir) + /Ar
r Ernt
1371
Ej, ctn 0,
where 0 and r refer to the axial and radial components in circular cylindrical coordinates, respectively .
By neglecting the second order term, we have
=K
1-
1 -i 2 y., Ka
Ur,~A 2:ra
y=K 1 -
1 ( U mA 2
( 1 _, ,,,,A
2 7ra
ra
The phase and attenuation constants of each mode are the real and imaginary parts of y, respectively. ( )2[, 1 - 1 U n,A rA + Im y A 2 2.ra r~ a ) 2 A2 U"n, ~T Rc(v ) . amn = 2r
'
For guides made of glass, where v is usually real and independent of A, the phase and attenuation constants become 27r ßnm = A [ 1 "I ) 2 _ U" anm 2r
1 A..
U _ A `' ( 2ra
(10)
a 3 vn'
Using A=10.6Wm,a=5 mm, v=1.5andU =2.405 in eq. (10), we obtained all = 1.91 x 10 -4/m = 1 .66 x 10 - ' dB/m. (Note : 20 log(E/E (,) = dB = 20 log exp(a L) = a L x 8.686.) The attenuation of this mode is negligible compared with the gain of a free XI. FEL OPTICS
Mingchang Wang / Wareguide transmission of a FEL
2
Fig. 2. Attenuation of EH mode a" versus A (Lm) for a dielectric waveguide.
electron laser even in the low gain regime G = 0.37/m [5] . Eq. (10) shows that the attenuation constant is propotional to A~/a3. Consequently, the losses can be made arbitrarily small by choosing the radius of the tube, a, sufficiently large relative to the wavelength A. The attenuation of EH mode has been plotted in fig. 2 as a function of A for v =1.5, and a = 1, 5 and 10 mm. 4. The coupling loss of waveguide modes A laser resonator consists of two mirrors. The resonator mode must result in a reproduction of the original amplitude and phase distribution after a round trip propagation in the cavity . Consider the end of a guide as a source of radiation. If the phase front of the wave arriving at one of the mirrors matches the mirror surface profile, the coupling loss of the mode back to the waveguide is the smallest . At any position z from the end of the guide we can position a mirror of radius R. It might be expected that optimum coupling at position z would occur when the radius R of a quasi-Gaussian bean. is equal to the wavefront radius R' given by b
z '
where b = -rrwo/A is a confocal parameter of the quasi-Gaussian beam. For the case when R = R', each Gaussian mode is imaged back on the guide end. When the radius at the mode waist w() = 0.643a . the coupling loss is the lowest [6].
J.J. Degnan had discussed the coupling loss of finite-aperture waveguide-laser resonators [7] . There are three types of resonators with low coupling loss : (1) large radius of curvature mirrors close to the guide, (2) large radius of curvature mirrors whose center of curvature lies approximately at the guide entrance, (3) small radius of curvature mirrors whose focus is positioned close to the guide entrance. Clearly, 100% efficiency coupling is achieved by placing a flat mirror directly against the guide. This efficiency is degraded by replacing the flat mirror with a smaller radius of curvature mirror or by increasing its separation from the guide. The coupling loss of EH mode is given to a good approximation by the formula ( d ) 3/2 (12) a; = 6.05 Ka, where d is the separation of the flat mirror from the guide. In order to reduce this loss, we need to place the mirror close to the guide. However, the cavity of a free electron laser requires a space d for the electron beam to enter and exit from the waveguide . Type 1 resonator is not suitable . In type 2, the matching is more efficient for mirrors with large radius of curvature (small a) since the phase front becomes approximately spherical as shown in fig . 3. This is similar to a concentric cavity of a conventional laser where z = R. Type 3 is like a confocal cavity where z = R/2. Degnan derived the coupling loss of a EH waveguide mode: a P a 1 2QJ,(Uo, ) x X
I0tt -dw, exp i
pw ,
J (~ dwn exp i J t ` pwo 2~ l )(U04010) 2 2 XjO(O1~_ù_j()£01)~ 1
1
(13)
where p = ka2/R, Q = dIR and h --- c/a . R is the
2a
T
IMMEEZZZE, d Fig. 3. Geometry for calculating the coupling loss .
Mingchang Wang / Wareguide transmission ofa FEL mirror radius of curvature and c is the radius of mirrors . When the value of the product ph = Kac/R is equal to the second root of the Bessel function J (U) the mirror captures only the large central lobe of the EH far-field distribution (ph = 5.520). If the value of ph is equal to the third root of the Bessel function, the mirror captures the large central lobe and the first outer ring (ph = 8.853). The minimum coupling loss is 0.06 dB. 5. The mode loss in metallic waveguide
E°, =J,(Ki r) exp[i(yz-cut)], Hôl
_ -
Hö1 =
-i
Eo
) 1/2
luo Eo (_110)
Jt(Ki r) exp[i(yz -art )]
1/2
(14)
ut) i
exp[i(yz-cvt)] . Ka J,(Kir)
The attenuation constant for TEO, mode is Uo~ Re(v ), (15) ( 2,ir ) ~~ a where Re(v )=(v-'- 1) -1 / 2 , UO = 3.832, v=0.19i6.1 for gold at A = 1.0 W m, v = 1 .49 -122.2 at A = 4.0 pm and v = 7.41 -153.4 at A = 10.0 Wm. The circular electric mode has the lowest loss in the metallic waveguide, while the circular magnetic and hybrid modes are rapidly attenuated. The attenuation constant a O for the lowest loss TEO, mode is plotted in fig. 4 for wavelengths in the range 1 .0 gm < A < 10.0 pm for a = 1 .0, 5.0 and 10.0 mm. Those data show a considerably lower loss compared with the lowest loss mode EH ,, for the dielectric guide . The attenuation of TEO , mode for the gold guide with a radius a = 1 .0 mm and wavelength A = 1 .0 pm is only 5.37 x 10 -4 dB/m. For a radius of 5 mm, the minimum loss of TEO, mode is ao, = 4.29 x 10 -6 dB/m. The metallic waveguide is superior to the hollow dielectric waveguide for long distance optical transmisao, =
ar (dB/M ) 10-' r10-2
a=1mm
10-3
5mm
10-4 10-5
10 mm
10-6
The approximate field components of the TE , mode in a metallic waveguide are (
873
1
2
3
4
5
6
7
8
9 a(Ji)
10
Fig. 4. Attenuation of TE , mode a , versus A (wm) for a gold waveguide . sion. Because of the relatively large dielectric constant exhibited by gold at an optical frequency, the attenuation constant for the lowest loss mode TEo, is very small and less sensitive to the curvature of the guide axis. The conclusion is that the mode transmission of a free electron laser can be treated by waveguide laser theory. The transmission and coupling losses can be calculated easily. References [1] Mingchang Wang et al., Chinese J. Lasers 17(7) (1990) 386. [2] E.A.J. Marcatili et al., Bell Syst. Tech. J. 43 (1964) 1783. [3] Mingchang Wang et al., Chinese J. Lasers 6(6) (1979) 55; Mingchang Wang et al., Chinese J. Lasers 7(4) (1980) 19. [4] J.A. Stratton, Electromagnetic Theory (McGraw-Hill, New York/London, 1941) p. 524. [5] Mingchang Wang et al., Acta Optica Sinica 3(9) (1983) 779. [6] R.L. Abrams, IEEE J. Quantum Electron. QE-8 (1972) 838. [71 J.J. Degnan, IEEE J. Quantum. Electron. QE-9 (1973) 901 .
XI. FEL OPTICS