JLAB-TN-02-014
April 23, 2002
Design details of the IR 10 kW Upgrade Free-electron Laser Resonator
Stephen Benson,
Thomas Jefferson National Accelerator Facility
Abstract
The design of a high power laser resonator for use in a ten-kilowatt
class free-electron laser (FEL) with a wavelength range of 1–14 µm is
described. The resonator design chosen consists of a simple, nearlyconcentric resonator with dielectric mirrors and output coupling through
one of the mirrors. The upstream mirror will have an actively controlled
radius of curvature. The center of the optical cavity mode will not always
be in the geometric center of the optical cavity. The change in the center
position will be described.
1. Introduction
Jefferson Lab is in the process of building a free-electron laser in the mid-infrared
to demonstrate scalability of energy recovering FELs. Due to cost and schedule
constraints it was decided to use a resonator design similar to the one used in the IR
Demo FEL. This is a near concentric resonator with transmissive output coupling.
The customer (the Navy) requires a laser with approximately ten kilowatts of output power operating at some wavelength in the mid-infrared with excellent wavelength
and power stability, nearly transform limited bandwidth, and a nearly diffraction limited
optical mode (here "nearly" means as close to ideal as possible but no worse than a factor
of two larger than the ideal value). The full specifications are detailed in reference [1].
2. Requirements
Free-electron lasers have some characteristics which set them apart from conventional lasers and some which are in common with certain conventional lasers. The small
signal gain in a FEL is typically low (<100%). The low gain is advantageous in reducing
the number of transverse resonator modes that are above threshold, as noted in reference
[2]. A near concentric cavity is nearly degenerate however and so gain guiding will have
to be studied even at saturation. The higher the losses, the more important this is. We use
the general characteristics of FELs to derive the following requirements:
1.
The gain medium in a FEL is smaller than the optical mode. This leads to
the need for careful alignment to get the optical mode overlapping the
electron beam. It also leads to a spatial filtering effect that almost
guarantees excellent mode quality. The optical cavity and optical transport
system must be designed so that this excellent mode quality is not
degraded. One source of beam quality degradation is aberration in the
output coupler. To keep the mode quality better than two times the
diffraction limit we have limited the phase distortion from each cavity
mirror to less than 10% of a wave and the distortion of the transmitted
beam to less than 20% of a wave.
1
2.
3.
4.
5.
6.
7.
8.
9.
The saturation intensity in a FEL is very high, leading to very high power
loading on the optics surfaces unless the resonator mode is designed to be
large at all the optical surfaces. We have chosen limits of 100 kW/cm2 as
the engineering limit for the average intensity and 1.5 mJ/cm2/micropulse
for the micropulse fluence for optical cavity elements.
The electron beam is pulsed and the laser will only lase when the optical
cavity length is very close to the synchronous length, defined as the length
at which the round trip time in the cavity matches the arrival time of the
electron bunches. The mirror mounts must be designed to decouple the
cavity length from the optical mode position and angle. The optical
mounts must also be quite stable so that the optical cavity length does not
drift.
The wavelength range of the laser will be 1–14 µm. This range was
chosen to allow for the use of ZnSe windows in the optical transport. The
system should be designed so that nothing fundamentally forbids operation
at longer wavelengths.
The gain is optimized if the optical mode is wrapped tightly around the
electron beam. The optical power output is optimized when the maximum
number of electrons is enclosed in the optical mode. Typically, a
compromise is made between these two effects. The beam size and
divergence around an optical waist is determined by its Rayleigh range
which is defined as the distance from a waist at which a Gaussian mode
doubles in area. A good value for the Rayleigh range in the laser for most
designs is approximately one third to one half the active wiggler length
(200–350 cm in our case). Calculations of the FEL gain indicate that the
optimum for our design is close to 200 cm. At long wavelengths the
diffraction is rather large in the wiggler bore for this Rayleigh range so one
wants to use 350 cm instead. At short wavelengths, the gain and the
intensity on the mirrors are the most important factors, so the shorter
Rayleigh range should be used.
The interaction between the laser mode and the electrons is maximized
when the waist is centered on the wiggler center. In this design the waist
will stay in the wiggler center to a tolerance of one fifth of the Rayleigh
range.
The optical mode axis and the electron beam axis must be as collinear as is
reasonably achievable. Estimates of the electron beam stability indicate
that the electron beam mean position and angle can be maintained to an
rms tolerance of better than 50 µm and 100 µrad, respectively. The optical
mode position and angle should be maintained to this tolerance if possible
but to no larger than one sixth of the 1/e2 mode radius and divergence in
any case.
With the photocathode drive laser presently in use, the maximum
repetition rate of the electrons is 74.85 MHz. This means that the optical
cavity must be an integer multiple of 2.0026 m in length. As the cavity is
lengthened, the power loading on the mirrors will go down and the
sensitivity to mirror misalignments will grow. A compromise must be
found. In order to allow ease of frequency selection the cavity should be
2N times the minimum length.
The optimum saturated gain is approximately one third to one fourth of the
small signal gain. For a gain to loss ratio greater than 4, the efficiency
slowly increases with cavity Q but the intensity on the mirrors also
increases. This means that, for saturated gain less than one fourth the
small signal gain, the laser output will probably be limited in power by
mirror heating rather than extraction efficiency. For our standard design,
2
10.
11.
the small signal gain is approximately 50%. This means that the losses
should be close to 15%.
Extraneous cavity losses should be maintained as low as reasonably
achievable. The main sources of loss are scatter by the mirrors, absorption
by the mirrors, and diffraction by apertures in the cavity. To minimize the
diffraction, all apertures should be at least three waists of the optical mode
and four for high power operation. This will not only minimize the
diffractive losses but will also reduce phase front distortion in the optical
mode. Mirror substrates should be as smooth as reasonable and coatings
should use low loss materials.
In addition to the laser requirements there are requirements for the optics
as far as environmental effects are concerned. The accelerator tunnel is a
relatively harsh environment with great quantities of ionizing radiation and
the need for high vacuum mounting of the optical elements. Optics and
mounting hardware must be compatible with ionizing radiation, and high
vacuum. All controls must be remotely operated and all diagnostics must
be remotely monitored.
We have summarized the requirements that are implied by the FEL and electron
beam properties in Table 1. Other requirements must be derived from the properties of
the resonator itself. These will be derived in section 5.
Table 1. Requirements taken from FEL and electron beam properties
Parameter
Requirement
Comments
Resonator type
Stable, near-concentric
Produces excellent mode quality
and is straightforward to design.
Aberration
If there is little aberration in the
<l/20 P-V surface
high reflector, one may relax the
distortion
surface distortion limit.
<l/5 P-V transmitted
wavefront distortion.
Circulating Intensity <100 kW/cm2
Factor of two below the state of
the art.
2
Circulating fluence
<1.5 mJ/cm
Allows operation at 75 MHz
Length stability
~1 µm
Passive control preferred.
Wavelength Range
1-14 µm
Limited by ZnSe windows.
Rayleigh range
~200 cm for short
Gain is optimized for 200 cm.
wavelengths and ~350 cm
Diffraction is minimized for 350
for long wavelengths
cm.
Waist position
Within 0.2zR of cavity
May start away from center and
center.
cross through center as mirrors
heat up and distort.
2
Mode position and
If this is less than 50 µm and
<1/6 of the 1/e mode
angle stability
100 µrad then 50 µm and 100
radius and divergence
µrad can be used.
Cavity length
N*2.0026 m
N can be 4,8,16... More
discussion on choice below.
Output coupling
15%
Smaller output coupling needed
for commissioning.
3
Extraneous losses
Controls and
diagnostics
<1% of circulating power
from all sources but output
coupling.
Remotely controlled and
read out.
4
Should be as low as possible.
Even at 1% the extraneous loss
might be greater than 1 kW.
Should also be computer
interfaced.
3. Defining Equations
Let us assume that the resonator modes are given to a good approximation by the
Gauss-Laguerre or Gauss-Hermite modes. Assume a general two-mirror resonator for
now with radii of curvature R1 and R2. The variable L represents the cavity length.
Define the cavity stability parameters g1 and g2 as
g1  1 
L
L
and g2  1 
R1
R2
(1)
The Rayleigh range of the cavity can be calculated from these parameters. We
will work backwards and assume the Rayleigh range and calculate the g parameters and
therefore the radii of curvature. The mode size at the resonator mode waist is given by
w
z
A0  0  R
2
2
2
(2)
A resonator is defined as geometrically stable if 0  g1g2 1 (The term "stable
resonator" is misleading since an unstable resonator is, in many ways, more stable than a
stable resonator, but the terminology is still commonly used).
We will find in our design that the product g1g2 for our resonator is quite close to
unity and therefore that the cavity is nearly unstable. Since we will be able to control the
radius of curvature of one of the mirrors and not the other the cavity will not be
symmetric. It will not be far from symmetric however. The following formulas from
Siegman [3] are useful.
g1 g2 (1  g1g2 )
zR 

 L  (g1  g2  2g1 g2 ) 2
2
(3)
The distance from the mode waist to the downstream mirror (defined here as
mirror 1) is given by:
z1
g2 (1 g1 )

(4)
L g1  g2  2g1g2
The mode area at the downstream mirror divided by the mode area at the waist is
given by
A1 w1 2
L
 2 
A0 w0
zR
g2
g1 (1  g1 g2 )
(5)
If one exchanges the subscripts 1 and 2, one finds the formula for the mode area
on the upstream mirror.
Cavity misalignments in a symmetric resonator can be represented as a linear
combination of two modes. The first mode is a rotation of both mirrors by equal angles.
The second is a rotation of both mirrors by equal and opposite angles. The first case will
cause the cavity mode to rotate about its center. The second will lead to a displacement
of the optical mode. In the first case, the cavity mode will rotate by an angle
5
2  (g1  g2 )
m
(6)
1  g1 g2
is the angle that both mirrors rotate. The displacement in the second case is

where  m
given by
x 
L(1  (g1  g2 ) 2)
m
1 g1 g2
(7)
4. Derived Equations
Let us define a parameter M  1 L 2zR  . In our case we also have M1 and M2,
which are the magnifications of the mode from the waist to mirrors 1 and 2. These are
defined according to equation (5).
2
Let us define a mean stability parameter g and an offset variable  so that
 g1g2  g so g1  g(1   ) and g2  g/(1  )
(8)
It is possible to rewrite equation (3) in terms of M as
1 (g1  g2 )  (g1  g2 )2 (4g 2 )
M
2
1g
(9)
Plugging equation (8) into equation (9) we find
M  Ms 1  where 
2
2(1   )
(10)
and where Ms  2 (1  g) is the magnification a symmetric resonator with the same mean
stability parameter g. The quantity  is second order in the resonator asymmetry and so
is quite small. For example, if g1/g2=1.5 we find that  is only 0.02. For most
calculations it is reasonable to use the formulas for M and Ms interchangeably.
The offset of the optical waist from the cavity center can be found from
substituting equation (8) into equation (4) and subtracting half the cavity length:
zoff 
L 
     L 
2 1  g  
2 1g
(11)
Note that, if desired, one can start with a negative  and move to a positive 
thereby moving from a positive zoff to a negative zoff,, always staying close to the center.
To satisfy the requirements the waist position must be within a certain fraction of a
Rayleigh range of the center. This is true if
zoff
zr

z 
L 
 M

  
2zr 1  g 2 M  1 zR li m
(12)
If we assume that we start with a negative and move to positive  as the mirror
heats up and we assume that we keep the Rayleigh range constant by distorting the high
reflector we find that:
6
2g 1
c1Pl
(13)


L
R 8MzRF
where the second relation is derived from equation (12) of TN-97-005. We use the same
definition of F as in that technical note:
kth
(14)
F
 e (h B   s  s t c )
where kth is the thermal conductivity, tc is the output coupler transmission, B is the bulk
absorption per cm, s is the coating absorption, e is the expansion coefficient, and h is
the mirror thickness. This figure of merit depends explicitly only on measurable
properties of the mirror coatings and substrate. Since the bulk and surface absorption are
wavelength dependent, the figure of merit will depend implicitly on the wavelength as
well. The units for F are in power per unit length. For this technical note we choose to
use kilowatts/micron to make the values small.
We can use equations (12) and (13) to derive a limit for the power due to waist
movement:
Pl 
16(M  2) 
z
13.7(M  2) 
z 
F   
F 
c1 M
zR li m
M
zR li m
(15)
For an asymmetric cavity the magnifications from the waist to each mirror will be
different. Substituting equation (8) into equation (5) we have
M1 
L
zR
g2
Ms
and M2  Ms (1   )
2 
g1(1 g ) 1 
(16)
We can now see that Ms is just the geometric mean of the magnification to each of
the two mirrors. When the output coupler distorts it will move the waist away from itself.
This will increase the magnification on the output coupler. It will also decrease the
magnification on the high reflector but the power will be reduced by the output coupling
so the intensity on the mirrors will be approximately balanced.
If both mirrors are allowed to freely distort, the power limit given in TN-97-005
applies [4]. This limit, that the Rayleigh range must not change by more than a factor of
two, is given by
32 M  1
27.4 M  1
F 
F
(17)
c1nm M
nm M
where nm is the number of mirrors that are distorting (1 or 2). In fact the limit must be
calculated self-consistently since the distorted mode will have a different magnification
and therefore a different change in Rayleigh range. When the limit is found self
consistently, it is found that the real limit is 75% of the one in equation (17).
Pl 
Note that the power limit in equation (17) is inversely proportional to the square
root of M for large M. This argues for a smaller magnification. The need to keep the
intensity on the mirrors less than 100 kW/cm2 argues for a larger magnification.
If we hold the Rayleigh range constant using deformable mirrors, this tradeoff no
longer exists. Nevertheless, the power is limited by the aberrations induced by the mirror
distortion, both in the mirror surface and in the transmitted radiation. From TN-97-005
and TN-98-015 [5] we find the power limits for aberration as:
7
Pl  41.2 ab F and Pl 
20.6abt
(18)
F

1 dn 
(n  1) 

 e dT 


where ab is the surface aberration limit per mirror and abt is the transmission aberration
limit. If both mirrors are distorting ab is usually 1/20. If only one mirror is distorting, it
may be as large as 1/10. The quantity abt is usually 1/5 but may be larger if aberration
correction in the form of a deformable mirror is included in the optical transport system.
The quantity in brackets in the denominator of equation (18) depends only on measured
properties of the mirror substrate. It is dimensionless and weakly dependent on
wavelength. Smaller numbers are preferable. For sapphire at 3 µm the denominator is
2.28. For calcium fluoride it is –0.003. For zinc selenide it is 9.4 to 9.6 depending on the
wavelength.
The power limit produced from an intensity limit on the mirrors is given by:
Ilim Mz R
(19)
2Q
where Ilim is the intensity limit on the mirror and Q is the resonator Q. Since M is
inversely proportional to zR2 for large M, one wants a small Rayleigh range, long cavity
length, and a low cavity Q to get the highest power. This means that the magnification
will get rather large.
Pl 
One cannot make the magnification arbitrarily large when using a near concentric
resonator. The magnification is limited by the angular tolerance in the mirrors.
Substituting equation (8) and using equation (10) in equations (6) and (7) we find
 M  2 
  1  s
M s m and
 Ms  1 2 

x 
(20)
L Ms 

1 (Ms  2)  m
2 Ms  1 
2 
For large Ms and small  these are closely approximated by   Ms m and
x  Lm 2 . The magnification is limited by the engineering limit of how well the
mirrors can be stabilized in angle. The limit is given by:
1  1
.
(21)
6 zR  lim
where lim is the limit of the mirror angular feedback system. It is worth noting here that
the factor of 1/6 is a conservative estimate of the angular tolerance and is not based on a
rigorous study of the power or gain vs. the angle of the resonator mode. The value based
on a more thorough study may be larger than 1/6.
M
8
5. Derived requirements
Modeling shows that the IR upgrade might be capable of extracting 20 kW from a
150 MeV, 10 mA electron beam. For the extremely low loss coatings envisioned for the
IR upgrade, the losses in the output coupler may be dominated by the bulk absorption.
For example, if the bulk absorption is 0.1%/cm, the coating absorption is less than 100
parts per million, the output coupling is 15%, and the mirror thickness is 1 cm, the bulk
losses will be 30% larger than the coating losses. For higher bulk absorption the ratio
will be even higher. The figure of merit F for sapphire at this wavelength is around 5
kW/µm for these parameters.
Cavity Length (32.0416 m): Using equation (19) we find that, for 15% losses, a
Rayleigh range of 200 cm, and 20 kW power output at 3 µm, the magnification must be
greater than 45. This means that the optical cavity must be at least 26.5 meters long. If
we use multiples of the IR Demo length of 8.0104 meters, the closest one larger than this
is 32.0416 meters. For this design we have therefore chosen a Rayleigh range of 200 cm
and a length of 32.0416 meters. The magnification M is therefore 65.17 and the mean
stability parameter g is –0.9693. The required angular stability from equation (21) is 1
µrad. for operation at 1 µm.
Mirror radii of curvature (see table 2): To reduce the cost and increase the
availability of the output coupler we propose to use an even number of meters for the
output coupler radius of curvature for the 200 cm Rayleigh range cavity and use a rather
lenient tolerance of 1% for the mirror. This means that the high reflector must have an
adjustment range to compensate for the possible range of radii of curvature of the output
coupler from 1584 to 1616 cm. It must also be able to handle the change in the radius of
curvature due to mirror distortion. Assuming that the mirror distortion is limited by the
surface distortion aberration (equation (18)), one can use equation (13) to show that this
should be limited to R R  0.301 M  1 . This is 3.8% for zR =200 cm. Since the
distortion is unidirectional, one only needs range in the shorter direction. The design cold
cavity radius of curvature for the high reflector should therefore be 1654 cm+1%/-4.8%
or 1671 to 1575 cm for zR = 200 cm. The range of radii of curvature for the high reflector
can be made smaller if the tolerance for the output coupler is tightened. The central value
can be changed by changing the central value of the output coupler, e.g. if the output
coupler is 1600 cm+0/-1%, the HR range will be 1654+1%/-3.8% or 1671 to 1591 cm for
zR = 200 cm. Note that this satisfies the desire to start with a negative  and move to a
positive  as the output coupler distorts. For the radius of curvature of 16 meters, 
moves from –0.034 to 0.039 as the mirror surface aberration grows from zero to 5% of a
wave. For zinc selenide the distortion is limited by the allowable distortion in the output
coupler. If we assume that we compensate the output coupler aberration to within a
factor of three, the maximum internal aberration is only 3% of a wave P-V. This means
that the range for the ZnSe mirrors can be smaller and the short end of the range is 1600
cm instead of 1575 cm.
For zR =350 it is better to use a larger radius of curvature of 16.5+0.17/-0 m for
the design value. The longer Rayleigh range will be used for wavelengths longer than 9
µm so that the output from the laser is not too large for the optical transport. In this
wavelength range the output coupler substrate will be zinc selenide. For this substrate the
power limit will again be due to the transmitted wavefront error. The change in the radii
of curvature should still be less than 3%. The range of the radius of curvature for the high
9
reflector is therefore 1700-1624 cm. This can be covered using a deformable mirror with
a starting radius of curvature of 17 meters.
Mirror diameters (5.08 for <4 µm, 7.62 cm for >4 µm): A good rule of
thumb for the mirror diameter is that the useful aperture of the mirror be greater than four
waists in diameter at the longest wavelength used. For a Rayleigh range of 350 cm and a
wavelength of 25 µm, the optical waist on the mirror is 24.7 mm. The mirrors must
therefore be greater than 9.9 cm in diameter. We choose to use a 4-inch diameter mirror
for the longest wavelengths. At 3.9 microns with a 200 cm Rayleigh range the mirror
should be greater than 5.1 cm diameter so a 2-inch mirror will do. For operation out to
~9 µm a 3 inch mirror will do. From 9 to 14 microns we can use a 350 cm Rayleigh
range and the waist size will range from 1.5–1.9 cm and so a 3 inch mirror can still be
used.
For broadband operation we will use hole coupling to outcouple the laser light.
This is a standard method for output coupling in broadband cavities. It has two
disadvantages. The first is that the output coupling efficiency is typically less than 50%.
About half of the power loss is in absorbed and scattered light. Since the broadband
cavity will only be used at “low” power levels of approximately 1 kW, this should not be
a critical problem. The second disadvantage is that the output coupling is a function of
wavelength. Ming Xie has shown that the wavelength can vary over a range of a factor of
two with little change in the output coupling [6]. Since we have the ability to change the
Rayleigh range, we can keep the output coupling constant over an even large range. If we
choose to operate at 14 microns with a 350 cm Rayleigh range and change the Rayleigh
range proportional to the wavelength down to 175 cm at 7 microns, we find that the spot
size on the mirror barely changes over the entire range. The output coupling should
therefore be approximately constant over that range. The output coupling will then
increase as the wavelength is increased from there (though one could also decrease the
Rayleigh range even more at that point) but should only increase slightly from that point
since the mode tries to avoid the hole as the output coupling gets larger. If we make the
hole 6 mm in diameter, the output coupling will be approximately 5% over the 7 to 14
micron wavelength range, growing slightly for shorter wavelengths. The total losses
should be approximately 12%. These are rough numbers and will have to be confirmed by
more detailed simulations. Ming Xie used a variable aperture on the high reflector to keep
the output coupling efficiency constant. With a variable aperture and variable Rayleigh
range we may be able to operate over a very large wavelength range. Note that, if
windows with the right transparency range are used, the laser can be used to even longer
wavelengths. The output coupling will drop slowly to 3% at 25 µm. This is still a very
usable number. The size of the broadband mirror should therefore be sufficiently large to
operate out to 25 microns. This means that is must be at least 10 cm in diameter.
10
Table 2. Derived specifications for the IR Demo resonator.
Cavity magnification M
65.17 (29.52 for long )
Mean cavity parameter g
-0.969 for zR=200 cm
-0.909 for zR=350 cm
Mirror tilt tolerance(@1 µm)
1.0 µrad
Mirror radii of curvature
1600 cm±1% for OC, zR=200 cm
1650+1%/-0% for zR = 350 cm
1575–1700cm for HR
Mirror diameters
5.08 cm for <4 µm
7.62 cm for <9 µm (zR=200 cm)
7.62 cm for 9–14 µm (zR=350 cm)
10.16 cm for broadband
The cold cavity waist position will be 27.5±16.2 cm for zR = 200 cm. It will be
30-18/+0 for zR =350 cm. For the sapphire cavity, the waist will move ~0.3zR=60 cm as
the mirror heats up to operating temperature. The movement is negative so the waist will
nominally move from about 30 cm downstream of the cavity center to 32 cm upstream of
the cavity center. This should be negligible in a 6-meter wiggler. For zR = 350 cm the
waist will move by 70 cm.
In table 3 we show the power limits calculated assuming a 20th wave aberration
limit on the surface distortion and a 5th wave aberration on the optical wavefront
distortion. The allowed displacement of the waist from the cavity center is one fifth of a
Rayleigh range and it is assumed that one starts one fifth of the Rayleigh range upstream.
The mirror intensity limit is 100 kW/cm2.
Let us assume that the high reflector has radius of curvature control that allows
the Rayleigh range to be held constant as the mirrors heat up. This means that the power
limit from the change in zR can be ignored. The power limit at 1 µm is due to the intensity
at the mirror. With a different wiggler, which would allow a shorter Rayleigh range, one
could increase this even further before running into a limit of tolerable angular stability.
At 3 µm the ultimate power limit of 28.6 kW will be from surface aberration. Before that
limit is reached, however, the output beam quality will degrade below desirable limits.
This can, in principle, be corrected using a deformable mirror in the optical transport. At
6 µm the ultimate power limit is 16.6 kW but the output beam quality will be quite poor
at that power. To operate at 6 µm at 10 kW with decent mode quality will require a
deformable mirror or lower losses in the output coupler. Since the bulk absorption at 6
µm is not well known, this may be possible. At 10 µm the ultimate power limit is quite
high at 38.7 kW. The limit before output beam quality is poor is much lower but close to
10 kW at 8.3 kW. It is interesting to note that calcium fluoride can provide up to seven
times the power obtained in the IR Demo due to the higher output coupling, the longer
cavity, and the radius of curvature control in the high reflector.
11
Table 3. Power limits for some different resonator materials and
wavelengths. Values in boldface are the ultimate power limits. Values in italics
are limits due to output beam quality. The differences in F vs. wavelength for a
given substrate are due to differences in the bulk and/or coating absorption.
Substrate
Sapphire
Sapphire
ZnSe
ZnSe
CaF2
1
3
6
10
6
Bulk absorption(cm-1)
0.001%
0.1%
0.1%
0.05%
0.05%
Surface absorp.(ppm)
50
100
100
100
100
Thickness(cm)
0.625
0.625
1
1
1
F (kW/µm)
16.5
4.62
1.35
1.88
0.41
Pl (kW) change in zR
20.9
17.5
10.2
23.7
30.7
Pl (kW) change in z
43.8
36.8
21.4
49.8
6.5
Pl (kW) surf. aberr.
34.7
28.6
16.6
38.7
5.0
Pl (kW)Intensity limit
9.8
29.3
58.7
97.8
58.7
Pl (kW) trans. aberr.
29.7
25.4
3.5
8.3
333.9
Angular stab.(µrad)
1.02
1.77
2.50
3.23
2.50
Wavelength(µm)
It is interesting to turn around the power limit and define a limit for absorbed
power in the mirror. If we assume that the aberration limit in equation 18 is the
determining limit for maximum laser power, it is possible to show that the maximum
absorbed power is given by:
k
Pabs  41.2 ab  th  Pli m
(22)
e
For sapphire the absorbed power per micron of wavelength is Plim = 13.3 W/µm.
For zinc selenide it is Plim = 4.9 W/µm. For calcium fluoride, it is Plim = 1.1 W/µm.
Aberration from the output coupler yields separate limits of 11.4 W/µm for sapphire and
1.04 W/µm for ZnSe. As noted above the aberrations can, in principle, be compensated.
Beam collimation
Since the focus of the output beam will change as the mirrors heat up, the
collimator will have to change its focus to compensate. We will need a collimator to
match the beam into the long optical transport to the user labs. The aperture of the
transport is 6.4 cm. We would like to have a reflective collimator as we did in the IR
Demo. Assume that the beam is kicked sideways 1.2 m downstream of the OC to a
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deformable mirror 1 meter away. The reflected beam is sent to a third mirror, which
deflects the beam to the ceiling. Polarization order is s-n-p-p-s-p-p.
The output beam for the sapphire mirror will be expanding rapidly, which is good
from the standpoint of making the beam fill the aperture of the transport mirrors. One
must merely collimate the beam in the collimator to obtain good beam transport to the
end of line. For ZnSe however the beam is already larger at the exit of the optical cavity
than the 6.4 cm aperture of the optical transport. It is therefore better to focus the beam at
the output coupler and defocus the beam in the collimator. The effect is that of a Galilean
telescope. This focusing can be accomplished by putting a convex 8 meter radius of
curvature on the output coupler. As the mirror heats up it will focus the beam more and a
deformable mirror in the collimator must defocus the beam more.
For sapphire, the focal length of the output coupler varies from -22.5 m in the cold
cavity to -28.1 m for a 20% transmitted aberration. A deformable mirror with a focal
length range of 11–12 m will collimate the beam.to a 4-waist dia. less than 6.1 cm at 3.6
µm, which is the longest wavelength where we can use sapphire.
For zinc selenide, a Rayleigh range of 200 cm, and a 8 m backside radius of
curvature, the focal length of the output coupler varies from 11.2 m cold cavity to 10.1 m
for 20% aberration. A deformable mirror with a focal length range of -150 m to -30 m
will collimate the beam to a 4-waist dia. less than 6.3 cm at 9 µm.
For zinc selenide, a Rayleigh range of 300 cm and an 8 m backside radius of
curvature, the focal length of the output coupler varies from 10.9 m to 9.5 m. The same
deformable mirror will collimate the beam to a 4 waist dia. Less than 6.3 cm at 14 µm.
Since we do not have a design for a deformable mirror that corrects spherical
aberration we need a way to compensate for aberrations induced in the zinc selenide
output coupler.
One possible way to do this is to have an absorbing calcium fluoride mirror that
distorts dynamically with the power level. Since the beam is being reflected from the
mirror, the optical phase distortion is opposite in sign from the phase distortion of the
output coupler. If the spot size is the similar, the radial dependence of the distortion will
match as well. If the absorption can be tailored to be just the right amount, the phase
distortion of the collimator mirror can exactly compensate the phase distortion of the
output coupler. If the absorption is approximately correct, one might trim the power
absorption by having a CO2 laser incident on the backside of the collimator mirror.
This is a very ambitious but elegant way of correcting the output coupler
aberrations but it may not work. A simpler solution is to insert a diamond-turned
correction plate that has a phase error machined into it that is double the size and opposite
in sign to a 0.2 wave aberration from the output coupler. This would be withdrawn at low
power. When the aberration reached 20% of a wave, it would be inserted and change the
aberration to –20%. The mirror would then continue to heat up until the 20% aberration
is reached again. At this point the laser power level is 3 times as high as without the
correction plate but has the same beam quality. Of course one can continue this by
adding more correction plates and get as high as one wants. This assumes that the
aberrations are exactly cancelled though and this will only be true at one wavelength. At
other wavelengths, the spot size on the mirror will be different than the one assumes in
making the correction plate. Also note that the correction plate must have a very effective
anti-reflection coating on it over the wavelength range in which it must work. When the
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wave plate is inserted it will also provide a defocusing of the beam that the collimator
must compensate
The broadband cavity output is basically a top-hat profile in the near field, which
transforms into an Airy pattern in the far field. If a focussing element is placed at least 10
meters from the cavity, it can collimate this Airy pattern and produce a nice optical mode
for the rest of the beam transport. The closest we can come to this is to use the second
and third mirror cassettes to focus the beam. The mean distance of these two mirrors
from the output coupler is 14 meters. If two spherical mirrors with 60 meter radii of
curvature are used, the net focal length of the pair is 14.8 meters (assuming 3 meter
separation). This is enough to collimate the beam. At the longer wavelengths, only the
central lobe of the Airy pattern survives. At shorter wavelengths, the outer rings will also
propagate.
Miscellaeous features:
All the discussion of collimation and aberration compensation assumes that the
output coupler is the only transmissive optic in the laser. To this end, we have designed
the system to have a gradual transition from beamline vacuum to transport vacuum over
the first few meters of optical transport. No window will be in the beam transport with
the exception of the vacuum windows in the labs. During commissioning we may have
some optical beam position monitors which use near-Brewster-angle, calcium fluoride
plates as pickoffs. These limit the wavelength range to around 7 to 8 microns or less.
When a new OBPM design has been commissioned, these calcium fluoride pick-offs will
be removed.
Since the angular stability of the optical cavity will be as small as 1 microradian,
it is important that the resonator have a feedback system on the resonator mirror steering.
Since we have to hold the Rayleigh range constant as the mirrors distort, a feedback
system that compensates for the change in the radius of curvature is also necessary.
Finally, as we have found in the IR Demo, active feedback in the optical transport is also
necessary.
6. Conclusion
We have derived a simple design for a high power laser resonator for a ten
kilowatt free-electron laser. Unlike the IR Demo FEL, the cavity parameters must be
measured and maintained real time. This implies the existence of good diagnostics for
measuring the radii of curvature (or the Rayleigh range) and the mirror steering. A
power output of over 10 kW will be possible without mirror limitations for operation near
3 microns. For operation from 4 to 14 microns, laser output in excess of 10 kW should
be easily achievable but the mode quality will have to be corrected in the collimator. It
should be possible to correct the aberrations for up to 10 kW of laser output.
14
References
1. “Engineering Design for the Upgrade FEL optical cavity” Michelle Shinn, JLABTN-02-013, April 22, 2002.
2. “Optical modeling of the Jefferson Laboratory IR Demo FEL”, Stephen V.
Benson, Paul S. Davidson, Ravi Jain, Peter K. Kloeppel, George R. Neil, Michelle
D. Shinn, Nucl. Inst. and Meth., A407 (1998) 401–406.
3. “Lasers”, A. E. Siegman, University Science Books, Mill Valley CA, 1986.
4. “Heat induced mirror distortions in the IR Demo Free-Electron Laser resonator”
Stephen Benson, Paul Davidson, Peter Kloeppel, George Neil, Michelle Shinn,
JLab-TN-97-005.
5. “Focusing effects due to mirror heating in the IR demo optical resonator,” S.
Benson, M. Shinn, JLab-TN-98-015.
6. “Performance of Hole Coupling Resonator in the Presence of Asymmetric Modes
and FEL Gain”, M. Xie and K.-J. Kim, Nucl. Inst. and Meth., A318 (1992) 877884.
15