**User Requirements:** The laser must have adequate gain for
achieving laser action with reasonable output coupling efficiency.
Past experience indicates that a design gain of approximately
50% is reasonable. The laser should produce at least one kilowatt
of laser power at some wavelength in the mid or near infrared.
If the near infrared cannot be reached, the wavelength is not
critical. Operation at approximately 6.4 µm is desired for
materials processing. We have chosen a wavelength of 3 µm
at full energy so that third harmonic lasing in the 1 µm
range is possible.

**Primary Specifications:**

**Wiggler gap (>10 mm):** Minimum clear aperture inside
the vacuum chamber is given by:

For this design we assume b_{max} 100 cm, e_{max}=20 mm-mrad, and g_{min} =55. This
yields a minimum clear aperture of 8 mm. Accounting for a 1 mm
thick wall, we therefore need a wiggler gap of at least 10 mm.
The wiggle plane dimension can be larger with no penalty.

**Wiggler wavelength (2.7±0.005 cm)** In order to get
as much laser power as possible it is important to operate the
driver accelerator with as much beam power as possible. The maximum
beam energy available with a single cryomodule and one cryounit
is estimated to be near 42 MeV. The laser wavelength is given
approximately by

where = laser wavelength wiggler wavelength, g is the dimensionless electron beam energy, and . A conservative estimate of wiggler parameter which can be produced using commercially available technology as a function of the gap to period ratio is given by the equation:

The wiggler parameter *K* must be at least 0.7 to achieve
reasonable gain (the gain varies as *K2 *).
If we assume a wavelength of 3 µm and an energy of 42 MeV
then . Figure 1 shows the required *K2*
versus the wiggler wavelength range with this assumption as well
as the gap required assuming the equation above. From this figure
we see that the wiggler wavelength must be less than 2.7 cm so
that the *K2* is greater than 0.5 and
greater than 2.45 so that the gap is greater than 10 mm.

Figure 1. Gap and wiggler parameter squared versus wiggler wavelength. A laser wavelength of 3 µm and an electron beam energy of 42 MeV was assumed.

A design from the APS of 2.7 cm wavelength exists and provides the largest gap for a 3 µm wavelength, so that wavelength was chosen.

**Wiggler type (hybrid, permanent magnet): **We need a very
large gap for a given wiggler field in order to minimize halo
losses. We also need an inexpensive design which can be quickly
built. This narrows the choice of wiggler type to the hybrid,
permanent magnet, wiggler. This is a simple, straightforward
design and several vendors have experience building such devices.
Other design types are a superferric wiggler, a pure permanent
magnet wiggler, and a wedged pole hybrid wiggler. The superferric
wiggler is not available commercially and could not be produced
in time. The pure permanent magnet wiggler has a lower field
for a given wiggler gap, and the wedged pole wiggler would we
more expensive and would take longer to build for only a small
increase in the available gap.

**Number of wiggler periods (40):** The maximum current we
are expecting to get is 5 mA. The leads to 210 kW of electron
beam power at 42 MeV. Assuming that 90% of this current participates
in the laser interaction and that 90% of the extracted power exits
the laser cavity, this means that the laser efficiency must be
at least 0.6%. The efficiency of a free-electron laser operated
with good optical beam quality is equal to 1/4N where N is the
number of wiggler periods. This implies that the number of periods
must be less than 42. The number of periods in a conventional
wiggler should be at least 40 in order to reduce the exhaust energy
spread from the laser to a value we feel can be accepted by the
electron beam transport We have chosen this value for the effective
number of periods so that we have a little overhead in the power
output.

**Wiggler parameter (0.707,1.0?):** To reduce the cost and
schedule of the wiggler we have decided to use a fixed gap wiggler.
The gap will be set by spacer blocks. A few different gaps can
be chosen, but changing the gap will not be easy. Since the electron
beam energy, the FEL wavelength, and the wiggler wavelength are
already chosen, the wiggler parameter must be 0.707. It may be
possible to operate at longer wavelength and still achieve the
minimum gap specification depending on the wiggler performance.
A wiggler parameter much higher than unity is not much better
and leads to too small a gap. The APS 2.7 cm undulator has an
rms *K2*=1.0 at a gap of approximately
12 mm. We therefore chose to operate at a 12 mm gap and will
use whatever wiggler parameter is available there.

**Field direction (horizontal):** The laser polarization is
determined from the wiggler polarization. It is more convenient
to build an optical cavity with all s-plane reflections and having
all mirrors in a horizontal plane. This can only be done with
a vertical polarization in the optical beam.

**Phase error(<5° rms):** In most wigglers the wiggler
quality can be quantified by the rms phase error of a copropagating
plane wave with the electrons as they follow the trajectory through
the wiggler. This is rigorously defined in reference [http://www.jlab.org/accel/fel/documentation/feldoc9-4-1.pdf].
If the rms phase jitter is less than 5° rms then the gain
will not be reduced by more than 1% from that of an ideal wiggler.
The third harmonic gain will not be reduced by more than 7% from
an ideal wiggler. The phase error is produced by two sources:
trajectory errors, which produce low frequency phase drifts due
to path lengthening, and field strength errors, which causes high
frequency jitter due to variations in the wiggle amplitude (see
specifications for trajectory and pole-to-pole field variation
below). We do not want either of these to dominate, so we limit
each of them separately but the primary specification is the phase
error.

**Integrated horizontal field at each end(200 G-cm):** We would
like the integrated field error must be small enough to be corrected
with air core correctors which can be placed close to the ends
of the wiggler. This can be done if the integrated field is less
than 200 G-cm. This field error is also small enough to cause
a small offset in the beam position before the corrector can be
reached. The bend angle for a 200 G-cm error at 42 MeV is approximately
1.5 mrad. The "ends" here are defined as the first
and last four poles of the wiggler.

**Integrated horizontal and vertical field (400 G-cm):** The
sum of the errors at each end of the wiggler and the core should
be less than 400 G-cm for the same reasons as for the previous
point.

**Second vertical and horizontal field integral (<5,000 G-cm2
): **The wiggler may displace the electron beam due to imperfections
in the field leading to non-zero second integral. We want the
beam displacement from a straight line trajectory to be less than
0.5 mm so that the beam is not strongly kicked by the output quadrupoles.
This is satisfied at the lowest planned energy of 27 MeV by a
field integral of less than 5000 G-cm2.

**Second horizontal field integral at either end (As small as
reasonable):** The displacement of the beam at either end of
the wiggler should be as small as possible. Second field integral
errors at either end can lead to a rotation or displacement of
the optical mode. To speed commissioning it is useful to make
this as small as possible. Since this is quite difficult in practice,
we have specified it as a best achievable specification.

**Harmonic content (<20% on any harmonic):** Hybrid wigglers
may have harmonic content at all the odd harmonics. The presence
of higher harmonics can enhance or degrade the gain at harmonics,
but the effect is not important for relative harmonic content
less than 20%. The effect of the harmonics on the fundamental
is minimal except to reduce the maximum available rms *K2*.
Note that it is the rms *K2* and not
the peak *K2* or even the fundamental
field component which must be measured when characterizing the
magnet.

**Rolloff of rms field strength versus vertical position (<0.5%
over central one cm):** For operability, one would like the
rms field strength to be independent of vertical position over
a reasonable range of transverse position. A field uniformity
of 0.1% leads to a wavelength dependence of up to 0.1% over
the central 1 cm of the wiggler.

**Variation of rms field strength versus horizontal position
(Unimportant):** The rms field strength must have a minimum
near the horizontal center of the wiggler and will increase away
from the center. Thus the field has an inhomogeneity versus the
horizontal position. The field profile will tend to focus the
beam and steer it onto the horizontal axis. The inhomogeneity
in the field is canceled by the angular dependence introduced
by the focusing so the net effect is not important.

**The integrated multipole fields:** The integrated multipole
fields through the entire wiggler shall be less than the following
tolerances at the nominal gap.

n | Normal Component (bn)
| Skew Component (an) |

1 (Quadrupole) | 50 Gauss | 50 Gauss |

2 (Sextupole) | 200 Gauss/cm | l 00 Gauss/cm |

3 (Octupole) | 300 Gauss/cm2 | 300 Gauss/cm2 |

These are similar to the values for the APS undulator A and have
been shown to be acceptable by Douglas CEBAF Tech Note No. 96-018

.

**Derived specifications:**

**Field strength (0.28±0.01, 0.4±0.01 T rms?):**
From the equation for the wiggler parameter *K* and the specification
of the wiggler parameter range we find that the rms magnetic field
strength must be 0.0.28 T for the large gap and ~0.4 T at a
gap of 12 mm.

**Pole spacing uniformity (.01 cm):** One possible source
of phase error is a variation in the wiggler wavelength. If the
wiggler wavelength is maintained to 100 µm precision (average,
non-cumulative) this will be a small contributor to the phase
error.

**Trajectory errors (<±100 µm and <±500
µrad at 42 MeV):** The trajectory, averaged over the wiggler
motion, should be a straight line over the central 40 periods
of the wiggler to within ±2AW where AW is the peak-to-peak
wiggle amplitude. This ensures that the electron beam stays in
the central part of the optical mode. Trajectory wander leads
to angular errors which lead to phase errors. The angular errors
must be less than ±500 µrad so that they do not dominate
the phase error. At lower energies the allowable errors increase
inversely with the so that the specification at maximum energy
is the most critical.

**References:**

1. CEBAF Spec. 09510-S-002 "Technical Specifications for the IR Demo FEL Wiggler":

http://www.jlab.org/accel/fel/documentation/feldoc9-4-1-2.pdf

2. CEBAF Technical note 96-018 "Effect of Multipoles in the IR wiggler" D. Douglas

http://www.jlab.org/~douglas/FEL/technote/CEBAFTN96018.ps