Synchrotron radiation occurs when a charge moving at relativistic speeds follows a curved trajectory. In this section, formulas and supporting graphs are used to quantitatively describe characteristics of this radiation for the cases of circular motion (bending magnets) and sinusoidal motion (periodic magnetic structures).

We will first discuss the ideal case, where the effects due to the angular divergence and the finite size of the electron beam—the emittance effects—can be neglected.

The angular distribution of radiation emitted by electrons moving through a bending magnet with a circular trajectory in the horizontal plane is given by

_{}
^{(1)}

where

Š_{B} = photon flux (number of photons per second)

*q* = observation angle in the horizontal plane

*y* = observation angle in the vertical plane

*a* = fine-structure constant

*g* = electron energy/*m*_{e}*c*^{2}
(*m*_{e}
= electron mass, *c* = velocity of
light)

*w* = angular frequency of photon (*e*
= *Ó**w*
= energy of photon)

*I* = beam
current

*e *= electron charge = 1.602
´ 10^{–19 }coulomb

*y* = *w*/*w*_{c} = *e*/*e*_{c }

*w*_{c} = critical
frequency, defined as the frequency that divides the emitted power into equal
halves,

= 3*g*^{3}*c*/2*r*

*r* = radius of instantaneous curvature of the electron
trajectory (in practical units,

*r*[m]
= 3.3 *E*[GeV]/*B*[T])

*E* = electron
beam energy

*B* = magnetic
field strength

*e*_{c} = *Ó**w*_{c} (in practical units,

*e*_{c} [keV] = 0.665 *E*^{2} [GeV] *B*[T])

*X* = *gy*

*x* = *y*(1
+ *X*^{2})^{3/2}/2

The subscripted *K’*s
are modified Bessel functions of the second kind. In the horizontal direction
(*y*
= 0), Eq. (1) becomes

_{}
(2)

where

_{}
(3)

In practical units [photons·s^{–1}·mr^{–2}·(0.1% bandwidth)^{–1}],

_{}

The function *H*_{2}(*y*)
is shown in Fig. 2-1.

**Fig. 2-1**. The functions G_{1}(y) and H_{2}(y), where y is the ratio of photon energy to critical photon
energy.

The distribution
integrated over *y* is
given by

_{}
(4)

where

_{}

(5)

In practical units [photons·s^{–1}·mr^{–1}·(0.1% bandwidth)^{–1}],

_{}

The function *G*_{1}(*y*)
is also plotted in Fig. 2-1.

Radiation from a bending magnet is linearly polarized when observed in the bending plane. Out of this plane, the polarization is elliptical and can be decomposed into its horizontal and vertical components. The first and second terms in the last bracket of Eq. (1) correspond, respectively, to the intensity of the horizontally and vertically polarized radiation. Figure 2-2 gives the normalized intensities of these two components, as functions of emission angle, for different energies. The square root of the ratio of these intensities is the ratio of the major and minor axes of the polarization ellipse. The sense of the electric field rotation reverses as the vertical observation angle changes from positive to negative.

Synchrotron radiation occurs in a narrow cone of nominal
angular width ~1/*g*. To provide a more specific measure of this angular width,
in terms of electron and photon energies, it is convenient to introduce the
effective rms half-angle *s** _{y}* as follows:

_{}
(6)

**Fig.
2-2.** Normalized intensities of
horizontal and vertical polarization components, as functions of the vertical
observation angle y, for different
photon energies. (Adapted from Ref. 1.)

** **

**Fig. 2-3.** The function C(y). The limiting slopes, for
e/e_{c} << 1 and e/e_{c} >> 1, are indicated**.**

where *s** _{y}* is given by

_{}
(7)

The function *C*(*y*) is plotted in Fig. 2-3. In terms of
*s** _{y}*, Eq. (2) may now be
rewritten as

_{}
(2a)

In a wiggler or an undulator, electrons travel through a periodic
magnetic structure. We consider the case where the magnetic field *B* varies sinusoidally and is in the vertical
direction:

*B*(*z*) = *B*_{0} cos(2*pz*/*l*_{u}) ,
(8)

where *z* is the distance
along the wiggler axis, *B*_{0} the peak magnetic field, and *l*_{u} the magnetic period. Electron motion is
also sinusoidal and lies in the horizontal plane. An important parameter characterizing
the electron motion is the deflection parameter *K* given by

_{}
(9)

In terms of *K*, the
maximum angular deflection of the orbit is *d *= *K/**g*. For _{}, radiation from
the various periods can exhibit strong interference phenomena, because the
angular excursions of the electrons are within the nominal 1/*g* radiation
cone; in this case, the structure is referred to as an undulator. In the case
*K* >> 1, interference effects are
less important, and the structure is referred to as a wiggler.

In a wiggler, *K* is
large (typically _{}) and radiation from
different parts of the electron trajectory adds incoherently. The flux distribution
is then given by 2*N* (where *N* is the number of magnet periods) times the appropriate formula for
bending magnets, either Eq. (1) or Eq. (2). However, *r* or *B* must be taken at the point of the electron’s trajectory tangent
to the direction of observation. Thus, for a horizontal angle *q*,

_{}
(10)

where

*e*_{cmax }= 0.665*
E*^{2}[GeV]* B*_{0}[T]* .*

When *y* = 0, the radiation is linearly polarized
in the horizontal plane, as in the case of the bending magnet. As *y*
increases, the direction of the polarization changes, but because the elliptical
polarization from one half-period of the motion combines with the elliptical
polarization (of opposite sense of rotation) from the next, the polarization
remains linear.

In an undulator, *K*
is moderate (_{}) and radiation from
different periods interferes coherently, thus producing sharp peaks at harmonics
of the fundamental (*n* = 1). The wavelength of the fundamental
on axis (*q* = *y* = 0) is given by

_{}
(11)

or

_{}

The corresponding energy, in practical units, is

_{}

The relative bandwidth at the *n*th harmonic is

_{}
(12)

On axis the peak intensity of the *n*th harmonic is given by

_{}_{,} _{(13)}

where

_{}
_{(14)}

Here, the *J’*s are
Bessel functions. The function *F** _{n}*(

_{}

The angular distribution of the *n*th harmonic is concentrated in a narrow cone whose half-width is
given by

_{}
(15)

**Fig. 2-4.** The function F_{n}(K) for different values of n, where K is the deflection
parameter**.**

Here *L* is the
length of the undulator (*L* = *N**l*_{u}).
Additional rings of radiation of the same frequency also appear at angular
distances

_{}
(16)

The angular structure of undulator radiation is illustrated in Fig. 2-5 for the limiting case of zero beam emittance.

We are usually interested in the central cone. An approximate formula for the flux integrated over the central cone is

_{}

(17)

or, in units of photons·s^{–1}·(0.1% bandwidth)^{–1},

_{}

The function *Q** _{n}*(

**Fig. 2-5.** The angular distribution of fundamental (n
= 1) undulator radiation for the limiting case of zero beam emittance. The
x and y axes correspond to the observation angles q and y (in radians), respectively,
and the z axis is the intensity in photons·s^{–1}·A^{–1}·(0.1 mr)^{–2}·(1% bandwidth)^{–1}.
The undulator parameters for this theoretical calculation were N = 14,
K = 1.87, l_{u} = 3.5 cm,
and E = 1.3 GeV. (Figure courtesy of R. Tatchyn, Stanford University.)

**Fig. 2-6. **The function Q_{n}(K) for different values of n.

_{}
(13a)

Away from the axis, there is also a change in wavelength:
The factor (1 + *K*^{2}/2) in Eq. (11) must be replaced by [1
+ *K*^{2}/2
+ *g*^{2}
(*q*^{2} + *y*^{2})].
Because of this wavelength shift with emission angle, the angle-integrated
spectrum consists of peaks at *l** _{n }*superposed on a continuum. The peak-to-continuum
ratio is large for

The total power radiated by an undulator or wiggler is

_{}
(18)

where *Z*_{0} = 377 ohms, or, in practical units,

_{}

The angular distribution of the radiated power is

_{}
(19)

or, in units of W·mr^{–2},

The behavior of the angular function *f** _{K}*(

Electrons in storage rings are distributed in a finite area of
transverse phase space—position ´
angle. We introduce the rms beam sizes *s** _{x}* (horizontal) and

_{}
(20)

and

_{}
(21)

for bends and undulators, respectively. For bending magnets, the electron beam divergence effect is usually negligible in the horizontal plane.

**Fig. 2-7**. The angular function f_{K},
for different values of the deflection parameter K, (a) as a function of the
vertical observation angle y when
the horizontal observation angle q
= 0 and (b) as a function of q when
y = 0**.**

**Fig. 2-8**.** **The function G(K).

TRANSVERSE COHERENCE

For experiments that require a small angular divergence and a
small irradiated area, the relevant figure of merit is the beam brightness
B, which is the photon flux
per unit phase space volume, often given in units of photons·s^{–1}·mr^{–2}·mm^{–2}·(0.1% bandwidth)^{–1}. For an undulator, an approximate formula
for the peak brightness is

_{}
(22)

where, for example,

_{}
(23)

and where the single-electron radiation from an axially extended source of finite wavelength is described by

_{}
(24

Brightness is shown in Fig. 2-9 for several sources of synchrotron radiation, as well as some conventional x-ray sources.

That portion of the flux that is transversely coherent is given by

_{}
(25)

A substantial fraction of undulator flux is thus transversely
coherent for a low-emittance beam satisfying *e*_{x}*e*_{y}
_{} (*l*/4*p*)^{2}.

Longitudinal coherence is described in terms of a coherence length

_{}

For an undulator, the various harmonics have a natural spectral
purity of D*l*/*l* = 1/*nN* [see Eq. (12)]; thus, the coherence
length is given by

_{}
(27)

which corresponds to the relativistically contracted length
of the undulator. Thus, undulator radiation from low-emittance electron beams
[*e*_{x}*e*_{y}
_{} (*l*/4*p*)^{2}]
is transversely coherent and is longitudinally coherent within a distance
described by Eq. (27). In the case of finite beam emittance or finite angular
acceptance, the longitudinal coherence is reduced because of the change in
wavelength with emission angle. In this sense, undulator radiation is partially
coherent. Transverse and longitudinal coherence can be enhanced when necessary
by the use of spatial and spectral filtering (i.e., by use of apertures and
monochromators, respectively).

The references listed below provide more detail on the characteristics of synchrotron radiation.

**Fig. 2-9**. Spectral brightness for several synchrotron
radiation sources and conventional x-ray sources. The data for conventional
x-ray tubes should be taken as rough estimates only, since brightness depends
strongly on such parameters as operating voltage and take-off angle. The indicated
two-order-of-magnitude ranges show the approximate variation that can be expected
among stationary-anode tubes (lower end of range), rotating-anode tubes (middle),
and rotating-anode tubes with microfocusing (upper end of range).

REFERENCES

1. G. K. Green,
“Spectra and Optics of Synchrotron Radiation,” in *Proposal for National Synchrotron Light Source*, Brookhaven National
Laboratory, Upton, New York, BNL-50595 (1977).

2. H. Winick,
“Properties of Synchrotron Radiation,” in H. Winick and S. Doniach, Eds.,
*Synchrotron Radiation Research* (Plenum,
New York, 1979), p. 11.

3. S. Krinsky,
“Undulators as Sources of Synchrotron Radiation,” *IEEE Trans. Nucl. Sci.* **NS-30**,
3078 (1983).

4. D. F. Alferov,
Yu. Bashmakov, and E. G. Bessonov, “Undulator Radiation,” *Sov. Phys. Tech. Phys. ***18**,
1336 (1974).

5. K.-J. Kim,
“Angular Distribution of Undulator Power for an Arbitrary Deflection Parameter
*K*,” *Nucl.
Instrum. Methods Phys. Res. ***A246**,
67 (1986).

6. K.-J. Kim,
“Brightness, Coherence, and Propagation Characteristics of Synchrotron Radiation,”
*Nucl. Instrum. Methods Phys. Res. ***A246**,
71 (1986).

7. K.-J. Kim,
“Characteristics of Synchrotron Radiation,” in *Physics of Particle Accelerators*, AIP Conf. Proc. 184 (Am. Inst. Phys.,
New York, 1989), p. 565.

8. D. Attwood,
*Soft X-Rays and Extreme Ultraviolet Radiation:
Principles and Applications* (Cambridge Univ. Press, Cambridge, 1999);
see especially Chaps. 5 and 8.