3.3.8 Polarization of Bend Magnet Sources
3.3.8.1 Linear and Circular Polarization
A circularly polarized X-ray has oscillating electric and magnetic fields that are
90 degrees out of phase with each other. In the convention of Born and Wolf [2], the
instantaneous electric field E rcp for a right circularly polarized photon propagating in
the z direction resembles a right-handed screw:
E rcp E 0 sin ωt À kz þ ϕ 0
½
b i þ cos ωt À kz þ ϕ 0
½
b j
n
o
ð3:27Þ
In this equation, ω ¼ 2πν is the angular frequency, k ¼ 2π/λ is the wave number,
where λ is the wavelength, ϕ 0 is an arbitrary phase shift, and b i and b j are unit vectors
along the x and y axes, respectively. With the above definition, it turns out that left
circularly polarized photons carry +ħ angular momentum. Although the Born and
Wolf convention is standard for optics and chemistry literature, most physics
literature uses the opposite definition, and one should check how the polarization
is defined if the sign of a dichroism effect is to be meaningful. The papers of deGroot
and Brouder generally use the Born and Wolf convention, while those of Thole, van
der Laan, and Carra use the physics or “Feynman” definition [71]. The pitfalls of
describing circular polarization are cogently described in Kliger et al. [72].
For an observer in the plane of the bend magnet trajectory, the observed particle
acceleration is purely horizontal and perpendicular to the particle velocity. Therefore, the resulting electric field is also purely horizontal, and the radiation is linearly
polarized along the x-axis. As the observer moves above or below the orbit plane, the
observed acceleration acquires a vertical component, and the resulting radiation
becomes elliptically polarized. The polarization asymptotically approaches pure
right circular polarization at large angles above the storage ring (assuming a clockwise orbit when viewed from above) and left circular polarization at large angles
below the ring.
To derive a quantitative expression for the polarization, we need to go back to the
x- and y-components of the electric field. As described by Kim [67,73], from a bend
magnet, these relative field amplitudes are of the form:
E x
E y
/
K 2=3 η
ð Þ
iγψ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ γψ
ð Þ
2
q
K 1=3 η
ð Þ
0
B
@
1
C
A
ð3:28Þ
where, as a reminder, X ¼ γψ and η was defined in Eq. 3.15.
The imaginary character of the E y (vertical) field component implies that it is 90
out of phase with the E x (horizontal) field component. This further implies that when
48
3 Synchrotron Radiation Fundamentals
3.3.8.1 Linear and Circular Polarization
A circularly polarized X-ray has oscillating electric and magnetic fields that are
90 degrees out of phase with each other. In the convention of Born and Wolf [2], the
instantaneous electric field E rcp for a right circularly polarized photon propagating in
the z direction resembles a right-handed screw:
E rcp E 0 sin ωt À kz þ ϕ 0
½
b i þ cos ωt À kz þ ϕ 0
½
b j
n
o
ð3:27Þ
In this equation, ω ¼ 2πν is the angular frequency, k ¼ 2π/λ is the wave number,
where λ is the wavelength, ϕ 0 is an arbitrary phase shift, and b i and b j are unit vectors
along the x and y axes, respectively. With the above definition, it turns out that left
circularly polarized photons carry +ħ angular momentum. Although the Born and
Wolf convention is standard for optics and chemistry literature, most physics
literature uses the opposite definition, and one should check how the polarization
is defined if the sign of a dichroism effect is to be meaningful. The papers of deGroot
and Brouder generally use the Born and Wolf convention, while those of Thole, van
der Laan, and Carra use the physics or “Feynman” definition [71]. The pitfalls of
describing circular polarization are cogently described in Kliger et al. [72].
For an observer in the plane of the bend magnet trajectory, the observed particle
acceleration is purely horizontal and perpendicular to the particle velocity. Therefore, the resulting electric field is also purely horizontal, and the radiation is linearly
polarized along the x-axis. As the observer moves above or below the orbit plane, the
observed acceleration acquires a vertical component, and the resulting radiation
becomes elliptically polarized. The polarization asymptotically approaches pure
right circular polarization at large angles above the storage ring (assuming a clockwise orbit when viewed from above) and left circular polarization at large angles
below the ring.
To derive a quantitative expression for the polarization, we need to go back to the
x- and y-components of the electric field. As described by Kim [67,73], from a bend
magnet, these relative field amplitudes are of the form:
E x
E y
/
K 2=3 η
ð Þ
iγψ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ γψ
ð Þ
2
q
K 1=3 η
ð Þ
0
B
@
1
C
A
ð3:28Þ
where, as a reminder, X ¼ γψ and η was defined in Eq. 3.15.
The imaginary character of the E y (vertical) field component implies that it is 90
out of phase with the E x (horizontal) field component. This further implies that when
48
3 Synchrotron Radiation Fundamentals