synchrotron radiation 53
area A s and the emitted radiation has angular opening angles
of ΔΦ and Δψ, as shown in Fig. 3.8.
FIGURE 3.8
For illustration of brilliance or brightness. Here A s is the emitting
area and ΔΨ and ΔΦ are opening angles of emitted photons.
The first concept to make note of is flux, which is expressed as the number of photons emitted per units of time
and per unit of bandwidth:
Flux = Photons/ (s · BW )
(3.34)
The brilliance (or brightness) is then defined as flux per
unit of emitting area and product of angles:
Brilliance = Flux/(A s · ΔΦ · Δψ)
and is usually expressed in units of [Photons/(s · mm · mrad 2 ·
BW )].
In a typical case of Gaussian distributions, the definition
of brilliance is based on the total effective sizes and divergences
flux
brilliance =
(3.35)
4π 2 Σ x Σ x ' Σ y Σ y '
where the total effective sizes include contributions from
electrons as well as photons:
)
)
Σ x =
x,e + σ
2
σ x = ε x β x + (D x
(3.36)
σ
2
ph,e
σ ε )
2
)
)
2
' = σ
2 + σ
'
' =
+ (D x
' σ ε )
2
Σ x
x ' ,e
ph,e
σ x
ε x β x
and similarly for the other plane.
3.3.4 Ultimate brightness
As we have seen, brilliance is defined by the overall effective emittance, which convolves electron and photon distributions:
)
)
ε ef f = σ e
2 + σ
2
σ
2
' + σ
2
(3.37)
ph
e
ph '