but we will just take it as a given. This conserved area (within a factor of π) is called
the emittance, and there are generally different values for the horizontal and vertical
emittances—ε x and ε y . At the point of tightest focus of the beam, the emittance can
also be described as the product of the physical size of the beam in the horizontal (σ x )
or vertical (σ y ) direction, and its angular divergence (σ x´ or σ y´ ) in that dimension.
Thus, at the beam waist:
ε x ¼ σ x σ x 0
ð2:13Þ
ε y ¼ σ y σ y 0
ð2:14Þ
The horizontal emittance ε x is increased by the synchrotron radiation caused by
the bend magnets, and it is often ~10–100 times larger than the vertical emittance ε y .
The ratio ε x /ε y is called the coupling ratio.
2.3.7 The Diffraction Limit
From the Heisenberg uncertainty principle, a light source has a natural limit to the
product of the source size and divergence, in both x and y directions, known as the
diffraction limit. If we represent the source sizes and divergences respectively in
either the horizontal or vertical planes using σ x,y ¼ σ r /√2 and σ x´,y´ ¼ σ r´ /√2, where
the subscript r refers to the radial source parameters, as defined in Wiedemann, then
we can write the following equation for the photon beam emittance:
ε photon,r ¼ σ r σ r 0 ≌
λ
2π
ð2:15Þ
For an undulator source, for each dimension, this reflects to the following
diffraction limited photon beam divergence and diffraction limited source size:
Fig. 2.10 The Twiss β x and β y parameters vs. orbit positions for the Sirius ring [25]
2.3 The Storage Ring: Inside the Shield Walls
23
the emittance, and there are generally different values for the horizontal and vertical
emittances—ε x and ε y . At the point of tightest focus of the beam, the emittance can
also be described as the product of the physical size of the beam in the horizontal (σ x )
or vertical (σ y ) direction, and its angular divergence (σ x´ or σ y´ ) in that dimension.
Thus, at the beam waist:
ε x ¼ σ x σ x 0
ð2:13Þ
ε y ¼ σ y σ y 0
ð2:14Þ
The horizontal emittance ε x is increased by the synchrotron radiation caused by
the bend magnets, and it is often ~10–100 times larger than the vertical emittance ε y .
The ratio ε x /ε y is called the coupling ratio.
2.3.7 The Diffraction Limit
From the Heisenberg uncertainty principle, a light source has a natural limit to the
product of the source size and divergence, in both x and y directions, known as the
diffraction limit. If we represent the source sizes and divergences respectively in
either the horizontal or vertical planes using σ x,y ¼ σ r /√2 and σ x´,y´ ¼ σ r´ /√2, where
the subscript r refers to the radial source parameters, as defined in Wiedemann, then
we can write the following equation for the photon beam emittance:
ε photon,r ¼ σ r σ r 0 ≌
λ
2π
ð2:15Þ
For an undulator source, for each dimension, this reflects to the following
diffraction limited photon beam divergence and diffraction limited source size:
Fig. 2.10 The Twiss β x and β y parameters vs. orbit positions for the Sirius ring [25]
2.3 The Storage Ring: Inside the Shield Walls
23