Total reflection occurs when the refracted ray is predicted to be parallel to the
surface:
cos θ cg ¼ n 2 =n 1
For transmission from vacuum, n 1 ¼ 1, and by employing the approximation for
small angle that cos θ ffi 1 À θ
2 /2, a useful formula is then:
θ cg ffi
ffiffiffiffiffi
2δ
p
ð4:15Þ
If we plug in some typical values (Table 4.1), we find that for gold at 35 Å, the
critical angle is about 6
, dropping to less than 0.5
(5–7 mrad) at 1 Å.
4.3.2 Less than Total External Reflection
So far, we have ignored the detailed shape of the reflectivity curve for angles above
the critical angle. This will depend on the complex index of refraction and the
surface roughness of the mirror. In Fig. 4.4 we illustrate (using Eqs. 4.10 and
4.11) the effects of different values for δ, β, and σ on the reflectivity as a function
of glancing angle.
In the absence of absorption, stronger refraction leads to a higher angle for total
external reflection. At most energies, higher Z elements will have larger values for δ.
For this reason, SiO 2 X-ray mirrors are often coated with a metal such Ni or Au. On
the other hand, increased absorption (larger β) diminishes the reflectivity. Both
effects are shown in Fig. 4.4.
The third factor influencing mirror performance is surface roughness, which can
be characterized by the rms surface displacement—σ. An approximate formula for
the loss of reflectivity due to roughness is:
R ¼ R 0 exp À 4πσθ i =λ
ð
Þ
2
h
i
ð4:16Þ
Fig. 4.4 Left: mirror p-reflectivity vs. glancing angle for different refractive index coefficients δ.
Middle: p-reflectivity vs. glancing angle for different absorption coefficients β. Right:
reflectivity vs. glancing angle for different amounts of surface roughness σ
4.3 Reflection: X-ray Mirrors
75
surface:
cos θ cg ¼ n 2 =n 1
For transmission from vacuum, n 1 ¼ 1, and by employing the approximation for
small angle that cos θ ffi 1 À θ
2 /2, a useful formula is then:
θ cg ffi
ffiffiffiffiffi
2δ
p
ð4:15Þ
If we plug in some typical values (Table 4.1), we find that for gold at 35 Å, the
critical angle is about 6
, dropping to less than 0.5
(5–7 mrad) at 1 Å.
4.3.2 Less than Total External Reflection
So far, we have ignored the detailed shape of the reflectivity curve for angles above
the critical angle. This will depend on the complex index of refraction and the
surface roughness of the mirror. In Fig. 4.4 we illustrate (using Eqs. 4.10 and
4.11) the effects of different values for δ, β, and σ on the reflectivity as a function
of glancing angle.
In the absence of absorption, stronger refraction leads to a higher angle for total
external reflection. At most energies, higher Z elements will have larger values for δ.
For this reason, SiO 2 X-ray mirrors are often coated with a metal such Ni or Au. On
the other hand, increased absorption (larger β) diminishes the reflectivity. Both
effects are shown in Fig. 4.4.
The third factor influencing mirror performance is surface roughness, which can
be characterized by the rms surface displacement—σ. An approximate formula for
the loss of reflectivity due to roughness is:
R ¼ R 0 exp À 4πσθ i =λ
ð
Þ
2
h
i
ð4:16Þ
Fig. 4.4 Left: mirror p-reflectivity vs. glancing angle for different refractive index coefficients δ.
Middle: p-reflectivity vs. glancing angle for different absorption coefficients β. Right:
reflectivity vs. glancing angle for different amounts of surface roughness σ
4.3 Reflection: X-ray Mirrors
75