92
3.6 Basic Fluorescence Microscopy Illumination Modes
(3.39)
n
b
c
b
c
b
c
λ
λ
λ
λ
λ
( ) ≈+
−
+
−
+
−
1
2
1
2
1
2
2
2
3
2
3
To understand how the evanescent field is generated, we can first apply Snell’s law of refrac
tion, from glass to water (Figure 3.5d):
(3.40)
sin
sin
forTIRF
θ
θ
w
g
w
ng
n
=
>
(
)
1
where
θg is the angle of incidence through the glass
θw is the angle of refraction through the water
Thus, by rearrangement
(3.41)
cos
sin
g
θ
θ
w
w
w
n
n
ib
=
−
=
1
2
where b is a real number in the case of TIRF; thus the angle of refraction into the water is a
purely imaginary number. The E-field in water in the 2D cross-section of Figure 3.3d can be
modeled as a traveling wave with distance and wave vectors of magnitude r(x, z) and wave
vector kw, respectively, with angular frequency ω after a time t, with z parallel to the optic axis
of the objective lens and xy parallel to the glass–water interface plane:
(3.42)
E
E
i k
r
t
E
i k x
k z
t
evanescent
w
w
=
⋅
−
=
+
−
(
)
0
0
exp
)
exp
sin
cos
w
w
w
(
ω
θ
θ
ω
By substitution from Equation 3.41, we get the evanescent wave equation:
(3.43)
E
E
bk z
i k n
n x
t
E
bk
evanescent
w
w
w
=
−
−
=
−
0
0
exp
)exp
sin
exp
w
g
(
θ
ω
wz
i k x
t
(
)
−
(
)
exp
*
ω
The intensity I(z) of the evanescent field at x = 0 as a function of z (i.e., as it penetrates deeper
into the solution) decays exponentially from the glass–water interface, characterized by the
depth of penetration factor d:
(3.44)
I z
I
z
d
( ) = ( )
−
0 exp
where
(3.45)
d
n
n
g
g
w
=
−
λ
π
θ
4
2
2
2
sin
A typical range of values for d is ~50–150 nm. Thus, after ~100 nm, the evanescent field
intensity is ~1/e of the value at the microscope glass coverslip surface, whereas at 1 μm depth
from the coverslip surface (e.g., the width bacteria), the intensity is just a few thousandths of
a percent of the surface value. Thus, only fluorophores very close to the slide are excited into
significant fluorescence, but those in the rest of the sample or any present in the surrounding