Chapter 3 – Making Light Work in Biology  93

solution are not. In effect, this creates an optical slice of ~100 nm thickness, which increases

the contrast dramatically above background noise to permit single-​molecule detection. It is

possible to adjust the angle of incidence θg to yield d values smaller than 100 nm, limited by

the numerical aperture of the objective lens (see Worked Case Example 3.1) and also larger

values, such that d → ∞ as θg → θc.

The intensity of the evanescent field is proportional to the square of the E-​field amp­

litude, but also depends on the polarization of the incidence E-​field in the glass and has

different values for the two orthogonal polarization components in water. Incident light that

is polarized parallel (p) to the xz plane of incidence as depicted in Figure 3.5e generates

an elliptically polarized evanescent field consisting of both parallel and perpendicular (s)

polarized components. Solving Maxwell’s equations generates the full solutions for the elec­

tric field vector components of the evanescent field as follows:

(3.46)

E

E

i

n

evanescent x

p

p

g

g

w

,

,

=

−

+



^



^



^



^

−

0

2

2

2

2

2

exp

cos

sin

/

δ

π

θ

θ

n

n

n

n

n

g

w

g

g

g

w

g

(

)

(

)

(

)

+

−(

)

2

4

2

2

2

/

cos

sin

/

θ

θ

(3.47)

E

E

i

n

n

y

s

s

g

evanescent

w

g

exp

cos

/

,

,

=

−





−(

)

0

2

2

2

1

δ

θ

(3.48)

E

E

i

n

n

evanescent,z

p

p

g

g

g

g

=

−





(

)

+

,0

2

4

2

2

exp

cos

sin

/

cos

si

w

δ

θ

θ

θ

n

/

2

2

θg

w

g

n

n

−(

)

where Ep,0 and Es,0 are the incident light E-​field amplitudes parallel and perpendicular to the

xz plane. As these equations suggest, there are different phase changes between the incident

and evanescent E-​fields parallel and perpendicular to the xz plane:

(3.49)

δ

θ

θ

p

g

g

w

g

g

n

n

n

n

=

−(

)

(

)

−

tan

sin

/

/

cos

w

1

2

2

(3.50)

δ

θ

θ

s

g

w

g

g

n

n

=

−(

)

−

tan

sin

/

cos

1

2

2

As these equations depict, the orientation of the polarization vector for s-​polarized incident

light is preserved in the evanescent field as the Ey component; as the supercritical angle of

incidence gets closer to the critical angle, the Ex component in the evanescent field converges

to zero and thus the p-​polarized evanescent field converges to being purely the Ez compo­

nent. This is utilized in the p-​TIRF technique, which uses incident pure p-​polarized light

close to, but just above, the critical angle, to generate an evanescent field, which is polarized

predominantly normal to the glass coverslip–​water interface. This has an important advan­

tage over subcritical angle excitation in standard epifluorescence illumination, for which the

angle of incidence is zero, since in p-​TIRF the polarization of the excitation field is purely par­

allel to the glass–​water interface and is unable to excite a fluorophore whose electric dipole

axis is normal to this interface and so can be used to infer orientation information for the