Differential Interference Contrast Microscopy (DIC) - Biomedical Optical Phase Microscopy and Nanoscopy - page 32

second DIC prism, and the output ray A 3 has coordinates ( 2 1, 2 1). However, the second
DIC prism moves the Y -polarized ray B 2 so that the second output ray B 3 has coordinates
(1,1). Hence, the output beam has a shear direction 1 45 .

In both the cases, the shears are the same, which is equal to the shear introduced by a single

DIC prism multiplied by p : However, the shear directions are mutually orthogonal. The

median axis of two beams passes through point (0,0) in both the cases. Therefore, there is
no misalignment between the images.

The intensity distribution in the image will be described by the transformed formula (2.2) :

h i

p dγðx; yÞ cos

π
λ

4

I i ðx; yÞ 5 I ~ sin 2

i

Γ 1 θðx; yÞ 2 ð 2 1 Þ 1 I c ðx; yÞ (2.19)

where i 5 1, 2 corresponds to the first or second state of shear direction, 2 45 or 1 45 .

In order to find the two-dimensional distribution of magnitude and azimuth ,we

capture two sets of raw DIC images at shear directions 2 45 and 1 45 with negative, zero,
and biases: 2 Γ

γ and θ

, 0, and 1 Γ [6,20] . The following group of equations represents these six

DIC images:

h i

p dγðx; yÞ cos

π
λ

4

I ij ðx; yÞ 5 I ~ sin 2

i

jΓ 1 θðx; yÞ 2 ð 2 1 Þ 1 I c ðx; yÞ (2.20)

where j 52 1, 0, 1.

Initially two terms are computed ( i 5 1, 2):

I i; 1 ð x ; y Þ 2 I i; 2 1 ð x ; y Þ

tan πΓ
λ

A i ð x ; y Þ 5 (2.21)

I i; 1 ð x ; y Þ 1 I i; 2 1 ð x ; y Þ 2 2 I i; 0 ð x ; y Þ

Using Eq. (2.20) , we can show that

! !

2 p π

λ

Þ 1 4

A 1 ð x ; y Þ 5 tan d γð x ; y Þ cos θð x ; y

! !

2 p π

λ

(2.22)

Þ 1 4 A 2 ð x ; y Þ 5 tan d γð x ; y Þ sin θð x ; y

Using the obtained terms, we can calculate the quantitative two-dimensional distributions of
the gradient magnitude and azimuth of optical paths in the specimen as:

q

ð

2

2 2

γðx; yÞ 5

p π

arctan A 1 ðx; yÞÞ 1 ð arctan A 2 ðx; yÞÞ

d

!

(2.23)

arctan A 2 ð x ; y Þ

2 4

θðx; yÞ 5 arctan

arctan A 1 ð x ; y Þ