2.1 Holographic Microscopy for the Study of Phase, Diffusive …
73
where P is the pupil function, ρ i is the distance from the output pupil to the image
plane, k is the constant.
Equation (2.8) shows that during the formation of a defocused image, frequency
components transmitted by the objective (the first factor) are added with some phase
shift proportional to d z and frequency square.
Let us show the influence of defocusing with several typical examples.
Plane mirror. For example, as such an object we can use a mirror of the heterojunction laser during the study of its thermal deformation [56, 57, 65] if its size
is higher than the microobjective resolution limit. When the plane wave falls on
the mirror parallel to the optical axis, reflected plane wave will have the spectrum
G 01
f
= δ
f
, and the spectrum U 02 = U 01 exp(−i2kd z ) will be different only
in phase factor G 02
f
= δ
f
exp(−i2kd z ), where δ
f
is the two-dimensional
delta function. Then, interference part included in (2.3) and being of utmost interest
for us will be equal to:
Re
V 1 V
∗
2
= 2Re
F
−1
H (
f , 0)δ(M
f )
F
−1∗
H (
f , d z )δ(M
f ) exp(−i2kd z )
=
2λ
4
ρ
4
i
M 4 cos 2kd z .
(2.9)
Essentially, the described system is the holographic variant of the Michelson
interferometer tuned for the infinite width lines, and in the image plane, there will be
cyclic blackening and lightening of all field of view as d z increases. The defocusing
does not influence the interference pattern form because plane wave is localized
neither in the object domain nor in the image area.
Diffusely reflected surface. If light is scattered by rough surface, then the image
of this surface V l ( x i ), as it was noticed, will have a speckle-structure. In the result
of coherent addition of two random fields V 1 ( x i ) and V 2 ( x i ), there will be a new
speckle-field
V ( x i ) = V 1 ( x i ) + V 2 ( x i ) =
{U 01 ( x 0 )ρ 1 ( x 0 )h( x i − M
x 0 , 0)
+U 02 ( x 0 )ρ 2 ( x 0 )h( x i − M
x 0 , d z )}d
2
x 0 ,
(2.10)
where U ol describes the wave reflected from macrorelief (smooth surface), and ρ l ( x 0 )
describes phase chance variation made by the object microrelief in the initial (l = 1)
and deformed (l = 2) states.
In order to detect field macromodulation or interference pattern, it is necessary
to eliminate speckle-structure, i.e., to average diffuser ensemble ρ, which further
will be marked as . . .. Let us analyze interference part
V 1 ( x i )V
∗
2 ( x i )
one more
time considering that
ρ 1 ( x 0 )ρ 1
x
0
= δ
x 0 − −
x
0
and assuming that in the result of
deformation U 02
x 0 +
d
= U 01 ( x 0 ) exp
iφ( x o )
, where
d is the two-dimensional
shift vector in the plane − → x 0 ; φ( x 0 ) =
k
d;
k =
k 2 −
k 1 is the change of wave
73
where P is the pupil function, ρ i is the distance from the output pupil to the image
plane, k is the constant.
Equation (2.8) shows that during the formation of a defocused image, frequency
components transmitted by the objective (the first factor) are added with some phase
shift proportional to d z and frequency square.
Let us show the influence of defocusing with several typical examples.
Plane mirror. For example, as such an object we can use a mirror of the heterojunction laser during the study of its thermal deformation [56, 57, 65] if its size
is higher than the microobjective resolution limit. When the plane wave falls on
the mirror parallel to the optical axis, reflected plane wave will have the spectrum
G 01
f
= δ
f
, and the spectrum U 02 = U 01 exp(−i2kd z ) will be different only
in phase factor G 02
f
= δ
f
exp(−i2kd z ), where δ
f
is the two-dimensional
delta function. Then, interference part included in (2.3) and being of utmost interest
for us will be equal to:
Re
V 1 V
∗
2
= 2Re
F
−1
H (
f , 0)δ(M
f )
F
−1∗
H (
f , d z )δ(M
f ) exp(−i2kd z )
=
2λ
4
ρ
4
i
M 4 cos 2kd z .
(2.9)
Essentially, the described system is the holographic variant of the Michelson
interferometer tuned for the infinite width lines, and in the image plane, there will be
cyclic blackening and lightening of all field of view as d z increases. The defocusing
does not influence the interference pattern form because plane wave is localized
neither in the object domain nor in the image area.
Diffusely reflected surface. If light is scattered by rough surface, then the image
of this surface V l ( x i ), as it was noticed, will have a speckle-structure. In the result
of coherent addition of two random fields V 1 ( x i ) and V 2 ( x i ), there will be a new
speckle-field
V ( x i ) = V 1 ( x i ) + V 2 ( x i ) =
{U 01 ( x 0 )ρ 1 ( x 0 )h( x i − M
x 0 , 0)
+U 02 ( x 0 )ρ 2 ( x 0 )h( x i − M
x 0 , d z )}d
2
x 0 ,
(2.10)
where U ol describes the wave reflected from macrorelief (smooth surface), and ρ l ( x 0 )
describes phase chance variation made by the object microrelief in the initial (l = 1)
and deformed (l = 2) states.
In order to detect field macromodulation or interference pattern, it is necessary
to eliminate speckle-structure, i.e., to average diffuser ensemble ρ, which further
will be marked as . . .. Let us analyze interference part
V 1 ( x i )V
∗
2 ( x i )
one more
time considering that
ρ 1 ( x 0 )ρ 1
x
0
= δ
x 0 − −
x
0
and assuming that in the result of
deformation U 02
x 0 +
d
= U 01 ( x 0 ) exp
iφ( x o )
, where
d is the two-dimensional
shift vector in the plane − → x 0 ; φ( x 0 ) =
k
d;
k =
k 2 −
k 1 is the change of wave