50 when it is inserted on the optical path. When the intensity image of the interference light is acquired by the imaging unit 50, the shield 36 is removed from the optical path of the object light. These functions will be described later.

In the observation apparatus 1 described above, a moving speed of the observation object 2 is set to V, a wavenumber component in the x axis direction of the object light before interacting with the observation object 2 is set to k_xⁱⁿ, and a wavenumber component in the x axis direction of the object light after interacting with the observation object 2 is set to k_x^out. In this case, a Doppler shift amount Δf of the optical frequency of the object light generated by the interaction with the observation object 2 is represented by the following Formula (1).

[

Formula
⁢

1

]



Δ
⁢
f

=

(

k
x

o
⁢
u
⁢
t

-

k
x

i
⁢
n

)

·

V

2
⁢
π

(
1
)

A numerical aperture of the lens 22 at the final stage of the irradiation optical system 20 is set to NA_in, a numerical aperture of the lens 31 at the first stage of the imaging optical system 30 is set to NA_out, and a wavelength of the light is set to λ. In this case, a maximum value Δf_maxof the Doppler shift amount Δf is represented by the following Formula (2), and a minimum value Δf_minof the Doppler shift amount Δf is represented by the following Formula (3). Therefore, the Doppler shift amount Δf of the optical frequency of the object light reaching the imaging plane of the imaging unit 50 is distributed in a range of [−Δf_max, Δf_max].

[

Formula
⁢

2

]



Δ
⁢

f
max

=

(

N
⁢

A

i
⁢
n

+

N
⁢

A

o
⁢
u
⁢
t

)

·

V
λ

(
2
)

[

Formula
⁢

3

]



Δ
⁢

f
min

=

-

(

N
⁢

A

i
⁢
n

+

N
⁢

A

o
⁢
u
⁢
t

)

·

V
λ

=

-
Δ

⁢

f
max

(
3
)

The time series data of the complex amplitude image of the object light on the imaging plane of the imaging unit 50 is set to U_obj(t). The time series data of the complex amplitude image of the reference light on the imaging plane of the imaging unit 50 is set to U_ref(t). U_ref(t) is represented as follows using the heterodyne frequency f₀.

U_ref(t)=U₀·exp(−i2πf₀t)

The time series data I(t) of the intensity image of the interference light acquired by the imaging unit 50 is represented by the following Formula (4) and Formula (5) (Formulas (5a) to (5d)). In addition, i is an imaginary unit, π is a circular constant, and t is a time variable. The notation of a variable representing a position in each image is omitted.

[

Formula
⁢

4

]



I
⁡
(
t
)

=

❘
"[LeftBracketingBar]"

U
obj

(
t
)

+

U
ref

(
t
)

❘
"[RightBracketingBar]"

2

=

I
obj

(
t
)

+

I

r
⁢
e
⁢
f

(
t
)

+

I

c
⁢
r
⁢
o
⁢
s
⁢
s

(
t
)

+

I

c
⁢
r
⁢
o
⁢
s
⁢
s

*

(
t
)

(
4
)

[

Formula
⁢

5

]



I

o
⁢
b
⁢
j

(
t
)

=

❘
"[LeftBracketingBar]"

U

o
⁢
b
⁢
j

(
t
)

❘
"[RightBracketingBar]"

2

(

5
⁢
a

)

I

r
⁢
e
⁢
f

(
t
)

=

❘
"[LeftBracketingBar]"

U

r
⁢
e
⁢
f

(
t
)

❘
"[RightBracketingBar]"

2

(

5
⁢
b

)

I

c
⁢
r
⁢
o
⁢
s
⁢
s

(
t
)

=

U
obj
*

(
t
)

·

U

r
⁢
e
⁢
f

(
t
)

(

5
⁢
c

)

I

c
⁢
r
⁢
o
⁢
s
⁢
s

*

(
t
)

=

U
obj

(
t
)

·

U

r
⁢
e
⁢
f

*

(
t
)

(

5
⁢
d

)

I_obj(t) (Formula (5a)) includes frequency components in a range of [−2Δf_max, 2Δf_max]. I_ref(t) (Formula (5b)) includes only DC components when U₀does not change with time. I_cross(t) (Formula (5c)) includes frequency components in a range of [−Δf_max+f₀, Δf_max+f₀]. I_cross*(t) (Formula (5d)) includes frequency components in a range of [−Δf_max−f₀, Δf_max−f₀].

When the frequency range of I_cross(t) or I_cross*(t) does not overlap with the frequency range of I_obj(t), the time series data of I_cross(t) or I_cross*(t) can be measured. Further, the time series data U_obj(t) of the complex amplitude image of the object light can be obtained from Formula (5c) or Formula (5d).

In order to prevent the frequency range of I_cross(t) or I_cross*(t) from overlapping with the frequency range of I_obj(t), it is necessary to set the heterodyne frequency f₀to f₀≥3Δf_max. In addition, when f₀=3Δf_max, the maximum frequency of the time series data I(t) of the intensity image of the interference light is 4Δf_max. Therefore, from the Nyquist sampling theorem, when the time series data I(t) of the intensity image of the interference light is acquired by the imaging unit 50 at a frame rate of twice the maximum frequency (8Δf_max), the time series data U_obj(t) of the complex amplitude image of the object light can be reproduced with high accuracy.

In the observation apparatus and the observation method described below, the frame rate when the imaging unit 50 acquires the time series data I(t) of the intensity image of the interference light can be made lower than 8Δf_maxby improving the content of processing by the analysis unit 60.

A function β(t) represented by the following Formula (6) is defined using the time series data U_obj(t) of the complex amplitude image of the object light on the imaging plane of the imaging unit 50 and the time series data U_ref(t) of the complex amplitude image of the reference light on the imaging plane of the imaging unit 50. Further, a function χ(t) represented by the following Formula (7) is defined using the function β(t).

[

Formula
⁢

6

]



β
⁡
(
t
)

=

U
obj

(
t
)

U
ref

(
t
)

(
6
)

[

Formula
⁢

7

]



χ
⁡
(
t
)

=

log

[

1
+

β
⁡
(
t
)

]

(
7
)

Re[χ(t)] being a real part of the function χ(t) is represented by the following Formula (8). As shown in Formula (8), Re[χ(t)] can be obtained based on the time series data I(t) of the intensity image of the interference light on the imaging plane of the imaging unit 50 and the time series data I_ref(t) of the intensity image of the reference light on the imaging plane of the imaging unit 50. Since both I(t) and I_ref(t) can be measured, Re[χ(t)] can be obtained from the measured values.

[

Formula
⁢

8

]



Re

[

χ
⁡
(
t
)

]

=

log
⁢

❘
"[LeftBracketingBar]"

1
+

β
⁡
(
t
)

❘
"[RightBracketingBar]"

=

log
⁢

❘
"[LeftBracketingBar]"

1
+

U
obj

(
t
)

U
ref

(
t
)

❘
"[RightBracketingBar]"

=

1
2

·
log

⁢

❘
"[LeftBracketingBar]"

U
obj

(
t
)

+

U

r
⁢
e
⁢
f

(
t
)

U

r
⁢
e
⁢
f

(
t
)

❘
"[RightBracketingBar]"

2

=

1
2

⁢

{

log
⁢

I
⁡
(
t
)

-

log
⁢

I

r
⁢
e
⁢
f

(
t
)

}

(
8
)

The time series data I_ref(t) of the intensity image of the reference light can be measured, for example, by inserting the shield 36 on the optical path of the object light in the interference optical system as illustrated in FIG. 2 before or after the measurement so that only the reference light out of the object light and the reference light reaches the imaging plane of the imaging unit 50, when the data does not change with time during and before and after the measurement. In addition, in Formula (8), the intensity image of the reference light corresponding to the time series data I(t) of the intensity image of the interference light is referred to as the time series data I_ref(t) of the intensity image of the reference light which does not change with time.

Further, when the time series data I_ref(t) of the intensity image of the reference light changes with time, for example, as in a configuration of an observation apparatus 1A illustrated in FIG. 3, a beam splitter 75 may be provided on the optical path of the reference light in the interference optical system, a part of the reference light may be split and extracted by the beam splitter 75, and the time series data I_ref(t) of the intensity image of the extracted reference light may be acquired by another second imaging unit 51.

The time series data I_ref(t) of the intensity image of the reference light acquired by the second imaging unit 51 may be equal to the time series data I_ref(t) of the intensity image of the reference light acquired by the imaging unit 50 when only the reference light reaches the imaging unit 50. The second imaging unit 51 may have a configuration similar to that of the imaging unit 50.

Im[χ(t)] being an imaginary part of the function χ(t) and Re[χ(t)] described above have a relationship represented by the Kramers-Kronig relations (hereinafter referred to as the “KK relations”) of the following Formula (9) and Formula (10). Therefore, by using Formula (10), Im[χ(t)] can be obtained from Re[χ(t)]. Thus, since the function χ(t) can be obtained, the time series data U_obj(t) of the complex amplitude image of the object light can be obtained from the function χ(t) using Formula (6) and Formula (7).

[

Formula
⁢

9

]



Re

[

χ
⁡
(
t
)

]

=

1
π

⁢

p
.
v
.

∫

-
∞

∞

Im

[

χ
⁢

(

t
′

)

]

t
′

-
t

⁢

dt
′

(
9
)

[

Formula
⁢

10

]



Im

[

χ
⁡
(
t
)

]

=

-

1
π

⁢

p
.
v
.

∫

-
∞

∞

⁢

Re

[

χ
⁡
(

t
′

)

]

t
′

-
t

⁢

dt
′

(
10
)

In addition, it will be described that Re[χ(t)] and Im[χ(t)] have the relationship represented by the above Formula (9) and Formula (10). When β(t) satisfies the following Formula (11), χ(t) can be Taylor-expanded as in the following Formula (12). That is, χ(t) can be expanded by a Dower of β(t).

[

Formula
⁢

11

]



❘
"[LeftBracketingBar]"

β
⁡
(
t
)

❘
"[RightBracketingBar]"

<
1

(
11
)

[

Formula
⁢

12

]



χ
⁡
(
t
)

=

∑

n
=
0

∞

(

-
1

)

n

n
+
1

⁢

β
⁡
(
t
)

n
+
1

(
12
)

In order to satisfy the above Formula (11), a transmittance and a reflectance of the beam splitter 71 or the beam splitter 72 may be set to appropriate values. Further, in order to satisfy the above Formula (11), as illustrated in FIG. 2, the object light may be attenuated by the neutral density filter 35 provided on the optical path of the object light in the interference optical system. The neutral density filter 35 is preferably provided on the optical path where the object light is collimated (for example, between the lens 32 and the lens 33 or between the beam splitter 71 and the lens 21).

In general, the convolution theorem holds for two functions f₁(t) and f₂(t). That is, when a Fourier transform of f₁(t) is set to F₁(ω) and a Fourier transform of f₂(t) is set to F₂(ω), a product of f₁(t) and f₂(t) is equal to a Fourier transform of convolution of F₁(ω) and F₂(ω).

From the convolution theorem and the above Formula (12), when the Fourier transform B(ω) of β(t) satisfies B(ω<0)=0, the Fourier transform X(ω) of χ(t) satisfies X(ω<0)=0. That is, when B(ω)=0 in the range of ω<0, X(ω)=0 in the range of ω<0. The inverse Fourier transform of the function X(ω) satisfying X(ω<0)=0 is χ(t), and thus, Re[χ(t)] and Im[χ(t)] have the relationship of the KK relations represented by the above Formula (9) and Formula (10).

In order to satisfy B(ω<0)=0, the heterodyne frequency f₀set by the modulation unit 40 may be set to Δf_maxor more. That is, as can be seen from the above Formula (6), when the frequency range [−Δf_max, Δf_max] of U_obj(t) is shifted by the heterodyne frequency f₀of U_ref(t), a negative frequency component may not be present after the shift.

When f₀is set to less than 3Δf_max, the imaging unit 50 acquires the time series data I(t) of the intensity image of the interference light at a frame rate lower than that in the conventional art, and the time series data U_obj(t) of the complex amplitude image of the object light can be reproduced with high accuracy. For example, when f₀=Δf_max, the maximum frequency of the time series data I(t) of the intensity image of the interference light is 2Δf_max, and thus, the frame rate of the imaging unit 50 may be 4Δf_max. This is one half of the frame rate conventionally required.

In addition, for each of I(t) and I_ref(t) acquired by the imaging unit 50, it is preferable to perform a Fourier transform, a padding process of enlarging a frequency domain after the Fourier transform and giving a predetermined value (for example, 0) to the enlarged domain, and an inverse Fourier transform after the padding process, and use the data after these processes in Formula (8).

Further, for U_obj(t) obtained from χ(t), it is preferable to perform a Fourier transform, a cropping process for extracting a part of a frequency domain after the Fourier transform, and an inverse Fourier transform after the cropping process, and use the data after these processes in subsequent processes.

Next, simulation results will be described. In this case, the configuration of the observation apparatus illustrated in FIG. 1 and FIG. 2 is assumed, and an original image (FIG. 4 and FIG. 5) parallel to the xy plane is moved in the x axis direction. The maximum value Δf_maxof the Doppler shift amount of the object light reaching the imaging plane of the imaging unit 50 is set to 40 kHz, and the heterodyne frequency f₀by the modulation unit 40 is also set to 40 kHz. FIG. 4 is a diagram showing the real part of the original image, and FIG. 5 is a diagram showing the imaginary part of the original image. The number of pixels of the original image is 256×512.