### Fourier Optics

```Fourier Optics
The methods of harmonic analysis have proven to be useful in
describing signals in many disciplines. In this experiment you will
explore the concepts of Fourier analysis as applied to the field of
optical
imaging.
propagation
of
Fourier
light
optics
waves
provides
based
on
a
description
harmonic
analysis
of
the
(Fourier
decomposition). Harmonic analysis is based on the expansion of an
arbitrary function () as a superposition (a sum or an integral) of
harmonic functions of different spatial frequencies,  . The harmonic
function ( )  −   , which has a spatial frequency  and an amplitude
( ) is the building block of the theory. Several of those functions,
each
with
its
own
spatial
frequency
and
amplitude
are
to
construct the function (). The function ( ) is called the Fourier
transform of (). The two functions are related by the following
Fourier transformations:
+∞
() = ∫
−∞
( ) = ∫
( )  −
+∞
()  +
−∞
In this experiment you will learn how to decompose an object,
() into its Fourier components
described by a certain function
( )  −   , propagate those spatial frequencies through an optical
system composed of multiple lenses, and create a magnified  image of
the object,  (), and an image of the Fourier transform ( ) of the
object. You will also measure the spatial frequencies present in an
1
object and determine relevant feature sizes of the object. You will
learn how to filter spatial frequencies and verify its impact in the
image of an object.
Experiment:
A He-Ne laser beam at 0 = 632.8 nm, which out of the
laser head typically has a beam diameter of 1 mm, has been
expanded to about 20-30 mm in diameter by a combination of
two lenses, one with a short focal length (a microscope
objective) and another one with a long focal length. The
distance between the two lenses has been adjusted to create
an expanded optical beam with a flat wavefront (i.e., an
expanded collimated beam). The flat wavefront is confirmed
by a shear plate interferometer.
Next, the light wave with a large diameter size and a
flat wavefront is ready to illuminate objects of interest.
As objects to be illuminated in this experiment you will use
several meshes with different grid sizes. Start with the
mesh that is marked with 120, which is supposed to have a
grid size of 120 µm according to the vendor. Place the mesh
at about 100 mm in front of the first lens of the optical
system that will come after the object.
The
optical
system
placed
after
the
object
is
illustrated in the Figure below and is composed of two
lenses, each with a focal length of  = 100  and they are
2
separated by about 200 mm, which is 2 = 200 . When the
object is placed at about the object focal point of the
first lens, 100 , and image of the object will appear at
the image focal point at about 100  of the second lens.
This configuration is known in the literature as 4 system.
At a distance  from lens 1, each spatial frequency coming
out of the object will converge to a point. This plane is
known as the Fourier plane. Put a white piece of paper to
observe the light pattern at this plane.
Now, as light propagates through the optical system,
lens 2 sees an object (the Fourier pattern) at its object
focal plane and will form an image of this pattern at a
screen at infinity (at very large distance from it). You
will take measurements to describe the Fourier pattern at a
3
screen at a large distance from lens 2. Measure the distance
form lens 2 and the screen used for observation,  . The
transverse
magnification
given by || =
of the Fourier pattern is
then
( −)

Sketch a drawing in your notebook of the pattern you
observe at the screen. Consider the brightest spot in the
pattern as a reference point and take measurements for the
location of the several other bright spots you observe in
image of the Fourier pattern at the screen.
Bring a new lens (lens 3) to the setup to create a
magnified image of the real object at the screen. Find the
location to form a sharp image of the object on the distant
screen. Move the lens 3 in and out of the optical path. You
will observe the magnified object pattern (, ) when the
lens is in the optical path, and the magnified Fourier
pattern ( ,  ) when the lens is out of the optical path.
You are performing a two dimensional Fourier transformation
by moving the lens in and out of the optical path.
Insert a slit (with the blades oriented in the vertical
direction) at approximately the Fourier plane. Because the
opening of the slit is set along the vertical direction, it
will
remove
horizontal
some
spatial
direction,
.
frequencies
Slowly
move
the
slit
in
in
the
the
horizontal direction and observe what happen at the screen.
Bring lens 3 in and out of the optical path. You should be
4
able to remove the horizontal lines from the magnified image
Analysis:
Consider the bright spots in one line in the vertical
direction. The spatial frequencies in the vertical direction
will then be given by: (, ) =
∆
⁄
,

where ∆
is the
verticalistance to the reference point for each bright spot
marked
by
integer
= ±1, ±2, ±3, …
and
the
positive
and
negative signs correspond to above and below the reference
point, respectively. As the image at the distant screen of
the Fourier pattern has been magnified, the corresponding
spatial frequency , in the back focal plane of lens 1
(Fourier plane in the Figure) is given by: , =  (, ) =
2
0
(, ). Given a mesh with a regular vertical grid size of
Λ , the spatial frequency in the vertical direction will be
given by , =
2
Λ
=
2
0
(, ).
Create a table of  and
=
1
0
1
0
(, ). Plot  =  versus
(, ). The slope of a linear fit of  versus  should
give you the vertical grid size Λ .
Repeat your analysis above for the horizontal direction
and determine Λ .
5
Questions to consider:
1. What would happen if the grid size was not constant along the
illuminated pattern?
2. What would happen if Λ was smaller than the laser wavelength?
3. Mathematically describe the two dimensional object with a regular
square
pattern
along
in
x-y
plane.
Calculate
the
Fourier
transformation of this two dimensional pattern.
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7
```