An Introductory Study of Fourier Optics
Briana Abrahms
Advanced Physics Laboratory
Professor Fraden
Introduction
The study of Fourier optics has surged over the past decade; its applications are
broad and germane to everyday life. The basic principle of Fourier optics lies in the fact
that the Fourier transform converts light diffracted by an object into frequency space, and
by filtering certain frequencies, the image can be altered and reconstructed. This can be
done either by physically imposing a filter on an optics system or by numerical methods
using, in the case of this experiment, the computer program Mathematica. The purpose of
this lab was to observe the diffraction pattern of a grating and explore the effects of
frequency filtering in its Fourier transform plane in order to study the properties of
Fourier optics.
Theory
All electromagnetic waves propagating through space have the general form
Ψ = A*sin[kx – ωt + φ], where A = amplitude, k = wave number, ω = frequency, and φ =
phase angle. When diffracting, two or more waves interfere with each other and combine
constructively or destructively depending on their relative phase differences. The light
scattered can be described as a sum of waves with different frequencies, which is the total
wave function. The Fourier transform F(k) of a function f(x) shows the amplitude, or
contribution, to the summation wave of particular wave frequencies; the Fourier
transform in this way can represent many different functions as a summation of sine
functions. Figure 1. below shows this process for a square wave. As one can see, the low
frequency components contribute the most to form the wave whereas the high frequency
components give the function its sharpness.
amplitude
Square
Wave
and
Fourier
Synthesis
1
0.5
x
0.2
0.4
0.6
0.8
-0.5
-1
Figure 1. A square wave with its Fourier components.
1
The Fourier transform has the complex form F(k) = ∫f(x)eikx with limits from negative
infinity to infinity. The diffraction pattern of two waves is based on the intensity of the
resultant interference, which can be found by squaring the amplitude of the combined
wave’s Fourier transform. In multiple slit diffraction, which is the case in this lab, the
equation sin θ = λm/a relates the angle of diffraction ‘θ’, the wavelength ‘λ’, the distance
between slits ‘a’, and the order of the angle ‘m’. A Fourier transform can be converted
back to the original function by performing the Inverse Fourier transform on it,
f(x) = F-1{F(k)}.
Since the Fourier transform of a wave function is based on an infinite sum of
frequencies, filters can be introduced to remove certain frequencies from passing through.
As can be seen by the mathematics of Fourier analysis, removing particularly high
frequencies will cause a loss in the detail of the image, whereas removing low
frequencies may lose some of the main image. Thus, one can manipulate an image by
filtering its Fourier transform in the FT plane before it is refocused onto the image plane.
In Low Pass filters, only low frequencies are allowed to pass through, so the filter is
completely black except for a circle in the center where light can pass. For High Pass
filters, the exact opposite is true, and since higher frequencies have a higher angle of
diffraction, the outer part of the filter is white while an inner circle is blacked out. Band
Pass filters block out a certain band of frequencies from passing; Gaussian filters are
tapered rather than having a sharp cutoff so the images appear fuzzier. See Results for
examples of the filters used in this experiment.
In order to see an object’s diffraction pattern as described above, there must be a
very specific optical setup that is based on the physics of optics. At this point is it
necessary to note that the type of diffraction dealt with in this lab is Fraunhofer, or FarField, diffraction, in which the diffraction object is at a large distance from the light
source and detection screen. If this distance is large enough, it ensures that the light
arriving at the diffraction object and screen is parallel, a necessary condition. An example
of the optics of image formation is shown in Figure 2. After light is diffracted from an
object, an objective lens Lt is needed to focus the light on the image plane Σi. A screen
placed one focal length away at Σt will show the diffraction object’s Fourier transform,
and therefore the objective lens also serves a transform lens. To bring the image in from
infinity to a convenient distance, an imaging lens Li which performs an inverse Fourier
transform is necessary.
Figure 2. Image Formation (Optics, Hecht)
Experimental Setup
See Figure 3. below to refer to the visual setup. The light source (1), a 1.2 mW
helium-neon laser with a 633 nm wavelength, initially passed through a polarizer so that
the intensity of the beam could be modulated (2). It was then reflected off a mirror (3)
into a spatial filter from which the light diffracted (4). To reduce the light to a plane
wave, it passed through a collimating lens (5) before entering the square grid grating (6).
After diffracting from the grating, the light was focused by a lens one focal length, or 10
cm, away (7). Another polarizer was placed at this point (8). The lens focused parallel
light one focal length away, creating the Fourier transform where a Liquid Crystal Device
was placed so that filters could be used (9). 10 cm from this was a lens to focus the light
diffracted by the LCD (10) and again a polarizer was placed to vary the light’s intensity
(11). Had this lens been replaced with a lens with a 5 cm focal length, the Fourier
transform would be seen instead of the image. One focal length from this final lens was a
digital camera connected to a computer to record the image (12).
10 cm
1
10 cm
10 cm
10 cm
2
3
4
5
6
7 8
9
10 11
12
Figure 3. Experimental Setup for Fourier Optics Lab.
For numerical analysis, the computer program Mathematic 5.0 was used. In this,
the unfiltered square grid was imported, and then numerically converted into its Fourier
transform. In Fourier space, a computer-generated filter was multiplied with the
transform, and then the inverse-Fourier transform was performed on this product,
creating a final filtered image.
Results
Grid
Real FT
Mathematica FT
Although Mathematica’s Fourier transform is more well defined than the experimental
result, one can see that each diffracted light wave has focused on exactly one point.
Low Pass 40 Filter
Real
Mathematica
Low Pass 80 Filter
Real
Mathematica
Gaussian Low 40 Filter
Real
Mathematica
Gaussian Low 80 Filter
Real
Mathematica
As discussed above, the effect of the low pass filters are to lose detail of the image, since
it blocks out high, image-defining frequencies. The Gaussian filters add an additional
element of fuzziness.
High Pass 40 Filter
Real
Mathematica
High Pass 80 Filter
Real
Mathematica
Gaussian Hi 40 Filter
Real
Mathematica
Gaussian Hi 80 Filter
Real
Mathematica
One can see that the high pass filters, while maintaining clarity and detail in the image,
block out major features of the object.
Mathematica
would not function
for this filter.
Band Pass 40 Filter
Real
Mathematica
would not function
for this filter.
Band Pass 80 Filter
Real
As to be expected, the band pass filters combine elements of both detail and structure
loss, although not to the extremity as do the low pass and high pass filters.
Conclusions
As is evident, the optical setup served well to study the principles of Fourier
optics. Both the physical and numerical filters behaved according to theoretical
prediction: the low pass filters resulted in fuzzier images because they blocked high
frequencies from passing; the high pass filters resulted in structure loss of the image
because the high-contributing low frequencies were blocked; the Gaussian filters, since
they lacked a sharp cutoff, gave softer, fuzzier images; finally, the band pass filters
resulted in a combination of the two major types of filtering involved.
Now knowing the effects of Fourier optics and filtering, one can see the possible
applications are immense. The most obvious may be image processing: cleaning and/or
altering images by blocking out very specific frequencies. Other applications include
microscopy, x-ray crystallography, and biological imaging, to name a few. The
manipulation of Fourier optics is a powerful tool, and this lab served as an effective
introduction to the quickly expanding utilization of this basic physical principle.
Bibliography
Crawford, Frank S. Waves. McGraw-Hill: New York, NY 1968.
Goodman, JW. Introduction to Fourier Optics. McGraw-Hill: San Francisco, CA 1968.
Hecht, Eugene. Optics, 3rd Edition. Addison-Wesley Publishing: Reading, MA 1998.
Example of Mathematica numerical procedure:
image = Import["C:\Fourier Optics\1object.jpg"];
imagedata = image[[1,1]];
dimi = Dimensions[imagedata];
filter = Import["C:\Fourier Optics\1filter.jpg"];
filterdata = filter[[1,1]];
dimf = Dimensions[filterdata];
For[newimage={};j=1,j<dimi[[1]]+1,j++,
For[newline={};i=1,i<dimi[[2]]+1,i++,
newline=Append [newline,imagedata[[j]][[i]][[1]]]
];
newimage=Append[newimage,newline]
];
For[newfilter={};j=1,j<dimf[[1]]+1,j++,
For[newline={};i=1,i<dimf[[2]]+1,i++,
newline=Append [newline,filterdata[[j]][[i]][[1]]]
];
newfilter=Append[newfilter,newline]
];
ListDensityPlot[newimage, PlotRangeAll, MeshFalse,FrameFalse,
AspectRatioAutomatic]
ListDensityPlot[newfilter, PlotRangeAll, MeshFalse,FrameFalse,
AspectRatioAutomatic]
ft = Fourier[newimage];
dim = Dimensions[newimage];
topleft=Take[ft,dim[[1]]/2,dim[[2]]/2];
botleft=Take[ft,-dim[[1]]/2,dim[[2]]/2];
topright=Take[ft,dim[[1]]/2,-dim[[2]]/2];
botright=Take[ft,-dim[[1]]/2,-dim[[2]]/2];
ft=BlockMatrix[{{botright,botleft},{topright,topleft}}];
ftf = ft * newfilter;
ListDensityPlot[Abs[ft]^(1/8), PlotRangeAll, MeshFalse,FrameFalse,
AspectRatioAutomatic]
ListDensityPlot[Abs[ftf]^(1/8), PlotRangeAll, MeshFalse,FrameFalse,
AspectRatioAutomatic]
filteredimage = InverseFourier[ftf];
dim = Dimensions[ftf];
finalgraphics = ListDensityPlot[Abs[filteredimage], PlotRangeAll,
MeshFalse,FrameFalse, AspectRatioAutomatic]
Export["C:/Fourier Optics/dog gauss high pass.jpg", finalgraphics,
ImageSize{640,480}]
C:/Fourier Optics/dog gauss high pass.jpg