PH 481
Physical Optics
Laboratory #9
Week of March 12
Read:
Do:
Winter 2012
pp. 519-529, 539-544 of "Optics" by Hecht
1. Experiment IX.1: : Fourier Transform of a Ronchi Ruling
Experiment IX.1: Fourier Transform of a Ronchi Ruling
The goal of this lab is to analyze the Fraunhofer diffraction pattern of an image and
compare to the two-dimensional Fourier transform of the same image. The field distribution in
the Fraunhofer diffraction pattern is the Fourier transform of the field distribution across the
aperture.* Figure 1 shows a diagram of the apparatus that will be used in the laboratory exercise.
Ronchi
Ruling
CCD
Camera
Iris
Polarizer
Microscope objective
& Pinhole
Converging
Lens
Recollimating
Lens
He-Ne Laser
DMD
Figure 1: Diagram of optical Fourier analysis setup.
The apparatus consists of a series of lenses and mirrors that allow for experimental
Fourier analysis using a Digital Micro-mirror Array (DMA). The laser light is collimated using a
beam expanding telescope consisting of a microscope objective, iris, and converging lens. A
spatial filter (in this experiment, a Ronchi ruling) is placed in the object plane to produce an
optical pattern. A converging lens then places the Fraunhofer diffraction pattern, which is also
the optical Fourier transform, onto the focal plane. Keeping in mind that each point on the focal
plane corresponds to a spatial frequency in the object plane, the DMA is used as a spatial filter
in the focal plane to analyze the intensity of any particular spatial frequency. The selected spatial
frequency is recollimated using another converging lens and viewed using a CCD camera.
For this experiment, a Ronchi ruling will be used as the object plane spatial filter.
Recall that the position for each spatial frequency component
and
in the Fourier transform
plane is found with
where
is the optical wave number and is the focal length of the lens (for this experiment,
From the pre-lab homework problem, you have already calculated the distance
for each intensity maxima in the viewing plane; use the separation of each mirror on the DMA
(
) to find the distance of each intensity maxima from the DMA’s center in
pixels.
Now that we have found the location of each intensity maxima on the DMA, we
seek to measure the intensity of each maxima experimentally. The correct spatial filter must be
selected to transmit the desired spatial frequency component. In the D4100 GUI, Click the
“Load” tab on the left and press “open image.” A library of usable spatial filters is now available
to you; you may choose to use either an annuli, two dots, or two horizontal bars as your filter
shape. Which shape did you choose and why?
First, select the “all-black” image. This will transmit all light incident on the DMA
to the CCD camera. In the “load” tab, choose “load and reset” and “global” as the loading
arguments and click “Add.” Click the “control” tab and add a delay time command of ~5000
ms. Click the “run” button in the top control bar to run the script. Open the IC Capture software
and take a picture. Save the image; this will be used as a reference to the individual spatial
frequency intensities.
We will now experimentally measure the intensity of each frequency component
produced by the Ronchi ruling using the DMA. Using your calculated spatial frequency
locations, repeat the above process with a new image that will transmit only the zero harmonic in
the focal plane (that is, the central bright spot of the Fourier transform). How does the image
produced by the zeroth order harmonic filter differ from that produced by the all-transmitting
filter?
Repeat the process for next three neighboring spatial frequency components in the
Fourier transform plane. Be sure to save images for each spatial frequency!
We will now open the images for each spatial frequency and analyze the maximum
intensity using ImageJ. First, open the image of the zeroth order harmonic and use the “analyze”
feature to determine the maximum intensity measured by the CCD. Repeat this operation with
each image collected for the spatial frequencies. Plot the intensity as a function of spatial
frequency ( . How do the relative values compare with the max amplitudes predicted in the
theoretical model?
NOTE: Our diagram should state explicitly where the Focal plane and Object plane are in our
experimental setup.
Equipment needed:
Item
Helium-Neon Laser
Al mirror
Polarizer
200 mm lens
500 mm lens
150 mm lens
Magnetic Lens Post
Mounting posts
Translation Stage
Iris
Ronchi Ruling
Digital Micromirror D.
CCD Camera
Power supply
Qty
1
3
1
2
1
1
1
8
2
1
1
1
1
1
Source (part #)
Melles Griot 05 LHP 121
Newport 10D10ER.1
Edmund A38,396
Newport KPX106
Newport KPX118
Newport KPX100
Newport MB-2
Thor Labs P3
Newport 423
Newport
Texas Instruments
Imaging Center DMK21AU04
Uniphase 1205
Pre-Lab Homework Problem
A planar electric field filtered with a Ronchi ruling is functionally represented in the object plane
as a series of rectangle functions of the form
(
∑
(
(
)
where a is the width of the rectangle, is the number of transmitting bars in the ruling, and
is the center of each bar as shown in Figure 1.
𝑥
𝑎𝑁
𝑎𝑁
𝑎𝑛
𝑎𝑁
𝑎
Figure 1: Diagram of a Ronchi ruling.
a. Compute the spatial Fourier transform of the electric field in the object plane.
b. Find the intensity (
of the electric field resulting from the Fourier transform.
Compare this solution to the intensity pattern produced by multiple slits in the Fraunhofer
regime after making the change of variables
where is the position in the viewing plane and is the distance between the object
plane and the point in the viewing plane (Note: recall that
.
c. Sketch the intensity pattern for N=4 as a function of
.
d. Using your answer from part b) find the position of the first four maxima when
e. Find the ratio between the central maximum intensity and each of the next three maximum
intensities.
Solution Data:
1000
Theoretical
900
Intensity (# photons/pixel)
800
Average Intensities
700
600
Average Local Intensities
500
400
Theoretical Local
300
200
100
0
-100
0
500
1000
1500
2000
2500
3000
3500
Distance from Center (µm)
Figure 1: This is an average of the data that was found experimentally. The difference in theoretical data and
experimental should be explained using apodization.