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LAB 4 - DIFFRACTION
FALL 2008
Objective We begin an analysis of monochromatic waves, this week in the context of diffraction. We will
investigate the diffraction of optical waves through apertures.
Background This week we will consider the diffraction pattern produced when light propagates through
apertures. In summary, the far-field diffraction pattern produced by a plane wave propagating through an aperture is found to be the spatial Fourier transform (FT) of the aperture function. For example, a rectangular aperture would give rise to a diffraction pattern that is a Sinc^2 (remember the intensity) function in both transverse directions. The width of the Sinc function (or diffraction pattern in general) is inversely proportional to the width of the aperture. For example if the rectangular aperture is oriented such that the dimension in the x-direction is wider than that of the y, then the diffraction pattern will be longer in the y direction than in the x. Propagation of the wave through a lens will bring the far field pattern to the focus of the lens.
The HeNe’s we are using have a low-quality Gaussian beam. In this week’s lab, we will perform spatial manipulations of the HeNe beam in order to obtain an approximate plane wave. We will first analyze the spatial filter, and how it works to clean up the starting beam. Then, we will collimate an expanded beam that has a low divergence angle, giving rise to a good plane wave approximation near the center of the beam. We will use this source to perform Fourier Optics experiments next week.
PRELAB
- Sketch the expected diffraction (intensity) pattern for a plane optical wave passing through each of the following apertures:
PART I LEVELING THE LASER
Discussion It is very useful, and sometimes necessary to level a laser beam to the surface of an optical table.
The leveled beam will then define an axis, and all other optical components can be referenced to this axis.
Procedure If it is not already there, place the laser on top of the two supports (one fixed and one a lab jack) at
one end of the optical table, with the beam pointing towards a wooden beam stop. The fixed height support should be at the front of the laser and the jack should be at the rear. Be sure that the wooden beam block is in place to ensure the beam does not leave the table. Turn the laser on. Place a small iris (in a holder) at the front of the laser. Adjust the height and the size of the aperture until the beam just passes through the center. (Be careful not to look at the reflection too long.) Place the iris at the far end of the table and adjust the jack until the beam passes through the iris. Now move the iris back to the laser and check if the jack adjustment changed the height of the beam (to within ~1 mm ). If not, adjust the iris and repeat the procedure. When your laser is level, you should be able to slide the iris along the entire table and have the beam pass through it at all times.
PART II DIFFRACTION FROM A SLIT
Discussion We will make a slit using electrical tape and a microscope slide. You will use diffraction to
determine the slit width.
Procedure
1) Using electrical tape, a microscope slide, and the exacto-knife, create a slit-like rectangular aperture.
2) Mount your slit in front of the laser beam and try to obtain a diffraction pattern on your white screen.
3) Measure the distance from the center of the diffraction pattern, and record the distance from the slit to the screen. For your writeup, calculate the width of your slit.
4) Try to make another slit with a different width. How does the diffraction pattern change as a function of the slit width? (i.e. does it get wider or narrower with increasing slit width?)
HeNe Laser
iris
HeNe Laser
iris
Now placing the screen in front of the spatial filter (such as a business card from the supplies drawer), slowly move the horizontal and vertical adjustments until some light appears on a business card beyond the pinhole. (This part takes time so be patient!) You may have to rotate the base to keep the beam passing through the center of the entrance iris. If no light appears after several trials ask the TA for help. Each person in each group should try this.
Once there is some light through the pinhole, move the horizontal and vertical screws until a pattern of concentric rings is formed. This is the diffraction pattern of the pinhole. Show the pattern to the TA and proceed to Part IV. You will adjust the focus in Part V.
Note: You should be very careful not to bump the laser after you have the spatial filter aligned!!!
PART IV FAR FIELD (FRAUNHOFER) DIFFRACTION
Discussion We will now use the spatial filter to study far field (Fraunhofer) diffraction from a circular
aperture. In lecture we will show that this pattern is the Fourier transform of the aperture which in this case is a pinhole (a circ function).
Procedure
1) Place a screen with ruled graph paper (1- mm spacing) about 1 m from the laser and record the distance. Center the graph paper on the beam so that measurements can be easily made.
2) Now align the spatial filter with the microscope objective defocused until you see the earlier ring pattern that is the diffraction from the pinhole. (NOTE: Be very careful not to look at the back reflection from the pinhole for long periods of time.) This pattern should be centered about the beam. Sketch the diffraction pattern and record the distance from the center of the spot where the pattern goes to zero for the first time.
3) The pattern is called an Airy pattern and the first zero is given by sin θ = 1.22 λ /d, where d is the
diameter of the pinhole, θ is the angle from the pinhole to the first zero (half-angle, see figure
below) and λ is the wavelength of light. Using this formula, calculate the pinhole diameter in the
spatial filter. (λ=0.6328 μm).
Defocused θ
Spatial Filter Laser
First zero of diffraction pattern
PART V COLLIMATION
Discussion A lens placed at exactly one focal length from the pinhole will image the pinhole at infinity.
Because the light is imaged at infinity, the cross-sectional area does not change and thus the beam is collimated (i.e. forms a column of light which is approximately a plane wave.) We will use collimated light to study the concepts of Fourier Optics.
Procedure
1) Continue turning the focusing ring on the spatial filter in the direction that makes the center spot larger and use the other adjustments to keep the pattern visible and centered. Eventually, the center spot becomes very large and bright and the horizontal and vertical adjustments become very sensitive. When the brightness and size of the center spot is at a maximum, the pinhole is in focus. (If the ring is turned too far, you will pass through the focus and the spot will become smaller.)
2) Roughly determine the focal length of the lens on the table by forming an image of an overhead light on the floor. Measure the distance from the floor to the center of the lens with a meter rule. From a previous lab exercise we know that this height is roughly its focal length.
3) Now place the lens about 1 focal length away from the spatial filter. By moving the lens, you should be able to make the size of beam near the lens the same size as the beam at the end of the table. You can verify this by sliding the iris along the beam path. You can also use a white piece of paper taped to the wooden beam block. If it does not pass through the iris all along the beam path, check to be sure the lens is properly aligned and then use the tilt adjustments on the spatial filter to correct for the tip. After this final adjustment, you should be able to slide the iris all along the beam.
QUESTIONS/PROBLEMS FOR WRITE-UP
1) When everything is exactly aligned, one may see flickering in the output of the laser. What is it caused from, and how can it be prevented?
2) What is the transverse spatial distribution of the collimated light given that the transverse spatial distribution of the laser light at the input of the spatial filter is Gaussian. Give reasons for your answer.
3) Why do you defocus the pinhole before you place the spatial filter in the setup?
4) (i) Explain how the spatial filter works in terms of Fourier optics. (ii) Why is the spatial filter defocused for the Fraunhofer diffraction part of the lab?
