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Chapter 5: Interference and Diffraction
5.1 Introduction
Interference and diffraction are common phenomena intrinsic to wave propagation. Interference
refers to the effects caused by the coherent addition of wave amplitudes that travel different
paths. If such waves are in phase, the light intensity is enhanced; conversely if they are out of
phase, the light is attenuated. Diffraction is the result of wave propagation that spreads a beam
of light from a straight linear path. The origin of the two phenomena is, in any case, exactly the
same. The following experiments are arranged in order of increasing complexity. This may blur
the distinction in your mind between the two phenomena of interference and diffraction. That’s
not entirely a bad idea, as they are so closely related.
Figure 5.1 “Snapshot” of a wave
Figure 5.2 The amplitude of two waves, 180
o
out of phase.
For most of the following exercises, you will observe diffraction and interference patterns
generated by a He-Ne laser beam incident on a variety of targets. These come in two forms: 2” x
2” squares such as 35-mm mounted film transparencies and rotating plastic disks imprinted with
a number of selectable patterns. These are listed with dimensions in Table 5.1 and Table 5.
below. The laser operating wavelength is 632.8 nm. With these numbers, you can compare
experimental observations with theoretical estimates. A list of targets is given in Table 5.
Note that most of the photographic targets are available as complementary pairs of positive and
negative, i.e. , where the positive target is transparent, the negative one is opaque and vice-versa.
The diffraction patterns from either of these are indistinguishable except for the presence or
absence of the undiffracted beam. One reason for providing this two-fold multiplicity is to
convince you that the patterns you see are quite distinct from geometric shadowing.
Table 5.1 Photographic film diffraction and interference targets.
10 micron single slit
25 micron single slit
50 micron single slit
25 micron diameter circular aperture
50 micron diameter circular aperture
100 micron diameter circular aperture
21 1 dimensional diffraction grating (+)
22 1 dimensional diffraction grating (-)
23 2 dimensional diffraction grating (+)
24 2 dimensional diffraction grating (-)
25 51 ring Fresnel zone plate (+)
26 51 ring Fresnel zone plate (-)
diffraction grating replica
Table 5.2 Rotating wheel diffraction and interference targets.
PASCO Single Slit Disk PASCO Multiple Slit Disk
20 μ width single slit 40 μ width/250 μ spacing double slit
40 μ width single slit 40 μ width/500 μ spacing double slit
80 μ width single slit 80 μ width/250 μ spacing double slit
160 μ width single slit 80 μ width/500 μ spacing double slit
20 – 200 μ variable width single slit 40 μ width/125 - 750 μ variable spacing double slit
square pattern 40 μ width single slit vs. 40 μ/250 μ double slit
hexagonal pattern 40 μ/250 μ double slit vs. 40 μ/500 μ double slit
60 μ diameter random opaque dots 40 μ/250 μ double slit vs. 80 μ/250 μ double slit
60 μ diameter random holes 40 μ/125 μ double slit vs. 40 μ/125 μ triple slit
80 μ width line 40 μ width/125 μ spacing double slit
40 μ width line/slit comparison 40 μ width/125 μ spacing triple slit
200 μ diameter circular aperture 40 μ width/125 μ spacing quadruple slit
400 μ diameter circular aperture 40 μ width/125 μ spacing quintuple slit
The direct unscattered laser beam is sometimes bright enough to obscure the diffraction pattern;
you can use a small patch of black tape over the beam spot to kill most of this light. It is a good
idea to take a careful look at the targets with a good light behind to understand what each is. (A
magnifying glass or optical comparator will help.) Also, note the sequence on the wheels, as the
labels are hard to read when they are mounted. The targets on the wheels are hard to align, so
it’s best to do all you can with the slides first.
Figure 5.5: An intensity minimum occurs at an angle given by a sin" =!
where a is the slit
width. If this condition is met, the amplitudes will cancel in pairs, as in the previous figure. The
intensity is therefore a minimum.
5.3 Single Slit Diffraction and Babinet’s Theorem
Parallel light, passing through a narrow slit, diverges in a distinctive pattern. The intensity
recorded at a distant screen by a slit of width, a , is given by the following expression:
I = I
0
sin!
2
where I 0
is the maximum intensity at the center and α is a quantity related to the location within
the pattern given by:
$ sin
a
At 0°, the intensity is at a maximum and falls to half this value at an angle given by:
! a
sin # 1. 39155
1 / 2
Dark bands are centered at:
a
n
!
0
sin (5.4)
where n is any integer. Thus, the narrower the slit, the wider the diffraction pattern! Equation
5.1 also predicts there will be a series of completely dark fringes every time sinα is zero, i.e.
when ( a /λ)sinθ takes on an integer value. One way of understanding the reason for these
θ θ
minima is shown in Figures 5.4 and 5.5. Assume we are observing light diffracted by a slit at an
angle given by a sinθ=λ. The amplitude contributions from separate parts of the wavefront will
cancel in pairs leading to a total of zero. The variation of the amplitude with angle is shown as
the dashed curve in Fig. 5.6 with x! " a sin # / $.
- Use the He-Ne laser as a light source and several different slide-mounted slits as targets.
Tape a piece of white paper to the aluminum screen to mark the position of the diffraction
minima (the screen itself does not reflect enough light to show the pattern very well).
Measure the angles of several diffraction minima for each of the 50, 25, and 10 micron
slits. Make a table comparing the measured angles with those predicted from Eq. 5.4.
- In the absence of any slit, the screen would only be illuminated by the direct undiffracted
beam. If the slit target is replaced by an obstructing line of equal width, a new diffraction
pattern will be formed. However, since the sum of the amplitudes from these two
configurations must be exactly zero, it follows that the obstructing line pattern must be
the same as for the equal width slit. A generalization of this principle is called Babinet’s
theorem. Verify that this is correct.
- Try measuring the diameter of a human hair using the methods outlined above.
5.4 Diffraction from a Circular Aperture
We have seen in Section 5.3 the diffraction pattern formed by a long parallel slit. A geometry
with more practical application is the diffraction of a plane wave by a circular aperture since this
sets the resolution limits of most optical instruments. For a hole diameter, d , the diffracted
intensity is given by:
2
1
0
J '
I I (5.5)
where J 1
is a sinusoidal-like curve called a Bessel function and δ is defined by:
!
"
$ sin
d
distance to some distant point. This extra distance is b sin! for a slit separation, b , and the
associated phase difference is ( 2 # b / ")sin!. The total amplitude is the sum of two terms:
sin )
sin( ) sin(
0 0
t
b
A A kz t A kz
TOT
"!
$
which can be rewritten as:
= t
b
kz
b
A A
TOT
2 cos sin sin sin
0
Since light intensity is proportional to the square of the amplitude, maxima and minima will be
observed for b sin " = n!
and b sin " = ( n + 1 / 2 )! respectively where n is any integer. The
intensity distribution is easily obtained from Equation 5.8:
2
0
2
0
sin
sin 2
4 cos
I I ' I (5.9)
where $ =(# b / ")sin !. Note that this function is exactly periodic. This analysis has not
included the single slit diffraction effects described by Equation 5.1. The combined modulation
of both phenomena is described by:
2 2
0
sin
sin sin 2
I I (5.10)
Figure 5.7: Interference from two parallel slits. The figure on the right shows the intensity pattern
for slits of width a= 80 micron and spacing d =250 micron, with θ in degrees.
The right-hand side of Fig. 5.7 shows the expected intensity for one of the double slits. Note that
the double-slit interference pattern is modulated by the coarser single-slit diffraction pattern.
- Observe the interference pattern from the two 40 μ and the two 80 μ double slits.
Measure the angles of the intensity minima for both the fine structure and the coarse
structure.
θ
- Compare the coarser structure with that for single slits with similar widths.
- Make a table comparing both the fine structure and coarse structure with the predicted
minima. Use the Excel spreadsheet program provided on the lab computers to plot the
intensity patterns expected and mark where your minima appear.
5.6 Multiple Slit Interference
The same basic procedure used to calculate the intensity for two parallel slits can be easily
generalized to N slits. The result is:
2 2
0
sin
sin sin
( N
I I (5.11)
where $ =(# a / ")sin! and $ =(# b / ")sin! with the definitions for a and b as given in the
previous sections. The effect of adding more slits is to make the bright bands narrower with a
width proportional to 1/N. Try to verify the general predictions of Eq. 5.11 using various
numbers of parallel slits. Compare your results with the Excel spreadsheet predictions.
5.7 Interference Effects with CD-ROM Disks
- CD-ROMs store information in grooves etched in aluminized plastic. Measure the
spacing of these grooves by using the disk as a reflective diffraction grating. (The setup
and equation are similar to that in Sect. 5.11 for lines on a ruler.)
- The optical device used to detect the etch pattern in current CD-ROM readers is a solid
state laser operating at a wavelength of 675 nm. A good deal of money has been spent on
developing a blue laser with a wavelength of 450 nm for such purposes. Can you explain
why there is such commercial interest?
5.8 Two-dimensional Diffraction
A slide with a periodic two-dimensional pattern is available (see Figure 5.11). Examine the
diffraction pattern with the He-Ne laser. How does it differ from the one-dimensional arrays of
straight lines?
Use the incandescent lamp as a source with an achromatic 200 mm focal length lens just
downstream of the mask as shown in Figure 5.8. Make the lamp to screen distance as large as
possible. Adjust the position of a focusing screen to image the lamp filament. Look for a
“rainbow” image of the filament on either side.
- Sketch the pattern observed and compare with the target mask. Are the shortest distances
in the diffraction pattern along the same axis as the narrowest structure in the mask?
5.10 Fresnel Zone Plate
Imagine light, traveling from left to right, incident on an opaque mask. We would like to find a
way of using interference effects to produce an enhanced light intensity on the optic axis. The
trick is to allow only those portions of the wave front which will arrive approximately in phase to
pass through the mask. The geometry is shown in Figure 5.9.
We need to arrange the inner and outer radii of opaque regions to satisfy the equations:
r
2 n + 1
2
2
! f = ( n +
) ", 0 # n < $; f %
r
2 n + 1
2
2( n + 1 /2) "
r
2 n + 1
2
2
! f = ( n + 1 ) ", 0 # n < $;
The result is that light diffracted toward the focal point, f , will all be within 90
o
of the same
phase and the intensity will be enhanced (Note that there are secondary focal length where
combinations of zones satisfy the above conditions.) The pattern produced by this prescription is
called a Fresnel zone plate. Such diffractive optical techniques are particularly useful for
focusing low energy X-rays where normal refractive lenses cannot be constructed. More
recently, these techniques are also finding application at visible wavelengths in conjunction with
refractive components. Adding diffractive focusing simplifies lens design when high quality
images are required.
Figure 5.12 has been photographically reduced and you can demonstrate that it behaves like a
lens. Find the focal length for the red and blue wavelengths transmitted by the appropriate
interference filters and compare with what you would expect from Equations 5.12. Use the
incandescent lamp as a source and mount the Fresnel zone plate downstream of the light baffle at
a distance of about 0.5 meter or so. Look for the sharpest image of the filament when the
position of the focusing screen is varied.
- Measure the image distance for several object distances and calculate the focal length
from the lens formula. Use the optical comparator to measure the diameter of the 51
st
ring on the slide ( n =25). Compare the measured focal lengths with the values calculated
from Eq. 5.12.
Figure 9: Fresnel zone plate geometry
5.11 Using Light to Measure the Spacing of Lines on a Ruler;
X-ray Diffraction for Structure Determination: The Double Helix
One of the most important uses for diffraction these days is the determination of molecular
structure. Drug companies and biomedical researchers depend on x-ray diffraction, and our
nation has spent billions of dollars providing bright x-ray sources for this purpose. X-ray
diffraction is just a simple application of the multiple slit diffraction effects you have been
investigating.
To get a feel for how researchers use diffraction to determine structure, you can do two simple
experiments:
The structure of lines on a ruler: The lines on a ruler form a periodic array with some nontrivial
structure, with patterns of thicker or longer lines interspersed with thinner or shorter lines. But
suppose the only information you had about the pattern of lines came from the results of
diffraction experiments! Could you figure out the pattern? Let’s find out. The geometry is
shown in the figure below. The laser light is scattered off the lines in the ruler. As usual, for
constructive interference, the adjacent paths must differ in length by an integral number of
wavelengths. This gives
m! = d (cos " # cos $
m
where d is the spacing of the lines.
- Arrange your HeNe laser beam so that it reflects from the surface of a steel ruler at near
grazing incidence in the region with the most closely spaced lines. Arrange the ruler and
laser so that the interference pattern appears on the wall. Identify the spot due to the
reflected beam and use it to determine α. Measure several other nearby spots to get β m
for several m. Use Eq. 5.13 to determine the spacing of lines on the ruler; compare with
the actual spacing. [Suggestion: Measure the height of the spots and of the ruler off the
floor and the distance to the wall to estimate α and β
m
.]
- Move the ruler so that the light only hits the longer lines. How does the pattern change?
Determine the spacing of these lines.
The structure of DNA:
- Send the HeNe beam through the tight coil of a helix provided for this purpose. Record
the diffraction pattern. From the pattern, determine
(1) the pitch angle of the helix, which is the angle of inclination of the wire with respect
to a plane perpendicular to the spring axis; and
(2) the spacing between successive turns of the wire.
Figure 5.11: Two-dimensional diffraction grating
Figure 5.12: Fresnel zone plate
Experiment 5 - Interference and diffraction Apparatus list
- He-Ne laser
- Optical bench with lamp, light baffle, screen
- Two microscope slides
- Box of 27 photographic film diffraction and interference targets
- Booklet with target image originals
- Replica diffraction grating
- Five pinhole apertures, various diameters
- CD-ROM fragment
- Achromatic lens and holder
- Interference filters
- Hand-held spectroscope (to be taken home)
- Hewlett-Packard model E3632A low voltage DC power supply
- Tungsten lamp and holder
- Metric tape measure
- Spectral lamps
- Slide holder
- Optical comparator (for measuring distances on slides)
