Download Lab 4: Diffraction of Light

Lab 4: Diffraction of Light
1
Introduction
Refer to Appendix D for photos of the apparatus
Diffraction and interference are important phenomena
that distinguish waves from particles. Thomas Young
first demonstrated diffraction and interference of light
waves in 1801. In this lab, you will reproduce the
following optical effects,
Figure 1: Single slit diffraction.
a barrier or obstacle cuts off a portion of a wavefront. Interference refers to superposition of two or
• The interference-diffraction pattern of multiple more waves that meet at one point in space.
slits
Single-slit diffraction: When a collimated (paral• Diffraction of light by a diffraction grating
lel) beam of light is incident on an obstacle, or an aperture, we normally expect to see nothing more than a
sharply defined shadow of the obstacle (or a bright
In the first part of the lab, you will use a Helium-Neon
spot that has the exact shape of the aperture). This
laser with a number of narrow slit arrangements to obseems to hold true for macroscopic apertures. Howserve and measure diffraction patterns. In the second
ever, there are a number of surprising effects that ocpart, you will use an instrument called a spectrometer
cur when the dimensions of an obstacle become comto study the emission spectrum of Mercury gas from
parable with the wavelength of light. The diffraction
a high-voltage discharge tube. The spectrometer uses
pattern of a single slit is a good example.
a diffraction grating to separate spectral components
of light. A diffraction grating is a surface etched with
To understand what a pattern produced by a single
a large number of closely spaced parallel slits.
slit looks like, we will first consider slits of different
size. If the slit width a is large, that is a >> λ, the
EXERCISES 1-7 PERTAIN TO THE EXPERdiffraction effects will not be noticeable and the size
IMENTAL SECTIONS.
of the bright spot that is produced, will be directly
proportional to the size of the slit.
• The diffraction pattern of a single slit
2
Background
If the slit is smaller, that is, on the order of a number of wavelengths in size, we have to use Huygens’
principle to determine what the resulting wavefront is
going to look like. According to Huygens’ principle,
we can approximate a particular wavefront by assuming every point on it is a point source of light. In case
of a wide slit, we will look at the wavefront as it is
just leaving the aperture.
The phenomena of diffraction and interference are
not particular to light or to electromagnetic radiation. Many examples of interference and diffraction of
waves exist in everyday life. Some analogies are, for
example, patterns formed by ripples on a smooth surface of a pond, or small water waves bending around
a rock in the middle of a calm lake.
To find out what pattern will be observed on the
screen, we will consider the intensity of light that is
In basic terms, diffraction can be described as the emitted from a slit of width a at a particular angle tobending of waves around corners that occurs when wards a distant viewing screen. This angle, θ, is mea4.1
tral maximum to the first diffraction minimum will
depend only on the width of the slit and the distance
between the slit and the screen. From figure 1, the
tangent of the angle θ is expressed as,
tan θ =
ym
D
(3)
where D is the distance from the slit to the screen
and ym is distance from the central axis to the mth
diffraction minimum.
Figure 2: Single slit diffraction pattern and intensity profile.
Assuming D >> a, for small angles, tan θ ≈ sin θ,
sured from the direction of the original light beam. D
is the distance from the slit to the screen.
sin θ ≈
ym
D
(4)
Firstly, as one would expect, there will be an intensity
maximum in the forward direction (θ ≈ 0). However,
intensity will not fall off smoothly with distance from Substituting equation 4 into equation 1, we get an
the center, but will have several additional maxima expression for distance from the center of the pattern
th
and minima (figure 1). For example, the intensity will to the m intensity minimum,
be brought to zero at angles corresponding to integer
values of m in the following formula (m can be both
mλD
positive or negative),
ym ≈
(5)
a
a sin θ = mλ
(1) Two-slit interference: Two coherent sources can be
produced by using the two slit arrangement shown in
The pattern in figure 2 can be explained using in- figure 3. If the slits S1 and S2 have a width a that is
terference and path difference arguments. Consider much smaller than the wavelength of light (a << λ)
two point sources on the wavefront at the front of the the slits can be considered as two point sources of coslit, which are a distance a/2 apart. The difference in herent light. If the screen is placed at a distance D
distances from each pair of these point sources to a from the slits as shown in figure3 (D >> d), interpoint on the screen D meters away will be (a sin θ)/2. ference fringes consisting of evenly spaced bright and
When this path difference equals an odd number m dark bands can be observed. The central fringe at
of half-wavelengths (mλ)/2, the waves emitted by the O is a bright fringe. Figure 3 shows the path of two
two sources will cancel, and no light will propagate in beams at a location P on the screen.
that direction (an intensity minimum).
The positions of the bright fringes are defined by the
In terms of displacement y from the central point on interference condition,
the viewing screen, the intensity can be expressed as,
I=
sin2 ( ayπ )
I0 ayπDλ2
( Dλ )
d sin θ = nλ
(6)
(2)
This condition corresponds to constructive interference between the rays r1 and r2 in figure 3. In equaThis equation is illustrated by the intensity curve tion 6 d is the spacing of the slits and n is the order
shown in figure 2.
of the fringe (n is an integer). Similarly the positions
of the dark fringes (destructive interference between
Additionally, for a given λ, the distance from the cen- r1 and r2 ) is defined by the interference,
4.2
P
4
r2
3
y
q
r1
2
S2
O
q
d
S1
Figure 4: Intensity pattern on the screen due to 2,3 and 4 inphase sources (slits).
b
D
d, then the two electric field vectors are nearly parallel
and the resultant amplitude is given by,
Figure 3: Two-slit interference.
δ
δ
ER = E1 + E2 = 2E0 cos( ) sin(ωt + )
2
2
(12)
1
d sin θ = (n + )λ
2
(7) The maximum possible value for the amplitude is
2E0 cos( 2δ ). Since the intensity is proportional to the
square of the amplitude of the electric field, the inFrom figure 3 the location of the nth maximum yn from tensity I at point P on the screen can be expressed
R
the central peak (located at θ = 0) can be expressed as,
as a function of the slit spacing d and the distance to
the screen D,
δ
IR = 4I0 cos2 ( )
(13)
2
yn
tan θ ≈ sin θ =
(8)
D
Here I0 is the intensity associated with a single source.
Substituting equation 8 into equation 6, the location
Equation 13 can be expressed as a function of the
of the nth maximum is located at,
angle θ by recalling the relationship between δ and
the path difference between the two rays r1 and r2 at
nλD
P. Referring to figure 3, the path difference between
yn =
(9)
δ
θ
the two sources S1 and S2 is d sin θ. Since 2π
= d sin
,
d
λ
equation 13 becomes,
Equations 6 and 7, describe the location of the maxima and the minima of the interference pattern. They
πd sin θ
do not describe the variation of the intensity as a funcIR = 4I0 cos2 (
)
(14)
λ
tion of position. The intensity at any point P on the
screen is proportional to the square of the amplitude
of the resultant electric field. The electric field com- The functional form of this curve is labeled number 2
ponents of the two sources at point P (in figure 3) are in figure 4.
described by,
Effect of multiple slits: When the number of slits
increases, the intensity pattern includes secondary
E1 = E0 sin ωt
(10) maxima (see curves labeled 3 and 4 in figure 4). However the location of the peaks remains unchanged.
The peak intensity of the primary maxima scales as
N 2 where N is the number of slits. The intensity of the
E2 = E0 sin(ωt + δ)
(11) secondary maxima are very small compared to the intensity of the primary maxima. When the number of
Here ω is the angular frequency of the waves and δ is slits becomes very large (N >> 1) there is effectively
the phase difference between the two waves. If D >> no light in the region between the primary maxima.
4.3
Interference maxima
given by,
dsin(q) = nl
Incident
plane
wave
Envelope
Grating
l2
2nd order
l1
n=2
Diffraction minima
given by,
asin(q) = ml
n=1
n=0
l1 < l2
l2
l1
1st order
All l
n = -1
l1
l2
n = -2
l1
l2
Equal mixture
of red and blue
l1 and l2
Figure 5: A two-slit diffraction pattern and intensity profile.
It is convenient to draw an envelope curve to understand the
pattern.
This fact is important for the understanding the principles of a diffraction grating.
The patterns shown in figure 4 are derived from the
assumption that the slits are point sources of light,
that is, they radiate light equally in all directions.
These idealized patterns repeat infinitely in all directions. As we have seen previously, in the case of a
finite single slit, the light is emitted in a non-uniform
way, described by the intensity profile in Figure 2.
For a distant screen case (D >> d), the pattern of
the idealized double slit case (the curve marked “2”
in Figure 4) will be modulated by the single-slit intensity profile (the curve shown in Figure 2) to produce
a profile, which will be similar to the one in Figure 5.
Figure 6: Diffraction grating.
light will be diffracted more than blue light. The
entire spectrum will repeat in high orders (higher
n).
• A diffraction grating should be designed so that
even the higher orders (|n| > 0) can be observed
within the first set of minima (primary lobe) of
the diffraction pattern shown in figure 5.
Thus, if a light beam with a number of wavelengths
strikes a diffraction grating, the wavelengths will separate so that they can be resolved at different angles.
This fact is used for wavelength measurements, i.e.,
for spectroscopy. For example, by passing the light
from a sample of electrically excited atoms through
a diffraction grating, one will observe distinct intensity maxima, called spectral lines. Each spectral line
Diffraction grating: A diffraction grating is a col- corresponds to a particular wavelength emitted by the
lection of a large number of regularly spaced narrow atoms. If the resolving power of the diffraction grating
slits. To better understand the properties of a diffrac- is sufficiently large, the spectral lines will be sharply
defined and well separated as shown in figure 6.
tion grating, remember that,
There exists a simple relationship between the angles
• The number of slits in the grating is very large, so and the wavelengths of the waves that are resolved by
the intensity maxima are very sharp and narrow. the diffraction grating,
The peak intensities are also much higher than in
the double slit case.
nλ
sin θ =
(15)
d
• The conditions for constructive interference are
the same as those given in equation 6. However
the source in the experiment is not monochro- Here, d is the line spacing (distance between two lines)
matic (mercury gas discharge tube) and will now of the diffraction grating, θ is the angle measured beemit a number of discrete wavelengths. The ze- tween the diffracted beam and the direction of the
roth order (n = 0) corresponds to constructive in- incident (undiffracted) beam, n is the order number
terference of all the wavelengths from the source. of the particular spectral line and λ is the wavelength.
Longer wavelengths will have greater diffraction
angles than shorter wavelengths. For example, red Note that for most diffraction gratings, d1 is a large
4.4
number (for example, 10,000 lines/inch). This means
that the diffraction gratings strongly separate the different wavelengths, and are therefore well suited for
spectroscopy. From equation 15, we can express the
wavelength of a particular diffracted beam as,
λ=
3
d sin θ
n
Suggested Reading
OPTICAL COMPONENTS SHOULD BE
BLOCKED BY USING PIECES OF CARDBOARD THAT ARE PROVIDED. BE PARTICULARLY CAREFUL WHEN YOU INSERT OR REMOVE LENSES INTO A
LASER BEAM.
(16) DO NOT TOUCH THE OPTICAL SURFACES OF LENSES AND MIRRORS. IF
THE SURFACES ARE UNCLEAN, PLEASE
BRING IT TO THE ATTENTION OF THE
TA IMMEDIATELY.
USE THE TRANSPARENT LENS TISSUES
TO DETECT THE BEAMS.
Refer to the chapters on Diffraction & Interference,
D. Halliday, R. Resnick and K. S. Krane, Physics MAKE SURE ALL MOUNTS ARE SE(Volume 2, 5th Edition, John Wiley, 2002)
CURELY FASTENED ON THE OPTICAL
TABLE.
4
Apparatus
5
• Helium - Neon laser on a stand, with power source
Experiment I:
Multiple Slits
Diffraction
from
• Glass plate with single and multiple slits
In the first experiment, you will determine the width
a and spacing d of a set of slits by observing their
diffraction patterns using a He-Ne laser. The equipment for these experiments consists of a laser light
source on a stand and a holder with an opaque glass
plate marked “Single and multiple slits”. The plate
is etched with a number of slits. The number of slits
varies between N = 1 and N = 6. However the slit
width and slit spacing remain unchanged. Note the
single slit aperture is marked “1”, the two-slit aperture is marked “2” and so on.
• Glass slide with single slits with different widths
• Diffraction grating
• Mercury gas discharge tube
• Spectrometer
• Black cloth to shield the spectrometer
• Ruler
• Meter stick
• Magnifying glass to read the angular Vernier scale
WARNING!!: KEEP TRACK OF YOUR
LASER BEAM AT ALL TIMES. NEVER
POINT THE BEAM AT PEOPLE, OR LOOK
IN THE APERTURE OF THE LASER OR BE
AT EYELEVEL WITH THE BEAM.
Before proceeding with your measurements, remember to minimize error by choosing sensible parameter combinations. For example, set up the slit and
the viewing screen as far apart as possible to increase
your precision, but without significantly diminishing
the brightness of the diffraction pattern.
Single slit diffraction: Set up the laser and the glass
KEEP EYES AWAY FROM DIRECT OR plate containing the slits on their respective mounts.
REFLECTED LASER BEAMS. OTHERWISE Make sure that the glass plate is horizontal and that
SERIOUS EYE DAMAGE WILL OCCUR.
the laser beam is perpendicular (normal) to the glass
plate by sending the reflected part of the beam close to
YOU SHOULD BE AWARE OF WHERE the laser cavity. Direct the beam through the center
THE LASER BEAMS STRIKE OPTICAL of the vertical aperture marked “1” (the single slit) toCOMPONENTS. REFLECTIONS FROM wards the viewing screen, located at the opposite side
4.5
of the table. Fine-adjust the position of the slit to get
a clear picture of the diffraction pattern. Normally,
the position of maximum brightness will be optimal
for observing the pattern.
Exercise 5: Slide each slit into the laser beam, and
measure the width of the central intensity maximum.
How does the single slit diffraction pattern change as
the width of the slit increases? Calculate the average value of the wavelength of the laser using your
Exercise 1: Sketch the pattern that you observe on measurements and the known slit widths.
the screen, and its intensity profile. Attach a sheet of
graph paper to the viewing screen, and mark the po- Exercise 6: Describe the effects of varying a, d, n
sition of the intensity minima and maxima. Find the and λ on the pattern observed.
width of the central intensity maximum by measuring
the distance between two minima closest to the cen- Exercise 7: Insert the 15000 lines/inch grating into
ter. One half of that number will give you the distance the holder and position it so that it is normal to the
between the center of the pattern and the first diffrac- laser beam. Sketch and describe the diffraction pattion minimum. Measure the distance D between the tern. Calculate the angle by which the first order
glass plate and the screen. Using equation 5, calcu- beam is diffracted. Is your measurement consistent
late the width of the slit. Propagate the errors in the with the angle that you expect based on equation 15?
measured quantities to find the error in the slit width. Repeat using the 7500 lines/inch grating.
Assume that the wavelength of the Laser is 632.8 nm.
Note: Once you have completed the experiInterference pattern of two and more slits: Slide ment, please remove all optical elements from
the glass plate on the holder so that the laser beam their mounts and place them on the table.
passes through the aperture labeled “2”. The pattern
you observe will be more complicated than in the previous case. It corresponds to a superposition of two
single-slit diffraction patterns.
6 Experiment II: Spectroscopy of
Exercise 2: Sketch the pattern that you observe
on the screen. Measure the spacing ∆y between the
bright spots on the screen. A good technique is to
count off as many as you can see in row, and measure the total length of the group. This is possible
because the screen is far away from the slits, and so
the maxima are evenly spaced. Find the spacing between the two slits, d, by adapting the formulae given
in the background section.
Exercise 3: Repeat the measurements for slits “3”,
“4” and “5” by simply sliding each slit in front
of the laser beam. Tabulate your results in a table.
Exercise 4: What are the similarities and differences
between patterns for the different number of slits?
State your observations based on your sketches of the
patterns. As the number of slits is increased, does the
position of the interference maxima change?
Interference pattern of different slit widths: In
this part you will be using a set of single slits with
different widths. With help from the TA, replace the
glass plate containing the slits with the glass slide containing the set of single slits.
Mercury Using a Diffraction Grating
In the second part, you will perform spectroscopy of
mercury vapor by setting up and aligning the spectrometer, and then measuring the angles at which the
mercury spectral lines are observed.
The spectrometer and the diffraction grating:
For this part of the experiment, you will set up and use
a diffraction grating spectrometer. The spectrometer
collimates the light from the mercury gas discharge
tube. The collimated beam passes through a diffraction grating. A telescope is used to observe the spectral lines at various angular positions. These angular
positions correspond to the locations of the intensity
maxima given by equation 15. The angular positions
can be noted on a Vernier scale.
Your objective is to measure the angles at which the
diffracted beams leave the diffraction grating. Due to
symmetry, the pattern of lines is going to be repeated
on both sides of the central maximum. You can use
this fact to simplify the measurement and reduce the
error: measure the angles between pairs of lines of the
same color and order (see figure 7). One-half of each
4.6
Mercury Lamp
Table 1:
Order (n)
Diffraction
Grating
Color
θ (o )
θ error
λ nm
λ error
Collimator
Table 2: Mercury spectral lines wavelengths in Angstroms Ȧ
Telescope
Wavelength in Angstroms
5791
5770
5461
4358
4046
q
First Order Lines
Second Order Lines
Figure 7. A diffraction grating spectrometer
• While looking through the telescope eyepiece,
maximize the brightness of the slit by carefully
moving the discharge tube in front of the slit.
Figure 7: A diffraction grating spectrometer.
of these angles will correspond to the diffraction angle
θ.
• Use the ring on the eyepiece of the telescope to
bring the crosshairs in focus.
You can get a feel for how a diffraction grating works
by taking it out of the spectrometer and observing
different light sources, including the discharge tube,
through the grating. Verify the well-known statement
that white light is in fact a mixture of different wavelengths, by observing one of the fluorescent ceiling
lamps.
• Find the colored spectral line images by turning
the telescope. By slightly rotating the diffraction
grating make sure the spectral lines are in the center of the view of the telescope. This ensures that
the lines of the grating are vertical
Prepare a table with columns similar to table 1. PropInsert the grating (with its rulings vertical) agate the uncertainties to find the error in the waveinto the grating holder of the spectrometer and length of the mercury line.
make sure that its surface is perpendicular to
the direction of incoming light from the colli- Exercise 8: Take measurements of the angles at
which the spectral lines are observed for all the ormator (see figure 7).
ders that you can see. Make sure to distinguish beAlign the spectrometer by carrying out the following tween lines of different order (n = 1, 2, 3, etc). Use
equation 15 and the value of d for the grating (you
steps,
have choice of two gratings with d1 = 15000 lines/inch
and d2 = 7500 lines/inch) to find the wavelength of
• Focus the telescope on a distant object, such as a each line. Average your results, and report the avwall on the other side of the lab.
erage wavelengths of the spectral lines in Angstroms.
Include the error with your measurement. Find the
• Open the slit of the collimator.
percentage error between your measurements and the
• Position the operating discharge tube near the col- wavelengths of the spectral lines given in table 2.
limator’s slit.
Your lab report should include:
• Find the bright, central image of the slit by positioning the telescope along the axis of the colli- Answers to exercises 1-8 with relevant data tables,
graphs, figures and qualitative comments.
mator.
• Use the translation screws on the collimator and Refer to Appendix C for Maple worksheets.
the telescope to make the image of the illuminated
slit as sharp as possible.
4.7