Download Laboratory Exercises in Physics 2

University of Oulu Student Laboratory of Physics
Laboratory Exercises in Physics 2
GRATING AND PRISM
1. Introduction
In this exercise two dispersive optical elements, a grating and a prism are studied.
With a dispersive component the light can be dispersed into a spectrum, i.e. to different colors or different wavelengths. In the measurements a simple optical spectrograph, where the light from a gas-discharge lamp is directed through a narrow slit and
a converging lens either to the grating or to the prism is used. The spectrum is investigated with a telescope, which can be moved around the grating or the prism. The telescope and the dispersive element are placed onto an optical table, which has an angle
scale for measuring diffraction and deviation angles.
The aim of this study is to measure for a glass prism a dispersion curve representing
the variation of its refraction index with the wavelength. The wavelengths needed for
the curve measurement are determined by first calibrating the grating spectrograph
with the known wavelengths and then measuring the unknown wavelengths with the
spectrograph. The refraction indices of the glass are determined by measuring the minimum deviation angles corresponding to these wavelengths with the prism spectrograph. Furthermore, the sensitivity of the eye for the red light and the resolving power
of the grating spectrograph are determined.
2. Theory
2.1 Diffraction grating
An array of a large number of parallel slits, which all have the same width and are
spaced at equal distances, form a diffraction grating. The important parameter is a so
called grating spacing d, which means the distance between the centers of the adjacent
slits. A transmission grating can be made by scratching many equally spaced slits on a
glass or quartz plate using a diamond point. A reflection grating is made by scratching
equally spaced grooves on a polished reflective metal screen. Fig. 1 presents a typical
transmission grating.
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GRATING AND PRISM
Figure 1 A transmission grating.
Figure 2 A plane wave strikes
perpendicular to the grating.
2.1.1 Grating equation
Diffraction grating can be considered with the Huygens principle. Fig. 2 represents an
idealized grating, which has a spacing d between the slits. When a plane wave with a
wavelength l is striking perpendicular to the grating each slit acts as a point-source
from which the light propagates in all directions. The diffracted light is thus composed
of the interfering waves coming from each slit of the grating. At a point P shown in
Fig. 1 the path length to each slit varies. If the path difference between the waves coming from the adjacent slits is equal to the wavelength l, the waves are interfering constructively and the intensity maximum occurs. When the light is normally incident onto the grating, the diffraction maxima are observed at angles q given by
(1)
d sin q = m l , m = 0,± 1,± 2, K ,
where integer m is called an order of the spectrum. Eq. (1) is known as the grating
equation for normal incidence.
If the grating is illuminated with white light including several wavelengths the value
m = 0 causes a non-diffracted white line in the middle of the pattern. Each value of
m forms its own spectrum symmetrically at both sides of the central white line. The
m = ±1 lines are called the first-order spectrum, the m = ±2 lines the second-order
spectrum etc. In these spectra different colors diffract at different angles. According
to Eq. (1) the angles q for longer wavelengths in the red end of spectrum are larger
than for shorter wavelengths at the violet end. Usually the spectra of different orders
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Laboratory Exercises in Physics 2
tend to overlap in the higher orders. Also the intensity of the spectral lines is decreasing when the order increases and so only the lower order spectra are useful when
measuring with simple instruments.
2.1.2 Resolving power of grating
The resolving power R of the grating spectrograph is defined as
R=
l
,
Dl
(2)
where Dl is the least resolvable wavelength
difference in the region whose mean wavelength is l. According to Rayleigh’s criteriFigure 3 Resolving of two specon two interference lines with equal intensitral lines.
ty can be just resolved if the principal maximum of the other coincide with the first minimum of the other. Fig. 3 shows the situation corresponding to the limit of the resolution.
It can be shown that the resolving power of the grating is increasing when the order m
and the number of lines N in the grating increase. Because the number of lines is the
width of the grating w divided with the grating spacing d the resolving power of the
grating is of the form
w
R = mN = m .
(3)
d
2.2 Prism
A prism is a transparent optical element with two nonparallel surfaces that refract
light. The most widely known prism is a triangular prism, which has a shape of equilateral triangle. When the light waves are refracting at the interface of two media the
waves are deviating from their original direction due to the refraction. This deviation
is described with the deviation angle d. Since the refraction indices of typical prism
materials such as glass vary with wavelength, the prism causes also dispersion, i.e. the
different wavelengths are refracted differently. Fig. 4 represents a triangular prism and
passing of the light through it. The angle e between the two refracting surfaces of the
prism is called the refracting angle of the prism.
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GRATING AND PRISM
A
n
e
B
q1
C
d
d1
d2
q1’
q2
D
’
q2
300 450
l (nm)
600 750
Figure 5 Dispersion curve.
Air
n=1
Air
n=1
Glass
n>1
Figure 4 A triangular prism and passing of light through it.
The dispersion curve of the prism represents its refractive index as a function of the
wavelength. An example of the dispersion curve is given in Fig. 5. Usually, as in the
case of Fig. 5 the refraction index decreases when the wavelength increases. In this
situation dn dl < 0 , and this is called normal dispersion, while if dn dl > 0 , the dispersion is said to be anomalous.
2.2.1 Deviation angle of prism
By using the geometry of Fig. 4 and Snell’s law an equation for the deviation angle of
the prism can be derived. The triangles ACD and BCD in Fig. 4 give
ACD : e + (90° - q1' ) + (90° - q 2' ) = 180° Þ e = q1' + q 2'
BCD :
d 1 + d 2 + (180° - d ) = 180° Þ d = d 1 + d 2
(4)
and from the points C and D we get
C : q1 = d1 + q1' Þ d1 = q1 - q1'
D : q 2 = d 2 + q 2' Þ d 2 = q 2 - q 2'
.
(5)
Combining these equations the deviation angle d may be written as
d = d1 + d 2 = q1 - q1' + q 2 - q 2' = q1 + q 2 - e .
(6)
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Laboratory Exercises in Physics 2
Snell’s law gives at the points C and D
C : sin q1 = n sin q1'
D : n sin q 2' = sin q 2
.
(7)
By using the equations (4) and (7) we get for the sine of the angle q2
sin q 2 = n sin q 2' = n sin( e - q1' ) = n sin e cos q1' - n cos e sin q1'
= sin e n 2 - n 2 sin 2 q1' - cos e (n sin q1' ) = sin e n 2 - sin 2 q1 - cos e sin q1 .
(8)
Þ q 2 = arcsin(sin e n 2 - sin 2 q1 - cos e sin q1 ).
The equations (6) and (8) together give for the deviation angle d
d = q1 + arcsin(sin e n 2 - sin 2 q1 - cos e sin q1 ) - e .
(9)
The deviation angle thus depends on the incident angle q1. It is also function of the refracting angle e and the refractive index n of the prism medium, which are constants
for a given prism. Because the refractive index depends on the wavelength the deviation angles are different for various colors and so the prism disperses the light into the
spectrum.
2.2.2 Minimum deviation
It can be shown that the deviation angle of the prism is at minimum when the light
passes symmetrically through the prism, i.e. when
q1 = q 2 = q and q1' = q 2' = q ' .
(10)
Hence, in the case of the minimum deviation the equations (4) and (6) give for the refracting angle e and the angle of minimum deviation dmin
e = q1' + q 2' = 2q ' Þ q ' =
e
2
1
and d min = q1 + q 2 - e = 2q - e Þ q = (d min + e )
2
(11)
and so Snell’s law (7) at point C gives
sin(
d min + e
) = n sin( e 2) Þ n =
2
d min + e
)
2
.
sin( e 2)
sin(
(12)
According to Eq. (12) the prism’s refractive indices for different wavelengths can thus
be determined by measuring the corresponding angles of the minimum deviation.
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GRATING AND PRISM
3. Optical spectrograph
Fig. 6 shows a simple optical spectrograph used for viewing the spectrum of a light
source. The source is a gas discharge lamp placed inside of a metallic shelter and it
emits a number of characteristic wavelengths. The light from the source goes through
a narrow slit, whose width can be adjusted with a screw, to the collimator, in which
the lenses direct the light to a dispersive element; a grating or a prism. In the case of
the grating a symmetrical diffraction pattern, consisting of the spectra of various orders, is formed to the both sides of the central position. The spectrum can be viewed
by rotating the telescope through various angles around the grating. The coarse tune
of the telescope can be made by rotating it with hands and the fine tune is made with a
screw.
The dispersive element is set to an underlay on an optical table, to which the telescope
is attached. The outline of the table is an angle plate with a vernier scale, with which
the position of the telescope can be observed. For measuring the diffraction or the deviation angle of the spectral line a vertical line of a crosshair of the telescope is moved
over the line and the respective reading on the angle scale is observed.
Screw
Figure 6 A grating spectrograph.
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Laboratory Exercises in Physics 2
7
4. Exercises
Solve the following exercises and attach the solutions to
your report:
1. Fig. 7 represents a situation where the observed angles at
the grating spectrograph’s angle scale at both sides of the
central line are q left and q right . Derive an expression for
the diffraction angle q .
2. Show that the absolute error limit of the grating spacing is
of the form
Dd £
360o
2q
q right
q left
Figure 7 Diffraction angle
and the angles observed at
the scale.
m
ml cos q
Dl +
Dq .
sin q
sin 2 q
5. Measurements
5.1 Calibration of grating spectrograph and resolving power of grating
1. Choosing lamps: Choose together with the tutor a suitable discharge lamp for the
calibration (for example a sodium or a krypton lamp) and make yourself familiar
with the handling of the lamps. The wavelengths of the calibration lines are given
in Table 1. Set the calibration lamp inside its shelter, switch it on and let it warm
up. Place the grating onto the underlay. Be careful when handling the optical components, avoid the fingerprints and other impurities and don’t touch the component
itself. The grating and the prism used in this study are both made of glass, so don’t
drop or damage them.
2. Tuning the spectrograph: Rotate the underlay and the telescope so that the light
comes perpendicular to the grating and the telescope is at the center. Find the line
corresponding to the order m = 0 and use it to adjust the slit width to a proper value. Exercise the observing of the angle values by measuring the place of the central line m = 0 . (You don’t need this value in the calculations, because all the angles of the studied lines are observed from both sides. So, the correct angle value,
which is not dependent on the position of the scale’s center, can be calculated from
the expression derived in exercise 1.)
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GRATING AND PRISM
3. Measuring the diffraction angles of the calibration lines: Find the first-order
spectrum from the left side of the center and measure the diffraction angles ( q left
in the measurement form) of the calibration lines. Then measure the corresponding angles ( q right ) from the right. If the intensity of the spectrum of the calibration
lamp is strong enough measure the angles of the calibration lines also in the second-order. For determining the resolving power of the grating, register also the
width of the grating to your measurement form.
Table 1 Wavelengths of the calibration lines.
Lamp
Na
Kr
Color
Yellow
Yellow
Yellow
Green
l (nm)
588.9951
589.5924
587.0916
557.0289
5.2 Sensitivity of the eye for the red light
4. Preparations: Switch off the calibration lamp and let it cool down before placing it
back to the drawer. Check that a neon lamp is placed inside to another shelter,
switch it on and set it as the source of the spectrograph.
5. Sensitivity measurements: When the neon lamp has been warmed up explore the
first-order spectrum. Adjust first the width of the slit so that you can find the red
line with the longest wavelength (the most reddish line). Then narrow the slit and
measure the corresponding diffraction angles from the left and right side.
5.3 Unknown wavelengths
6. Preparations: Switch off the neon lamp and let it cool down. Place the lamp chosen with the tutor to the other shelter, let it warm up and explore its first-order
spectrum using a wider slit. How many lines can you find? Which are the colors of
the lines? For the dispersion curve measurements a sufficient amount of the lines
(4 - 7) with the wide wavelength range, preferably from violet to red, is needed.
You can use either one lamp with sufficient number of lines or two lamps completing suitably one another.
7. Angle measurements: When the necessary lines have been found narrow the slit
width and measure the diffraction angles on the left side and on the right side.
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Laboratory Exercises in Physics 2
5.4 Dispersion curve
8. Preparations: Replace the grating with the prism. Adjust the slit width and find the
spectrum from the left side by rotating the prism and the telescope.
9. Minimum deviation angles: When the spectrum has been found find the minimum
deviation by rotating the prism so that the lines move towards the center. When the
lines turn back, the prism is in the position producing the minimum deviation.
Measure the minimum deviation angles for the all the lines investigated with the
grating. Repeat the measurements on the right side.
6. Analysis
6.1 Calibration of grating spectrograph and resolving power of grating
Calculate the diffraction angles of the calibration lines in the first-order from the expression derived in exercise 1 by using the angles measured on the both sides. If you
have observed also the second-order repeat the calculations for it, too. Determine the
grating spacing d from Eq. (1) for each calibration line. The final result is the mean
value of all the calculated grating spacings. Estimate the error limit of the grating
spacing by determining the maximum deviation from the mean value and by calculating the absolute error limits from the equation given in exercise 2. The error limit of
the final result is the maximum of all the calculated errors. (Note: If you have used the
sodium lamp in the calibration you have probably observed just one yellow line in the
first-order spectrum. In this case the wavelength of the calibration line is the mean
value of the wavelengths of the yellow lines, given in Table 1, and their difference is
the error limit of the calibration wavelength. The error limit of the other wavelengths
is 0.0005 nm.)
Find the resolving power of the used grating for the first-order from Eq. (3) by using
the grating width registered to the measurement form.
6.2 Sensitivity of the eye for the red light
Calculate the diffraction angle of the measured neon line by using the angles observed
on both sides. Determine the wavelength of the line from Eq. (1). This wavelength is
the sensitivity of your eye for the red light.
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GRATING AND PRISM
6.3 Unknown wavelengths
Calculate the wavelengths of the measured lines same way as in the case of the neon
lamp. Evaluate the error limit of the wavelength with the total differential method by
using the line chosen with the tutor.
6.4 Dispersion curve
Calculate the minimum deviation angles for the wavelengths investigated by using the
angles observed from both sides. Then determine the refractive indices of the wavelengths from Eq. (12). Evaluate also the error limit of the refractive index of the chosen line with the total differential method. Present the observed (l , n ) - pairs in table
and draw the corresponding points onto a graph. Draw by conforming to the points the
dispersion curve of the prism used in the study and attach the curve to your report.
UNIVERSITY OF OULU
Student: ___________________________
STUDENT LABORATORY IN PHYSICS
Date of measurements: ____ / ____ 20___
Laboratory Exercises in Physics 2
Tutor: _____________________________
MEASUREMENT FORM
GRATING AND PRISM
Grating spacing
Lamp
Color
m=1
q left
m=2
q right
q left
q right
Measurement accuracy of the diffraction angle Dq = _________
Resolving power of the grating
Width of the grating
w = __________ cm
Sensitivity of the eye
Lamp
Color
q left
Ne
The most reddish
m = 1
q right
The unknown wavelengths and the dispersion curve
Lamp
Grating
m=1
Color
q left
q right
Tutor’s signature _________________________________
Prism
Minimum deviation
d left
d right