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Saint Petersburg Electrotechnical University "LETI"
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А. М. VASILEVSKY G. A. KONOPLEV O. S. STEPANOVA
OPTICS AND OPTICAL MEASUREMENTS
IN SOLAR ENERGY
Tutorials for laboratory workshops
St. Petersburg
Publishing house ETU "LETI"
2019
УДК 535.3
ББК 22.34
В19
Vasilevsky A. M., Konoplev G. A., Stepanova O. S.
В19 Optics and optical measurements in solar energy: учеб.-метод. пособие.
СПб.: Изд-во СПбГЭТУ «ЛЭТИ», 2019. 45 с.
ISBN
The book includes tutorials for four laboratory workshops in the field of optics and optical measurements which belong to the master course «Optics and optical measurements in solar energy». It covers such topics as Fraunhofer diffraction,
measuring focal lengths, a wavelength calibration of CCD spectrophotometers, and
absorption spectroscopy. Each tutorial is preceded by a short theoretical introduction and a description of the relevant experimental methods.
The book is mainly intended for master students in «Electronics and
nanoelectronics» (11.04.04), also it may be of some interest for R&D professionals
in this area.
УДК 535.3
ББК 22.34
Рецензент: д-р физ.-мат. наук, проф. Н. Т. Сударь (СПбГПУ Петра Великого).
Approved by the editorial and publishing council of the university as a tutorial
ISBN
© СПбГЭТУ «ЛЭТИ», 2019
INTRODUCTION
Optical and spectroscopic methods are now widely used in the development
and characterization of materials and structures in optoelectronics and
photovoltaics. Laboratory workshops presented in this book are aimed to develop
competences in the fields of physical optics, absorption spectroscopy and modern
spectroscopic equipment. Each tutorial contains brief theoretical introduction, simplified layouts and specifications of used equipment, requirements for a report, test
questions, and a list of recommended textbooks. The appendix is dedicated to the
software used for a wavelength calibration of charge-coupled device (CCD) spectrophotometers.
A report has to be prepared for each workshop. It must contain following sections:
1) The aim of the workshop;
2) The description and layout of the experimental setup;
2) The results of the experiments in the form of tables and diagrams;
3) Mathematical expressions used for the calculations, examples of the calculations;
4) Conclusions which includes a critical analysis of the results.
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1. MEASURING OF THE FOCAL LENGTHS OF LENS AND SPHERICAL
MIRRORS
Aim of the workshop: studying various experimental methods for measuring
the focal lengths of the positive and negative lenses, concave spherical mirrors.
1.1. Overview
An optical element made of a transparent material which is bounded by two
refracting mostly spherical surfaces is called a lens. Lenses can be positive and
negative. For a single lens the Gaussian formula is valid:
1 1 1
  ,
f b a
(1.1)
where a and b – the distance from the first principal plane of the lens to the object
and from the second principal plane of the lens to the image respectively. In the
case of thin lenses a and b are the distances from the lens center to the object and
image. When applying the formula (1.1) the sign convention for lenses should be
followed.
If the image size of the segment y1 perpendicular to the optical axis is equal
to y2 , the linear magnification is equal to:
y
b
 2 .
y1
a
(1.2)
A spherical reflective surface is called a spherical mirror. The focal length of a
spherical mirror is equal to the half its radius of curvature.
If a and b – distances from the object and image to the mirror respectively,
then
1 1 1
  .
f b a
(1.3)
Linear magnification of a mirror is given by (1.2).
1.2. Experimental setup
The measurement setup is a horizontal optical bench, on which riders containing an object (an incandescent lamp), a screen, a collimator, and a test lens or mirror are installed. All the elements of the optical system must be installed perpendicular to the optical bench; their centers should be in a straight line parallel to the
4
guide. Use a ruler to measure the distances between the elements along the optical
bench.
Initial adjustment of the measurement setup is carried out as follows. Put the
riders with the screen, the object and the lens close to each other and set their centers at the same height. Then produce on the screen a sharp demagnified image of
the filament and mark the position of the image center. After that, produce a sharp
magnified image of the filament. If the center of the image has shifted, adjust the
object position in the horizontal and vertical direction to align the demagnified and
magnified images.
1.3. Tasks and guidance to implement them
Task 1. Measurement of the focal length and the optical power of a positive
lens.
Measurements must be carried out using different methods.
Method 1. Measuring the distance from the lens to the object and image.
It follows from (1.1) that
f 
ab
.
a b
(1.4)
Consequently, to find f using (1.4) it is necessary to measure the distance
from the object to the lens and from the lens to the image (Fig. 1.1).
y1
F2
F1
y2
a
b
Fig. 1.1. Measuring of the focal length f using the distance
from the lens to the object and image
Set the screen and the object (light source) at a sufficient distance from each
other, then put the lens between them and move it along the bench until a sharp
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demagnified image of the filament is produced on the screen. Measure the distances a and b with the ruler. Repeat the measurements at least five times with both
demagnified and magnified images of the object, and calculate the averaged value
of f.
Method 2: Measuring the sizes of the object and image.
From (1.1) и (1.2) it follows that
f a
y2
.
y1  y2
(1.5)
Therefore, to find f it is necessary to produce the image of the object, and
measure the linear sizes of the object and image.
Method 3. Direct measurement of the focal length f .
The back focal length of a thin lens is the distance at which initially collimated (parallel) rays are focused to a point, or alternatively the front focal length indicates how far in front of the lens a point source must be located to form a collimated beam.
Put the light source and the collimator on the optical bench. Adjust the position of the collimator to produce a parallel beam. Then, put the positive lens and
measure the distance f from the lens to the plane in which the image is formed.
Method 4. Bessel’s method.
When the distance l from the object to image is more than 4 f it is always
possible to find two lens positions at which the demagnified and magnified images
are produced on the screen. Moreover, these positions are symmetrical to the midpoint between object and image (Fig. 1.2).
According to the equation (1.4), the first position of the lens can be found as
follows:
f 
(l  c  b2 )(c  b2 )
,
l
and the second position
(l  b2 )b2
.
l
By equating the right-hand sides, we get
l c
b2 
.
2
f 
6
The last equation proves that both positions of the lens are symmetrical to the
midpoint between object and image.
2
1
Α1
1
2
c
2
1
2
 2
1
b2
l
Fig. 1.2. Determining the focal length by the method of Bessel
To find the expression for f , consider the lens in the first position. In this
case the distance from the object to the lens
a1  l  c  b2  l  c 
l c l c

,
2
2
and the distance from the lens to the image
b1  c  b2  c 
l c l c

.
2
2
Substituting these values into (1.4), we get:
7
l 2  c2
.
f 
4l
(1.6)
Therefore, to find f it is necessary to measure the distance l between the object and the screen and the distance c at which the lens is shifted to get the
demagnified image instead of the magnified one.
This method is the most accurate, as in all previous cases the distances are
measured from the middle of the lens, while they should be measured from the
principal planes. This is particularly important for thick lenses and lens systems. In
Bessel’s method this inaccuracy is eliminated due to the fact that only the shift of
the lens is measured instead of the absolute distance from it.
The procedure for measuring f by Bessel’s method is as follows. Find the
approximate value of f by producing the image a remote object, such as the Sun
or a room lamp; the image will be nearly at the focal plane of the lens. Set the object and the screen at a distance l  4 f and put the lens between them. Produce on
the screen magnified and demagnified images by shifting the lens. Repeat the
measurements at least five times; calculate the averaged value of f using the expression (1.6).
Task 2. Measurement of the focal length and the optical power of a negative
lens.
In the case of a negative lens, it is necessary to use an auxiliary positive lens.
Consider the positive lens L1 , which collects at the point A 2 the light beam originated from the point A1 . If we put the diverging (negative) lens L2 into the converging light beam, the image will shift to the point A2 (Figure 1.3). As the path
of rays is reversible, it can be assumed that if the object were at the point A2 , the
negative lens would form the image in the point A 2 . Therefore, the distance from
the lens L2 to the point A 2 can be marked a, and the distance to the point A2 – b.
Since for diverging lenses f  0 , it follows from the expression (1.1) that:
f 
ab
.
a b
(1.7)
Put on the bench the light source, the converging lens, and the screen. Produce on the screen the sharp image of the filament. Mark the position of the screen
(the point A 2 ). Then shift the screen and put the negative lens between the positive
lens and the A 2 point. Shift the screen again and produce the sharp image of the
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object (the point A2 ). Measure the distances between the lenses and from the lens
L2 to the point A2 . Calculate the distances a and b . Repeat the measurements at
least five times, and calculate the averaged value of f.
L1
L2
A1
A2
A2
a
b
Fig. 1.3. Determining the focal length of a negative lens
Task. 3. Measurement of the focal length of a concave mirror.
Take the light source, the concave mirror and the screen. For obvious reasons
the last two elements must be shifted from the optical axis. Produce the image of
the filament on the screen (see fig. 1.4).
y1
F
y2
f
b
a
Fig. 1.4. Determination of the focal length of a concave mirror
Measure the distances a and b and calculate f using the equation:
9
f 
ab
.
ab
(1.8)
Repeat the measurements at least five times, and calculate the averaged value
of f.
1.4. Content of the report
The report must include:
1. Optical schemes of all measurement setups where the distances between the
optical elements and rays trajectories are indicated.
2. Tables with the results of the measurements.
3. The results of the calculations.
1.5. Test questions
1. What is a lens?
2. What is the linear magnification of a lens, a spherical mirror?
3. How to measure the focal length of a positive lens, a negative lens, a concave and a convex mirror?
4. Prove that if the distance from the object to the image exceeds four-fold focal length, then the image on the screen can be obtained at two different positions
of the lens.
2. WAVELENGTH CALIBRATION OF CCD SPECTROPHTOMETER
Aim of the workshop: studying the methods of a wavelength calibration of
CCD spectrophotometers (dispersing element – a concave diffraction grating) using the emission lines of mercury and helium.
2.1. Overview
Absorption spectroscopy is a method of characterization, which is based upon
the analysis of the spectral composition of the optical radiation transmitted through
or reflected from a sample. Among other things absorption spectroscopy allows determining:
– Atomic, molecular, crystalline structure of various substances, their optical
properties;
– The structure of the polymers and their isotropic isomeric modification, as
well as the structure of the intermediates, such as chemical radicals and molecular
associations;
10
– The chemical composition.
Devices for measuring absorption spectra are called spectrophotometers.
2.1.1. The basic principles of absorption spectroscopy. Law of Beer–Lambert–
Bouguer
According to the law of Beer–Lambert–Bouguer the intensity of the monochromatic collimated beam I  passing through a layer of a homogenous isotropic
medium depends exponentially on the thickness of the layer d and the concentration of the attenuating species C :
I  I0 exp   d   I0 exp  Cd  ,
(2.1)
where I 0 – the intensity of incident monochromatic radiation;  – spectral absorption coefficient,  – spectral molar absorption.
Spectrophotometers can measure the spectral transmittance of a sample
I
T   ,
I 0
(2.2)
and the spectral absorption coefficient has to be calculated
 
1 I 0
ln
.
d I
(2.3)
In the practice of absorption analysis the dimensionless parameter called optical density is often used:
I
D  lg 0 .
I
(2.4)
It is obvious that D  0.4343 .
2.1.2. Principles of operation and layout of CCD spectrophotometers
Any spectrophotometer consists of an illuminating system, a sample compartment, a monochromator or polychromator, a photodetector, an electronic unit,
and a number of auxiliary elements. If a monochromator and a single-element
photodetector are used, samples can be installed in front of the entrance slit and the
exit slit; the scanning of the spectrum in this case is mechanical. If the design of
the device is based upon a polychromator and an array of photodetectors (e.g.
CCD, CMOS or PDA), sample are installed in front of the entrance slit (Fig. 2.1).
11
Light source
Condenser
Sample
CCD
Polychromator
Reference
Electronic unit
PC
Fig. 2.1. Block diagram of a single-beam spectrophotometer – polychromator
based on a concave diffraction grating and CCD photodetector
Spectrophotometer which is built according to the Rowland scheme with a
concave grating and multi-element CCD photodetector is used in this laboratory
workshop. It consists of (Fig. 2.2) an illuminating system, a polychromator, a CCD
photodetector, an electronic unit (controller) and a personal computer (PC).
Polychromator
6
Light sources
2
4
3
5
1
7
PC
Controller
Fig. 2.2. Block diagram of the multi-channel spectrum analyzer
The illuminating system includes two light sources, DDS-30 deuterium lamp
(1) and the incandescent lamp (2), and the illuminating spherical mirror (3). The
selection of light sources is carried out by turning of the mirror. Deuterium lamps
are designed for operation in the ultraviolet (UV) region; incandescent lamps are
used for the visible region.
The light beam focused by the spherical mirror (3) passes through the sample
(4) and directed into the entrance slit of the polychromator (5). Being reflected
from the plane mirror (6), the beam falls on the concave diffraction grating (7) and
decomposes into the spectrum. The spectrum is focused on the CCD linear sensor
(7) consisting of 512 pixels.
12
The output electric signal from the CCD is directed to the controller, which
amplifies and converts it into a digital code. The controller equipped with 10-bits
ADC with the conversion error U  U source 1024 . Then, the spectral data are
transferred to a PC with specialized software. Spectra are stored as ASCII files
with the extension .spe, which contains the boundaries of the working wavelength
spectral range and 512 discrete values of the transmission coefficient corresponding to all pixels of the photodetector.
The minimum time that is required to measure one spectrum is 8 ms. In the
real conditions to achieve the required level of the output signal the CCD exposure
are increased 10 ... 20 times the minimum value. In addition, to reduce the noise
the result is averaged over several (1 to 20) sequentially obtained representation of
the spectrum. Total time of a single measurement does not exceed 2 sec.
Basic technical characteristics of the spectrophotometer are presented in
tabl. 2.1.
Table 2.1
Technical characteristics of the spectrophotometer
Working spectral range
240…690 nm
Spectral resolution
1 nm
concave;
groove density N1 = 300 grooves/mm;
Diffraction grating
blaze angle  = 1.7;
radius of curvature R = 100 mm;
height H = 25 mm
UV-region
Deuterium lamp DDS-30
Light source
Visible region
Incandescent lamp
Photodetector
CCD linear sensor 512 pixels, uncooled
Time of recording of single spectrum
8 ms… 2 s
External interface
LPT
Power supply
220 V, 50 Hz
Before starting the measurements the dark signal of the CCD and the signal
corresponding to the level of 100% transmittance (reference signal) must be recorded and stored in the memory of the PC. The dark signal is measured when the optical shutter of the polychromator is closed, the reference signal – when the reference
is installed in the sample compartment of the device. The transmittance is estimated
according to the formula
I I
T   T  ,
I 0  IT 
13
(2.5)
where IT  – the intensity corresponding to the CCD dark signal; I  – the intensity
of the radiation transmitted through the sample; I 0 – the intensity of the radiation
transmitted through the reference.
2.1.3. Concave diffraction grating
Consider the optical scheme of a polychromator with a concave diffraction
grating. The entrance slit, grating and photodetector (CCD array) are placed on a
circle of the radius equal to half the radius of curvature of the grating, called Rowland circle (fig. 2.3).
Diffraction
grating
Entrance
slit
CCD
Fig. 2.3. Installation of a diffraction grating Rowland circle
The linear dispersion of the system
dl
is a parameter of the diffraction gratd
ing, and is given by the equation
dl
R2
,
 mN
d
S
(2.6)
where N  the number of grooves per a millimeter grating length; R  the radius
of curvature of the grating; S  distance from the image of the entrance slit on the
surface of the detector to the center of the grating.
The condition of the principal maxima of the diffraction grating has the form
b(sin   sin )  m ,
14
(2.7)
where b  the grating period; ,  – the angles of incidence and diffraction, respectively; m  1, 2, 3 ...  the order of the spectrum. In most spectrophotometers
only one of the orders (usually the first) are used. It is called a work order.
2.2. Wavelength calibration procedure
Wavelength calibration of a CCD spectrometer is the technical procedure to
establish a unique relation of the wavelength of the radiation incident on the multichannel detector after spectral decomposition by means of a dispersive element,
with the numbers of pixels.
A wavelength calibration generally can be performed by two methods:
1. The method of calculation of the trajectories of rays with different wavelengths using the optical scheme of a device and the parameters of a dispersive element (grating).
2. The calibration of the instrument using reference spectra.
Recommended for calibration reference spectra are usually divided into three
classes.
References of the first class are symmetrical and quite narrow individual
peaks in absorption or emission spectra, which have been measured with the instrument with a very high spectral resolution. Emission lines of mercury, helium,
and the rotational structure of the absorption bands of simple molecules: HCl near
2600 cm-1 and CO2 near 2000 cm-1 are examples of class I references.
References of the second class are peaks in absorption or emission spectra,
which do not satisfy one of the requirements for the first class references. Rotational structure of the absorption bands of H2O, CO2 and other gases and vapors
are examples of the second class references.
The third category of references includes such absorption bands, which do not
satisfy the requirements of class I or II normals, but may in some cases be used for
coarse calibration.
In this workshop for establishing the relationship between each spectral channel (pixel of the CCD array) with a certain wavelength the emission lines of mercury (table. 2.2) and helium (table. 2.3) are used. For this purpose, DRGS-12 mercury–helium lamp is installed in the device instead of DDS-30 deuterium lamp.
During calibration procedure the DRGS-12 emission spectrum is recorded
and *.spe file, which contains the signal levels in all spectral channels, is generated. The peaks observed in the emission spectrum of the lamp must be identified by
associating them with a particular characteristic line of helium or mercury. It is
15
usually guided by a sequence of lines in the spectrum and their relative intensities,
taking into account that they can be observed in the second or third-order of diffraction. Furthermore, some of the weak lines o are often not visible due to the
presence of background radiation.
Table 2.2
Mercury lines
Wavelength, nm
Intensity
253.7
62
265.2/265.4
1
275.3
0.3
280.6
0.3
289.4
0.5
292.5
0.1
296.7
6
302.2
2
312.6
8
313.2
14
334.1
1
Mercury lines
Wavelength, nm
Intensity
365.0
10
365.5
3
366.3
3
404.7
11
407.8
1
435.8
33
546.1
25
577.0
6
579.1
6
671.6
0.2
737.2
0.2
1014.0
0.4
Table 2.3
Helium lines
Wavelength, nm
Intensity
294.5
…
318.8
0.7
388.9
12
402.6
0.6
447.1
6
468.6
…
471.3
0.7
Helium lines
Wavelength, nm
Intensity
492.2
2
501.6
6
587.6
100
656.0
…
667.8
…
706.5
1
1083.0
22
Note. Intensities are given as a relative to the intensity of the strongest line of helium at 587.6 nm
(intensity of this line is 100).
From (2.7) it follows that at a constant angle of incidence  in a working order m each pixel corresponds to a certain angle of diffraction which is uniquely associated with a wavelength . For small angles we can write

m
m
   or n     .
b
b
(2.8)
It follows from (2.8)
(n)  a1n  a2 ,
(2.9)
where a1 and a2 are constants. If the wavelength is known, this equation allows
determining the number of the pixel to which optical radiation is directed.
16
The coefficients a1 and a2 can be estimated by the least squares regression
method from the pairs of values wavelength  – number of channel n found at the
previous stage. Then a calibration line is plotted and the precise boundaries of the
working spectral range corresponding to the first and last element of the
photodetector are determined. The use of linear approximation is quite justified,
since the CCD sensor is installed near the normal to the grating, and the linear dispersion can be considered constant in the entire working spectral range.
2.3. Tasks and guidance to implement them
Task 1. Recording of the emission lines of mercury and helium.
Install DRGS-12 lamp; turn the device on. Run the software. Open the shutter
and watch on the screen the spectrum of the lamp.
Adjust the charge accumulation time (exposure) of the CCD (sensitivity control in the bottom of the screen), in such a way that the most intense peak in the
spectrum is not «cutted». Save the spectrum to the file spectr1.spe.
To simplify the calibration procedure additional optical filters (normal class
III) – UVG-360 and the glass are used. The transmission spectra of these filters are
shown in fig. 2.4 (Т – transmittance). By turns put in front of the entrance slit the
filter UVG-360 and the glass filter and save the corresponding spectra to the file
spectr2.spe and spectr3.spe respectively.
Task 2. Analysis of the recorded spectra of mercury and helium. Preliminary
calibration.
Using data on the emission lines of helium and mercury (see. table. 2.2 and
table. 2.3) identify three or four most intense peaks in the recorded spectra (use
SA9 software for visualizing spectra, making various operations with them, and
finding peak wavelengths).
Create a table connecting the peak wavelengths and the channel numbers corresponding them (calibration table). Plot a diagram and approximate the dependence of the wavelength on the channel number by a straight line (n)  a1n  a2 ,
find the coefficients a1 and a2 . Use MS Excel or any other software for curve fitting (Mathcad, Statistica etc.)
Estimate the boundaries of the working spectral range corresponding to the
first and last elements of the CCD  n0 and  n511 .
17
100
90
80
Т, %
70
60
Glass filter
50
Filter UVG-360
40
30
20
10
0
200
250
300
350
400
450
500
λ, нм
Fig. 2.4. The spectral transmittance of glass and UVG-360 filters
Task 3. Final calibration.
Open spectr1.spe file in Notepad (or any other text editor) and replace the
first line with the wavelength of the first channel and the second line – with the
wavelength of the last channel estimated in the task 2. Save the file with the new
name spectr1a.spe. Perform similar actions with spectr2.spe and spectr3.spe files.
Identify as many helium and mercury peaks as possible and supplement the
calibration table. Plot the revised calibration line and find more accurate boundaries of the working spectral range. Make corrections to the spectra files according
to the revised calibration.
Task 4. Assessing the accuracy.
Make a table containing wavelengths for all the emission peaks observed in
the spectrum spectr1a.spe (use curve «capture» in SA9), compare them with the
data in tables 2.2 and 2.3, and find the discrepancy  between measured and real
wavelengths (calibration error) for each peak.
Find the emission lines that appear in the second order. To do this, use the
spectrum spectr3a.spe, find emission lines that disappear at this spectrum but present in the spectrum without filters, and estimate emission wavelengths of the second order and real emission wavelengths m2  2m1.
2.4. Content of the report
The report must include:
18
1. Principles of operation and layout of the CCD spectrophotometer;
2. Emission spectra of DRGS-12 lamp in the form of diagrams (without filter,
with glass filter and UVG-360 filters).
3. The results of a wavelength calibration:
– the table, containing pairs of values: wavelength – channel number and calibration error for the identified peaks (including the peaks observed in the second
order of diffraction);
– the calibration line and approximating function (n)  a1n  a2 ;
– the boundaries of the working spectral range.
2.5. Test questions
1. Basic principles of absorption spectroscopy.
2. What is the working principle of a CCD spectrometer?
3. What are the characteristics of a concave diffraction grating, what is Rowland circle?
4. What are so called overlapping diffraction orders?
5. How is wavelength calibration of CCD spectrophotometers carried out?
3. INVESTIGATION OF THE ABSORPTION SPECTRA
OF AMORPHOUS SEMICONDUCTORS
Aim of the workshop: experimental study of the optical absorption spectra of
amorphous and microcrystalline semiconductors; estimation of the optical band
gap.
3.1. Overview
3.1.1. The energy states of electrons in semiconductors
According to the quantum–mechanical theory stationary energy states of electrons in isolated atoms have discrete values. Solids are an ensemble of vast numbers of atoms, chemical bonding between which combines them into a crystal lattice. If the atoms do not interact with each other, the ensemble of N atoms would
have N-fold degenerate energy levels. The interaction removes the degeneracy and
discrete energy levels of individual atoms are split in energy bands. The solution of
the Schrödinger equation in alternative extreme cases – strong and weak coupling
approximation – gives qualitatively the same picture of the electron energy spectrum of the crystal. In both cases, allowed and forbidden bands alternate with each
other, and the number of single-electron states in the allowed bands multiple of the
19
number of atoms, e.g. macroscopically large. This gives the quasi-continuous distribution of energy levels within the allowed bands.
The highest completely filled at T = 0 K energy band in a semiconductor is
called the valence band, and the next following band is called the conduction band.
The energy gap between the top of the valence band EV and the bottom of the
conduction band Ec is called the band gap.
In pure single crystals with translational symmetry allowed energy bands have
sharp edges, and there are no localized states in the band gap. In amorphous and
microcrystalline semiconductors translational symmetry is broken or missing at all,
which leads to a blurring of the band edges (so-called tails of the density of states
are formed) and a large number of localized states are present in the band gap.
The band gap E g varies from few hundredths to a few electronvolts for semiconductors and exceeds 6 eV for dielectrics. Semiconductors with the band gap
less than 0.3 eV is called narrow-gap and semiconductors with the band gap more
than 3 eV – wide gap.
3.1.2. Methods of measuring the absorption spectra of semiconductor materials
in the form of bulk and thin film samples
When propagating through a semiconductor optical radiation is being weakened due to reflection and absorption. Consider the case where a collimated beam
of monochromatic radiation with intensity I 0 is incident at a plane-parallel plate
semiconductor (fig. 3.1).
I0
R I0
n 1
d
1  R  RI0 exp  d 
n
n 1
I  I 0 1  R  exp  d 
2
Fig. 3.1. Propagation of radiation through a semiconductor plate
The spectral transmittance of the plate T is a ratio of the intensity of radiation transmitted through the sample I  to the incident radiation intensity I 0 ; it
20
can be experimentally measured using a spectrophotometer. In accordance with the
Bouguer–Lambert law and taking into account Fresnel reflection from the boundaries the transmittance is
T 
where
2
n  1

R
 n  12
(1  R)2 exp   d 
(1  R 2 )exp  2 d 
,
(3.1)
– the coefficient of Fresnel reflection; n – the refractive index;
 – the absorption coefficient; d – sample thickness. In the spectral region of
strong absorption multiple reflections inside the sample can be ignored and the expression (3.1) is significantly simplified:
T  (1  R)2 exp   d  .
(3.2)
The absorption coefficient  is a material property and depends on the
wavelength of radiation. The dependence of the absorption coefficient on the photon energy    or wavelength of radiation     is an absorption spectrum. Expression (3.2) allows estimation of the absorption spectrum from the transmission
spectrum T    measured in experiment with a spectrophotometer
I0
R1I0
n 1
1  R1  R2 I0 exp  d 
nf
d
1  R1 1  R2  R3I0 exp  d 
ns
n 1
I  I0 1  R1 1  R2  1  R3  exp  d 
Fig. 3.2. Propagation of radiation through the structure film–substrate
In many cases, particularly for amorphous and microcrystalline materials,
preparing a bulk sample is impossible or impractical. Instead, sampleы in the form
of thin films are deposited on a transparent insulating substrate (Fig. 3.2). To find
21
the transmittance of such structures we must take into account Fresnel reflection
from three boundaries: air–film ( R1 ), film–substrate ( R2 ), substrate–air ( R3 ):
T  (1  R1)(1  R2 )(1  R3 )exp    d  ,
 n f  1
R1 
2
 n f  1
2
 n f  ns 
R2 
2
 n f  ns 
2
2
ns  1

R3 
,
2
 ns  1
(3.3)
where n f – refractive index of the film; ns – refractive index of the substrate. Expression (3.3) is written for the case when the substrate does not absorb the radiation in the working spectral range and multiple reflections can be neglected.
When the spectral equipment allows measuring both reflectance and transmittance it is advisable to use the following approach for the calculation of the absorption coefficient
T  (1  R )exp  d  ,
(3.4)
where T – spectral transmittance of the sample; R – spectral reflectance of the
sample. This equation is valid for both bulk and thin film samples.
3.1.3. The mechanisms of optical absorption in semiconductors
In semiconductors five basic mechanisms of optical absorption can be distinguished:
– Fundamental (intrinsic or interband);
– Exciton absorption;
– Free carriers absorption;
– Impurity absorption;
– Lattice absorption.
In the ultraviolet, visible and near infrared (IR) regions fundamental absorption caused by optical transitions of electrons from the valence band to the conduction band is dominant. Exciton absorption takes place due to the formation of
excitons – quasi-particles, bound states of electrons and holes; it is observed only
in very pure semiconductors at low temperatures. Free carriers absorption, impurity absorption and lattice absorption occur in the far infrared region. When electrons
in a semiconductor interact with electromagnetic radiation the laws of energy conservation and momentum conservations must be satisfied:
E  E  ,
p  p  k ,
22
where E, p – energy and quasi-momentum of an electron before the interaction
with a photon; E′, p′ – energy and quasi-momentum after the interaction;  –
photon energy; ħk –photon momentum. The momentum of a photon is very small
compared with the quasi-momentum of an electron, so it can be ignored p′ = p. It
means that only such optical transitions are possible when the quasi-momentum
(wave vector) of electron is preserved. These transitions are called vertical or direct.
Semiconductors can be divided into two groups in which the mechanisms of
interband transitions are significantly different. In direct-band semiconductors
extrema of the valence and conduction bands are located in the same point of the
Brillouin zone, usually at k = 0, which ensures that the law of conservation of quasimomentum is satisfied. Among direct-gap semiconductors are widely used materials
such as gallium arsenide, gallium nitride, indium antimonide and many others.
If the transition matrix element differs from zero in the first-order perturbation
theory, transitions are allowed. Fig. 3.3, a shows the direct allowed transitions. For
semiconductors, in which the conduction band and the valence band have spherical
symmetry, the intrinsic absorption coefficient for direct allowed transitions is given by
the following equation

   A   Eg

12
  0,
,
  Eg ,
(3.5)
  Eg .
In accordance with the law of conservation of energy direct interband transitions are possible only when the photon energy exceeds the band gap of a semiconductor, or when the wavelength that does not exceed a certain critical value
c  hc Eg which is called the fundamental absorption edge.
E
Е
Ei
Ec
ħω
Ev
2
E
1
0
[111]
[000]
[100] k
k
а
b
Fig. 3.3. Fundamental absorption for direct interband transitions (a); optical transitions in a semiconductor with complex energy bands structure (b)
23
In indirect-band semiconductors, i.e. silicon and germanium, the extrema of
the valence and conduction bands are located in different points of the Brillouin
zone (Fig. 3.3, b). Quasi-momentum conservation in this case is provided by the
participation of the third particle – a phonon. Indirect transitions can occur with
both absorption and emission of phonons; the dependence of the absorption coefficient on the energy of a photon in this case is divided into two branches:




2
2



E

E


E

E
g
p
g
p
   A 

exp E p kT  1 1  exp  E p kT




  A

  Eg  E p





,


  Eg  E p ,
(3.6)
2
exp E p kT  1
,
  Eg  E p .
(3.7)
The equations (3.2) (3.5) (3.6) are valid in a limited range of photon energies
corresponding to transitions from energy levels near the valence band ceiling to the
levels near the conduction band bottom, where E  k  can be approximated by a parabola.
a
α2
b
α1/2
1
2
Еg
Еg – Еp
hω
Еg
2Еp
hω, эВ
Еg + Еp
Fig. 3.4. Dependence  2   for direct allowed transitions (a) and indirect transitions (b)
To find the band gap E g from the absorption spectrum for direct allowed
transitions we need to plot the curve     f   and extrapolate the linear
2
segment to the intersection with the horizontal axis; the point of intersection is E g
(Fig. 3.4, a). In the case of indirect transitions we need to plot the curve
  1 2  f  
having two linear sections corresponding to transitions with the
24
absorption and emission of a photon, and extrapolate them to the intersection with the horizontal
axis. The band gap in this case is determined by
the midpoint of the interval limited by the intersection points (Fig. 3.4, b).
In amorphous and micro crystalline semiconductors translational symmetry is substantially
broken or missing altogether, resulting in a significant change in the character of dependency
  f   and distortion of the fundamental ab-
α, sm–1
105
104
1
103
102
10
1
0,1
2
1,5
2,0 2,5 ħω, eV
Fig. 3.5. The absorption spectrum
of the amorphous semiconductor
(1 – parabolic portion,
2 – Urbach tail)
sorption edge in comparison with corresponding
single crystal materials. Fig. 3.5 shows an example of the spectrum of an amorphous semiconductor.
In the range of energies higher than the band gap   Eg , there is a steeper
slope of the absorption curve compared to a single crystal, which is the consequence of the violation of the selection rules; the corresponding segment of the absorption curve is parabolic:

   B   Eg

r
,
  Eg ,
(3.8)
where r – index varying from 1 to 3 depending on a material (in most cases, r = 2).
When   Eg the dependence   f   is exponential
   Eg
  0 exp 
 E0

,

  Eg ,
(3.9)
where E0 – the characteristic Urbach energy, and the corresponding segment of
the curve is known as Urbach tail. This phenomenon is caused by the high density
of localized states in the band gap.
Compared with the corresponding single crystals absorption edge of amorphous semiconductors is shifted to shorter wavelengths (the smaller the degree of
order, the bigger the shift).
3.1.4. Methods to reduce Fresnel losses
The coefficient of Fresnel reflection from the boundary between two media
with refractive indices n1 and n2 can be estimated according to the formula
25
2
n1  n2 

. For example, the coefficient of reflection from the air–glass interR
2
 n1  n2 
face (n=1.5) is 4%, from air–silicon interface (n=3.5) – 30%.
As follows from (3.2), (3.3), unless special measures are taken, in optoelectronic devices based on semiconductor thin films Fresnel losses can reach tens of
percents. To reduce them dielectric interference coatings (antireflection coatings)
are used.
The simplest antireflective coating is a dielectric film with a thickness of a

quarter wavelength d  . In this case the reflectance of the interface:
4
2
 n2  n n 
R 0 1 2  ,
 n2  n n 
 0 1 2
(3.10)
where n0 – the refractive index of the coating; n1 , n2 – refractive indices of the
materials forming the initial boundary.
When the refractive index of the coating n0  n1n2 , light rays reflected
from its top and bottom sides will suppress each other due to destructive interference and the reflection coefficient will turn to zero. It is not always feasible in
practice, because it is difficult to choose a material with a predetermined refractive
index, which simultaneously meets all technological requirements, but even a single-layer coating can reduce losses of 3–4 times. The best effect can be achieved
with multilayer antireflection coatings.
3.2. Experimental setup
The experimental setup for measuring optical reflectance consists of the following elements (Fig. 3.6):
1. Personal computer;
2. Fiber-optic spectrometer AvaSpec;
3. Light source – AvaLight-DHc;
4. Adjustable table for samples;
5. Reflectometric fiber-optical probe;
6. The holder of the optical probe;
7. Reflection reference.
Reflectometer probe consists of two fiber-optical waveguides combined into a
single cable. One end of the cable is mounted to the holder above the adjustable
26
table. At the other end the optical fibers diverge, the first one is connected to the
light source while the second one is connected to the spectrometer. Optical radiation from the light source goes through the first waveguide (fiber 1) exits the optical probe and falls on the surface of a test sample. Radiation reflected from the surface of the sample enters the second waveguide (fiber 2) of the reflectometric
probe and goes to the spectrometer.
Optical
probe
Fiber 1
Sample
Fiber 2
Light source
Reference
Spectrometer
Table
PC
USB port
Fig. 3.6. Experimental setup for measuring reflection spectra
The optical table is equipped with micromechanical screws, allowing positioning the test and reference samples under the probe. The probe holder is also
equipped with a micromechanical screw, which allows changing the distance between the probe and the sample surface. This design eliminates the need for finetuning mirrors.
The experimental setup for measuring transmittance consists of the following
elements (Fig. 3.4):
1. Personal computer;
2. Fiber optic spectrometer AvaSpec;
3. Light source – AvaLight-DHc;
4. The sample holder with collimating lenses;
5. Two fiber optic waveguides.
27
Sample
Fiber 1
Fiber 2
Light source
Collimators
Spectrometer
PC
USBport
Fig. 3.7. Transmission spectra setup
Nonmonochromatic radiation from the light source enters the waveguide 1
(fiber 1), then passes through the collimating lens and falls on the surface of a
sample. Transmitted radiation enters are focused with the collimating lens at the
end of the waveguide 2 (fiber 2) and transferred to the spectrometer. Transmission
measurements do not require a reference sample.
The spectrometer is based on a CCD linear photodetector and works in the
spectral range 200…1100 nm with the spectral resolution less than 1 nm. In the
setups described above the spectral range is limited to 400…1000 nm because the
lights source equipped with an incandescent lamp only. The instrument is controlled by a computer using AvaSoft software supplied by the manufacturer.
3.3. Tasks and guidance to implement them
The objects of study are thin films of amorphous silicon deposited on optical
glass substrate and other samples of various thin films (on the instructions).
Task 3.1. Recording of transmission spectra of the samples.
Turn the spectrometer on; let the lamp to warm up for at least 15 minutes.
Connect the setup for transmission measurements to the spectrometer and the light
source. Run AvaSoft, chose transmission measuring mode, perform calibration.
One by one install test samples, measure the transmission spectra, save them.
Task 3.2. Recording of reflection spectra of the samples.
28
Connect the setup for reflection measurements to the spectrometer and the
light source. Switch to the reflectance measuring mode. Install reference (aluminum mirror), perform calibration. One by one install test samples, measure the reflectance spectra, save them.
Task 3.3. Estimation of the absorption spectra.
Export spectra to MS Excel. Plot transmittance T    and reflectance R   
spectra for all samples. Calculate absorption coefficient  at each wavelength.
Use the expression (3.4).
Plot    curves for all samples. To find the photon energy in electronvolts
use the formula 
1.24
, where λ – wavelength in micrometers.

Task 3.4. Estimation of the reflection losses on the interfaces.
Find transmittance of the optical glass sample in the spectral region where absorption is negligible to estimate the reflection losses on the interfaces. Find how
to decrease Fresnel losses with a single-layer antireflection coating NaF (n = 1.33).
Consider how to reduce the reflectance from the silicon surface (n = 3.51)
with a single-layer antireflection coating SiO2 (n = 1.45).
Task 3.4. Estimation of the band gap.
Plot the diagrams   
12
f
  , find the linear segments and extrapolate
them to the intersection with the horizontal axis. Determine the optical band gap in
accordance with the technique described above.
3.4. Content of the report
The report must include:
1. Experimental setups layouts.
2. Transmittance T    , reflectance R    and absorption spectra   
for all samples.
3. Estimation of Fresnel losses on the interfaces and parameters of antireflective coatings to reduce them.
4. Diagrams 1 2  f   and the results of band gap estimation.
3.5. Test questions
1. What are the different experimental methods for determining the bandgap?
29
2. How to find the absorption coefficient from the spectral transmittance
T    for a bulk sample?
3. How to find the absorption coefficient from the spectral transmittance
T    of a thin film on a transparent substrate?
4. How to find the absorption coefficient from the spectral transmittance and
reflectance?
5. What are the distinctive feature of fundamental absorption in direct and indirect bandgap semiconductors?
6. What are the distinctive feature of the optical absorption in amorphous
semiconductors?
7. Explain the algorithm for estimation of the bandgap from absorption spectra     for different types of semiconductors.
4. INVESTIGATION OF FRAUNHOFER DIFFRACTION FROM
VARIOUS OBSTACLES IN COHERENT AND INCOHERENT LIGHT
AND MEASURING THE DIAMETER OF A THIN WIRE
Aim of the workshop: investigation of the diffraction patterns from various
periodic and non-periodic structures produced by a He–Ne laser and an incandescent lamp; estimation of geometrical parameters of these structures and the laser
generation wavelength using the observed diffraction patterns.
4.1. Overview
Diffraction could be defined as the deviation from the rectilinear propagation
of electromagnetic waves in free space when it encounters obstacles such as iris
diaphragms or slits. These phenomena are direct evidence of the wave nature of
light. In the case of Fraunhofer diffraction an obstacle is illuminated by a plane
electromagnetic wave (collimated beam) which changes the direction of propagation after interacting with an obstacle but stays plane. It means that both the light
source and the screen where the diffraction pattern can be observed should be
placed at infinite distances from the obstacle. As an alternative it is possible to use
two converging lens (collimators); the first one produces collimated beam from a
point source located at the focal point while the second one projects the diffraction
pattern on the screen.
One of the solutions of the wave equation describing free-space light propagation is a monochromatic plane harmonic wave:
30
E  E0 cos  t  kr  ,
(4.1)
where E is electric field strength   2 T – circular frequency; k – wave vector,
k  2  ; r – radius vector to the observation point.
For weak light fields complex representation can be used:
E  E0 exp  j (kr  t )  E0 exp( jkr)exp( jt ) ,
(4.2)
where E  r   E0 exp  jkr  is a complex amplitude. It represents both the amplitude and the phase of a wave. The time factor exp   jt  retains its form for linear
operations on fields, therefore, in the subsequent equations, it will be omitted, and
only complex amplitudes will be used.
Such plane wave can only exist if unbounded in time and space. When limited
in time (in the direction of its propagation, specified by the vector k ) it is no longer strictly monochromatic but distributed in some range of frequencies  .When
limited in spatial domain by an obstacle it is no longer plane – some variations in
the directions of wave vectors occur. In terms of geometrical optics it means that a
collimated beam starts to expand instead of propagating strictly in one direction.
This phenomenon is called diffraction.
As follows from (4.2), a plane wave is characterized by two frequencies: temporal   2 T and spatial k  2  . By analogy with the temporal limitation it
can be stated that the spatial limitation produces new frequencies in a spatial spectrum which is equivalent to new directions of propagations. The analysis of diffraction in the mathematical sense is reduced to the application of the theory of
Fourier transforms.
If the function f (x) is not periodic, then it can be represented using the integral Fourier transform:
f  x 

 f   x  exp  j 2 x x  d  x ,
(4.3)

where
f x  

 f  x  exp   j 2 x x  dx .
(4.4)

The weight function f ( x ) in the integral representation (4.3) is called the
spectral density or Fourier image of the function. A variable  x is a coordinate in
31
the frequency space, called a frequency (temporal, spatial) depending on the type
of the main variable x in the signal space. Units  x are the inverse of unit x.
In this laboratory workshop, the phenomenon of diffraction is investigated using monochromatic (helium–neon laser) and non-monochromatic (incandescent
lamp) light sources.
To observe Fraunhofer diffraction, it is necessary to install a positive lens after the obstacle (diaphragm, slit, wire…) and put the screen in its back focal plane
(Fig. 4.1). Consider the process of image formation for plane waves propagating at
an angle φ when diffracted on a narrow long slit with parallel edges limiting the
beam in X direction; the width of the slit is b.
X
A

P
l
x

0
0
Z
f
B
Fig. 4.1. Fraunhofer diffraction on a slit
An incident plane monochromatic wave travels in Z direction; in case of coherent illumination this wave is formed in the far zone of a point monochromatic
source or generated by a laser. In the coordinate system XYZ , the wave vector of
the incident plane wave has only one nonzero component, k  0,0,k  , and the complex wave amplitude is described by the expression E  r   E0 exp  jkr  
 E0 exp  jkz  .
Because waves diffract in directions perpendicular to the slit the wave vectors
of the secondary waves will have non-zero X components. The elementary strips of
the wavefront dx in the plain of the slit have the form of thin narrow strips. The
32
amplitudes of the waves from different strips gathering at the point P are the same,
since they have the same area and slope. If   0 , the phases of all secondary
waves are the same, after the lens they are focused at the point 0 of the focal plane.
If   0 , the phase of the secondary wave from a particular strip depends on the
distance from the center of the slit x. The wave travelling in the direction  from
the strip located at a distance x from the center of the slit is shifted by phase with
respect to the wave of the same direction from the middle of the slit by
  k x x  kx sin  . The phase relations of secondary waves when focused at the
point P, will be the same as in plane AB. The amplitude of the electric field at the
point P, resulting from the secondary waves in the direction φ from the entire slit is
given by the expression
b 2
EP  K () E0'

exp( jkx sin )dx  K ()E0' b
b 2
sin U1
,
U1
(4.5)
E
where K  – slope coefficient; E0'  0 – amplitude per unit wavefront width;
b
U1   kb 2  sin    b   sin  .
IIφ
–2λ/b
2 b
–λ/bb

2
–2π

–π
0
λ/b
 b
22λ/b
 b
π
2π
2
φ
sin 
UU
Fig. 4.2. The relative intensity distribution in the focal plane of the collecting lens during
diffraction from the slit, size b
33
At small diffraction angles, the slope coefficient K  in (4.5) is almost independent of φ and can be replaced with the value at   0 . Then the intensity of
light after the slit at an angle φ is determined by the expression
 sin U1 
 sin  b sin    
I ()  I 0 

I

 ,
0
U

b
sin




 1 
2
2
(4.6)
where I 0 is the intensity at   0 .
The relative angular intensity distribution for diffraction by a slit is shown in
Fig. 4.2. The central distribution maximum corresponds to sin   0 ; condition of

minima sin   m , where m  1,  2, ... . The angular size of the central zone in
b
the diffraction pattern from one slit

sin    .
b
At small diffraction angles sin   tg  
(4.7)
l
f
  , therefore, the linear size of
the central zone of the diffraction distribution l0 is determined by the expression
l 0  2l   2

f.
b
(4.8)
Thus, the diffraction pattern from a single slit with coherent illumination in
the focal plane of a lens is an alternation of bright a dark strips. The narrowing of
the slit leads to the expansion of both the central zone and the entire pattern. Moving the slit along the axis X does not change the position and dimensions of the
diffraction distribution. When the slit is rotated, the pattern also turns. If an obstacle has the form of a non-transparent strip of width b (for example, thin wire) the
diffraction pattern from it is same as from the slit of the same width.
When illumination is incoherent, for example from a point nonmonochromatic source, the condition of the central maximum sin   0 is fulfilled
for all wavelengths, and the conditions of minima and non-zero maxima of the diffraction distribution depend on a wavelength.
In this laboratory workshop, the phenomenon of diffraction is investigated using monochromatic (helium–neon laser) and non-monochromatic (incandescent
lamp) light sources.
34
Consider the phenomenon of diffraction on a harmonic periodic structure,
transmittance of which varies according to harmonic law
( x) 
EP  E0



1
1  cos(2x / d ),    x  ; E  x     x  E0 ,
2
A
( x)exp(ikx sin )dx 
2
A
E 
2
(4.9)

 1  cos(2x / d )exp(ikx sin )dx ,


ikx sin 
dx;
 1  cos(2x / d ) e
E0 

A
,

1
cos(2 x / d )  (ei 2 x / d  ei 2 x / d ) ,
2
E  A(k sin ) 
A
A
(2 / d  k sin )  (2 / d  k sin ) .
2
2
(4.10)
Consequently, the diffraction pattern from a periodic harmonic structure consists of three lines:

  0, sin    .
d
(4.11)
Thus, only diffraction at 0 and ± 1 orders is observed.
Consider the phenomenon of diffraction on diffraction gratings – periodic
structures consisting of N slits or N mirror elements. A reflective diffraction grating is a set of narrow equally spaced parallel mirror strips with a triangular profile.
When a collimated beam with a wavelength λ at an angle α is incident on the grating, diffraction occurs on each strip (Fig. 4.3). The beams diffracted from all strips
are focused in the focal plane of a lens. These beams are coherent and interfere
with each other. Thus, the interference of N light beams is observed in the focal
plane of a lens.
The optical path difference Δ for rays reflected from the edges of two adjacent
mirror elements is determined by the ratio
  AC  AB  d sin   d sin   d (sin   sin ) ,
(4.12)
where d is a grating period.
The resulting intensity distribution I  from the whole grating, taking into account the interference of N beams for a diffraction angle  , has the form
35
 
 N 
sin 2    sin 2 

 
 


I  I 0 I1I 2  I 0
,
2
  
2  
   sin   
 
 
(4.13)
where: I 0 – the intensity of incident light; I1 –intensity distribution due to diffraction on a single element; I 2 – the factor considering interference of N rays;  –
optical path difference on a single element.


C
B
A
b
d
Fig. 4.3. Diffraction on a flat diffraction grating
Positions of maxima are defined by the grating equation
d (sin   sin )  m ,
(4.14)
where: m  0,  1,  2, ... – diffraction order. At maxima I 2  N 2 .
The distribution (4.13) is shown in Fig. 4.4, 4.5. The diffraction pattern from
a diffraction grating with coherent illumination in the focal plane of a lens is an alternation of bright a dark strips. Each bright strip corresponds to a specific diffraction order.
If a grating is illuminated with non-monochromatic light, then the condition
m  0 is fulfilled for all wavelengths, therefore a white strip appears in zero order.
On both sides of it, spectra of positive and negative orders will be observed. If the
grating is used to decompose optical radiation into a spectrum, one of the orders is
used as so-called working order.
36
I
I1
m0
(2  )   / 

m 1
2 
m2
/d
   2 / d
m
1
(2 )   / 
2
Fig. 4.4. The intensity distribution for diffraction on a reflective
grating ( 1 – diffraction angles corresponding to the extrema of the function I1,  2 – diffraction
angles corresponding to the extrema of the function I2)
I1
5
4
3
I2
2
1
0
1
2
m
Fig. 4.5. The intensity distribution in diffraction maxima
Thin metal meshes also can be used as periodic structures, then the diffraction
pattern will be two-dimensional.
4.2. Experimental setup
The optical setup for investigating diffraction from various obstacles consists
of an optical bench on which all necessary elements are installed: an incoherent
light source (incandescent lamp) or a coherent light source (laser), a collimating
lens, a set of objects (obstacles) and a screen.
When using a helium–neon laser as a light source, there is no need for lenses,
because the laser radiation has sufficient brightness. Moreover, the laser is a coher37
ent light source, and the difficulties associated with overlapping diffraction patterns from different wavelengths of a light source are eliminated. A laser is an almost perfect monochromatic point source of light.
4.3. Tasks and guidance to implement them
Tasks 1. Measuring the wavelength of the helium–neon laser.
Assemble a setup, which includes a laser, a test pattern (mira), and a screen.
The mira serves as a periodic harmonic structure; it has 9 zones with different periods (tabl. 4.1). The number of stripes per millimeter  and the period of the
structure d are related by the formula   1/ d .
Table 4.1
Zone number
–
 , mm 1
1
15.4
2
9.2
3
6.2
4
6
5
4
6
3.3
7
2.5
8
2
9
1.7
Turn on the laser, direct radiation to the mira. Attention: the laser is highvoltage equipment, so do not touch the tube and the power supply! Laser radiation
is dangerous for your vision, so it is prohibited to direct it into the human eye!
Set the laser so that the beam propagates along the optical axis. Produce on
the screen the diffraction patterns from all zones of the mira and sketch (or photograph) them. Measure the distance from the mira to the screen with a tape measure.
Make sure that after the mira the incident beam splits into three diffracted beams.
Measure the angular deviation of the diffracted beams for 1, 2 and 3 zones of the
mira and, using the expression d sin   m , where m  0,  1 , estimate the wavelength of the laser radiation λ for each zone and calculate the averaged value.
Tasks 2. Investigation of Fraunhofer diffraction on a single wire.
Replace the mira with a set of thin wires of different diameters. Turn on the
laser, direct laser radiation to one of the wires and observe a diffraction pattern on
the screen. Make sure that in this case the diffraction pattern is identical to the diffraction on a single slit (see Fig. 4.2). Sketch (or photograph) the diffraction patterns and estimate the diameter of each wire, assuming that the laser wavelength
  632.8 нм.
Tasks 3. Investigation of diffraction patterns on two-dimensional periodic
structures.
Install a mesh instead of the block of wires. Turn on the laser and direct laser
radiation to the mesh and observe a diffraction pattern on the screen. Make sure
that in this case the diffraction pattern corresponds to diffraction on a two38
dimensional periodic structure. Sketch (or photograph) the diffraction patterns and
estimate the period of the mesh.
Tasks 4. Investigation of diffraction pattern on a reflective diffraction grating.
Assemble a setup including a light source (incandescent lamp), a reflective
diffraction grating (number of lines / mm = 1000) and a screen.
Turn on the lamp and observe a diffraction pattern on the screen. Identify the
zero order and determine which of the orders should be used to decompose radiation into a spectrum.
4.4. Content of the report
The report must include:
1. Measurement schemes.
2. Results of measurements and calculations.
3. Drawings (or photographs) of diffraction patterns.
4.5. Test questions
1. Explain Fraunhofer diffraction phenomenon.
2. Explain the main features of the diffraction pattern on a single obstacle (slit
and wire).
3. What are the features of the diffraction pattern on a harmonic periodic
structure?
4. What are the features of the diffraction pattern on a two-dimensional periodic structure?
5. How can one determine the geometrical dimensions of the obstacles and
the wavelength of laser generation from the diffraction pattern?
RECOMMENDED TEXTBOOKS
DiMarzio C. A. Optics for engineers Boca Raton, CRC Press, 2012.
Matveev A.N. Optics / transl. from Russian. Moscow: Mir Publishers, 1988.
39
APPENDIX
SPECTRAL ASSISTANT 9.0 PROGRAM DESCRIPTION
General information
Spectral Assistant 9.0 (SA9) software is designed for processing and visualization of UV–Vis spectra measured with a CCD spectrophotometer and stored in
text files with the * .spe extension. The program runs on Windows XP/7/10 platforms and does not require installation.
Fig. A.1. Main window of SA9 program
The main window of the program consists of two frames (Fig. A.1). In the top
frame a list of currently loaded spectra is displayed. The following information is
shown for each spectrum: spectrum number (a unique identifier that is assigned to
the spectrum when it is downloaded from a file and does not change until it is removed from the list), file name, recording date and time, presentation (absorption
spectrum, transmission spectrum, or intensity), the cuvette thickness (for liquid
samples), and the dilution ratio (for solutions).
In the bottom frame, spectra can be viewed as a diagram, where the wavelength or channel number is plotted along the abscissa axis, and the absorption coefficient, transmittance, or signal intensity, depending on the spectrum representation, is plotted along the ordinate axis. Spectra can only be displayed in one repre40
sentation at a time. Spectral curves can be captured by double-click on it, in this
mode the precise coordinates of any point in the corresponding spectrum are
shown in the status bar.
File menu
Open (Ctrl+O). Loads a spectrum from a file and adds it to the list.
Save (Ctrl+S). Saves the spectrum to a file.
View in Notepad. Opens a spectrum file in Notepad to view the source text.
Properties (F5). Opens a dialog box where you can set the spectrum presentation, the thickness of the cuvette and the dilution ratio.
Edit menu
Delete from list (Del). Removes selected spectra from the list.
Clear the list (Ctrl+Del). Removes all spectra from the list.
Select all (Ctrl+A). Selects all the spectra in the list.
Sort. Set the rules of sorting the spectra in the list.
Spectrum menu
Plot
(F3). Plots the curves on the diagram for the selected spectra (this
action also can be performed by double-click on the spectrum number in the list).
Before plotting the graph, make sure that the spectrum representation displayed in
the list (transmission, absorption, or intensity) is true, and if necessary, correct the
information in the Properties menu.
Plot all (Ctrl+F3). Plots curves for all spectra in the list.
Hide
(F4). Remove the curves from the diagram for selected spectra (this
action also can be performed by double-click on the spectrum number in the list).
Hide all (Ctrl+F4). Removes curves for all spectra in the list.
Normalization
. Sets limits on the coordinate axes to make curves fully
visible (size by content).
Zoom in
. Increases the scale of the diagram.
Zoom out
. Reduces the scale of the diagram.
Fragment
. Enlarge the selected fragment of the diagram.
Backoff
. Returns the diagram to the previous scale.
Drag
. Switches program to the mode when moving the mouse in the diagram area while the left button is pressed, the fragment is not selected, but the
41
whole diagram is dragged. To return to the normal just reselect the menu item or
click the corresponding button on the toolbar.
Lables. Allows editing the axes labels and the legend.
Actions menu
In the Actions menu, the commands for processing spectra are grouped. Before selecting most menu items, it is necessary to choose the numbers of spectra in
the list. Where appropriate, batch processing (the implementation of any action
simultaneously to several spectra) is possible.
Convert to absorbance. Converts a transmission spectrum into absorption
(via natural logarithms).
Convert to OD. Converts a transmission spectrum to optical density (via the
decimal logarithms).
Convert to transmittance. Converts an absorption spectrum into transmission.
Curve smoothing. Performs the boxcar averaging of a spectrum. When
choosing this menu item enter the number of points by which smoothing is performed.
Addition. Addition of two or more spectra. When choosing this menu item
enter the numbers of the spectra (hereinafter, the numbers of the spectra are entered in a text line and separated by commas).
Subtraction. Subtraction of two spectra.
Multiplication, Division. Multiplication (division) of two spectra.
Multiplication by a constant, Division by a constant. Multiplication (division) of the spectrum by a constant. Enter the constant in the dialog box when selecting this menu item.
View menu
Scale. Choosing the scale of the diagram.
Axis dimension. Switches between the wavelength or channel number on the
diagram.
Export Data to MS Excel
Spectra saved in *.spe files can be easily exported to Excel using the Spectrum add-in. To use this add-in, copy spectrum.xla file to computer’s hard drive,
42
start MS Excel, select the menu item Tools\Add-ons, click the Browse button and
specify in the dialog box the path to the spectrum.xla file.
After these actions, a new Spectrum menu item will appear in the main window. Before starting work, enter the wavelength calibration of the spectrometer data by selecting the menu item Spectrum\Calibration data.
To import a spectrum file, choose Spectrum\Import a spectrum file (you
can import several spectra at the same time). As a result, one or more columns will
be formed containing, depending on the type of spectrum, the absorption, transmission, or intensity for each spectral channel. The file name is placed in the first cell
of the column.
Plotting a diagram is carried out using the Spectrum\Plot diagram command. A dialog box is displayed where you must specify the cell interval where the
spectrum file was imported (you can specify the entire column or several columns)
and the presentation mode. A diagram is formed on a separate sheet, where the selected spectra are displayed in graphical form.
43
TABLE OF CONTENTS
INTRODUCTION ..................................................................................................... 3
1. Measuring of the focal lengths of lens and spherical mirrors ............................... 4
1.1. Overview ......................................................................................................... 4
1.2. Experimental setup .......................................................................................... 4
1.3. Tasks and guidance to implement them .......................................................... 5
1.4. Content of the report ..................................................................................... 10
1.5. Test questions ................................................................................................ 10
2. Wavelength calibration of CCD spectrophtometer ............................................. 10
2.1. Overview ....................................................................................................... 10
2.2. Wavelength calibration procedure ................................................................ 15
2.3. Tasks and guidance to implement them ........................................................ 17
2.4. Content of the report ..................................................................................... 18
2.5. Test questions ................................................................................................ 19
3. Investigation of the absorption spectra ............................................................... 19
of amorphous semiconductors ................................................................................. 19
3.1. Overview ....................................................................................................... 19
3.2. Experimental setup ........................................................................................ 26
3.3. Tasks and guidance to implement them ........................................................ 28
3.4. Content of the report ..................................................................................... 29
3.5. Test questions ................................................................................................ 29
4. Investigation of Fraunhofer diffraction from various obstacles in coherent and
incoherent light and measuring the diameter of a thin wire ............................... 30
4.1. Overview ....................................................................................................... 30
4.2. Experimental setup ........................................................................................ 37
4.3. Tasks and guidance to implement them ........................................................ 38
4.4. Content of the report ..................................................................................... 39
4.5. Test questions ................................................................................................ 39
Recommended textbooks ........................................................................................ 39
APPENDIX Spectral Assistant 9.0 Program description....................................... 40
44
Василевский Александр Михайлович,
Коноплев Георгий Асадович,
Степанова Оксана Сергеевна
Optics and optical measurements in solar energy
Tutorials for laboratory workshops
Редактор И. И. Иванов
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