Optics: educational manual

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A.D. Muradov G.Sh. Yar-Mukhamedova G.K. Mussabek

OPTICS Educational manual

Almaty «Qazaq University» 2019


UDC 535 (075) LBC 22.3 я 73 M 96 Recommended for publication by the decision of the Academic Council of the Faculty of Physics and Technology, Editorial and Publishing Council of Al-Farabi Kazakh National University (Protocol №4 dated 16.04.2019); Educational and Methodological Association (PMG) for training specialists in the fields "Telecommunications" and "Engineering" on the basis of AUEC and EMA of REMC for students of the specialty "RET" (Protocol №1 dated October 23, 2019) Reviewers: Doctor of Technical Sciences, Professor D.U. Smagulov Doctor of Physical Sciences, Professor M.E. Abishev

M 96

Muradov A.D. Optics: educational manual / A.D. Muradov, G.Sh. YarMukhamedova, G.K. Mussabek. – Almaty: Qazaq University, 2019. – 154 p. ISBN 978-601-04-3998-6 The present educational manual is a guide for laboratory work implementation on "Optics" section of the general physics course. Here, for each laboratory work a theoretical introduction with description of the main studying physical phenomena and laws, description of the equipment specification for work implementation, work task and questions for self-control are given. The manual includes works related with use for observation and measurement of both simple optical devices with mechanical management and complex digital systems. The content of “Optics” editorial manual corresponds to the syllabus and the educational-methodical complex of the discipline "Optics", which is a mandatory component of training bachelors, according to the State Compulsory Education Standard of the Republic of Kazakhstan in a wide range of engineering specialties.

UDC 535 (075) LBC 22.3 я 73 © Muradov A.D., Yar-Mukhamedova G.Sh., Mussabek G.K., 2019 © Al-Farabi KazNU, 2019

ISBN 978-601-04-3998-6



The "Optics" is the fourth section of the general physics course, and the performance of laboratory works on the topic is an integral part of the general physical practice works. The issues referring to the importance of conducting the physical experiment itself, obtaining the measurement results and their mathematical processing, graphs building, etc., are not presented here, since they were considered in the previous sections of physical practice. However, in solution of optical problems there is a certain specificity, requiring a special attention. Firstly, it is necessary to mention non-contact methods for measuring linear parameters, nodes, temperatures, etc., that allow achieving greater accuracy than conventional instrumental measurements by the caliper, micrometer, goniometer and other instruments may provide. Optical observations of objects and phenomena are usually carried out using eyepieces, telescopes and microscopes. In general, the optical measuring system of such setups must be strictly centered, that is, all their parts should be placed on the same axis. Therefore, before starting the measurements, one should adjust the optical parts of the setup. In the present textbook, the questions of optical setups adjusting are properly discussed in contrast to widely available and well-known descriptions of the preparation of optical setups. This proper discussion is required since the conduct of practical physical classes inevitably leads to some more advance in performing laboratory works as compared to studying a theoretical course. At the beginning of each work, based on the specificity of their implementation, a short theoretical material, which describes the underlying physical phenomenon and the derivations of the basic relationships required to perform the experiment and determine the desired quantities are given. A specific feature of laboratory works performing is that a concentrated light flux enters the eye of the observer. Therefore, the observer should take appropriate precautions when performing measurements. 3

This textbook fully meets the requirements of the general physics course program in the "Optics" section for physical and technical faculties of universities and contains a description of fifteen laboratory works, performed at the Physical and Technical Faculty of the alFarabi KazNU. The authors express their sincere gratitude to M.B. Aitzhanov, who participated in the preparation of laboratory works descriptions.



1.1. Aim of the work The purpose of the work is mastering the techniques of work using an Abbe refractometer, learning basic methods of preparing the refractometer for measurements and studying the refraction of twocomponent mixtures, measurements of refractive indices (refraction) of liquids and verification of the Lorentz-Lorenz formula. In addition, in this work the method for calculating the polarizability of molecules and their effective radius will be considered, as well as the method of quantitative analysis of two-component mixtures.

1.2. Theory of the light dispersion laws 1.2.1. Laws of reflection and refraction of electromagnetic (light) waves Let us imagine an electromagnetic wave falling at an arbitrary angle onto the interface between two media. Then we can compose the equation for this wave propagating with velocity V1. We denote the current coordinates of the point on the surface by x, y, z, and the radius  vector of this point – by r . The  normal n to the surface coincides in direction with z' (Fig.1.1). By cos  , cos  , cos  we can define the direction cosines of the  Figure 1.1. Illustration to the normal n . Then for the wave proderivation of the equation of a pagating along the chosen direction plane wave propagating along an arbitrary direction z', we can write the expression: 5

   rn  x cos  y cos   z cos   . E  Re E exp i (t    Re E exp i (t   V1  V1   

Proceeding from it, we will compose expressions for incident, reflected and refracted waves. The interface between two media is the xy plane, which satisfies the condition z=0. We also assume that in the incident wave the normal  n lies in the plane zx, for which cos   0 (Fig. 1.2). Restrictions on the direction of  the normals n1 (for the reflec-

ted wave) and n2 (for the refracted wave) are not imposed. The propagation velocity of the electromagnetic wave in the second medium becomes equal to V2.

Figure 1.2. Illustrative scheme of reflected and refracted incident waves propagation at the interface between two media

Then for the incident wave the equation takes the form:   x cos   z cos  E  Re E00 exp i  t  V1  

  , 

for the reflected wave:   x cos  1  y cos 1  z cos  1   , E1  Re E 01 exp i1  t  V1   

for a refracted wave:   x cos  2  y cos  2  z cos  2   . E 2  Re E02 exp i 2  t  V2    6

In the case of the boundary condition of z=0, the equality of the tangential components of the electric field strength will be posed as:

E  E 1  E 2 . The equality condition for the tangential components of the strengths is satisfied at any moment in time t and for any х,у coordinates. This equation can be rewritten taking into account the expressions for the equations of each wave, as:     x cos   x cos 1  y cos 1    E01 exp i1  t    E00 exp i  t  V V1 1      

= E02 exp i2  t  x cos  2  y cos  2  .   



The written identity is valid only if a number of conditions are fulfilled: 1.   1  2 .This result is trivial for linear problems, which we consider. 2. cos  / V1  cos  2 / V2  0 . According to this condition, assu  ming that the normal n to the incident wave E lies on the zx plane, we conclude that the normals to the reflected and refracted waves   ( n1 and n2 ) also lie on this plane. 3. cos / V1  cos1 / V1  cos 2 / V2 . Let us analyze these relations in two following stages: 3.1. cos  cos1 , for the first optical medium. Consequently,   1 . Thus, the law of reflection of electromagnetic (light) waves, which states that the incidence angle of the wave is equal to its reflection angle, is obtained and    1 . 3.2. cos / cos 2  V1 / V2 , for the interface between two media. We should pay attention to      / 2 and  2   2   / 2 . Substitution from these ratios the expressions for α and α2 angles into this condition allows to get the law of refraction of electromagnetic waves, sin  / sin  2  V1 / V2 . The last expression can be represented in the familiar form knowing that V1  c / n1 and V2  c / n2 (с – is the 7

propagation speed of electromagnetic (light) waves in a vacuum). We get the final expression:

V n sin   1  1  n21 . sin  2 V2 n2


The resulting relation in optics is called Snell's law. Here n1 and

n2 are the absolute refractive indices of the first and second medium, and n21 is the relative refractive index of the second medium with respect to the first one. In the case of reverse propagation of light (i.e., when it falls at an angle  2 from the second medium to the first one) the incident and refracted rays exchange places (here we obtain the property of reversibility of light rays). In this case, the relation n21  1/ n12 is true, where n12 is the relative refractive index of the first medium with respect to the second one. The value of the reflected light energy increases with the increase in the incidence angle, when light propagates from optically less dense medium to optically denser medium, and the value of the refracted ray energy decreases. In the opposite case, when light propagates from optically denser medium to optically less dense medium (in optically less dense medium the absolute refractive index is smaller), the proportion of reflected light energy increases with the increase in the angle of incidence. However, going closer to a certain value of the incidence angle  0 , the total energy of the light ray is reflected from the interface and the light does not fall from the first medium into the second one (we observe the phenomenon of its total internal reflection). The angle  0 is called the critical angle of total internal reflection. It can be determined by taking into account that    and 2

n1  n2 . Then according to (1.1) we obtain that:

sin  пр  n2 / n1 ]. 8


1.2.2. Dispersion of light In the case of refraction, diffraction, or interference of light, white light decomposes into the spectrum. Such phenomena are generally called the dispersion of light. From this series of phenomena one can single out a narrow sense of the dispersion of light concept. This is the dependence of the phase velocity of the electromagnetic (light) wave on its frequency, or the dependence of the refractive index of a substance on the frequency (wavelength) of light, i.e.

n  f ( 0 ) .


Where λ0 is the wavelength of light in vacuum. The quantitative characteristics of the dependence of the refractive index of a given substance on the wavelength are the average and relative dispersion. The relation determines the average dispersion of a substance: n  (n2  n1 ) /( 1  2 ) . 


The determination of the dispersion through the average dispersion is not entirely convenient, different wavelength intervals are usually used. Therefore, in practice, as the dispersion measure either an average dispersion nF  nc or a relative dispersion is accepted.

(nF  nc ) /( nD  1) ,


where (nF , nc , nD ) – refractive index values for Fraunhofer spectral lines F, C, D with wavelengths of F  486,1 nm, C  656,3 nm and D  589,3 nm correspondingly. In spectroscopy the value that is inverse to the relative dispersion is used and that is called the dispersion coefficient.

N  (nD  1) /( nF  nc ) . 9


The value of the function (1.3), for all transparent colorless substances will be (dn / d )  0 . The nature of such dispersion behavior is called normal. If the substance absorbs light, then in the absorption region and near it the dispersion progress reveals an anomaly and the function (1.3) will show (dn / d )  0 . The phenomenon of light dispersion can be explained from the point of view of the interaction of the electromagnetic field of the light wave with the electric charges of the medium. The electric field of a light wave polarizes a dielectric (medium). The measure of polariza  tion of the medium is estimated by the polarization vector P  Nex ,  where N is the number of dipoles per unit volume, and ex is the electric moment of the dipole. It is known, that the value of the polarization vector is directly proportional to the value of the electric field intensity 𝐸⃗ of the light wave   P  0E , where χ is the electrical susceptibility of the substance, which is related to the permittivity of the substance by the relation   1   . The refractive index of the substance (n) according to Maxwell's theory is defined as n   , where ε is the permittivity of the substance, μ is the magnetic permittivity of the substance (and for most transparent bodies it can be considered that   1 ). According to the above mentioned, for the refractive index of the substance (n), we can write the following expression:

n 2  1  ( Ne /  0 E ) x ,


where x is the electron displacement. Let us compose the equation of motion of an electron in an atom of substance, under the exposure to electromagnetic field of a light 10

wave in order to determine its displacement x. To do this, we should consider the forces acting on a single electron: 1. The driving force acts from the side of the transmitted light wave electric field, that causes the electron to move from the equilibrium position to x. It is defined as f1  eE0 sin t ; 2. The return of the electron to the equilibrium position is accomplished by a quasi-elastic force, which is defined as f 2  m002 x , where ω0 is the proper frequency of an electron, m0– its mass; 3. The frictional force caused by the interaction of an electron with neighboring particles leads to damped oscillations and is defined as f 2  m0 x , where x – electron velocity, β – coefficient of oscillations damping. Taking into account the listed above forces, we can write down the equation of motion of an electron as:

m0 x  eE0 sin t  m002 x  m0 x .


The solution of this differential equation is:

x  A0 sin( t   ) ,


where A0  eE0 / m0 (02   2 )  4 2 2 and tg  2 /(02   2 ) . (1.10)

In the case of small frictional forces, i.e. when   0 , instead of the expression (1.9), we can write:

x  eE0 / m0 (02   2 ) sin t .


Substituting x from (1.11) into expression (1.7), we obtain that

n 2  1  Ne 2 /  0 m0 (02   2 ) . 11


The analysis of the relation (1.12) is shown as a graph n  f ( ) in Fig. 1.3, where AB and CD are the regions of normal dispersion. However, the refractive index does not have a definite value (dashed line), at ω = ω0. This is due to the assumption of the absence of frictional forces, i.e. no damping of oscillations, for which   0 . It is necessary to take into account that oscillations damping leads to BC area appearing. Within the BC segment, the refractive index decreases with the increase in frequency, i.e. (dn / d )  0 (such a change is called an anomalous dispersion). This is observed in the resonant absorption region (for the frequency ω close to ωо). In the equation (1.8), the actions of neighboring molecules and dipoles are not taken into Figure 1.3. Types of light dispersion account. Therefore, it is incomnear a single absorption band with frequency ωо plete. Lorenz and Lorentz considered this factor. They obtained the following dependence for the refractive index in the case of nonassociated liquids with nonpolar molecules: (n 2  1) /( n 2  2)  (4 / 3) N

and [(n 2  1) /( n 2  2)]M /   (4 / 3) N A ,


where N is the number of particles per unit of volume, α is polarizability of molecules, n is the refractive index, NА is the Avogadro number, ρ is the substance density, and M is the molar mass. The Clausius-Mossotti formula, which is essentially the equation (1.13), is known from the "Electricity" course (if just go from n to ε, using n   relation, which characterizes the electronic polarization of dense dielectrics). 12

The value [( n 2  1) /( n 2  2)]M /   R is called molecular refraction. The additivity rule of molecular refractions is satisfied for solutions, i.e. refraction of the solution is composed of the refractions of its components R   ck Rk , where ck is the concentration of the k k

component in molar fractions. In the case of a solution with a uniform mixture of two components with the number of molecules per unit of volume N1 and N2, respectively, the formula (1.13) will be written as: (n 2  1)(n 2  2)  (4 / 3)( N11  N 2 2 ) .

Here: п is the solution refractive index, N1  (

1 M1


) N A is the num-

ber of the first component particles, and their density is equal to 2 M1 1  ) N A is the number of the second com, N2  ( 1

(V1  V2 )



ponent particles and their density is  2 

M2 , where (V1  V2 )

V1 and V2 are

the volumes of the first and the second components correspondingly, М1 and М2 are their molar masses .  M1   and  V1 

Let us denote the density of pure components as 01  

M  02   2  , and the bulk density as   [V1 /(V1  V2 )] . Then the rela V2 

tion can represent the expression (1.14):  01  02  01 n22  1  n22  1  4 n 2  14 4 N    N  ( 1   )   N     A 1 A 2 A 1 M1 3 M2 M 1 n22  2  n 2  23 n22  2  3

(1.15) or finally Clausius-Mossotti formula will take the next form: n22  1 n22  1  n12  1 n22  1  .     n22  2 n22  2  n12  2 n22  2  13


1.2.3. Determination of refractive indices of substances The existing methods for determining the refractive indices of gaseous, liquid and solid bodies can be represented by the following three methods: by means of measuring the smallest deflection angle of the beam in the prism; by observing the critical refraction rays when light passes through the interface between two media with different refractive indices and by comparing the optical density of the studied substances with the standard one. The first method is carried out using spectrometers (goniometerspectrometers). The second method is based on work with devices called refractometers. The third one is based on determining the path difference of coherent rays during propagation in the media with different refractive indices. Apparatus for this method application is called interferometer (interferometric refractometer). Here only the second method will be discussed in detail. According to this method, measurements are carried out using refractometers, the most common among which are the Abbe system instruments. These are refractometers of PL, RPL-3, IRF-22 type and other refractometers. The device is based on a complex prism, which consists of two rectangular prisms with glass refractive indices greater than 1.7 (Fig. 1.4). Fig. 1.4 (left) shows the upper illuminating prism (its hypotenuse edge is a matt surface scattering incident rays). The bottom prism is a measuring prism. The matt surface of the hypotenuse edge of the illuminating prism scatters an incident light ray. Then the ray passes through a thin planeparallel layer of the studied liquid with a refractive index п and falls on the diagonal edge of the lower (measuring) prism at all possible angles ranging from 0° to 90°. A sliding light beam incident at 90° will be refracted on the hypotenuse edge of the measuring prism. Then it will go further into prism at the critical refraction angle  r and come out from it at an angle i. For all other rays of the scattered beam the angle i is minimal. As a result, in the eyepiece field of the telescope Оk, there will be observed a clear boundary between the bright and dark fields (dark in 14

the upper part and bright in the lower part). In the case when the primary light beam directed from below to the large leg of the measuring prism (Fig. 1.4. right), the positions of the dark and bright fields in the eyepiece will change places (in the upper part there will be a bright field, and in the lower one – dark).

Figure 1.4. Scheme of the rays’ path in the prisms when using the methods of: a sliding beam (left), total internal reflection (right)

Knowing the refractive index of the measuring prism п0 and its geometry, one can easily determine (taking into account the law of refraction) an analytical formula for the relationship between the refractive index of the considered liquid n with known parameters

n  sin  n02  sin 2 i  cos sin i .


This formula underlies the design of refractometers and the calibration of their reference scales.

1.3. How do RPL-3, RL and IPF-22 refractometers work? Optical schemes of refractometers (at the top for the RPL-3 and RL and at the bottom for the IRF-22) are shown in fig. 1.5. The main part of the refractometer includes: a measuring head, which includes an illuminating 2 and measuring 3 prisms, a compensator 4 and a telescope (details 5,6,7,8 and 9). 15

The light beam is directed from the illuminator 1 (or mirrors 1 for RL and IRF-22 refractometers) to a measuring head, which consist of a double prism (2 and 3). A thin layer of the testing liquid is placed between the diagonal planes of the measuring head. The emerged beam follows through the dispersion compensator 4, the telescope objective 5, the prism 6, the grid with sight lines 7 (in the RPL-3 and RL these are three parallel dashes, and in the IRF-22 – a crosshair), scale 8 and through the eyepiece 9 enters to the observer’s eye.

Fig. 1.5. The principal scheme of beam paths in RPL-3, RL and IRF-22 refractometers (the top one is for RPL, RL, the bottom one is for IRF-22): 1 – illuminator (mirror), 2 – illuminating prism, 3 – measuring prism, 4 – dispersion compensator, 5 – objective of the telescope, 6 – rotating prism, 7 – grid with sight lines, 8 – refractive index values scale, 9 – telescope eyepiece, 10 – a mirror for illuminating the scale, 11, 13 and 14 – a system of rotary prisms, 12 – a micro-objective lens for focusing the scale 8 on a grid with sight lines 7, 15 – a protective glass


Figure 1.6. General view of the refractometer RPL-3 (RL): 1 – base, 2 – column, 3 – device case, 4 – dispersion compensator scale, 5 – the rotation screw of the compensator, 6 – the lower chamber with measuring prism, 7 – illuminator with red light filter (or mirror for RL refractometer), 8 – the upper chamber with a lighting prism, 9 – the instrument scale, 10 – a handle designed to combine the grid line with the light and shade boundary (move up and down), 11 – eyepiece

Dispersion compensator 4 is used to eliminate the spectral coloring of the border of light and shade. The compensator consists of Amici prism (a direct vision prism that transmits yellow rays with λ = 589.3 nm without deviation). In the IRF-22 refractometer, two compensating prisms form an optical system with variable dispersion. By turning the prisms around the direction of the beam, the correct position of the compensator is selected. Reading on the scale of the refractive index is made by combining the sighting line of the grid with the boundary of light and shade. Depending on the refractometer’s operation type, the viewfinder could be aligned with the interface between bright and dark fields in two ways: 1. In RPL-3 and RL refractometers it could be done by turning the telescope (together with the compensator) around the axis perpendicular to the plane of the scheme where the beam goes out of the measuring prism; 17

2. In IRF-22 by turning the measuring head together with the device scale relative to the same axis. Fig. 1.7 shows how in IRF-22 refractometer the scale 8 is illuminated by the mirror 10 and is projected by the micro-lens 12 through the system of prisms 11, 13 and 14 into the focal plane of the eyepiece 9. The refractometers measuring head is made as two chambers with an illuminating 2 and a measuring 3 prisms connected with a hinge to each other (the upper chamber could be slightly opened relative to the lower one). The chambers have internal channels with external fittings for the supply and removal of thermostatic liquid. The temperature of the test liquid is estimated using a thermometer installed in the mounting fitting. In both head chambers light beams are directed through the windows. In the case of colorless and weakly colored liquids, light is sent to the upper (illuminating) prism. When measuring the refractive indices of intensely colored liquids that strongly absorb light, a window of the lower (measuring) prism is used.

Figure 1.7. General view of the IRF-22: 1 – base of instrument, 2 – device case, 3 – mirror for illumination of investigated substance, 4 – illuminating prism chamber, 5 – measuring prism chamber, 6 – scale for estimating the average dispersion of the substance, 7 – handwheel for rotating prisms of the dispersion compensator, 8 – eyepiece of the instruments telescope, 9 – backlight mirror of the refractive index scale, 10 – handwheel, rotation of which tilts the measuring head to align the boundaries of light and shade with the grid intersection.


The RPL-2 and RL refractometers have a second scale with values of sugar concentration in percents, which is situated to the right of the scale with refractive indices. Naturally, this part of the total scale can be used when determining the concentration of a sugar solution. The limits of measurement of the refractive index nD for refractometers RPL-3, PL are 1.3000 to 1.5400 and for IRF-22 nD = 1.7000. The error margin of the refractive indices nD scale in repeated measurements is ± 2*10-4.

1.4. Work assignment and experimental technique 1.4.1. Read the specification and description of operation principle of the RPL-3 refractometer (IRF-22). 1.4.2. Prepare the measuring head of the refractometer following the next steps: a) open the upper chamber of the measuring head; b) rinse the planes of the upper and lower chambers with distilled water; c) remove traces of liquid by applying filter paper to the prism surface. Attention, do not rub the surface of the prisms! 1.4.3. Check the setup on the refractometers zero-point. To do this, by dropper or melted glass rod apply one or two drops of distilled water on the plane of the measuring prism and close the upper chamber of the measuring head. The eyepiece handle of the RPL-3 refractometer is turned down to the lower position and moved until the light and shade boundary appears in the field of view. The same effect is achieved in the IRF-22 refractometer by turning the corresponding handwheel 10 (Fig. 1.7). Observing the field of view, rotate the rim of the diopter tip of the eyepiece until a sharp image of scale division and the line of sight appears there. The most contrasting illumination of the field of view could be achieved by moving the illuminator up and down in front of the instrument input window (and in the IRF-22 refractometer, by changing the orientation of the mirror). Turning the scale sector 4, in 19

Fig. 1.6. (handwheel 7, in Fig. 1.7.) by rotating the prism of the dispersion compensator one can eliminate the color lightshade boundary. The reading of the refractive index on the scale is carried out by combining the sight line of the grid with the light and shade boundary. Correct setting of the device zero is when the point of the lightshade boundary at 20 °C corresponds to combination of the sight line of the grid with refractive index scale at nD = 1.33299. In case of readings deviation from this value, the difference between the measured and the given (nD = 1,33299) values should be taken into account in all subsequent measurements. 1.4.4. Measure the refractive indices of one-component and a number of two-component liquids with different concentrations according to the measurement technique similar to described in p. 1.4.3. 1.4.5. Based on the measurement data of the refractive index for one-component liquids according to p.1.4.4, calculate the polarizabi3 lity α and the effective radius of the molecules 𝑟 = √𝛼 using formula (1.13). 1.4.6. Check the Lorentz-Lorentz formula (1.15). Verification should be performed numerically and graphically by relation [( n 2 – 1 )/(n 2 + 2)] = f(δ) and using the measurement data from p. 1.4.3. 1.4.7. Measure the refractive index for a two-component solution with an unknown concentration. Determine the concentration value according to the diagram, obtained in p. 1.4.6. 1.4.8. Assess the accuracy of the refractive index measurements for two possible methods of illuminating the tested liquid layer through the upper and lower windows of the measuring head of the instrument. Questions for self-control 1. What are the absolute and relative refractive indexes of a substance? 2. What is the phenomenon of total internal reflection of light? 3. Explain the operation principle of the Abbe refractometer (the formation of a definite border between the bright and dark fields). 4. Explain the need and operation principle of the dispersion compensator in the refractometer. 5. What is meant by specific, atomic and molecular refraction?


Literature 1. Landsberg G.S. Optics. – M.: Nauka, 2003. 2. Saveliev I.V. The course of general physics. “Lan”, 2011. 3. Sivukhin D.V. General course of physics. Volume IV. Optics, 2005. 4. Akhmetov E.A., Sarsembinov Sh.Sh., Ronzhin V.V. Koshkimbayeva A.Sh. General physical practicum. Optics. – Almaty, 1999.



2.1. Aim of the work To learn the measuring method of the refractive index of a planeparallel plate by determining its thickness using an optical microscope. To compare the accuracy of plate thickness measurements made in using a microscope and micrometer. To analyze the causes of measurement errors.

2.2. Theoretical introduction The laws of reflection and refraction of electromagnetic (light) waves, as well as the nature of the phenomenon of light dispersion can be found in p.1.2.1 and p.1.2.2 of the description of the laboratory work No.1 of this tutorial. Here we will consider a narrower problem, which allows to understand the principle of the method for determining the refractive index of a glass plate using an optical microscope.

2.2.1. Refraction at a flat surface Let us show that in the general case, when a homocentric light beam is refracted at the flat interface between two transparent substances with different refractive indices (n1 and n2), it becomes astigmatic. The resulting beam image will be blurred (not clear). Fig. 2.1 shows a flat interface between two transparent media, which coincides with the yoz plane (the oz axis is perpendicular to the drawing plane). The refractive indices of the media are n1 and n2, respectively, and n1 > n2. Let us suppose that the point light source S is located on the ox axis, and the beam from it falls on the point A1 at the interface between two media. According to the law of light refraction 22

n1sin i1 = n2 sin i2,


where i1 is the angle of incidence, and i2 is the angle of refraction. We will continue the refracted beam А1В1 in the opposite direction until it intersects the axis ох at point S′. Let us denote coordinates of the points S and S′ on the x-axis, as х and х′ respectively, and, the coordinate of the point А1 on the y-axis as y.

Figure 2.1. Light refraction scheme at the flat interface between the media (beam astigmatism)

From right triangles SA1О and S′A1О we have: 𝑠𝑖𝑛𝑖1 = 𝑦/√𝑥 2 + 𝑦 2 and 𝑠𝑖𝑛𝑖2 = 𝑦/√𝑥′2 + 𝑦 2 .


Substituting expression (2.2) into (2.1) and solving the equation with respect to х′, we get that: 𝑛2


𝑥 ′ = (𝑛2 ) √𝑥 2 + [1 − (𝑛12 )] 𝑦 2 . 1



From expression (2.3) one can see that the direction of the SA1 beam (at a given position x of the light source S) affects the position of the point S′, i.e. depends on the location of the point A1. This means that different rays are not refracted in the same way as they pass from 23

one medium to another, and their extensions intersect the ох axis in different places as well. Let us consider the second SA2 beam, which forms a narrow homocentric light beam with the first beam SA1. After refraction, SA2 beam will propagate in A2B2 direction. The reverse continuation of the SA2 intersects the axis ox at the point S″. Reverse continuations of А1В1 and A2B2 rays intersect at the point S1. The other rays emanating from the point source S which lies between SA1 and SA2 rays, after their refraction will give rays, the reverse extensions of which will also intersect at point S1 and cross the ox axis between S′ and S″ points. Let us mentally rotate Fig. 2.1 around the ox axis at an angle dα in order to isolate a spatial rays beam emanating from the point source S within an infinitely narrow solid angle dΩ1. After refraction, this beam will turn into another beam with a solid angle dΩ2, the section of which by the xoy plane is determined by the lines A1B1 and A2B2. The reverse extensions of all the rays of the dΩ2 beam will cross the ox axis within the interval S′S″. The S′S ″ line is one of the focal lines of the astigmatic beam dΩ2. Another focal line will pass through the point S1. It will be a small part of the arc, which is formed when you rotate the pattern at an angle dα around the ox axis. Taking an infinitely small angle dα, instead of an arc, we obtain a small segment of a straight line belonging to the beam dΩ2 that intersects this segment. It will be a focal line. The focal line S′S″ lying in the drawing plane is called sagittal, and the focal line S1 perpendicular to the drawing plane is meridional line. When the angle of incidence of the light beam changes, the positions of both focal lines change. The geometric locations of the S1 points shown by the dotted line in fig. 2.2 takes the caustic form. Sagittal focal lines S′S″ are located along the ox axis from point S′ to point S″. In the case of a normal incidence (i1 = 0) of a narrow light beam emanating from point S onto the interface between media, both focal lines will emerge together into one point S′. In this case, the refracted light beam will remain homocentric. Putting y = 0 in the relation (2.3), we can find the coordinate of the point S′, then: x’=(n2/n1)x. (2.4) 24

The image of the light source S (object) in this case remains clear, since the astigmatism in such conditions is small. The condition when the angle of incidence tends to the critical one (from which the total internal reflection occurs) corresponds to the limiting positions of the focal lines corresponding to the points A and 0 (or C and 0 for the incidence angle i1 with the opposite sign).

Figure 2.2. Caustic formation scheme

2.2. Measuring the refractive index of a glass plate by means of microscope An optical density of a plane-parallel layer of a transparent substance (glass plate) is greater than of air. Therefore, the subject viewed through this layer seems closer to us. This follows from the analysis of formula (2.4). The apparent approximation of the object S′S is shown in fig. 2.2, where S′S = a. Let us estimate it and then it will be possible to calculate the refractive index nglass of the glass plate relative to air (the refractive index for air is taken as n2 = 1), according to formula (2.4) nglass=(n1/n2)=[h/(h−a)]=(h/hк), 25


where h = x is the true thickness of the plane-parallel glass plate; hk = х′ = h is its apparent thickness. The measurements should be carried out in the following order: a) the tested plate is placed on the microscope stage; b) first focus the microscope on the upper surface of the plate, and then on the bottom. Then the difference in the micrometric screw scale, with the help of which the microscope tube is moved during focusing, is equal to the apparent thickness of the plate hk. Measuring hk, a refractive index is calculated nglass according to the formula (2.5). For the convenience of assessing microscope focusing, fixed points are drawn in the form of ink streaks, on the upper and lower surfaces of the studied plate.

2.3. Microscope specification A microscope is an optical device designed to obtain significantly enlarged images of small objects. Optical and mechanical components of most microscopes of all types, except of highly specialized ones, in general terms are arranged in the same way. A general view and the principle optical scheme of the MBU-4 microscope are shown in fig. 2.3. The main parts of the optical system of a microscope are a short-focus lens and an eyepiece. They are located at the distance significantly greater than their focal lengths. The studied object usually is placed near the front focus of the objective lens. Behind the lens a real, inverse and magnified image of the object is formed. Looking into the eyepiece like into a magnifying glass, the observer sees this image even more enlarged, imaginary and direct. The microscope gives an image which is the opposite with respect to the object at the distance of the best (clear) view D = 25 cm. The microscope consists of the main mechanical units: the base 1, the tube holder 9, the tube 5, which includes the lens 4 and the eyepiece 6, as well as an object table 3, a mirror 2 with a fork-shaped holder for directing light on the studied object. The microscope is focused by moving the tube using the screws of coarse 7 and micrometric 8 mechanisms. The micrometric mechanism has a drum with a scale of 50 divisions. The division tick 26

is 0.002 mm. Attention! The micrometric mechanism has a locking device, which limits the motion of the tube within two strokes of the guiding line, or in 20-25 turns of the hand-wheel mechanism.



Figure 2.3. Schematic diagram of the light rays course in a microscope (a) and a general view of the microscope MBU-4 (b)

2.4. Work assignment 2.4.1. Study the specification of the microscope and the method of measuring the thickness of the glass plate using this microscope. 2.4.2. Using the microscope, measure the apparent thickness of the plate hk, as well as its actual thickness h. The number of measurements should be sufficient to estimate the error by an approximate statistical method. 27

2.4.3. Calculate the refractive index of the glass plate using formula (2.5). 2.4.4. Measure the plate thickness with a micrometer. 2.4.5. Compare the accuracy of the plate thickness measurements in accordance with S. 2.4.4 and S. 2.4.4. Questions for self-control 1. Formulate the basic laws of geometric optics. 2. Explain the course of the light rays in the microscope. 3. Why does the object appear closer when viewed through a flat glass plate or a layer of liquid? 4. Under what conditions formula (2.4) may be used? Literature 1. Landsberg G.S. Optics. – M.: Nauka, 2003. 2. Saveliev I.V. The course of general physics. “Lan”, 2011. 3. Akhmetov E.A., Sarsembinov Sh.Sh., Ronzhin V.V., Koshkimbayeva A.Sh. General physical practicum. Optics. – Almaty, 1999.



3.1. Aim of the work To learn the method of coherence of light sources according to the principle of division of the wave front. Qualitative assessment of the influence of the source (slit) size on the clarity of the interference pattern. Mastering the basic techniques of bringing the setup into a centered optical system by its adjustment. Mastering the measurement of linear parameters of the optical system using an optical micrometer. Mastering an interference method of measuring the lengths of light waves by passing them through light filters. Assessment of the consistency of accuracy of linear parameters measurements for various versions of experimental settings.

3.2. Brief theoretical introduction to the interference of light The phenomenon of energy redistribution in space occurring in the imposition of wave processes is called interference. What does this definition mean? Let us assume that in the points S1 and S2 the sources of some oscillations distant from the point of observation A at distances r1 and r2 are situated (Fig. 3.1). Waves propagating in space from the first and second excitation sources reach the observation point A at some point in time, and their equations can be written in the following form:

x1  a1 sin( t  kr1 )


x2  a2 sin( t  kr2 ) ,



where а1, а2 are the amplitudes of oscillations, k  2 is the wave  number, λ is the wavelength. At point A, the resulting oscillation x can be determined by jointly solving two wave equations (3.1) and (3.2). In general, it can be represented by the following expression:

x  a sin( t   ) .


Here a is the resulting amplitude of oscillation, which is related to the initial amplitudes by the ratio

a 2  a12  a22  2a1a2 cos 0 ,


where φо = k(r1 – r2) is the phase difference of the oscillations. The resulting oscillation phase φ can be determined from the ratio:

  arctg

a1 sin kr1  a2 sin kr2 . a1 cos kr1  a2 cos kr2

Figure 3.1. Wave formation pattern

The value of the resulting amplitude (eq. 3.4) depends on phase difference φо = k (r1 – r2) of the oscillations or the path difference ∆ = r2 – r1. In optics, when considering the light interference, the optical path difference is usually taken as ∆ = n2 r2 – n1 r1, where n1 and n2 are the refractive indices of media in which light propagates from their separation points to the imposition points. Waves are called coherent if during a long observation time the phase difference remains constant, i.e. φо = const. If this condition is 30

met, two features of the resulting oscillations are identified for consideration: 1) where the phase difference is φо = 0, 2π, 4π, …., 2mπ or    2m  m . When this condition is met, the resulting amplitude 2 is determined as a  a12  a22  2a1a2 and the energy of the resulting oscillation (intensity) will be equal to 2 J  J 1  J 2  2 J 1 J 2 > ( J 1  J 2 ) , because J ~ a .

It means that at the observation point A, the resultant oscillation intensity will increase. The resulting value of the energy J is greater than the sum of the energies J1 and J2 generated by each of the excitation sources separately at the initial points. 2) φо = π, 3π, 5 π, …., (2m+1)π or   (2m  1)  . In this case, 2 the resulting oscillation intensity will be equal to J  J1  J 2  2 J1 J 2 < (J1  J 2 ) ,

therefore, at the observation point A there is a decrease in intensity. These two features show that there is energy redistribution in space in the imposition of coherent waves. The maximum intensity (energy or amplitude) is observed in places where the path difference from two excitation points to the observation point is equal to an even number of half-waves (an integer number of waves):   2m

 2

 m


and the minimum intensity is observed in places where the path is equal to an odd number of half-waves:   (2m  1) 31

. 2


This is the phenomenon of wave interference. For incoherent waves, the phase difference φо changes many times during the observation period, passing all values from 0 to  . In the case of their imposition, the positions of the minima and maxima will shift in space very quickly. Therefore, it is not possible to observe even a short-term picture of interference, since the phase shift changes rapidly and the entire interference pattern is blurred. For incoherent waves at each point A, measurements of the amplitude of the resulting wave will give its average value during the observation time: a  a12  a22  2a1a2 cos 0 ,

where is the average value of the cosφ0 function during the observation time. The average value of the cosφ0 function for one period is = 0. Consequently, the resulting intensity J at the observation point will be equal to the sum of the intensities generated by each of the wave excitation sources J = J1 + J2, i.e. there is a simple or independent imposition of waves, otherwise known as superposition of waves. Observation of the light interference from two separate natural radiation sources is impossible, since waves coming from them will not be coherent. This is due to the following fact. It is known, that atoms of an excited substance in one radiation event, lasting about ~ 10-8 s, emit energy in the form of waves. Then it can begin to re-emit energy, but the initial phase of the new waves will be different. Therefore, in separate emission acts, the same atom does not produce coherent waves, and two independent radiating bodies consist of a mass of atoms (for example, the filament of incandescent bulbs) which will never produce coherent waves. In practice, coherent light waves are obtained in the following way. Radiation from one source is divided into two fluxes, and then, after passing through various paths, they are forced to meet again. This ensures that all elementary radiation events occurring in the main emission source will simultaneously be repeated in both fluxes, i.e. such light fluxes will be coherent. As a result, the path difference 32

between the interfering waves has a value that lies within the propagation of a single emission event wave. Two methods are used to obtain coherent waves in optics: a) by dividing the amplitudes of oscillations (interference in thin films), for example, Newton's rings; b) by dividing the wave front, for example, using the Fresnel biprism, in the Jung’s experiment, etc.. The details of specific ways for receiving coherent waves will be considered in the course of performance of corresponding laboratory works.

3.3. Brief theory of the method Fresnel biprism is an optical component in the form of two prisms with folded bases, which have a small refractive angle (about half a degree). Usually a biprism is made from a single piece of glass. Let us consider the course of light rays through a biprism (Fig. 3.2). Let a point monochromatic light source be located at point S in the form of a narrow infinite slit parallel to the refracting edge of the biprism (perpendicular to the plane of the drawing). Then the incident on a biprism wave front of the radiation will be divided into two light beams. According to the law of refraction in geometrical optics, each of the beams behind the biprism will deviate to the optical axis of the SO system. Therefore, these light beams are imposed on each other when exiting a biprism, and they seem to come from imaginary sources S1 and S2.

Figure 3.2. Schematic diagram of the light rays’ course through the biprism


Sources of light waves S1 and S2 are the images of the same radiation source S. Therefore, both of these beams are coherent. At any point in the beams imposition, one can observe the interference effect, which depends on the path difference. In Fig. 3.2, the interference observation area is shown shaded. Let us define the relationship between the geometry of the system and the local characteristic of the interference pattern. To do this, we will use the drawing shown in Fig. 3.3. Let us denote by S1 and S2 imaginary light sources obtained in rays passing through the biprism, and by ВВ1 the screen on which the interference pattern is observed. The distance between imaginary sources is assumed to be t = S1S2, and the distance from sources S1 and S2 to the screen is AB = l and the segment on the screen from the optical axis of the system to the observation point is BB1= ym. If the path difference ∆ = S2В1 – S1В1 = mλ, then at the point B1 there will be a maximum illumination (light line perpendicular to the plane of the drawing or parallel to the refracting edge of the biprism). Based on the similarity of the triangles S1S2С and АВВ1, it will be possible to write y down that S 2C  BB1 , or knowing that l >> t then   m . S1C AB t l Then the m-th maximum will be separated from point B at the distance ym  l . The distance between two adjacent maxima is t determined from the expression:

y  y m1  y m 

l . t


Figure 3.3. The diagram for calculating the localization of interference lines


Equation (3.7) is the main calculation formula for determining the wavelength of a radiant flux using a biprism, since δy, l and t values can be directly measured at the appropriate optical setup. However, it remains to clarify the question of measuring the distance between imaginary sources S1S2 and the distance from imaginary sources to the screen S1В1 and S2В1 (observation point). As it will be shown later, there is nothing complicated. Fig. 3.4 shows the layout of optical parts for calculating the distance between imaginary light sources t = S1S2. At the point O between the biprism and the screen (in the experiment, an optical micrometer) a lens the power of which is about 5 diopters is placed. By moving the lens to the right and left along the system’s optical axis, one can achieve an appearance of a clear image of two slits S1 and S 2 in the focal plane of the optical micrometer (on the screen). Depending on the type of the laboratory setup, we can get both enlarged and reduced images of slits S1 and S 2 slits. In our experimental setup, due to its optical-geometrical parameters, as well as the possible greatest measurement accuracy, only a reduced image of the slits can be obtained.

Figure 3.4. Scheme of optical parts placing to calculate the distance between imaginary light sources

The distance between the images of the two slits t   S1S2 can be measured using an optical micrometer. Let us additionally measure the size of the segments between the slit and lens a, as well as between the lens and the optical micrometer a'. From the similarity of the triangles ∆S1ОS2 and S1OS2 the desired distance t = S1S2 can be found: 35

t  t

a . a


Strictly, the actual source of radiation (slit) S and the imaginary light sources S1 and S2 do not lie in the same plane perpendicular to the optical axis. However, such a displacement at small refractive angles of the biprism is not large compared to the distance between the source and the screen l. Therefore, the distance between the source (slit) and the screen (an ocular micrometer in experimental setup) is taken as the value of l. Combining equations (3.7) and (3.8), we obtain the final calculation formula for finding the radiation wavelength after passing through the light filter:


at  y . al


3.3. Description of the laboratory setup Fig. 3.5 shows all the units of the laboratory setup assembled on an optical bench 9, where the following parts are fixed with the help of raters 8: illuminator 1 with a light filter 2 (in recent setups, the revolving head with replaceable light filters is fixed in front of the optical micrometer (along the beam)), slit 3, biprism 5, lens 6 and optical micrometer 7. It should be noted that the lens 6 is set on the optical bench only when the measurement of the distance between imaginary sources of coherent waves is conducted. All raters are equipped with locking screws. The rater fixing on the optical bench is carried out using two lower ones. The upper screw is used to secure the rack of the individual components of the system (slit, biprism, etc.). By loosening the bottom screws, one can freely move the raters along the optical bench. The scale 10, installed near the base of the optical bench, is used to measure the distances between the various units of the setup (for example, between the slit and optical micrometer). 36

Figure 3.5. General view of the setup for determining the wavelength using the Fresnel biprism: 1 – lantern illuminator OI-19, 2 – light filter, 3 – slit, 4 – screw for adjusting the slit width, 5 – biprism in the rim, 6 – lens in the rim, 7 – ocular micrometer, 8 – rater, 9 – optical bench, 10 – scale, 11 – toggle switch, 12 – rheostat handle, 13 – step-down transformer

The lighter OI-19 is used as a light source. It consists of a lantern with a light bulb (8 V, 20 W) and a two-lens collector with an iris diaphragm. The light bulb of the illuminator is connected to the electrical network through a step-down transformer 13. In the transformer case there is a rheostat with a handle 12, for adjusting the heat of the light bulb. Toggle 11 switches the current. Slit 3 of the setup is sliding. The slit width can be changed by rotating the handwheel 4. Biprism 5 is fixed in the frame. Together with the rim, it can be rotated around the direction perpendicular to the rim plane (around the horizontal axis). The slit provides the same kind of movement. The interference pattern in the device is observed in the focal plane of the ocular micrometer, with the help of which the Figure 3.6. General view of values of δу and t 'can be measured. the MOV-1-15x ocular Mentioned values are included in the micrometer: 1 – eyepiece calculation formula (3.9). There the screw corolla, 2 – micrometric screw ocular micrometer MOV-1-15х (Fig. 3.6) drum, 3 – micrometer clamp, 4 – locking screw is used. 37

The main element of the MOV-1-15x ocular micrometer is a 15-fold compensation eyepiece 1 with a diopter pickup and readout device. The ocular micrometer is fixed on the rack with a clamp 3 by means of a fixing screw 4. A clear image of the reading device is obtained by turning the eyepiece corolla 1. Limits of aiming ± 5 diopters. The reading device is located in the focal plane of the eyepiece. It consists of a fixed scale with the graduation interval of 1 mm and a movable grid with the crosshairs and an index in the form of two scratches at the level of the fixed scale against the crosshairs (Fig. 3.7). The interference pattern and the image from imaginary sources, as well as the readout device, are also located in the focal plane of the eyepiece. Therefore, in the field of view of the adjusted eyepiece, the picture of the object of observation (interference pattern and image from imaginary sources) and the readout device are clearly visible. The movable grid of the reading device in the form of the crosshairs is connected with an exact micrometer screw in such a way that, as the corrugated drum of the screw rotates, it and the risks jointly move in the field of view of the eyepiece relative to the main fixed scale. Scratches and the crosshairs in the eyepiece field of view are moved to one division of the main fixed scale, when Figure 3.7. View of the the drum of the micrometric screw 2 readout device in the field х (Fig. 3.6) is rotated one full turn (the of view of the MOV-1-15 ocular micrometer division value of the main scale is 1 mm). The screw drum is divided into 100 parts, the step of their division is 0.01 mm (since the step of division of the main scale is 1 mm). A complete calculation with an ocular micrometer is done by adding the readings on a fixed scale and readings on the screw drum. The position of scratches in the eyepiece field of view relative to the zero of the fixed scale indicates an integer number of counting. The fractions of readings are determined by the scale division of the drum, which falls against the index of the fixed nozzle of the screw. 38

3.5. Preparing setup for measurements The installation is considered to be adjusted, if all elements of the optical system have a common optical axis parallel to the optical bench (centered system). To do this, perform the following actions: the illuminator, the slit and the eyepiece should be fixed on the optical bench with the help of raters. The slit should be oriented in the horizontal direction. By means of the ruler or tapering the eyepiece close to the slit, the same height of the slit and the input window of the eyepiece with respect to the optical bench are selected. Then place the eyepiece and the slit at the maximum possible distance. Next, turn on the lamp of the fixture and check the light beam entering into the eyepiece with a sheet of white paper. In the case of a light beam passing above the eyepiece, the illuminator is raised relative to the optical bench. If it comes from below, then the illuminator is lowered until a ray of light hits the input window of the eyepiece. The installation of a white target disk on the input window of the eyepiece is very useful for alignment. After this, a biprism is installed on the optical bench between the slit and the eyepiece (approximately in the middle). Further, observing the interference fringes through the eyepiece, select the height of the biprism so that the fringes are located in the middle of the field of view of the eyepiece. Achieve the clearest interference pattern by adjusting the slit width and orientation of the biprism. The brightness of the illuminator lamp filament is also selected to provide a clear picture of the studied phenomenon. The brightness should not be excessively high in order to avoid injury of observer's eye. The accuracy of the adjustment is finally checked as follows: watching through the eyepiece, move the biprism with the rider along the optical bench towards yourself. The installation is adjusted if the interference pattern remains in the center of the field of view. It often happens that when a biprism moves, the pattern of interference smoothly moves up or down. In this case, return the biprism to its original position and slightly reduce the height of its setting and move it to yourself along the optical bench again. If the 39

movement of the interference pattern in the field of view has decreased in comparison with the first attempt, it means that we are going right and that we should lower the biprism a little. To avoid moving of the interference pattern beyond the limits of the field of view, we should also lower below the eyepiece. If, in repeated testing, the magnitude of the displacement of the pattern of interference did not decrease, but increased, then you should not lower the biprism, but put it somewhat higher. These procedures are performed until the interference pattern remains in the center of the eyepiece field of view in the course of movement of the biprism along the optical bench. This corresponds to the fact that the central or zero maximum of interference all the time remains within the average scratches of the ocular micrometer’s main scale. In this case, the installation is considered sufficiently prepared for measurements.

3.6. Work assignment On the prepared installation, the measuring part of work is performed in the following order: a) install one of the light filters on the light beam course. By means of red, green and blue light filters change the transmittable range of light wavelengths; b) place the lens between the biprism and the eyepiece micrometer in such a position that it would be possible to obtain the image of two coherent radiation sources (two slits) in the focal plane of the eyepiece. Next, measure the distance between the image of imaginary sources t'. To do this, keeping observation through the eyepiece, rotate the drum of the micrometric screw and bring the crosshair of the eyepiece down until it coincides with the image of the first selected slit. Then do the counting on the scale of the ocular micrometer. Then, continuing to observe in the eyepiece, rotate the drum in the same direction and bring the center of the crosshair to the image of the second slit. Repeat counting on micrometer scale. The difference between these two counting data is the value of t'; 40

c) without changing the position of the optical nodes of the installation, measure the distances between the lens and the slit – a, between the lens and the focal plane of the eyepiece – a', and the distance between the source (slit) and the eyepiece – l. It is convenient to perform the measurement of the distances a, a ', and l using a rightangled triangle. To do this, tightly apply a triangle with one cathetus to the scale near the base of the optical bench. The other cathetus must be directed to some optical assembly of the device (the plane of the slit, the main plane of the thin lens, the focal plane of the ocular micrometer). Against to the right angle of the triangle (or acute angle), the counting is taken on a scale 10 for the corresponding node (Fig. 3.5). The desired distance between the positions of the various optical units of the installation will give the difference between such two counting. d) Having determined the values of а, а', l by the indicated method, remove the lens from the optical bench. An interference pattern will appear in the ocular micrometer’s field of view. Using an ocular micrometer, measure the distance δу between two adjacent interference bands. It should be noted, that in order to increase the accuracy of measuring δу, first the width of dark stripes k is measured, and then the distance between the adjacent bands is calculated. Substitute all found values of δу, t ', a and a' into equation (3.9) and determine the desired wavelength of light. The number of measurements carried out for δу, t ', a, a' should be sufficient for estimating errors by an approximate statistical method. Questions for self-control 1. What is the basis of interference? 2. Why is the interference pattern observed only with a small distance between coherent sources and a limited path difference? 3. What color will be the zero (central) interference maximum created by the imposition of two coherent waves from white light sources? 4. What view will the interference pattern have when the light filter is removed? 5. How will the interference pattern change if the refractive angle of the biprism is increased?


6. Why is the horizontal orientation of the gap more efficient in a laboratory installation? 7. What sign indicates the correct alignment of the installation? Literature 1. Landsberg G.S. Optics. – M.: Science, 2003. 2. Godzhaev N.M. Optics. – M.: High School, 1977. 3. Akhmetov E.A., Sarsembinov Sh.Sh., Ronzhin V.V., Koshkimbaeva, A.Sh. General Physical Workshop. Optics. – Almaty, 1999.



4.1. Aim of the work Mastering the method of implementing the coherence of light sources according to the principle of dividing the amplitude (thin films, bands of equal thickness). Obtaining skills of working with a microscope: assessing the linear magnification of image obtained with optical device, determining division value of eyepiece micrometer and measuring the linear dimensions of an object with it. Determination of the wavelength of light, transmitted through the light filter by interference method. Mastering the method of measuring the radius of curvature of the lenses using Newton's rings.

4.2. Brief theory of the method Newton's rings are a variation of the interference phenomenon in thin films, which is also called bands of equal thickness. A setup for observing Newton's rings allows obtaining coherent light beams by dividing amplitudes of light vector oscillations. The division principle can be represented as follows: a small area is singled out at the front of the light wave incident on a thin plate. In the case of a reflected beam belonging to this section of front from the upper and lower medium interfaces at the thin plate, energy is divided into two parts. Two beams resulting from the energy separation, will have smaller amplitudes of oscillations than in the incident beam. When two beams are brought together at the same point, after passing through different ways, they will give one or another interference pattern depending on the difference in path difference between them. Let us consider in more detail the method of Newton’s rings obtaining. Let there be plane-convex lens of a small curvature and placed on plane-parallel glass plate (Fig. 4.1). Between the lens and 43

plate there is an air wedge, which gradually thickens from the lens center to the edges of the air layer. Usually, there is no optical contact between the lens and the plate throughout the air wedge. Although there is an air layer with the smallest thickness (OO'n2>n3; б) n1