Laboratory work No. 43

Section 5.Optics

Topic 5.2.Wave properties of light

Lab title: Determining the wavelength of light using a diffraction grating

Learning objective: obtain a diffraction spectrum, determine the wavelengths of light of different colors

Learning Objectives: observe the interference pattern, obtain first- and second-order spectra, determine the visible boundaries of the spectrum of violet light and red light, and calculate their wavelengths.

Safety regulations: rules for conducting a practical lesson in the office

Standard time: 2 hours

Educational results declared in the Federal State Educational Standard of the third generation:

The student must

be able to: measure the wavelength of light, draw conclusions based on experimental data

know: diffraction grating structure, grating period, conditions for the formation of maxima

Occupation availability

Guidelines for performing laboratory exercises

Laboratory notebook, pencil, ruler, device for determining the wavelength of light, stand for the device, diffraction grating, light source.

Procedure for conducting the lesson: individual work

Theoretical background

A parallel beam of light, passing through a diffraction grating, due to diffraction behind the grating, propagates in all possible directions and interferes. An interference pattern can be observed on a screen placed in the path of interfering light. Light maxima are observed at points on the screen. For which the condition is met:  = n (1)

 - wave path difference;  - light wavelength, n – maximum number. The central maximum is called zero: for it  = 0. To the left and right of it are maxima of higher orders.

The condition for the occurrence of a maximum (1) can be written differently: n = dSin

Picture 1

Here d is the period of the diffraction grating,  is the angle at which

light maximum (diffraction angle). Since the diffraction angles are small, then for them we can take Sin  = tan , and tan  = a/b Figure 1, therefore n = dA/b (2)

This formula is used to determine the wavelength of light.

As a result of measurements, it was found that for red light λcr = 8 10-7 m, and for violet light - λph = 4 10-7 m.

There are no colors in nature, there are only waves of different wavelengths

Analysis of formula (1) shows that the position of the light maxima depends on the wavelength of monochromatic light: the longer the wavelength. The further the maximum is from zero.

White light is complex in composition. The zero maximum for it is a white stripe, and the maxima of higher orders are a set of colored

bands, the totality of which is called the spectrum  and  Figure 2

Figure 2

The device consists of a bar with a scale 1, a rod 2, a screw 3 (the bar can be adjusted at different angles). Along the bar in the side grooves, you can move the slider 4 with the screen 5. A frame 6 is attached to the end of the bar, into which a diffraction grating is inserted, Figure 3

Figure 4

Figure 3 diffraction grating

Diffraction grating decomposes light into a spectrum and allows you to accurately determine the wavelengths of light

Figure 5

Work order

Assemble the installation, Figure 6

Install a light source and turn it on.

Looking through the diffraction grating, point the device at the lamp so that the lamp filament is visible through the window of the device screen

Install the screen at the greatest possible distance from the diffraction grating.

Measure the distance b from the instrument screen to the diffraction grating using the bar scale.

Determine the distance from the zero division (0) of the screen scale to the middle of the violet stripe both on the left “a l” and on the right “a p” for spectra of order , Figure 4 and calculate the average value, a sr

Repeat the experiment with a spectrum of  order.

Perform the same measurements for the red bands of the diffraction spectrum.

Using formula (2), calculate the wavelength of violet light for spectra of  and  orders, the wavelength of red light of  and  orders.

Enter the results of measurements and calculations into table 1

Draw a conclusion

Table No. 1

Questions to reinforce theoretical material for the laboratory lesson

Why is the zero maximum of the diffraction spectrum of white light a white stripe, and the maximum of higher orders a set of colored stripes?

Why are the maxima located both to the left and to the right of the zero maximum?

At what points on the screen are , ,  maxima obtained?

What is the appearance of the interference pattern in the case of monochromatic light?

At what points on the screen is the light minimum obtained?

What is the difference in the path of light radiation ( = 0.49 µm), giving the 2nd maximum in the diffraction spectrum? Determine the frequency of this radiation

Diffraction grating and its parameters.

Definitions of interference and diffraction of light.

Conditions for maximum light from a diffraction grating.

At the end of the practical work, the student must submit:- Completed laboratory work in accordance with the above requirements.
Bibliography:

V. F. Dmitrieva Physics for professions and technical specialties M.: Publishing House Academy - 2016

R. A. Dondukova Guide to conducting laboratory work in physics for secondary vocational education M.: Higher school, 2000

Laboratory work in physics with questions and assignments

O. M. Tarasov M.: FORUM-INFA-M, 2015

DETERMINING THE WAVELENGTH OF LIGHTBY USING

DIFFRACTION GRATING

GOAL OF THE WORK: Determine the wavelength of red and violet light.

EQUIPMENT: 1. A device for determining the wavelength of light,

2. light source, 3. diffraction grating.

THEORY: A parallel beam of light, passing through a diffraction grating, due to diffraction behind the grating, propagates in all possible directions and interferes. An interference pattern can be observed on a screen placed in the path of interfering light. Light maxima are observed at points on the screen for which the following condition is met:  =n, where D is the wave path difference,n– maximum number,l- light wavelength. The central maximum is called zero; for it  = 0. To the left and right of it are maxima of higher orders.

Diffraction Screen

lattice

The condition for the occurrence of a maximum can be written differently:

n = dsin

Whered– period of the diffraction grating,j– the angle at which the light maximum is visible (diffraction angle).

Since diffraction angles are, as a rule, small, for them we can take

sin  = tan ,Atan  = a/b

Therefore n×l = d×a/b

White light is complex in composition. The zero maximum for it is a white stripe, and the maximum of higher orders is a set of seven colored stripes, the totality of which is called the spectrum, respectively 1 th , 2 th , ... order, and the longer the wavelength, the further the maximum is from zero.

The diffraction spectrum can be obtained using a device to determine the wavelength of light.

ORDER OF WORK:

Place the lamp on the demonstration table and turn it on.

Looking through the diffraction grating, point the device at the lamp so that the lamp filament is visible through the window of the device screen.

Install the instrument screen at a distance of 400 mm from the diffraction grating and obtain a clear image of the spectra on it 1 th and 2 th orders of magnitude.

Determine the distance from the zero division “0” of the screen scale to the middle of the purple stripe, as to the left side “a” l ", and to the right "a P ", for first order spectra and calculate the average value "a sr.f »

A sr.f1 = (a l + a P ) / 2

diffraction grating

screen

Repeat the experiment with a second order spectrum. Determine a for him sr.f2

Perform the same measurements for the red bands of the diffraction spectrum.

Calculate the wavelength of violet light, the wavelength of red light (for 1 th and 2 th orders) according to the formula:

= ,

Whered = 10 -5 m – constant (period) lattice,

n – spectrum order,

b– distance from the diffraction grating to the screen, mm

8. Determine the average values:

λ f = ; λ cr =

9. Determine measurement errors:

absolute –Δ λ f = |λ sr.f. - λ tab.f. | ; Whereλ tab.f = 0.4 µm

– Δ λ cr = |λ Wed cr. - λ tab.cr. | ; Whereλ tab.cr = 0.76 µm

relative –δ λ f = %; δ λ cr = %

10. Prepare a report. Enter the results of measurements and calculations into the table.

spectrum

spectrum edge

violet. colors

spectrum edge

red colors

light wavelength

op.

« A l »,

mm

« A P »,

mm

« A Wed »

mm

« A l »,

mm

« A P »,

mm

« A Wed »

mm

f ,

cr ,

11. Draw a conclusion.

CONTROL QUESTIONS:

1. What is light diffraction?

   What is a diffraction grating?

   At what points on the screen are the 1st, 2nd, 3rd maximums obtained? How do they look?

   Determine the diffraction grating constant if, when illuminated with light with a wavelength of 600 nm, the second-order maximum is visible at an angle of 7

   Determine the wavelength if the first-order maximum is 36 mm from the zero maximum, and a diffraction grating with a constant of 0.01 mm is located at a distance of 500 mm from the screen.

   Determine the wavelength incident on a diffraction grating with 400 lines on each millimeter. The diffraction grating c is located at a distance of 25 cm from the screen, the third-order maximum is 27.4 cm away from the zero maximum.

USING NEWTON'S RINGS

Goal of the work: observe experimentally the interference of light in a thin film (in the air layer between the lens and the plate) in the form of Newton’s rings and determine the wavelength of light using Newton’s rings.

Devices and accessories: a plano-convex lens placed with its convex side on a plane-parallel plate and fixed to it; microscope; Light source; ruler with millimeter scale.

Note: the theory of the method and a description of the installation are given in work No. 2.

1. Determining the division value of the ocular scale

Note: the task is performed in the same way as in work No. 2.

2.Determining the wavelength of light

The diameter of Newton's ring can be directly measured in ocular scale divisions. Multiplying this result by the quantity b, expressed in mm/div., we get the diameter in mm.

Radii i th and n -th dark rings in accordance with formula (2.5)

r t, i = ,r t, n = , (3.1)

Squaring these expressions and subtracting one from the other, we get

. (3.2)

Formula (3.2) is also valid for light rings. Since the center of the ring is established with a large error, in the experiment it is not the radius that is measured, but the diameter of the ring D . Then formula (3.2) takes the form

, (3.3)

where we get the formula for calculating the wavelength of light

. (3.4)

The radius of the lens is given in table. 3.1, the lens number is indicated on the lens holder. To simplify calculations, we denote the value by T . Then

l = . (3.5)

Table 3.1

Completing of the work

2.1. See paragraph 2.1 in work No. 2.

2.2. See paragraph 2.2 in work No. 2.

2.3 See paragraph 2.3 in work No. 2.

2.4. Using formula (3.5), determine < l>.

,

Where D T find using a formula similar to formula (2.7).

2.6. Enter the results of measurements and calculations into the table. 3.2. Record the final result as a confidence interval indicating the reliability and relative error.

Table 3.2

CONTROL QUESTIONS

1. The phenomenon of light interference.

2. Coherence.

3. Optical path length and optical path difference.

4. Conditions for maxima and minima during interference.

5. Phenomena that occur during reflection:

a) from a medium that is optically denser;

b) from a medium that is optically less dense.

6. Lines of equal thickness. Newton's rings.

7. Derivation of the calculation formula.

8. Progress of an experiment to determine the radius of curvature of a lens or the wavelength of light using Newton’s rings.

9. Calculation of measurement errors.

LABORATORY WORK No. 4

DETERMINING THE WAVELENGTH OF LIGHT

USING A DIFFRACTION GRATING

Goal of the work: determine the characteristics of a diffraction grating; measure the wavelength of light using a diffraction grating.

Devices and accessories: experimental setup, diffraction grating.

Information from theory

Diffraction light refers to phenomena caused by a violation of the integrity of the wave surface. Diffraction manifests itself in a violation of the straightness of propagation of vibrations. The wave goes around the edges of the obstacle and penetrates the geometric shadow area. Diffraction phenomena are inherent in all wave processes, but appear especially clearly only in cases where the wavelengths of the radiation are comparable to the size of the obstacles.

From the point of view of the concepts of geometric optics about the rectilinear propagation of light, the boundary of the shadow behind an opaque obstacle is sharply outlined by rays that pass by the obstacle, touching its surface. Consequently, the phenomenon of diffraction is inexplicable from the standpoint of geometric optics. According to Huygens' wave theory, which considers each point of the wave field as a source of secondary waves propagating in all directions, including into the region of the geometric shadow of an obstacle, the appearance of any distinct shadow is generally inexplicable. Nevertheless, experience convinces us of the existence of a shadow, but not a sharply defined one, as the theory of rectilinear propagation of light states, but with blurred edges.

Huygens-Fresnel principle

The peculiarity of diffraction effects is that the diffraction pattern at each point in space is the result of the interference of rays from a large number of secondary Huygens sources. The explanation of these effects was carried out by Fresnel and was called the Huygens-Fresnel principle.

The essence of the Huygens-Fresnel principle can be represented in the form of several provisions:

1. The entire wave surface excited by any source S 0 area S , can be divided into small areas with equal areas dS , which are a system of secondary sources emitting secondary waves.

2. These are secondary sources equivalent to the same primary source S 0 , are coherent. Therefore, waves propagating from the source S 0 , at any point in space must be the result of the interference of all secondary waves.

3. The radiation powers of all secondary sources - sections of the wave surface with the same areas - are the same.

4. Each secondary source with area dS emits predominantly in the direction of the outer normal n to the wave surface at this point; amplitude of secondary waves in a direction of s n corner a, the smaller, the larger the angle a, and is equal to zero at a³p/2.

5. The amplitude of secondary waves that have reached a given point in space depends on the distance of the secondary source to this point: the greater the distance, the smaller the amplitude.

The Huygens-Fresnel principle makes it possible to explain the phenomenon of diffraction and provide methods for its quantitative calculation.

Fresnel zone method

The Huygens-Fresnel principle explains the straightness of light propagation in a homogeneous medium free of obstacles. To show this, consider the action of a spherical light wave from a point source S 0 at an arbitrary point in space P (Fig. 4.1). The wave surface of such a wave is symmetrical relative to a straight line S 0 P . Amplitude of the desired wave at a point P depends on the result of the interference of secondary waves emitted by all sections dS surfaces S . The amplitudes and initial phases of secondary waves depend on the location of the corresponding sources dS relative to point P .

Fresnel proposed a method of dividing the wave surface into zones (Fresnel zone method). According to this method, the wave surface is divided into ring zones (Fig. 4.1), constructed so that the distances from the edges of each zone to the point P differ by l/2(l - wavelength of light). If we denote by b distance from the top of the wave surface 0 to the point P , then the distances b + k (l/2) form the boundaries of all zones where k - zone number. Vibrations coming to a point P from similar points of two adjacent zones are opposite in phase, since the path difference from these zones to the point P equal to l/2. Therefore, when superimposed, these oscillations mutually weaken each other, and the resulting amplitude will be expressed by the sum:

A = A 1 - A 2 +A 3 - A 4 + ... . (4.1)

Amplitude value A k depends on area D.S. k k th zone and angle a k between the outer normal to the surface of the zone at any point and the straight line directed from this point to point P .

It can be shown that the area D.S. k k th zone does not depend on the zone number in the conditions l<< b . Thus, in the considered approximation, the areas of all Fresnel zones are equal in size and the radiation power of all Fresnel zones - secondary sources - is the same. At the same time, with an increase k angle increases a k between the normal to the surface and the direction to the point P , which leads to a decrease in radiation intensity k th zone in a given direction, i.e. to a decrease in amplitude A k compared to the amplitudes of previous zones. Amplitude A k also decreases due to an increase in the distance from the zone to the point P with growth k . Eventually

A 1 > A 2 > A 3 > A 4 > ... > A k > ...

Due to the large number of zones, the decrease A k is monotonic in nature and we can approximately assume that

. (4.2)

Rewriting the resulting amplitude (4.1) in the form

we find that, according to (4.2) and taking into account the small amplitude of the remote zones, all expressions in brackets are equal to zero and equation (4.1) is reduced to the form

A = A 1 / 2. (4.4)

The result obtained means that the vibrations caused at the point P spherical wave surface, have an amplitude given by half of the central Fresnel zone. Therefore, the light from the source S 0 exactly P propagates within a very narrow direct channel, i.e. straight forward. As a result of the interference phenomenon, the effect of all zones except the first is destroyed.

Fresnel diffraction from simple obstacles

Action of a light wave at a certain point P reduces to the action of half of the central Fresnel zone if the wave is unlimited, since only then the actions of the remaining zones are mutually compensated and the action of the remote zones can be neglected. For a finite section of the wave, the diffraction conditions differ significantly from those described above. However, here too, the use of the Fresnel method makes it possible to predict and explain the features of the propagation of light waves.

Let's consider several examples of Fresnel diffraction from simple obstacles.

Diffraction by a circular hole . Let the wave from the source S 0 meets an opaque screen with a round hole on the way B.C. (Fig. 4.2). The result of diffraction is observed on the screen E , parallel to the plane of the hole. It is easy to determine the diffraction effect at a point P screen located opposite the center of the hole. To do this, it is enough to build waves on the open part of the front B.C. Fresnel zones corresponding to the point P . If in the hole B.C. fits k Fresnel zones, then the amplitude A resulting oscillations at a point P depends on whether the number is even or odd k , as well as on how large the absolute value of this number is. Indeed, from formula (4.1) it follows that at the point P amplitude of the total oscillation

(the first equation of the system for odd k , the second - when even) or, taking into account formula (4.2) and the fact that the amplitudes of two neighboring zones differ little in value and can be considered A k-1 approximately equal Ak, we have

where plus corresponds to an odd number of zones k , fitting on the hole, and the minus is even.

With a small number of zones k amplitude A k little different from A 1 . Then the result of diffraction at the point P depends on parity k : if odd k a diffraction maximum is observed, and a minimum is observed when the diffraction is even. Minimums and maximums will be more different from each other the closer they get A k To A 1 those. the less k . If the hole opens only the central Fresnel zone, the amplitude at the point P will be equal A 1 , it is twice as large as that which occurs with a completely open wave front (4.4), and the intensity in this case is four times greater than in the absence of an obstacle. On the contrary, with an unlimited increase in the number of zones k , amplitude A k tends to zero (A k<< A 1 ) and expression (4.5) turns into (4.4). In this case, light actually spreads in the same way as in the absence of a screen with a hole, i.e. straight forward. This leads to the conclusion that the consequences of wave concepts and concepts of rectilinear propagation of light begin to coincide when the number of open zones is large.

Oscillations from even and odd Fresnel zones mutually weaken each other. This sometimes leads to an increase in light intensity when part of the wave front is covered by an opaque screen, as was the case with an obstacle with a round hole on which only one Fresnel zone is placed. The light intensity can be increased many times by making a complex screen - the so-called zone plate (a glass plate with an opaque coating), which covers all even (or odd) Fresnel zones. The zone plate acts like a converging lens. Indeed, if the zone plate covers all even zones, and the number of zones k = 2m , then from (4.1) it follows

A = A 1 + A 3 +...+ A 2m-1

or with a small number of zones, when A 2m-1 approximately equal A, A = mA 1 , i.e. intensity of light at a point P at 2 m ) 2 times more than with unhindered propagation of light from the source to the point P , wherein A = A 1 / 2, and intensity accordingly / 4 .

Diffraction by a circular disk. When placed between the source S 0 and a screen of a round opaque disk NE one or several first Fresnel zones closes (Fig. 4.3). If the disk closes k Fresnel zones, then at the point P sum wave amplitude

and, since the expressions in brackets can be taken equal to zero, similarly to (4.3) we obtain

A = A k +1 / 2. (4.6)

Thus, in the case of a round opaque disk in the center of the picture (point P ) for any (both even and odd) k it turns out to be a bright spot.

If the disk covers only part of the first Fresnel zone, there is no shadow on the screen, the illumination at all points is the same as in the absence of an obstacle. As the radius of the disk increases, the first open zone moves away from the point P and the angle increases a between the normal to the surface of this zone at any point and the direction of radiation towards the point P (see Huygens-Fresnel principle). Therefore, the intensity of the central maximum weakens as the disk size increases ( A k+1 << A 1 ). If the disk covers many Fresnel zones, the light intensity in the region of the geometric shadow is almost everywhere equal to zero and only near the observation boundaries there is a weak interference pattern. In this case, we can neglect the phenomenon of diffraction and use the law of rectilinear propagation of light.

Fraunhofer diffraction

(diffraction in parallel rays)

In the case of spherical waves, the result of diffraction depends on three parameters: the wavelength of the radiation emitted by the source S 0 , the geometry of the obstacle (dimensions of the slot, hole, etc.) and the distance from the obstacle to the observation screens. Under Fraunhofer diffraction conditions, a transition to plane waves occurs, which eliminates the dependence of the diffraction result on the third quantity (the distance from the obstacle to the observation screen), and the geometric dimensions of the obstacle can be taken into account in advance. In the case of a hole of unchanged shape and size, the result of diffraction depends only on changes in the spectral composition of the radiation given by the source S 0 . Therefore, diffraction phenomena in parallel beams can be used for spectral analysis of the composition of the radiation of the substances under study.

A schematic diagram of the observation of plane waves (Fraunhofer diffraction) is shown in Fig. 4.4.

Light from a point source S 0 turns into a lens L 1 into a beam of parallel rays (a plane wave), which then passes through a hole in an opaque screen (circle, slit, etc.). Lens L 2 collects at various points of its focal plane, where the observation screen is located E , all rays passing through the hole, including rays deviated from the original direction as a result of diffraction.

Diffraction from a single slit. In practice, the gap appears as a rectangular hole, the length of which is much greater than the width. In this case, the image of the point S 0 (Fig. 4.4) will stretch into a strip with minima and maxima in the direction perpendicular to the slit, because light diffracts to the right and left of the slit (Fig. 4.5). If we observe the image of the source in the direction perpendicular to the direction of the generating slit, then we can limit ourselves to considering the diffraction pattern in one dimension (along X ).

Since the slit plane coincides with the front of the incident wave, then, in accordance with the Huygens-Fresnel principle, the slit points are secondary sources of waves oscillating in the same phase.

Let us divide the area of ​​the slit into a number of narrow strips of equal width parallel to the generatrix of the slit. The phases of waves from different strips at the same distances are equal, the amplitudes are also equal, because the selected elements have equal areas and are equally inclined to the direction of observation.

If, when light passed through a slit, the law of rectilinear propagation of light was observed (there would be no diffraction), then on the screen E , installed in the focal plane of the lens L 2 , the image of a slit would be obtained. Therefore, the direction j = 0 defines an undiffracted wave with amplitude A 0 , equal to the amplitude of the wave sent by the entire slit.

Due to diffraction, light rays deviate from the rectilinear direction at an angle j. Deviation to the right and left is symmetrical relative to the center line OC 0 (Fig. 4.5). To find the action of the entire slit in the direction determined by the angle j, it is necessary to take into account the phase difference characterizing the waves reaching the observation point C j from various strips (Fresnel zones).

Let's draw a plane FD , perpendicular to the direction of the diffracted rays and representing the front of the new wave. Since the lens does not introduce an additional difference in the path of the rays, the path of all rays from the plane FD to the point C j is the same. Consequently, the total path difference of the rays from the slit F.E. is given by a segment ED . Let us draw planes parallel to the wave surface FD , so that they divide the segment ED into several sections, each of which has a length l/2 (Fig. 4.5). These planes will divide the gap into the above-mentioned strips - Fresnel zones, and the path difference from neighboring zones is equal to l/2 in accordance with the Fresnel method. Then the result of diffraction at the point C j will be determined by the number of Fresnel zones that fit into the slits (see Fresnel diffraction by a round hole): if the number of zones is even ( z = 2k ), at point C j diffraction minimum is observed if z- odd ( z = 2k + 1), at point C j- maximum diffraction. Number of Fresnel zones placed on the slits F.E. , is determined by how many times in the segment ED contained l/ 2 i.e. . Line segment ED , expressed in terms of slit width A and diffraction angle j, will be written as ED = a sin j .

As a result for the situation maximums diffraction we obtain the condition

a sin j = ±( 2k + 1)l / 2,(4.7)

For minimums diffraction

a sin j = ± 2 k l /2,(4.8)

Where k = 1,2,3.. - integers. Magnitude k , taking the values ​​of numbers in the natural series, is called the order of the diffraction maximum. The ± signs in formulas (4.7) and (4.8) correspond to light rays diffracting from the slit at angles + j And - j and converging at side focuses of the lens L 2 : C j And C- j, symmetrical relative to the main focus C 0 . In the direction j = 0 the most intense central maximum of zero order is observed.

The position of the diffraction maxima according to formula (4.7) corresponds to the angles

, , etc.

In Fig. Figure 4.6 shows the light intensity distribution curve as a function sin j. Position of the central maximum ( j = 0) does not depend on the wavelength and, therefore, is common to all wavelengths. Therefore, in the case of white light, the center of the diffraction pattern will appear as a white stripe. From Fig. 4.6 and formulas (4.7) and (4.8) it is clear that the position of the maxima and minima depends on the wavelength. Therefore, simple alternation of dark and light stripes occurs only in monochromatic light. In the case of white light, diffraction patterns for waves with different l shift according to wavelength. The central maximum of white color has a rainbow color only at the edges (one Fresnel zone fits within the width of the slit). The side maxima for different wavelengths no longer coincide with each other; closer to the center there are maxima corresponding to shorter waves. Long-wavelength maxima are further apart (j = arcsin l/2) than shortwave. Therefore, the diffraction maximum is a spectrum with the violet part facing the center.

Diffraction grating

A diffraction grating is a system of a large number of slits of equal width and parallel to each other, lying in the same plane and separated by opaque spaces equal in width. A diffraction grating is made by applying parallel lines to the surface of the glass. The number of lines per 1 mm is determined by the region of the spectrum of the radiation being studied and varies from 300 mm -1 in the infrared region to 1200 mm -1 in the ultraviolet.

Let the lattice consist of N parallel slots with the width of each slot a and the distance between adjacent slots b (Fig. 4.7). Sum a + b = d is called the period or constant of the diffraction grating. Let a plane monochromatic wave be incident normally on the grating. It is required to investigate the intensity of light propagating in a direction making an angle j with the normal to the lattice plane. In addition to the intensity distribution due to diffraction at each slit, there is a redistribution of light energy due to the interference of waves from N slits of coherent sources. In this case, the minima will be in the same places, because the condition for the minimum diffraction for all slits (Fig. 4.8) is the same. These minima are called major. Main minima condition a sin j = ± k l coincides with condition (4.8). Position of main minima sin j = ± l /a , 2l /a ,... shown in Fig. 4.8.

However, in the case of many slits, to the main minima created by each slit separately are added minima resulting from the interference of light passing through the various slits. In Fig. As an example, Fig. 4.8 shows the intensity distribution and the location of maxima and minima in the case of two slits with a period d and slot width a.

In the same direction, all slits emit vibration energy of the same amplitude. And the result of interference depends on the difference in the phases of oscillations emanating from similar points of neighboring slits (for example, C And E , B And F ), or from the optical path difference ED from similar points of two adjacent slits to the point C j. For all similar points this path difference is the same. If ED = ± k l or, since ED = dsi n j ,

d sin j = ± k l , k = 0,1,2..., (4.9)

vibrations of adjacent slits mutually reinforce each other, and at the point C j diffraction maximum is observed at the focal plane of the lens. The amplitude of the total oscillation at these points of the screen is maximum:

A max = N A j ,(4.10)

Where A j - amplitude of vibration sent by one slit at an angle j. Light intensity

J max = N 2 A j 2 = N 2 J j.(4.11)

Therefore, formula (4.9) determines the position of the main intensity maxima. Number k gives the order of the main maximum.

The position of the main maxima (4.9) is determined by the relation

. (4.12)

There is one zero-order maximum and is located at the point C 0 , maximums of the first, second, etc. orders of two and they are located symmetrically relative to C 0 , what the sign indicates + . In Fig. Figure 4.8 shows the position of the main maxima.

In addition to the main maxima, there are a large number of weaker secondary maxima, separated by additional minima. Side maxima are much weaker than the main ones. The calculation shows that the intensity of the side maxima does not exceed 1/23 of the intensity of the nearest main maximum.

At the main maxima the amplitude is N times, and the intensity is N 2 times the amplitude given at the corresponding location by one slit. Lines clearly localized in space with increased brightness are easily detected and can be used for spectroscopic studies.

As you move away from the center of the screen, the intensity of the diffraction maxima decreases (the distance from the sources increases). Therefore, it is not possible to observe all possible diffraction maxima. Note that the number of diffraction maxima produced by the grating on one side of the screen is determined by the condition ½ sin j½ £ 1 (j = p/ 2 - maximum diffraction angle), from where, taking into account (4.9)

At the same time, we should not forget that k - an integer.

The position of the main maxima depends on the wavelength l. Therefore, when the diffraction grating is illuminated with white light, all maxima except the central one ( k = 0), will be decomposed into a spectrum with the violet end facing the center of the diffraction pattern. Thus, a diffraction grating can serve to study the spectral composition of light, i.e. to determine the frequencies (or wavelengths) and intensities of all its monochromatic components. The instruments used for this are called diffraction spectrographs, if the spectrum under study is recorded using a photographic plate, and diffraction spectroscopes, if the spectrum is observed visually.

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Goal of the work. Study the phenomena of diffraction and interference of light waves, the use of these phenomena in medical and biological research. Learn to determine the wavelength of light using a diffraction grating.

Relevance. Interferometers, which are based on the phenomenon of light interference, are widely used in medicine; in particular, with the help of an interferometer, refractive indices can be determined with an accuracy of up to the sixth decimal place. Interference methods are used to determine the coefficients of linear and volumetric expansion, refractive indices of gases and vapors with a very high degree of accuracy. Instruments based on this principle are used to monitor the air composition in mines, mines, and industrial premises. The same method is used in medicine to study changes in blood composition in some difficult to recognize diseases. Using interferometers, wavelengths, short distances, and the quality of optical surfaces are determined with a high degree of accuracy.

The use of a diffraction grating in optical devices makes it possible to increase their resolution. Diffraction of monochromatic X-rays in polycrystalline bodies makes it possible to perform X-ray diffraction qualitative and quantitative analyses. Using this method, J. Watson and F. Crick established the structure of DNA (1962).

Since the conditions for reflection and absorption of electromagnetic waves by bodies depend, in particular, on the wavelength, this feature of holography allows it to be used as a method of intravision (introscopy).

Devices and accessories: diffraction grating, screen, ruler.

Theoretical part

Interference of light. Light interference is a phenomenon that occurs when light waves are superimposed and are accompanied by their strengthening or weakening. A stable interference pattern arises when coherent waves are superimposed. Coherent waves are waves with equal frequencies and identical phases or having a constant phase shift. The amplification of light waves during interference (maximum condition) occurs in the case where Δ contains an even number of half-wavelengths:

Where k – maximum order, k=0,±1,±2,±,…±n;

λ – light wavelength.

Attenuation of light waves during interference (minimum condition) is observed if the optical path difference Δ contains an odd number of half-wavelengths:

Where k – minimum order.

The optical difference in the path of two beams is the difference in distances from the sources to the observation point of the interference pattern.

Interference in thin films. Interference in thin films can be observed in soap bubbles, in a spot of kerosene on the surface of water when illuminated by sunlight.

Let beam 1 fall on the surface of a thin film (see Fig. 2). The beam, refracted at the air-film boundary, passes through the film, is reflected from its inner surface, approaches the outer surface of the film, is refracted at the film-air boundary and the beam comes out. We direct beam 2 to the exit point of the beam, which runs parallel to beam 1. Beam 2 is reflected from the surface of the film, superimposed on beam, and both beams interfere.

When the film is illuminated with polychromatic light, we get a rainbow picture. This is explained by the fact that the film is not uniform in thickness. Consequently, path differences of different magnitudes arise, which correspond to different wavelengths (colored soap films, iridescent colors of the wings of some insects and birds, films of oil or oils on the surface of water, etc.).

Light interference is used in devices called interferometers. Interferometers are optical devices that can be used to spatially separate two beams and create a certain path difference between them. Interferometers are used to determine wavelengths with a high degree of accuracy over short distances, refractive indices of substances, and determine the quality of optical surfaces.

For sanitary and hygienic purposes, the interferometer is used to determine the content of harmful gases.

The combination of an interferometer and a microscope (interference microscope) is used in biology to measure the refractive index, dry matter concentration, and thickness of transparent microobjects.

Huygens–Fresnel principle. According to Huygens, every point in the medium that the primary wave reaches at a given moment is a source of secondary waves. Fresnel clarified this position of Huygens, adding that secondary waves are coherent, i.e. when superimposed they will produce a stable interference pattern.

Diffraction of light. Diffraction of light is the phenomenon of deviation of light from rectilinear propagation.

Diffraction in parallel rays from a single slit. Let the target width V a parallel beam of monochromatic light falls (see Fig. 3):

A lens is installed in the path of the rays L , in the focal plane of which the screen is located E . Most rays do not diffract, i.e. do not change their direction, and they are focused by the lens L in the center of the screen, forming a central maximum or a zero-order maximum. Rays diffracting at equal diffraction angles φ , will form maximums 1,2,3,…, on the screen n – orders of magnitude.

Thus, the diffraction pattern obtained from one slit in parallel beams when illuminated with monochromatic light is a light stripe with maximum illumination in the center of the screen, then there is a dark stripe (minimum of the 1st order), then there is a light stripe (maximum of the 1st order order), dark band (2nd order minimum), 2nd order maximum, etc. The diffraction pattern is symmetrical relative to the central maximum. When the slit is illuminated with white light, a system of color stripes is formed on the screen; only the central maximum will retain the color of the incident light.

Conditions max And min diffraction. If in the optical path difference Δ an odd number of segments equal to , then an increase in light intensity is observed ( max diffraction):

Where k – order of maximum; k =±1,±2,±…,± n;

λ – wavelength.

If in the optical path difference Δ an even number of segments equal to , then a weakening of the light intensity is observed ( min diffraction):

Where k – minimum order.

Diffraction grating. A diffraction grating consists of alternating stripes that are opaque to the passage of light with stripes (slits) of equal width that are transparent to light.

The main characteristic of a diffraction grating is its period d . The period of the diffraction grating is the total width of the transparent and opaque stripes:

A diffraction grating is used in optical instruments to enhance the resolution of the device. The resolution of a diffraction grating depends on the order of the spectrum k and on the number of strokes N :

Where R - resolution.

Derivation of the diffraction grating formula. Let us direct two parallel beams to the diffraction grating: 1 and 2 so that the distance between them is equal to the grating period d .

At points A And IN rays 1 and 2 diffract, deviating from the rectilinear direction at an angle φ – diffraction angle.

Rays And focused by lens L onto the screen located in the focal plane of the lens (Fig. 5). Each grating slit can be considered as a source of secondary waves (Huygens–Fresnel principle). On the screen at point D we observe the maximum of the interference pattern.

From the point A on the beam path drop the perpendicular and get point C. consider the triangle ABC : right triangle, ÐBAC=Ðφ like angles with mutually perpendicular sides. From Δ ABC:

Where AB=d (by construction),

CB = Δ – optical path difference.

Since at point D we observe maximum interference, then

Determining the wavelength of light using a diffraction grating

1. DIFFRACTION OF LIGHT

Diffraction of light is the phenomenon of light bending around obstacles encountered in its path, accompanied by spatial redistribution of the light wave energy - interference.

Calculation of the light intensity distribution in the diffraction pattern can be carried out using the Huygens-Fresnel principle. According to this principle, each point on the front of a light wave, i.e., the surface to which light has propagated, is a source of secondary coherent light waves (their initial phases and frequencies are the same); the resulting oscillation at any point in space is caused by the interference of all secondary waves arriving at this point, taking into account their amplitudes and phases.

The position of the light wave front at any moment in time is determined by the envelope of all secondary waves; any deformation of the wave front (it is caused by the interaction of light with obstacles) leads to a deviation of the light wave from the original direction of propagation - the light penetrates into the region of the geometric shadow.

2. Diffraction grating

A transparent diffraction grating is a glass plate or celluloid film on which, at strictly defined distances, narrow rough grooves (strokes) that do not transmit light are cut with a special cutter. The sum of the width of the unbroken, transparent gap (slit) and the width of the groove is called the lattice constant or period.

Let a plane monochromatic light wave with a wavelength fall on the grating (let us consider the simplest case - the normal incidence of the wave on the grating). Each point of the transparent spaces of the lattice to which the wave reaches, according to Huygens’ principle, becomes a source of secondary waves. Behind the bars, these waves propagate in all directions. The angle of deflection of light from the normal to the grating is called the diffraction angle.

Let us place a collecting lens in the path of the secondary waves. It will focus all secondary waves propagating at the same diffraction angle in the appropriate place on its focal surface.

In order for all these waves to maximize each other when superimposed, it is necessary that the phase difference of the waves coming from the corresponding points of two adjacent slits, i.e. points located at equal distances from the edges of these slits, be equal to an even number or the difference the course of these waves was equal to an integer m wavelengths. From Fig. 1 it is clear that the difference in the path of waves 1 and 2

for point P is equal to:

Consequently, the condition for the maximum intensity of the resulting light wave during diffraction from a diffraction grating can be written as follows:

, (2)

Where The plus sign corresponds to a positive path difference, the minus sign to a negative one.

Maxima satisfying condition (2) are called principal, number m called the order of the main maxima or the order of the spectrum. Meaning m=0 corresponds to a maximum of zero order (central maximum). There is one maximum of the zeroth order; there are two maximums of the first, second and higher orders to the left and to the right of the zero.

The position of the main maxima depends on the wavelength of the light. Therefore, when the grating is illuminated with white light, the maxima of all orders except zero, corresponding to different wavelengths, are shifted relative to each other, i.e., they are decomposed into a spectrum. The violet (short-wavelength) boundary of this spectrum faces the center of the diffraction pattern, the red (long-wavelength) boundary faces the periphery.

3. Description of installation

The work is carried out on a GS-5 spectrogoniometer with a diffraction grating installed on it. A goniometer is a device designed to accurately measure angles. The appearance of the GS-5 spectrogoniometer is shown in Fig. 2.

Fig.2

Collimator 1, equipped with a spectral slit adjustable by a micrometric screw 2, is mounted on a fixed stand. The slit faces the (mercury lamp). A transparent diffraction grating 4 is installed on the object stage 3.

The diffraction pattern is observed through eyepiece 5 of telescope 6.

The purpose of the work is to study the diffraction grating, find its characteristics and use it to determine the length of light waves in the emission spectrum of mercury vapor.

In the laboratory of the physical workshop of the Department of Physics of USTU-UPI, a mercury lamp is used as a source of a line spectrum in laboratory work No. 29, in which, during an electric discharge, a line spectrum of radiation is generated, which, having passed the collimator of the GS-5 spectrogoniometer, falls on a diffraction grating (photo of GS-5 is shown on the title file). The experimenter determines the diffraction angle with an accuracy of up to a few seconds by pointing the sighting line of the eyepiece at the corresponding line of the spectrum, then, using the method described above, calculates the wavelength of the selected line.

In the computer version of this work, the experimental conditions are modeled quite accurately. An eyepiece is displayed on the display screen, the sighting line of which should be aimed at any selected spectral line, more precisely at the middle of the color strip, which increases the accuracy of angle measurements to several arc seconds.

Like the real spectrum of mercury vapor, the computer work also “generates” the four brightest visible lines of the spectrum: violet, green and two yellow lines. The spectra are located mirror symmetrically relative to the central (white) maximum. Below under the eyepiece, for better orientation, all the lines of the spectrum of mercury are shown on a thin black strip. Moreover, two yellow lines merge into one. The fact is that these lines are located nearby and have similar wavelengths - the so-called doublet, but on a good diffraction grating they are separated (resolved), which is visible in the eyepiece. In this work, one of the tasks is to determine the resolution of the diffraction grating.

So, by hovering the cursor over “Measurements” and pressing the left mouse button, you can start taking measurements. You can “rotate” the eyepiece in four different modes, both left and right, until a colored vertical line appears in the eyepiece’s field of view. You should point the black vertical sighting line of the eyepiece at the central part of the color strip, while the diffraction angle values ​​are displayed on the digital display with an accuracy of several arc seconds. The spectral lines range from approximately 60 to 150 degrees. At the same time, the accuracy of the numerical values ​​of the angles and, as a consequence, the correctness of the results obtained depend on the thoroughness of the experiments. The experimenter is given the opportunity to choose the sequence of measurements himself.

The measurement results must be entered into the appropriate report tables and the necessary calculations must be made.

4.1. Determination of the wavelength of spectral lines of mercury vapor.

Measurements are carried out for first order spectral lines (m=1). The lattice constant is d=833.3 nm, its length (width) is 40 mm. The value of the sine of an angle can be determined using the appropriate tables or using a calculator, however, it should be borne in mind that arc seconds and minutes must be converted to decimal places of degrees, i.e. 30 minutes are equal to 0.5 degrees, etc.

The measurement results are entered in Table 2 of the report (see Appendix). The wavelength value is obtained using formula (2):

4.2.Calculation of diffraction grating characteristics.

Maximum order value m diffraction spectra for any diffraction grating can be determined in the case of normal incidence of light on the grating using the following formula:

Meaning m max is determined for the longest wavelength - in this work for the second yellow line. The highest order of the spectra is equal to the integer part (without rounding!) of the ratio.

Resolution R A diffraction grating is characterized by its ability to separate (resolve) spectral lines that differ little in wavelength. A-priory

where is the wavelength near which the measurement is made;

The minimum difference in wavelengths of two spectral lines perceived separately in the spectrum.

The value is usually determined by the Rayleigh criterion: two spectral lines and are considered allowed if the maximum is of the order of m one of them (with a longer wavelength), determined by the condition

,

coincides with the first additional minimum in the spectrum of the same order m for another line determined by the condition:

.

From these equations it follows that

,

and the resolution of the grating turns out to be equal

(6)

Thus, the resolution of the grating depends on the order m spectrum and from the total number N the strokes of the working part of the grating, i.e. the part through which the radiation under study passes and on which the resulting diffraction pattern depends. Using formula (5) the resolving power is found R diffraction grating used for the first order spectrum (m=1).

From (5) it follows that two spectral lines are resolved by the diffraction grating in the spectrum m- th order if:

. (7)

Using the found value R, formula (5) calculates (in nanometers) the linear resolution of spectral lines near the lines f, z, g of the spectrum

(9)

where is the angular distance between two spectral lines that differ in wavelength by .

Formula for D is obtained by differentiating relation (2): the left side by diffraction angle, and the right side by wavelength:

,

(10)

Thus, the angular dispersion of the grating depends on the order m of the spectrum, the constant d grating and on the diffraction angle.

Using formula (8), the angular dispersion of the diffraction grating used is found (in “/nm-arcseconds per nanometer) for diffraction angles corresponding to all measured wavelengths of the spectrum.

The results obtained are recorded in Table 2 of the report (see Appendix).

5. Security questions

1. What is the phenomenon of light diffraction?

2. Formulate the Huygens-Fresnel principle.

3. What is the resolution of a diffraction grating and what does it depend on?

4. How to experimentally determine angular dispersion D diffraction grating?

5. What is the appearance of the diffraction pattern obtained from a transparent grating?

APPLICATION

REPORT FORM

Title page:

U G T U - U P I

Department of Physics

REPORT

for laboratory work 29

Study of diffraction gratings. Determining the wavelength of light using a diffraction grating

Student______________________________

Group ______________________________

Date _________________________________

Teacher……………………….

On the inner pages:

1. Calculation formulas:

where is the wavelength;

m – spectrum order (m=1).

2. Radiation source – mercury lamp.

3. Ray path

4. Results of measurements of diffraction angles and wavelengths

spectral lines of mercury vapor. Table 1

5. Calculation of the required quantities.

Table 2 Characteristics of the diffraction grating

6. Estimation of wavelength measurement errors is calculated using the formula:

Table values ​​of wavelengths of spectral lines of mercury vapor:

Violet – 436 nm,

Green - 546 nm,

1 yellow – 577 nm,

2 yellow - 579 nm.