**WITH THE HELP OF NEWTON’S RINGS**

**The purpose of the work:** to 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 on it; microscope; Light source; ruler with a millimeter scale.

Note: The theory of the method and the description of the setup are given in paper No. 2.

1. **Determination of 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 divisions of the ocular scale. Multiplying this result by the value **b** , expressed in mm / div, we get the diameter in mm.

The radii of the ** i** -th and

**-th dark rings in accordance with the formula (2.5)**

*n* **r _{t,} _{i} = ,r _{t,} _{n} = ,** (3.1)

By 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 set 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)

whence we obtain 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. In order to simplify calculations, the value denote by ** T** . Then

**l** *= .* (3.5)

Table 3.1

Lens number | R, mm at a = 0.95 |
Lens number | R, mm at a = 0.95 |

98 + 2 |
76 + 3 |
||

64 + 1 |
110 + 1 |
||

107 + 3 |
74 + 3 |

**Completing of the work**

2.1. See paragraph 2.1 in work No. 2.

2.2. See item 2.2 in work No. 2.

2.3 See paragraph 2.3 in paper No. 2.

2.4. By formula (3.5) determine *<* **l** *>.*

2.5. Calculate the absolute error using the formula

,

where **D T** is found using a formula similar to formula (2.7).

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

Table 3.2

Ring number | x _{1} |
x _{2} |
D |
D2 ^{_} |
i – n | D ^{2} – _{i}D ^{2} _{n} |
T |
T – |
(T – ) ^{2} |

. . . | |||||||||

Sum | |||||||||

Wed value |

TEST QUESTIONS

1. The phenomenon of light interference.

2. Coherence.

3. Optical path length and optical path difference.

4. Conditions for maxima and minima in interference.

5. Phenomena occurring 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. The course of the experiment to determine the radius of curvature of the lens or the wavelength of light using Newton’s rings.

9. Calculation of measurement errors.

LAB #4

**LIGHT WAVE LENGTH DETERMINATION**

**USING A DIFFRACTION GRATING**

**Purpose of work** : to determine the characteristics of the diffraction grating; measure the wavelength of light using a diffraction grating.

**Instruments and accessories** : experimental setup, diffraction grating.

**Information from the theory**

**Diffraction** of light is a phenomenon caused by the violation of the integrity of the wave surface. Diffraction manifests itself in violation of the straightness of the propagation of oscillations. The wave goes around the edges of the obstacle and penetrates into the region of the geometric shadow. Diffraction phenomena are inherent in all wave processes, but they manifest themselves especially clearly only in those cases where the wavelengths of the radiations are comparable to the size of the obstacles.

From the point of view of the ideas of geometric optics about the rectilinear propagation of light, the boundary of the shadow behind an opaque obstacle is sharply outlined by the rays that pass by the obstacle, touching its surface. Consequently, the phenomenon of diffraction is inexplicable from the standpoint of geometric optics. According to the 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 sharply defined, as the theory of rectilinear propagation of light claims, but with blurry edges.

**Huygens-Fresnel principle**

A feature 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 some source ** S _{0}** with an area

**, can be divided into small sections with equal areas**

*S***, which are a system of secondary sources emitting secondary waves.**

*dS* 2. These secondary sources, which are equivalent to the same primary source ** S _{0}** , are coherent. Therefore, waves propagating from the source

**at any point in space must be the result of the interference of all secondary waves.**

*S*_{0}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 an area ** dS** radiates predominantly in the direction of the outer normal

**to the wave surface at this point; the amplitude of the secondary waves in the direction making an angle**

*n***a**with

**is the smaller, the larger the angle**

*n***a**, and is equal to zero for

**a**³ p / 2.

5. The amplitude of the 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 to give methods for its quantitative calculation.

**Fresnel zone method**

The Huygens-Fresnel principle explains the rectilinear propagation of light 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

**(Fig. 4.1). The wave surface of such a wave is symmetrical with respect to the straight line**

*P***. The amplitude of the desired wave at the point**

*S*_{0}P**depends on the result of the interference of secondary waves emitted by all sections**

*P***of the surface**

*dS***. The amplitudes and initial phases of the secondary waves depend on the location of the corresponding sources**

*S***with respect to the point**

*dS***.**

*P* Fresnel proposed a method for dividing a wave surface into zones (the method of Fresnel zones). According to this method, the wave surface is divided into annular zones (Fig. 4.1), constructed in such a way that the distances from the edges of each zone to the point ** P** differ by

**l**/ 2 (

**l**

*is*the wavelength of the light wave). If we denote by

**the distance from the top of the wave surface 0 to point**

*b***, then the distances**

*P*

*b**+*(

**k****l**/2) form the boundaries of all zones, where

**is the zone number. Oscillations coming to point**

*k***from similar points of two neighboring zones are opposite in phase, since the path difference from these zones to point**

*P***is equal to**

*P***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)

The magnitude of the amplitude ** A _{k}** depends on the area

**-th zone and the angle**

*DS*_{k}of the*k***a**between the outer normal to the zone surface at any point and the straight line directed from this point to the point

_{k}**.**

*P* It can be shown that the area ** DS of _{the} k** -th zone does not depend on the zone number under the conditions

**l**

*<<*

**. Thus, in the considered approximation, the areas of all Fresnel zones are equal and the radiation power of all Fresnel zones – secondary sources – is the same. At the same time, as**

*b***increases, the angle**

*k***a**between the normal to the surface and the direction to the point

_{k}**increases, which leads to a decrease in the radiation intensity of the**

*P***-th zone in this direction, i.e. to a decrease in the amplitude**

*k***in comparison with the amplitudes of the previous zones. The amplitude**

*A*_{k}**also decreases due to an increase in the distance from the zone to the point**

*A*_{k}**with increasing**

*P***. Eventually**

*k* *A* _{1} *> A* _{2} *> A* _{3} *> A* _{4} *> … > A* _{k} *> …*

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

. (4.2)

Rewriting the resulting amplitude (4.1) in the form

, (4.3)

we find that, according to (4.2) and taking into account the smallness of the 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 oscillations caused at point ** P** by the spherical wave surface have an amplitude given by half of the central Fresnel zone. Consequently, the light from the source

**to the point**

*S*_{0}**propagates within a very narrow direct channel, i.e. straightforward. As a result of the phenomenon of interference, the action of all zones is destroyed, except for the first.**

*P* **Fresnel diffraction from the simplest obstacles**

The action of a light wave at some point ** P** is reduced to the action of half of the central Fresnel zone if the wave is infinite, since only then the actions of the remaining zones are mutually compensated and the action of 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 application of the Fresnel method makes it possible to foresee and explain the features of the propagation of light waves.

Consider several examples of Fresnel diffraction from simple obstacles.

** Diffraction by a circular hole** . Let a wave from a source

**encounter an opaque screen with a round hole**

*S*_{0}**on its way (Fig. 4.2). The result of diffraction is observed on the screen**

*BC***, parallel to the hole plane. It is easy to determine the diffraction effect at point**

*E***of the screen, located opposite the center of the hole. To do this, it suffices to construct on the open part of the wave front**

*P***the Fresnel zones corresponding to the point**

*BC***. If**

*P***Fresnel zones fit in the hole**

*k***, then the amplitude**

*BC***of the resulting oscillations at the point**

*A***depends on the evenness and oddness of the number**

*P***, as well as on how large the absolute value of this number is. Indeed, from formula (4.1) it follows that at the point**

*k***the amplitude of the total oscillation**

*P*(the first equation of the system for odd *k**,* the second – for even) or, taking into account formula (4.2) and the fact that the amplitudes of two neighboring zones differ little in magnitude and we can consider ** A _{k-1}** approximately equal to

**we have**

*A*_{k},, (4.5)

where plus corresponds to an odd number of zones ** k** that fit on the hole, and minus to an even number.

With a small number of zones ** k** , the amplitude

**differs little from**

*A*_{k}**. Then the result of diffraction at the point**

*A*_{1}**depends on the parity of**

*P***: for odd**

*k***, the diffraction maximum is observed, and for even k, the minimum. The minimums and maximums will be the more different from each other, the closer**

*k***is to**

*A*_{k}*i.e. the smaller*

**A**_{1}**is. If the hole opens only the central Fresnel zone, the amplitude at point**

*k***will be equal to**

*P***, it is twice that which occurs with a fully 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**

*A*_{1}**, the amplitude**

*k***tends to zero**

*A*_{k}*(*and expression (4.5) turns into (4.4). Light in this case actually propagates in the same way as in the absence of a screen with a hole, i.e. straightforward. This leads to the conclusion that the consequences of wave representations and representations of rectilinear propagation of light begin to coincide when the number of open zones is large.

**A**)_{k}<< A_{1} Oscillations from even and odd Fresnel zones cancel each other out. This sometimes leads to an increase in the light intensity when a part of the wave front is covered by an opaque screen, as was the case with a barrier with a round hole, on which only one Fresnel zone fits. The intensity of light can be increased many times over if a complex screen is made – 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 is *k**=* 2 *m**,* 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}* is approximately equal to

*A*,*A**=*, i.e. the intensity of light at point

**mA**_{1}**is (2**

*P***)**

*m*^{2}times greater than with unimpeded propagation of light from the source to point

**, while**

*P**2, and the intensity, respectively*

**A = A**/_{1}*4 .*

^{}/ **Diffraction on a circular disk.** When placed between the source ** S _{0}** and the screen of a round opaque disk

**one or several first Fresnel zones are closed (Fig. 4.3). If the disk covers**

*SW,***Fresnel zones, then at point**

*k***the amplitude of the total wave**

*P*and, since the expressions in parentheses 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)

*, a bright spot is obtained.*

**k** 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

**a**between the normal to the surface of this zone at some point and the direction of radiation towards the point

**increases (see the Huygens-Fresnel principle). Therefore, the intensity of the central maximum weakens with increasing disk size (**

*P**). 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 does a weak interference pattern take place. In this case, we can neglect the phenomenon of diffraction and use the law of rectilinear propagation of light.*

**A**<<_{k+1}**A**_{1} **Fraunhofer diffraction**

**(diffraction in parallel beams)**

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 (the size 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 excludes 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 constant shape and size, the result of diffraction depends only on the change in the spectral composition of the radiation given by the source

**. Therefore, diffraction phenomena in parallel beams can be used for spectral analysis of the radiation composition of the studied substances.**

*S*_{0}The schematic diagram of the observation of plane waves (Fraunhofer diffraction) is shown in fig. 4.4.

Light from a point source * S _{0}* is converted by lens

**into a beam of parallel rays (a plane wave), which then passes through a hole in an opaque screen (circle, slit, etc.). The lens**

*L*_{1}**collects at various points of its focal plane, where the observation screen**

*L*_{2}**is located, all the rays that have passed through the hole, including the rays deviated from the original direction as a result of diffraction.**

*E* Rays diffracting at one angle, the lens ** L _{2}** collects at one point of the focal plane

*E.* Since the observation of diffraction by the Fraunhofer method is carried out in the place where the light is collected by the lens ** L _{2}** , the phenomenon gains significantly in brightness and the observation of the diffraction pattern is facilitated.

Let us consider several cases of Fraunhofer diffraction.

**Diffraction from a single slit.** In practice, the slot is represented by 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 the light diffracts to the right and left of the slit (Fig. 4.5). If we observe the image of the source in a direction perpendicular to the direction of the generatrix of the slit, then we can restrict ourselves to considering the diffraction pattern in one dimension (along

**).**

*x*Since the plane of the slot coincides with the front of the incident wave, in accordance with the Huygens-Fresnel principle, the points of the slot are secondary sources of waves oscillating in one phase.

Let us divide the area of the slot into a number of narrow strips of equal width, parallel to the generatrix of the slot. The phases of the 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 the law of rectilinear propagation of light was observed during the passage of light through the slit (there would be no diffraction), then on the screen ** E** installed in the focal plane of the lens

*L*_{2}*,*an image of the slit would be obtained. Therefore, the direction

**j**

*=*0 defines an undiffracted wave with an amplitude

**equal to the amplitude of the wave sent through the entire slit.**

*A*_{0} Due to diffraction, light rays deviate from the rectilinear direction by an angle **j** *.* Deviation to the right and left is symmetrical about the center line ** OC _{0}** (Fig. 4.5). To find the effect of the entire slot 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

**from different stripes (Fresnel zones).**

*C*_{j} Let us draw the 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 ray path difference, the path of all rays from the

**plane to the point**

*FD***is the same. Therefore, the total path difference of rays from the slit**

*C*_{j}**is given by the segment**

*FE***. We draw planes parallel to the wave surface**

*ED***so that they divide the segment**

*FD***into several sections, each of which has a length of**

*ED***l**/ 2 (Fig. 4.5). These planes will divide the slot 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 point

**C****is determined by the number of Fresnel zones that fit into the slot (see Fresnel diffraction by a round hole): if the number of zones is even (**

_{j}

*z**=*2

**), at point**

*k***there is a minimum of diffraction, if**

*C*_{j}*z*is odd (

*z**=*2

*k**+*1), at point

**is the diffraction maximum. The number of Fresnel zones that fit into the slot**

*C*_{j}**is determined by how many times**

*FE***l/**2 is contained in the segment

**, i.e. . Segment**

*ED***, expressed in terms of the slit width**

*ED**and the diffraction angle*

**a****j**, will be written as

*ED = a sin*j*.* As a result, for the position of the diffraction **maxima** , we obtain the condition

*and sin* j*=* ±( *2 k +* 1)

**l**/ 2,(4.7)

for diffraction **minima**

*and sin* j*=* ± 2 ** k l** /2,(4.8)

where ** k** = 1,2,3.. are integers. The value

**, which takes the values of the numbers of 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 +**

*k***j**and –

**j**and converging at the side foci of the lens

**:**

*L*_{2}**and**

*C*_{j}

**C**_{–}**, which are symmetric with respect to the main focus**

_{j}**. In the direction**

*C*_{0}**j**

*=*0, the most intense zero-order central maximum is observed.

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

, , etc.

On fig. 4.6 shows the light intensity distribution curve in the function ** sin j** . The 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, a simple alternation of dark and light bands occurs only in monochromatic light. In the case of white light, the diffraction patterns for waves with different

**l**shift according to the wavelength. The central white maximum has a rainbow color only at the edges (one Fresnel zone fits within the width of the slit). Lateral maxima for different wavelengths no longer coincide with each other; closer to the center are the maxima corresponding to shorter waves. The long-wavelength maxima are farther apart

*(*

**j**

*=*

**arcsin****l**/2) than the short-wavelength ones. Therefore, the diffraction maximum is the spectrum with the violet part facing the center.

**Diffraction grating**

A diffraction grating is a system of a large number of slits, identical in width and parallel to each other, lying in the same plane and separated by opaque gaps equal in width. A diffraction grating is made by applying parallel strokes to the glass surface. The number of strokes per 1 mm is determined by the region of the spectrum of the studied radiation 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

**and the distance between adjacent slots**

*a***(Fig. 4.7). The sum**

*b***is called the period or constant of the diffraction grating. Let a plane monochromatic wave be normally incident on the grating. It is required to investigate the intensity of light propagating in the direction making an angle**

*a + b = d***j**with the normal to the grating plane. In addition to the distribution of intensity due to diffraction at each slit, there is a redistribution of light energy due to the interference of waves from

*slits of coherent sources. In this case, the minima will be in the same places, because the condition for the minimum diffraction for all slots (Fig. 4.8) is the same. These minima are called principal. The condition of principal minima*

**N****coincides with condition (4.8). The position of the main minima**

*a sin*j*=*±*k*l

*sin*j*=*±

**l**

*/a**,*2

**l**,… is shown in fig. 4.8.

*/a* However, in the case of many slits, the main minima produced by each slit individually are supplemented by minima resulting from the interference of light transmitted through different slits. On fig. 4.8, for example, shows the intensity distribution and the location of the maxima and minima in the case of two slits with a period ** d** and a slit width

*a*. In the same direction, all slots radiate vibration energy of the same amplitude. And the result of interference depends on the phase difference of oscillations coming from similar points of adjacent slots (for example, ** C** and

*E**,*

**and**

*B***), or on the optical path difference**

*F***from similar points of two adjacent slots to point**

*ED*

*C*_{j}*.*For all similar points, this path difference is the same. If

*±*

**ED**=

**k****l**or, since

*ED**=*

**d si**

*n*j*,* **d sin j = ± k l , k**

*=*0,1,2…, (4.9)

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

**A _{max} = NA _{j} ,** (4.10)

where *A* _{j}*is* the amplitude of the oscillation sent by one slot at an angle **j** . light intensity

**J _{max} u003d N ^{2} A _{j} ^{2} u003d N ^{2} J _{j}** . (4.11)

Therefore, formula (4.9) determines the position of the main intensity maxima. The number ** k** gives the order of the principal maximum.

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

. (4.12)

There is only one zero-order maximum and is located at the point ** C _{0}** , the maxima of the first, second, and so on. orders of two and they are located symmetrically with respect to

**, as indicated by the sign**

*C*_{0}__+__. On fig. 4.8 shows the position of the main maxima.

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

In the main maxima, the amplitude is ** N** times, and the intensity is

**times greater than the amplitude given in the corresponding place by one slit. Lines with increased brightness clearly localized in space are easily detected and can be used for spectroscopic studies.**

*N*^{2} As the distance from the center of the screen increases, 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** given by the grating on one side of the screen is determined by the condition

. (4.13)

It should not be forgotten that ** k** is 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 for the central one ( ** k** = 0), decompose into a spectrum with the violet end turned towards 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 intensity 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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