5.104. A plane light wave with wavelength



and intensity

falls normally on a large glass plate whose opposite side serves as

an opaque screen with a round aperture equal to the first Fresnel

zone for the observation point

In the middle of the aperture

there is a round recess equal to half the Fresnel zone. What must

the depth

of that recess be for the intensity of light at the point

to be the highest? What is this intensity equal to?

5.105. A plane light wave with wavelength = 0.57 um falls

normally on a surface of a glass (n = 1.60) disc which shuts one

and a half Fresnel zones for the observation

point

What must

the minimum thickness of that disc be for the intensity of light

at the point

to be the highest? Take into account the interference

of light on its passing through the disc.

5.106. A plane light wave with wavelength = 0.54 um goes

through a thin converging lens with focal length

f =

50 cm and

an aperture stop fixed immediately after the lens, and reaches

a screen placed at a distance

b =

75 cm from the aperture stop.

At what aperture radii has the centre of the diffraction pattern

on the screen the maximum illuminance?

5.107. A plane monochromatic light wave falls normally on

a round aperture. At a distance

b =

9.0 m from it there is a screen

showing a certain diffraction pattern. The aperture diameter was

decreased ii = 3.0 times. Find the new distance

at which the

screen should be positioned to obtain the diffraction pattern similar

to the previous one but diminished ri times.

5.108. An opaque ball of diameter

D = 40

mm is placed between

a source of light with wavelength ? = 0.55 um and a photographic

plate. The distance between the source and the ball is equal to

a =

12 m and that between the ball and the photographic plate

is equal to

b =

18 m. Find:

(a)

the image dimension y' on the plate if the transverse dimension

of the source is y = 6.0 mm;

(b)

the minimum height of irregularities, covering the surface

of the ball at random, at which the ball obstructs light.

Note.

As calculations and experience show, that happens when

the height of irregularities is comparable



with the width of the Fresnel zone along



iAi



which the edge of an opaque screen passes.



5.109.

A point source of monochromatic 0



light is positioned in front of a zone plate



at a distance

a =

1.5 m from it. The image



of the source is formed at a distance

b =

1.0 m from the plate. Find the focal

length of the zone plate.



5.110.

A plane light wave with wave-



Fig. 5.22.

length ? = 0.60 um and intensity I, falls

normally on a large glass plate whose side view is shown in

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5.104. A plane light wave with wavelength and intensity Io

falls normally on a large glass plate whose opposite side serves as an opaque screen with a round aperture equal to the first Fresnel

zone for the observation point P. In the middle of the aperture

there is a round recess equal to half the Fresnel zone. What must

the depth h^ of that recess be for the intensity of light at the point

P to be the highest? What is this intensity equal to?

5.105. A plane light wave with wavelength = 0.57 um falls normally on a surface of a glass (n = 1.60) disc which shuts one

and a half Fresnel zones for the observation point P. What must

the minimum thickness of that disc be for the intensity of light

at the point P to be the highest? Take into account the interference

of light on its passing through the disc. 5.106. A plane light wave with wavelength = 0.54 um goes

through a thin converging lens with focal length f = 50 cm and

an aperture stop fixed immediately after the lens, and reaches

a screen placed at a distance b = 75 cm from the aperture stop.

At what aperture radii has the centre of the diffraction pattern on the screen the maximum illuminance? 5.107. A plane monochromatic light wave falls normally on

a round aperture. At a distance b = 9.0 m from it there is a screen

showing a certain diffraction pattern. The aperture diameter was

decreased ii = 3.0 times. Find the new distance b' at which the

screen should be positioned to obtain the diffraction pattern similar to the previous one but diminished ri times.

5.108. An opaque ball of diameter D = 40 mm is placed between

a source of light with wavelength? = 0.55 um and a photographic plate. The distance between the source and the ball is equal to

a = 12 m and that between the ball and the photographic plate

is equal to b = 18 m. Find:

(a) the image dimension y' on the plate if the transverse dimension of the source is y = 6.0 mm; (b) the minimum height of irregularities, covering the surface of the ball at random, at which the ball obstructs light.

Note. As calculations and experience show, that happens when

the height of irregularities is comparable with the width of the Fresnel zone along which the edge of an opaque screen passes. iAi

5.109. A point source of monochromatic 0^^0 o

light is positioned in front of a zone plate^0^00 0 1 /

at a distance a = 1.5 m from it. The image

of the source is formed at a distance

b = 1.0 m from the plate. Find the focal

length of the zone plate.

5.110. A plane light wave with wave- Fig. 5.22.

length? = 0.60 um and intensity I, falls normally on a large glass plate whose side view is shown in

4A

(^11 ) % /A /)// 5

Fig. 5.22. At what height h of the ledge will the intensity of light

at points located directly below be

(a) minimum;

(b) twice as low as /0(the losses due to reflection are to be neglect-

ed).

5.111. A plane monochromatic light wave falls normally on an

opaque half-plane. A screen is located at a distance b = 100 cm

behind the half-plane. Making use of the Cornu spiral (Fig. 5.19), find:

(a) the ratio of intensities of the first maximum and the neighbour-

ing minimum;

(b) the wavelength of light if the first two maxima are separated

by a distance Ax = 0.63 mm.

5.112. A plane light wave with wavelength 0.60 pm falls normally

on a long opaque strip 0.70 mm wide. Behind it a screen is placed

at a distance 100 cm. Using Fig. 5.19, find the ratio of intensities

of light in the middle of the diffraction pattern and at the edge of

the geometrical shadow.

5.113. A plane monochromatic light wave falls normally on a long

rectangular slit behind which a screen is positioned at a distance

b = 60 cm. First the width of the slit was adjusted so that in the

middle of the diffraction pattern the lowest minimum was observed.

After widening the slit by Ah = 0.70 mm, the next minimum was

obtained in the centre of the pattern. Find the wavelength of light.

5.114. A plane light wave with wavelength X = 0.65 p.m falls

normally on a large glass plate whose opposite side has a long rectan-

gular recess 0.60 mm wide. Using Fig. 5.19,

find the depth h of the recess at which the

diffraction pattern on the screen 77 cm

away from the plate has the maximum

illuminance at its centre.

5.115. A plane light wave with wave-

length X = 0.65 p.m falls normally on a

large glass plate whose opposite side has

a ledge and an opaque strip of width

a = 0.30 mm (Fig. 5.23). A screen is placed

at a distance b = 110 cm from the

plate. The height h of the ledge is such

that the intensity of light at point^2 of the^ Fig. 5.23.

screen is the highest possible. Making use

of Fig. 5.19, find the ratio of intensities at points 1 and 2.

5.116. A plane monochromatic light wave of intensity Io falls

normally on an opaque screen with a long slit having a semicircular

p o ///////////////////

0 (^2) J / / // /// // (^) ///

Fig. 5.24. Fig. 5.25.

220

= 0.65 pm. Find the angle of diffraction of third order for a wave

length X2 = 0.50 1.1.m.

5.126. Light with wavelength 535 nm falls normally on a diffrac-

tion grating. Find its period if the diffraction angle 35°corresponds

to one of the Fraunhofer maxima and the highest order of spectrum

is equal to five.

5.127. Find the wavelength of monochromatic light falling nor-

mally on a diffraction grating with period (^) d = 2.2 pm if the angle

between the directions to the Fraunhofer maxima of the first and

the second order is equal to AO = 15°.

5.128. Light with wavelength 530 nm falls on a transparent

diffraction grating with period 1.50 lila. Find the angle, relative

to the grating normal, at which the Fraunhofer maximum of highest

order is observed provided the light falls on the grating

(a) at right angles;

(b) at the angle 60° to the normal.

5.129. Light with wavelength X = 0.60 tim falls normally on

a diffraction grating inscribed on a plane surface of a plano-convex

cylindrical glass lens with curvature radius R = 20 cm. The period of the grating is equal to d = 6.0 [cm. Find the distance between

the principal maxima of first order located symmetrically in the

focal plane of that lens.

5.130. A plane light wave with wavelength X = 0.50 μm falls

normally on the face of a glass wedge with an angle 0 30°. On the

opposite face of the wedge a transparent diffraction grating with

period d = 2.00 pm is inscribed, whose lines are parallel to the

wedge's edge. Find the angles that the direction of incident light

forms with the directions to the principal Fraunhofer maxima of

the zero and the first order. What is the highest order of the spect

rum? At what angle to the direction of incident light is it observed?

5.131. A plane light wave with wavelength 1 falls normally on

a phase diffraction grating whose side view is shown in Fig. 5.26.

The grating is cut on a glass plate with refractive index n. Find

the depth h of the lines at which the intensity of the central Fraun-

hofer maximum is equal to zero. What is in this case the diffraction

angle corresponding to the first maximum?

T 0

a Fig. 5.26. Fig. 5.27.

5.132. Figure 5.27 illustrates an arrangement employed in obser-

vations of diffraction of light by ultrasound. A plane light wave

with wavelength? = 0.55 pm passes through the water-filled tank T

_0

in which a standing ultrasonic wave is sustained at a frequency v = 4.7 MHz. As a result of diffraction of light by the optically inhomogeneous periodic structure a diffraction spectrum can be

observed in the focal plane of the objective 0 with focal length

f = 35 cm. The separation between neighbouring maxima is Ax

= 0.60 mm. Find the propagation velocity of ultrasonic oscillations in water. 5.133. To measure the angular distance* between the components of a double star by Michelson's method, in front of a telescope's lens a diaphragm was placed, which had two narrow parallel slits

separated by an adjustable distance d.^ While diminishing^ d,^ the

first smearing of the pattern was observed in the focal plane of the

objective at d = 95 cm. Find *, assuming the wavelength of light

to be equal to X, = 0.55 pm.

5.134. A transparent diffraction grating has a period d = 1.50 pm.

Find the angular dispersion D^ (in angular minutes per nanometres)

corresponding to the maximum of highest order for a spectral line of wavelength X = 530 nm of light falling on the grating (a) at right angles; (b) at the angle 00= 45° to the normal. 5.135. Light with wavelength X falls on a diffraction grating at right angles. Find the angular dispersion of the grating as a function of diffraction angle 0. 5.136. Light with wavelength X = 589.0 nm falls normally on

a diffraction grating with period d = 2.5 pm, comprising N =

= 10 000 lines. Find the angular width of the diffraction maximum

of second order. 5.137. Demonstrate that when light falls on a diffraction grating at right angles, the maximum resolving power of the grating cannot

exceed the value //X, where 1 is the width of the grating and X is

the wavelength of light. 5.138. Using a diffraction grating as an example, demonstrate that the frequency difference of two maxima resolved according to Rayleigh's criterion is equal to the reciprocal of the difference of propagation times of the extreme interfering oscillations, i.e. 1:5v = = 1/St. 5.139. Light composed of two spectral lines with wavelengths 600.000 and 600.050 nm falls normally on a diffraction grating 10.0 mm wide. At a certain diffraction angle 0 these lines are close to being resolved (according to Rayleigh's criterion). Find 0. 5.140. Light falls normally on a transparent diffraction grating

of width 1 = 6.5 cm with 200 lines per millimetre. The spectrum

under investigation includes a spectral line with = 670.8 nm consisting of two components differing by 62‘, = 0.015 nm. Find: (a) in what order of the spectrum these components will be resolv- ed; (b) the least difference of wavelengths that can be resolved by this grating in a wavelength region X (^) 670 nm.

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Diffraction of Light: Problems and Solutions, Study Guides, Projects, Research of Physics

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