— Radius of the periphery of the $k$ th Fresnel zone:
\[
r_{k}=\sqrt{k \lambda \frac{a b}{a+b}}, k=1,2,3, \ldots,
\]
— Cornu’s spiral (Fig. 5.19). The numbers along that spiral correspond to the values of parameter $v$. In the case of a plane wave $v=x \sqrt{2 / b \lambda}$, where $x$
Fig. 5.19.
and $b$ are the distances defining the position of the element $d S$ of a wavefront relative to the observation point $P$ as shown in the upper left corner of the figure.
216
— Fraunhofer diffraction produced by light falling normally from a slit. Condition of intensity minima:
\[
b \sin \theta= \pm k \lambda, \quad k=1,2,3, \ldots,
\]
where $b$ is the width of the slit, $\theta$ is the diffraction angle.
— Diffraction grating, with light falling normally. The main Fraunhofer maxima appear under the condition
\[
d \sin \theta= \pm k \lambda, \quad k=0,1,2, \ldots,
\]
the condition of additional minima:
\[
d \sin \theta= \pm \frac{k^{\prime}}{N} \lambda,
\]
where $k^{\prime}=1,2, \ldots$, except for $0, N, 2 N, \ldots$.
— Angular dispersion of a diffraction grating:
\[
D=\frac{\delta \theta}{\delta \lambda}=\frac{k}{d \cos \theta} .
\]
— Resolving power of a diffraction grating:
\[
R=\frac{\lambda}{\delta \lambda}=k N,
\]
where $N$ is the number of lines of the grating.
— Resolving power of an objective
\[
R=\frac{1}{\delta \psi}=\frac{D}{1.22 \lambda},
\]
where $\delta \psi$ is the least angular separation resolved by the objective, $D$ is the diameter of the objective.
— Bragg’s equation. The condition of diffraction maxima:
\[
2 d \sin \alpha= \pm k \lambda \text {, }
\]
where $d$ is the interplanar distance, $\alpha$ is the glancing angle, $k=1,2,3, \ldots$.
5.97. A plane light wave falls normally on a diaphragm with round aperture opening the first $N$ Fresnel zones for a point $P$ on a screen located at a distance $b$ from the diaphragm. The wavelength of light is equal to $\lambda$. Find the intensity of light $I_{0}$ in front of the diaphragm if the distribution of intensity of light $I(r)$ on the screen is known. Here $r$ is the distance from the point $P$.
5.98. A point source of light with wavelength $\lambda=0.50 \mu \mathrm{m}$ is located at a distance $a=100 \mathrm{~cm}$ in front of a diaphragm with round aperture of radius $r=1.0 \mathrm{~mm}$. Find the distance $b$ between the diaphragm and the observation point for which the number of Fresnel zones in the aperture equals $k=3$.
5.99. A diaphragm with round aperture, whose radius $r$ can be varied during the experiment, is placed between a point source of light and a screen. The distances from the diaphragm to the source and the screen are equal to $a=100 \mathrm{~cm}$ and $b=125 \mathrm{~cm}$. Determine the wavelength of light if the intensity maximum at the centre of the diffraction pattern of the screen is observed at $r_{1}=1.00 \mathrm{~mm}$ and the next maximum at $r_{2}=1.29 \mathrm{~mm}$.
5.100. A plane menochromatie light wave with intensity $I$, falls normally on an epaque screen with a round aperture. What is the intensity of light $I$ behind the screen at the point for which the aperture
(a) is equal to the first Fresnel zone; to the internal half of the first rone;
(b) was made equal to the first Fresnel zone and then half of it was elosed (along the diameter)?
5.101. A plane monochromatic light wave with intensity $I_{\text {, }}$ falls normally on an opaque dise elosing the first Fresnel zone for the observation point $P$. What did the intensity of light $I$ at the point $P$ become equal to after
(a) half of the dise (along the diameter) was removed;
(b) half of the external half of the first Fresnel zone was removed (along the diameter)?
5.102. A plase mosochrematie light wave with intensity $I_{\text {, }}$ falls normally on the surfaces of the epaque screens shown in Fig. 5.20. Find the intensity of light $I$ at a point $P$
Fie. 5.30.
(a) located behind the corner points of sereons 1,3 and behind the edge of half-plane 4:
(b) for which the rounded-off edge of sereens $5-8$ coincides with the boundary of the first Fresnel zone.

Derive the general formula describing the results obtained for screens 1-4; the same, for screens $5-8$.
5.103. A plane light wave with wavelength $\lambda=0.60 \mathrm{pm}$ falls normally on a sufficiently large glass plate having a round recess on the opposite side (Fig. 5.21). For the observation point $P$ that recess corres. ponds to the first one and a half Fresnel zones. Find the depth $\boldsymbol{h}$ of the recess at which the intensity of light at the point $P$ is
(a) maximum;
(b) minimum:
(c) equal to the intensity of incident light.
5.104. A plane light wave with wavelength $\lambda$ and intensity $I_{\text {, }}$ falls normally on a large glass plate whose opposite side serves as an opaque sereen with a round apertare equal to the first Presnel zone for the observation point $\vec{P}$. 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 $\lambda-0.57 \mathrm{~mm}$ falls normally on a surface of a glass $(n-1.60)$ dise which shuts one and a half Fresnel zones for the observation point $\boldsymbol{P}$. What must the minimum thickness of that dise be for the intessity of light at the point $P$ to be the highest? Take into account the interference of light on its passing through the dise.
5.106. A plane light wave with wavelength $\lambda=0.54 \mu \mathrm{m}$ goes through a thin converging lens with focal length $f=50 \mathrm{~cm}$ and an aperture stop fixed immediately after the lens, and reaches a screen placed at a distance $b=75 \mathrm{~cm}$ from the aperture stop. At what apertare 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 aperiure. At a distance $b=9.0 \mathrm{~m}$ from it there is a screen shewing a certain diffraction pattern. The aperture diameter was decreased $\eta=\mathbf{3 . 0}$ times. Find the new distance $b^{\prime}$ at which the sereen should be positiened to obtain the diffraction pattern similar to the previous one but diminished $\eta$ times.
5.108. An opaque ball of diameter $D=40 \mathrm{~mm}$ is placed between a source of light with wavelength $\lambda=0.55 \mu \mathrm{m}$ and a photographic plate. The distance between the souree and the ball is equal to $a=12 \mathrm{~m}$ and that between the ball and the photographic plate is equal to $b=18 \mathrm{~m}$. Find:
(a) the image dimension $y^{\prime}$ on the plate if the transverse dimension of the source is $y=6.0 \mathrm{~mm}$;
(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 shew, 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.
5.109. A point source of monechromatic light is positioned is front of a zone plate at a distance $a=1.5 \mathrm{~m}$ from it. The image of the source is formed at a distance $b=1.0 \mathrm{~m}$ from the plate. Find the focal length of the zone plate.
5.110. A plane light wave with wavelength $\lambda=0.60 \mathrm{~mm}$ and intensity $I_{\text {, f falls }}$ normally on a large glass plate whose side view is shown in Fig. 5.22. At what height $h$ of the ledge will the inteasity of light at points located directly below be
(a) minimum;
(b) twice as low as $I_{0}$ (the losses due to reflection are to be neglected).
5.111. A plane monochromatic light wave falls normally on an opaque hall-plane. A screen is located at a distance $b=100 \mathrm{~cm}$ 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 neighbouring misimum:
(b) the wavelength of light if the first two maxima are separated by a distance $\Delta x=0.63 \mathrm{~mm}$.
5.112. A plane light wave with wavelength $0.60 \mathrm{~mm}$ falls normally on a long opaque strip $0.70 \mathrm{~mm}$ wide. Behind it a screen is placed at a distance $100 \mathrm{~cm}$. 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 sereen is positioned at a distance $b=60 \mathrm{~cm}$. First the width of the slit was adjosted so that in the middle of the diffraction pattern the lowest minimum was observed. After widening the slit by $\Delta h=0.70 \mathrm{~mm}$, the next minimum was obtained in the centre of the pattern. Find the wavelength of light.
5.114. A plane light wave with wavelength $\lambda=0.65 \mu \mathrm{m}$ falls normally on a large glass plate whose opposite side has a long rectangular recess $0.60 \mathrm{~mm}$ wide. Using Fig. 5.19, find the depth $\boldsymbol{H}$ of the recess at which the diffraction pattern on the sereen $77 \mathrm{~cm}$ away from the plate has the maximum illuminance at its centre.
5.115. A plase light wave with wavelength $\lambda=0.65 \mathrm{pm}$ falls normally on a large glass plate whose opposite side has a ledge and an epaque strip of width $a=0.30 \mathrm{~mm}$ (Fig. 5.23). A sereen is placed at a distance $b=110 \mathrm{~cm}$ from the plate. The height $h$ of the ledge is such that the intensity of light at point 2 of the sereen is the highest possible. Making use of Fig. 5.19, find the ratio of intensities at points $t$ and 2.
5.116. A plane menochromatic light wave of intensity $I$, falls nermally on an epaque sereen with a long slit having a semieireular
Fie. 5.24 .
cut on one side (Fig. 5.24). The edge of the cut coincides with the boundary line of the first Fresnel zone for the observation poiat $P$. The width of the slit measures 0.90 of the radius of the cut. Using Fig. 5.19, find the intensity of light at the point $\boldsymbol{P}$.
5.117. A plane monechromatie light wave falls aormally on an opaque screen with a long slit whose shape is shown in Fig. 5.25. Making use of Fig. 5.49, find the ratie of intensities of light at points 1,2 , and 3 located behind the screen at equal distances from it. For point $J$ the rounded-ofl edge of the slit coincides with the boundary line of the first Fresnel rone.
5.118. A plane monochromatie light wave falls normally on an epaque serees shaped as a long strip with a round hole in the middle. For the observation point $P$ the hole corresponds to half the Fresnel zone, with the hole diameter being $n=1.07$ times less than the width of the strip. Using Fig. 5.19, find the intensity of light at the point $P$ provided that the intensity of the incident light is equal to $I_{6}$.
5.119. Light with wavelength $\lambda$ falls normally on a long rectangular slit of width b. Find the angular distribution of the intensity of light in the case of Fraunhofer diffraction, as well as the angular position of minima.
5.130. Making use of the result obtained in the foregoing problem, find the conditions defining the angular position of maxima of the first, the second, and the third onder.
5.121. Light with wavelength $\lambda=0.50 \mu \mathrm{m}$ falls on a slit of width $b=10 \mu \mathrm{m}$ at an angle $\theta_{0}=30^{\circ}$ to its normal. Find the angular position of the first minima located en both sides of the central Fraunhofer maximum.
5.122. A plane light wave with wavelength $\lambda=0.60 \mu \mathrm{m}$ falls normally on the face of a glass wedge with refracting angle $\theta=15^{\circ}$. The opposite face of the wedge is opaque and has a slit of width $b=10 \mu \mathrm{m}$ parallel to the edge. Find:
(a) the angle $\Delta \theta$ between the direction to the Fraunhofer maximum of reroth order and that of incident light:
(b) the angular width of the Fraunhofer maximum of the reroth onder.
5.123. A monochromatie beam falls on a reflection grating with period $d=1.0 \mathrm{~mm}$ at a glaneing angle $\alpha_{2}=1.0^{\circ}$. When it is diffracted at a glaneing angle $\alpha=3.0^{\circ}$ a Fraunhofer maximum of second onder eccurs. Find the wavelength of light.
5.124. Draw the approximate difraction pattern originating in the case of the Fraunholer diffractien from a grating consisting of three identical slits if the ratio of the grating period to the slit width is equal to
(a) two;
(b) three.
5.125. With light falling normally on a diffraction grating, the angle of diffraction of second onder is equal to $45^{\circ}$ for a wavelength
$\lambda_{1}=0.65 \mu \mathrm{m}$. Find the angle of diffraction of third order for a wave length $\lambda_{3}=0.50 \mu \mathrm{m}$.
5.126. Light with wavelength 535 am falls normally on a diffraction grating. Find its period if the diffraction angle $35^{\circ}$ corresponds to one of the Fraushofer maxima and the highest order of spectrum is equal to five.
5.i27. Find the wavelength of mosochromatic light falling normally on a diffraction grating with period $d=2.2 \mu \mathrm{m}$ if the angle between the directions to the Fraubhof maxima of the first and the second order is equal to $\Delta \theta=15^{\circ}$.
5.128. Light with wavelength $530 \mathrm{~mm}$ falls on a transparent diffraction grating with period $1.50 \mathrm{\mu m}$. Fiad the angle, relative to the grating normal, at which the Fraushofer maximum of highest onder is observed provided the light falls on the grating
(a) at right angles;
(b) at the angle $60^{\circ}$ to the sormal.
5.129. Light with wavelength $\lambda=0.60 \mu \mathrm{m}$ falls normally on a diffraction grating inscribed on a plase surface of a plano-convex cylindrical glass leas with curvature radius $\boldsymbol{R}=20 \mathrm{~cm}$. The period of the grating is equal to $d=6.0 \mu \mathrm{m}$. Find the distance between the priacipal maxima of first order located symmetrically in the focal plase of that less.
5.130. A plane light wave with wavelength $\lambda=0.50 \mu \mathrm{m}$ falls normally on the face of a glass wedge with an angle $\theta=30^{\circ}$. On the opposite face of the wedge a transparent diffraction grating with period $d=2.00 \mu \mathrm{m}$ is inscribed, whose lises are parallel to the wedge’s edge. Find the angles that the direction of incideat light forms with the directions to the priscipal Fraunhofer maxima of the zero and the first onder. What is the highest order of the spectrum? At what angle to the direction of iscident light is it observed?
5.131. A plane light wave with wavelength $\lambda$ 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 Fraushofer maximum is equal to zero. What is in this case the diffraction angle corresponding to the first maximum?
Fig. 5.26 .
Fie. 5.27.
5.132. Figure 5.27 illustrates an arrangement employed in observations of diffraction of light by ultrasound. A plase light wave with wavelength $\lambda=0.55 \mu \mathrm{m}$ passes through the water-filled tank $T$
in which a standiag ultrasonic wave is sustained at a frequency $v=4.7 \mathrm{MHz}$. As a result of diffraction of light by the optically inhomogeneous periodic structure a diffraction spectrum cas be observed in the focal plane of the objective $O$ with focal length $f=35 \mathrm{~cm}$. The separation between neighbouring maxima is $\Delta x=$ $=0.60 \mathrm{~mm}$. 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 twe narrow parallel slits separated by an adjustable distance $d$. While diminishing $d$. the frst smearing of the pattern was observed in the focal plane of the objective at $d=95 \mathrm{~cm}$. Find $\downarrow$, assuming the wavelength of light to be equal to $\lambda=0.55 \mu \mathrm{m}$.
5.13. A transparent diffraction grating has a period $d=1.50 \mu \mathrm{m}$. Find the angular dispersion $D$ (in angular miautes per nasometres) cerresponding to the maximum of highest order for a spectral line of wavelength $\lambda=530 \mathrm{~nm}$ of light falling on the grating
(a) at right angles;
(b) at the angle $6,=45^{\circ}$ to the normal.
5.135. Light with wavelength $\lambda$ falls en a diffraction grating at right angles. Find the angular dispersien of the grating as a function of diffraction angle $\theta$.
5.136. Light with wavelength $\lambda=589.0 \mathrm{~nm}$ falls normally on a diffraction grating with period $\bar{d}=2.5 \mu \mathrm{m}$, comprising $N=$ — 10000 lines. Fiad 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 $I / \lambda$, where $I$ is the width of the grating and $\lambda$ 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 diflerence of propagation times of the extreme interfering oscillations, i.e. $\delta v=$ $=1 / 8 t$.
5.139. Light composed of two spectral lines with wavelengths 600.000 and $600.050 \mathrm{~nm}$ falls normally on a diflraction grating $10.0 \mathrm{~mm}$ wide. At a certain diffraction angle $\theta$ these lines are close to being resolved (according to Rayleigh’s criterion). Find $\theta$.
5.140. Light falls normally on a transparent diffraction grating of width $t=6.5 \mathrm{~cm}$ with 200 lines per millimetre. The spectrum under investigation includes a spectral line with $\lambda=6 \% 0.8 \mathrm{~nm}$ consisting of two components differing by $8 \lambda=0.015$ sm. Find:
(a) in what order of the spectrum these components will be resolved:
(b) the least difference of wavelengths that can be resolved by this grating in a wavelength region $\lambda \approx 670 \mathrm{~nm}$.
5.141. With light falling normally on a transparent diffraction grating $10 \mathrm{~mm}$ wide, it was found that the components of the yellow line of sodium ( 589.0 and $589.6 \mathrm{sm}$ ) are resolved beginaing with the fifth order of the spectrum. Evaluate:
(a) the period of this grating:
(b) what mest be the width of the grating with the same period for $=$ doublet $\lambda-460.0 \mathrm{sm}$ whose components difler by $0.13 \mathrm{~nm}$ to be resolved in the third order of the spectrum.
5.142. A transparent diffraction grating of a quarts spectrograph is $25 \mathrm{~mm}$ wide and has 250 lises per millimetre. The focal length of an objective in whose focal plane a photographic plate is located is equal to $80 \mathrm{~cm}$. Light falls on the grating at right angles. The spectrum under isvestigation includes a doublet with components of wavelengths 310.154 and $310.184 \mathrm{~nm}$. Determine:
(a) the distances on the photographic plate between the components of this doublet in the spectra of the first and the second order;
(b) whether these componeats will be resolved in these orders of the spectrum.
5.143. The ultimate resolving power $2 / 8$ of the spectrograph’s trihedral prism is determined by diffraction of light at the prism edges (as in the case of a slit). Whea the prism is orieated to the least deviation angle is accordasce with Rayleigh’s criterion,
\[
\lambda / \Delta \lambda=b|d n / d \lambda| \text {. }
\]
where $b$ is the width of the prism’s base (Fig. 5.28), and dn/di is the dispersion of its material. Derive this formula.
Fie. 5.28.
5.144. A spectrograph’s trihedral prism is manufactured from glass whose refractive index varies with wavelength as $n=A+B / \lambda^{2}$, where $A$ and $B$ are constants, with $B$ being equal to $0.040 \mathrm{~mm}^{2}$. Making use of the formula from the forezoing problem, find:
(a) how the resolving power of the prism depends on $\lambda$; calculate the value of $N 8 \lambda$ in the vieinity of $\lambda_{1}=434 \mathrm{~mm}$ and $\lambda_{2}=656 \mathrm{sm}$ if the width of the prism’s base is $b=5.0 \mathrm{~cm}$;
(b) the width of the prism’s base capable of resolving the yellow doublet of sodium ( 589.0 and $589.6 \mathrm{~nm}$ ).
5.145. How wide is the base of a trihedral prism which has the same resolving power as a diffraction grating with 10000 lines in the second order of the spectrum if $\left|d_{\mathrm{n}} / d \lambda\right|=0.10 \mathrm{\mu m}-1$ ?
226
226
1
5.146. There is a telescope whose objective has a diameter $D=$ $=5.0 \mathrm{~cm}$. Find the resolving power of the objective and the minimum separation between two points at a distance $l=3.0 \mathrm{~km}$ from the telescope, which it can resolve (assume $\lambda=0.55 \mu \mathrm{m}$ ).
5.147. Calculate the minimum separation between two poiats on the Moon which cas be resolved by a reflecting telescope with mirror diameter $5 \mathrm{~m}$. The wavelength of light is assumed to be equa! to $\lambda=0.55 \mu \mathrm{m}$.
5.148. Determine the misimum multiplication of a telescope with diameter of objective $D=5.0 \mathrm{~cm}$ with which the resolving power of the objective is totally employed if the diameter of the eye’s pupil is $d_{0}=4.0 \mathrm{~mm}$.
5.149. There is a microscope whose objective’s aumerical aperture is $\sin \alpha=0.24$, where $\alpha$ is the balf-angle subtended by the objective’s rim. Find the minimum separation resolved by this microscope when an object is illuminated by light with wavelength $\lambda=0.55 \mu \mathrm{m}$.
5.150. Find the minimum magnification of a microscope, whose objective’s numerical aperture is $\sin \alpha=0.24$, at which the resolving power of the objective is totally employed if the diameter of the eye’s pupil is $d_{0}=4.0 \mathrm{~mm}$.
5.151. A beam of $\mathrm{X}$-rays with wavelength $\lambda$ falls at a glancing angle $60.0^{\circ}$ on a linear chain of scattering centres with period $a$. Find the angles of incidence corresponding to all diffraction maxima if $\lambda=2 a / 5$.
5.152. A beam of X-rays with wavelength $\lambda=40$ pmilis hormally on a plane rectangular array of scattering centres and produces a system of diffraction maxima (Fig. 5.29) on a plane screen removed from the array by a distance $l=10 \mathrm{~cm}$. Find the array periods a and $b$ along the $x$ and $y$ axes if the

Fie. 5.29 . distances between symmetrically located maxima of second order are equal to $\Delta x=60 \mathrm{~mm}$ (along the $z$ axis) and $\Delta y=40 \mathrm{~mm}$ (along the $y$ axis).
5.153. A beam of $X$-rays impinges on a three-dimensional rectangular array whose periods are $a, 6$, and $c$. The direction of the incident beam coincides with the direction along which the array period is equal to a. Find the directions to the diffraction maxima and the wavelengths at which these maxima will be observed.
5.154. A narrow beam of $X$-rays impinges on the aatural facet of a $\mathrm{NaCl}$ single crystal, whose density is $p=2.16 \mathrm{~g} / \mathrm{cm}^{3}$ at a glancing angle $a=60.0^{\circ}$. The mirror reflection from this facet produces a maximum of second order. Find the wavelength of radiation. 5.155. A beam of $X$-rays with wavelength $\lambda=174 \mathrm{pm}$ falls on the surface of a single crystal rotating about its axis which is parallel to its surface and perpendicular to the direction of the incideat beam. In this case the directions to the maxima of second and third order from the system of plases parallel to the surface of the single crystal form as angle $a=60^{\circ}$ between them. Find the corresponding interplanar distance.
5.156. On transmitting a beam of $\mathrm{X}$-rays with wavelength $\lambda=$ $=17.8 \mathrm{pm}$ through a polycrystalline specimen a system of diffraction rings is produced on a screen located at a distance $l=15 \mathrm{~cm}$ from the specimen. Determine the radius of the bright ring corresponding to second order of reflection from the system of planes with interplanar distance $d=155 \mathrm{pm}$.