Wave optics theory. wave optics

Optics is a branch of physics that studies the propagation of light and its interaction with matter. Light is electromagnetic radiation and has a dual nature. In some phenomena, light behaves like an electromagnetic wave, in others it behaves like a stream of special particles of photons or light quanta. Wave optics deals with the wave properties of light, quantum - quantum.

Light is the photon flux. From the point of view of wave optics, a light wave is a process of oscillation of electric and magnetic fields propagating in space.

Optics deals with light waves, mainly infrared, visible, ultraviolet ranges. As an electromagnetic wave, light has the following properties (they follow from Maxwell's equation):

The vectors of the electric field E, magnetic field H, and wave propagation velocity V are mutually perpendicular and form a right-handed system.

The vectors E and H oscillate in the same phase.

The following condition is satisfied for the wave:

The light wave equation has , where is the wave number, is the radius vector, and is the initial phase.

When a light wave interacts with matter, the electric component of the wave plays the greatest role (the magnetic component outside magnetic media affects less), therefore E is called light vector and its amplitude denote A.

Equation (1) is a solution to the wave equation, which has the form:

(2), where is the Laplacian; V is the phase velocity V=c/n(3).

For non-magnetic media =1 =>. From (3) it can be seen that n=c/v. According to the type of wave surface, flat, spherical, elliptical, etc. are distinguished. waves.

For a plane wave, the amplitude of the light vector of equation (1) is constant. For a spherical one, it decreases with distance from the source according to the law .

The energy transfer of a light wave is characterized by the Pointig vector.

It represents the density of the energy flux and is directed in speed - in the direction of its transfer. The vector S changes very quickly with time, so any radiation receiver, including the eye, during an observation time much longer than the wave period, registers the time-averaged value of the Pointig vector, which is called light wave intensity., where. Taking into account (1) and the fact that Hono has the same form, we can write that (4)

If we average equation (4) over time, then the second term will disappear, then (5). From (5) it follows that I-(6).

IntensityI- this is the amount of energy transferred per unit time by a light wave through a unit area. The line along which the wave energy propagates is called beam. Another characteristic of a light wave is its polarization. The real source consists of a huge number of atoms that emit, being excited, during t=10 -8 s, while emitting a wave fragment λ=3m.

These waves have different directions of the vector E in space; therefore, different directions of the vector E occur in the resulting radiation during the observation time, i.e. direction E for a real source changes randomly in time, and the light from such a source is called natural (non-polarized). If the direction of oscillations of the vector E is ordered, then such light is polarized. Distinguish light plane polarized, polarized in a circle and ellipse.

WAVE OPTICS

WAVE OPTICS

Physical section optics, which studies the totality of phenomena in which waves appear. the nature of the world. Ideas about waves. The nature of the propagation of light goes back to the fundamental work of Goll. scientist 2nd floor. 17th century X. Huygens. Creatures. V.'s development about. received in the studies of T. Jung (Great Britain), O. Fresnel, D. Arago (France) and others, when fundamental experiments were carried out that made it possible not only to observe, but also to explain the phenomena of light interference, light diffraction, measure the length, establish the transverse light vibrations and reveal other features of the propagation of light waves. But to match the transverse light waves with DOS. V.'s idea about. about the propagation of elastic oscillations in an isotropic medium, it was necessary to endow this medium (world) with a number of requirements that are difficult to reconcile with each other. Ch. some of these difficulties were resolved in con. 19th century English physicist J. Maxwell in the analysis of ur-tions connecting fast-changing electric. and magn. fields. In the works of Maxwell, a new V. o., el.-magn. theory of light, with the help of which it turned out to be a very simple explanation of a number of phenomena, for example. polarization of light and quantities. relations during the transition of light from one transparent dielectric to another (see FRESNEL FORMULA). The use of el.-mag. theories in various V.'s tasks about. showed agreement with experiment. So, for example, the phenomenon of light pressure was predicted, the existence of which was proved by PN Lebedev (1899). Supplement el.-mag. theory of light model representations of the electronic theory (see LORENTZ - MAXWELL EQUATIONS) made it possible to simply explain the dependence of the refractive index on the wavelength (dispersion of light) and other effects.

Further expansion of the boundaries of V. o. occurred as a result of applying the ideas of special. theory of relativity (see RELATIVITY THEORY), experiment. substantiation of a cut was associated with thin optical. experiments, in to-ryh DOS. the role played relates. source and receiver of light (see MICHAELSON'S EXPERIENCE). The development of these ideas made it possible to exclude from consideration the world ether not only as a medium in which e-mags propagate. waves, but also as an abstract frame of reference.

However, an analysis of experimental data on equilibrium thermal radiation and the photoelectric effect showed that V. o. has a definition. application boundaries. The distribution of energy in the spectrum of thermal radiation was explained by him. physicist M. Planck (1900), who came to the conclusion that the elementary oscillates. the system radiates and absorbs energy not continuously, but in portions - quanta. The development of quantum theory by A. Einstein led to the creation of photon physics - a new corpuscular optics, edges, complementing the el.-mag. theory of light, fully corresponds to the generally accepted ideas about the dualism of light.

Physical Encyclopedic Dictionary. - M.: Soviet Encyclopedia. Editor-in-Chief A. M. Prokhorov. 1983 .

See what "WAVE OPTICS" is in other dictionaries:

Wave optics is a branch of optics that describes the propagation of light in terms of its wave nature. The phenomena of wave optics are interference, diffraction, polarization, etc. See also Wave optics in nature Links ... Wikipedia

A branch of physical optics that studies the totality of such phenomena as light diffraction, light interference, light polarization, in which the wave nature of light is manifested ... Big Encyclopedic Dictionary

wave optics- - [L.G. Sumenko. English Russian Dictionary of Information Technologies. M .: GP TsNIIS, 2003.] Topics information technology in general EN physical optics ... Technical Translator's Handbook

A branch of physical optics that studies a set of phenomena in which the wave nature of light is manifested, such as light diffraction, light interference, light polarization. * * * WAVE OPTICS WAVE OPTICS, a section of physical optics that studies ... ... encyclopedic Dictionary

wave optics- banginė optika statusas T sritis fizika atitikmenys: angl. wave optics vok. Wellenoptik, f rus. wave optics, f pranc. optique d'ondes, f; optique ondulatoire, f … Fizikos terminų žodynas

Physical section optics, which studies the totality of phenomena in which the wave nature of light is manifested, such as diffraction of light, interference of light, polarization of light ... Natural science. encyclopedic Dictionary

The style of this article is not encyclopedic or violates the norms of the Russian language. The article should be corrected according to the stylistic rules of Wikipedia. Contents ... Wikipedia

Quantum mechanics ... Wikipedia

Table "Optics" from the encyclopedia of 1728 O ... Wikipedia

Wave optics- a branch of physical optics that studies the totality of phenomena in which the wave nature of light is manifested. The first works of X. Huygens (1629 1695) 2nd half. 17th century Wave optics received significant development in the studies of T. Young (1773 1829), O. ... ... Concepts of modern natural science. Glossary of basic terms

Books

• Wave optics Fifth stereotypical edition, Kaliteevsky N. In the textbook by N. I. Kaliteevsky "Wave optics" the foundations of the electromagnetic theory of light are considered. Due attention is paid to the experiment. Presentation of the properties of electromagnetic waves ...

Wave optics – branch of optics that studies the totality
phenomena in which the wave nature of light is manifested.
Huygens' principle - each point reached by
wave serves as the center of secondary waves, and the envelope of these
waves gives the position of the wavefront in the next
point in time (wavefront - geometric place
points to which oscillations reach by time t).
This principle is the basis of wave optics.

Law of reflection

A plane wave is incident on the interface between two media
(wave front - plane AB), propagating
along direction I.
When the wave front reaches the reflecting surface
at point A, this point will begin to radiate a secondary wave.
For the wave to travel the distance BC, it is required
time t = BC/v.

Law of reflection

During the same time, the front of the secondary wave will reach the points
hemisphere, the radius AD of which is equal to v t = BC.
The position of the reflected wave front at this moment

plane DC, and the direction of propagation of this
waves - beam II. From the equality of triangles ABC and ADC
the law of reflection follows: the reflection angle i1/ is equal to the angle
falling i1

Law of refraction

Plane wave (wave front - plane AB),
propagating in vacuum along the direction I
the speed of light c, falls on the interface with the medium, in
which the speed of its propagation is equal to v.
If the time taken by the wave to travel the path
BC is equal to t , then BC = c t. During the same time, the wave front
excited by point A in the medium with speed v, will reach
points of the hemisphere, the radius of which is AD = v t.

Law of refraction

During the same time, the wave front excited by point A in the medium
with speed v, will reach the points of the hemisphere, the radius of which is AD =
vt. The position of the refracted wave front at this moment
time in accordance with the Huygens principle is given
plane DC, and the direction of its propagation - beam III.
It follows from the figure that

coherence

Correlation is called coherence.
(consistency) of several vibrational or wave
processes in time, which manifests itself when they are added.
Oscillations are coherent if their phase difference is constant during
time and when adding oscillations, an oscillation is obtained
the same frequency.
The classic example of two coherent oscillations is
two sinusoidal oscillations of the same frequency.
Wave coherence means that
at various spatial points
oscillation waves occur
synchronous, i.e. phase difference
between two points does not depend
from time.

INTERFERENCE OF MONOCHROMATIC LIGHT

Light interference is a special case of a general phenomenon
wave interference, which consists in the spatial
redistribution of energy of light radiation at
superpositions of coherent electromagnetic waves.

Stackable monochromatic light waves
(vectors of electric field strengths of waves E1 and
E2) at the observation point they oscillate along one
straight.
The amplitude of the resulting oscillation in
the point under consideration.

The intensity of the resulting wave
Intensity in case of in-phase
oscillations (phases f1 and f2 are the same or different
for an even number)
Intensity in case of antiphase
oscillations (phases f1 and f2 differ by an odd number)

Optical path length between two points of the medium -
distance over which light (optical radiation)
would spread in a vacuum during its passage
between these points
Optical path difference - the difference between the optical
length of paths light travels
Phase difference of two coherent light waves ()
Relationship between phase difference and optical path difference
.

Conditions for interference maxima and minima

OBTAINING COHERENT BEAMS BY WAVE-FRONT DIVISION

Young's method
The role of secondary coherent sources S1 and S2 is played by two
narrow slits illuminated by a single small-angle source
size, and in later experiments, light was passed through
narrow slot S, equidistant from
two other slots. The interference pattern is observed
in the area of ​​overlapping light beams emanating from S1 and S2.

Fresnel mirrors
Light from source S is incident by a divergent beam on two
flat mirrors A1O and A2O, located relative to each other
friend at an angle only slightly different from 180° (angle φ
small).
The source and its images S1 and S2 (angular distance between
equal to 2φ) lie on the same circle of radius r with
centered at O ​​(the point of contact of the mirrors).
Light beams, reflected from mirrors, form two imaginary
source images S1 and S2, which act as
coherent sources (obtained by splitting the same
wave front,
outgoing from S).
interference pattern
observed in the region of mutual
overlapping of reflected beams
(screen E is protected from direct
light entry damper 3).

Fresnel biprism
Formed by two identical folded bases
prisms with small refractive angles. light from
point source S is refracted in both prisms, in
as a result of which light propagates behind the biprism
rays, as if emanating from imaginary sources S1 and S2,
being coherent. In the shaded figure
areas - areas of intersection of refracted fronts -
an interference pattern is observed.

Lloyd's Mirror
The point source S is at a very close
distance to the surface of a flat mirror M, so the light
reflected by the mirror at an angle close to the sliding angle.
The coherent sources are the primary source S and
its imaginary image S1 in the mirror.

Interference pattern from two coherent sources

Two narrow slots S1 and S2 are located close to each other and
are coherent sources - real or
imaginary images of the source in some optical
system. The result of interference is at some point A
screen parallel to both slots and located from
them at a distance l(l > > d). The origin is selected at the point
Oh, symmetrical with respect to the slots.

Optical path difference (see construction and l > > d).
Intensity maxima (taking into account the condition
interference maximum).
Intensity minima (taking into account the condition
interference minimum).
Interference fringe width (distance between
two adjacent maxima (or minima)).

The emergence of interference maxima and minima from the point of view of wave theory

OBTAINING COHERENT BEAMS BY AMPLITUDE DIVISION

Monochromatic light from a point source S, incident
on a thin transparent plane-parallel plate (see Fig.
figure), is reflected by two surfaces of this plate:
top and bottom. To any point P located with that
same side of the plate as S, two beams come, which
give an interference pattern. On the record
there is a division of the amplitude, since the wave fronts on
it is preserved, changing only the direction of its
movement.

Interference from a plane-parallel plate
Beams 1 and 2 going from S to P (point P on the screen,
located in the focal plane of the lens) are generated
by one incident beam and after reflection from the top and
the lower surfaces of the plate are parallel to each other.
If the optical path difference between beams 1 and 2 is small in
compared with the coherence length of the incident wave, then
they are coherent, and the interference pattern
determined by the optical path difference between
interfering beams.

Optical path difference between interfering
rays from point O to plane AB

interference maxima
in reflected light correspond to
lows in the passing, and
vice versa (optical difference
move for passing and
reflected light
differs by 0/2).

Interference from a plate of variable thickness
On the wedge (angle a between the side faces
small) a plane wave is falling (let its direction
propagation coincides with parallel beams 1 and 2).
At a certain mutual position of the wedge and lens
rays 1" and 1" reflected from the top and bottom
surfaces of the wedge intersect at some point A,
which is the image of point B. Since the rays 1 "and 1"
coherent, then
they will
interfere.

Beams 2 "and 2", formed during the division of beam 2,
falling to another point of the wedge, are collected by a lens at the point
A". The optical path difference is already determined by the thickness
d". A system of interference fringes appears on the screen.
If the source is located far from the surface of the wedge, and
angle a is negligible, then the optical path difference between
interfering beams is calculated quite accurately
according to the formula for a plane-parallel plate
cant

Newton's rings
Observed when light is reflected from an air gap,
formed by a plane-parallel plate and
a plano-convex lens in contact with it
with a large radius of curvature.
A parallel beam of light is incident on a flat surface
lenses are normal; strips of equal thickness look like
concentric circles.

SOME APPLICATIONS OF INTERFERENCE

Enlightenment of optics
This is the minimization of reflection coefficients
surfaces of optical systems by applying
transparent films, the thickness of which is commensurate with the length
waves of optical radiation.
Film thickness d and refractive indices
films (n) and glasses (nc) are selected so that
interfering
beams 1" and 2"
extinguished each other.

INTERFEROMETERS

Optical instruments that can
spatially split a beam of light into two or more
number of coherent beams and create between them
a certain travel difference. Bringing these bundles together
observe interference.

Diffraction of light

Diffraction of light - a set of phenomena observed during
propagation of light through small apertures
boundaries of opaque bodies, etc. and due to wave
the nature of the world.
The phenomenon of diffraction, common to all wave processes,
has features for light, namely here, as a rule,
the wavelength is much smaller than the dimensions d of the barriers (or
holes).
So watch
diffraction can
just enough
large distances I from
barriers (I > d2/).

Huygens-Fresnel principle
The light wave excited by the source S can be
presented as the result of a superposition of coherent
secondary waves "radiated" by fictitious sources.

Huygens-Fresnel principle

Fraunhofer diffraction

Fresnel zones

Zone plates

In the simplest case, glass plates
the surface of which is applied according to the principle of location
Fresnel zones alternating transparent and opaque
rings with radii determined for given values
a, b and expression

If we place the zone
plate in strictly
certain place (at
distance a from the point
source and at a distance b from
observation points on the line,
connecting these two points), then
it is for wavelength light
will block the even zones and
leave free odd,
starting from the center.
As a result, the resulting
amplitude A = A1 + A3 + A5 + ...
should be more than
fully open wave
front. Experience confirms these
conclusions: zone plate
increases illumination,
acting like a gatherer
lens.

FRESNEL DIFFRACTION

Fresnel diffraction (diffraction in converging beams)
Refers to the case when an obstacle falls
spherical or plane wave, and the diffraction pattern
observed on a screen behind an obstacle on
finite distance from it.

Diffraction at a circular hole

there is a screen with a round hole.
The diffraction pattern is observed at point B of screen E,
lying on the line connecting S with the center of the hole.
The screen is parallel to the hole.

Analysis of results. The type of diffraction pattern depends on
the number of Fresnel zones that fit on the open part of the wave
surface in the plane of the hole. Amplitude of the resulting
oscillations excited at point B by all zones
(the "plus" sign corresponds to odd m, "minus" to even m).
If the hole opens an even number of Fresnel zones, then at point B
there is a minimum, if odd, then a maximum. Least
intensity corresponds to two open Fresnel zones,
maximum - one Fresnel zone.

Diffraction by a circular disk

On the path of a spherical wave from a point source S
there is a round opaque disc. Diffractive
the picture is observed at point B of screen E, which lies on the line
connecting S with the center of the disk. The screen is parallel to the disk.

Analysis of results. The section of the wave covered by the disk
front must be excluded from consideration and the Fresnel zone
build starting from the edges of the disk.
If the disk covers m Fresnel zones, then the amplitude
the resulting oscillation at point B is equal to
i.e. equal to half the amplitude due to the first
open Fresnel zone. Therefore, at point B always
there is a maximum - a bright spot, called
Poisson's spot, the brightness of which with increasing size
disk is reduced.

FRUNHOFER DIFFRACTION (PARALLEL BEAM DIFFRACTION)

Refers to the case where the light source and point
observations are infinitely distant from the obstacle,
causing diffraction. Practically enough for this
place a point light source at the focus of the collecting
lenses, and study the diffraction pattern in the focal
plane of the second converging lens installed behind
an obstacle.

Fraunhofer diffraction by a slit

normal to the plane of the slot of width a.
Parallel beams of rays emerging from a slit in
arbitrary direction φ (φ - angle
diffraction) are collected by a lens at point B.

Construction of Fresnel zones

The open part of the wave surface MN in the slot plane
divided into Fresnel zones, having the form of stripes,
parallel to the edge M and drawn so that the difference
travel from their respective points was /2.
Optical path difference between the extreme beams MN and
N.D.
The number of Fresnel zones that fit within the slit width.
The condition of the diffraction minimum at point B
(the number of Fresnel zones is even).
The condition of the diffraction maximum at point B
(the number of Fresnel zones is odd).

Diffraction spectrum

Dependence of the intensity distribution on the screen on the angle
diffraction. Most of the light energy is concentrated in
central maximum. With increasing diffraction angle
the intensity of side maxima sharply decreases
(relative intensity of maxima
I0:I1:I2: ... = 1: 0.047: 0.017: ...).
When illuminated with white light, the central maximum has
view of a white stripe (it is common for all wavelengths), lateral
the maxima are rainbow colored.

Influence of the Slit Width on the Diffraction Pattern

Decreasing
slot width
central
maximum expands
(see figure a), c
width increase
cracks (a>)
diffractive
stripes become narrower
and brighter (see figure b).

Diffraction at two slits

Plane monochromatic light wave is incident
normal to a screen with two identical slits (MN and
CD) width a, spaced from each other at a distance b;
(a + b) = d.

Diffraction pattern on two slits

between the two main maxima is an additional
minimum, and the maxima become narrower than in the case of one
cracks.

Diffraction grating

One-dimensional diffraction grating
A system of parallel slots (strokes) of equal thickness,
lying in the same plane and separated by equal
width at opaque intervals.
Constant (period) of a diffraction grating
The total width of the slot a and the opaque gap b
between the cracks.

Diffraction pattern on a grating

The result of mutual interference of waves coming from all
slots, i.e., multipath interference is carried out
coherent diffracted beams of light coming from all
cracks.

The more slots in
grating, the more
light energy will pass through
lattice, the more minima
formed between neighboring main
maxima, i.e. the maxima will be
more intense and sharper.
The maximum order of the spectrum,
given by a diffraction grating

SPATIAL GRID. X-RAY DIFFRACTION

Spatial formations in which elements
structures are similar in shape, have geometric
correct and periodically repeating arrangement,
as well as dimensions commensurate with the wavelength
electromagnetic radiation.
In other words, such spatial formations
must have a periodicity in three not lying in one
plane directions. As a spatial
lattices crystals can be used.
The distance between atoms in a crystal (10-10 m) is such that
they can show x-ray diffraction
radiation (10-12-10-8 m), since for observation
diffraction pattern requires commensurability
lattice constant with the wavelength of the incident radiation.

X-ray diffraction on a crystal

A beam of monochromatic X-ray radiation (on
the figure shows parallel beams 1 and 2) is incident on
crystal surface at the glancing angle (the angle between
incident beam and crystallographic plane) and
excites the atoms of the crystal lattice, which
become sources of coherent secondary waves 1" and 2",
interfering with each other. Result of interference
waves is determined by their path difference 2d sin (see figure).

Wulf-Bragg formula

Diffraction maxima are observed in those
directions in which all reflected atomic
planes, the waves are in the same phase (in
directions determined by the Wulf-Bragg formula)
.

RESOLUTION OF OPTICAL INSTRUMENTS

Because light has a wave nature,
created by an optical system (even an ideal one!)
the image of a point source is not a point, but
is a bright spot surrounded by
alternating dark and light rings (in the case of
monochromatic light) or iridescent rings (in
case of white light).
Therefore, a fundamentally unavoidable phenomenon
diffraction limits the possible resolution
abilities of optical instruments - abilities
optical instruments to give a separate image of two
close to each other points of the object.

Rayleigh criterion

Images of two nearby identical dots
sources or two nearby spectral lines with
equal intensities and identical symmetrical
contours are resolvable (separated for perception) if
central maximum of the diffraction pattern from one
source (line) coincides with the first minimum
diffraction pattern from another.

DIFFRACTION GRATING AS A SPECTRAL INSTRUMENT

The position of the main maxima in the diffraction grating
depends on the wavelength:
Therefore, when white light is passed through the grating, all
maxima, except for the central one (m = 0), expand into
spectrum, the violet region of which will be facing
center of the diffraction pattern, red - outward.
This property is used to study the spectral
composition of light (determining wavelengths and intensities
all monochromatic components), i.e. diffractive
grating can be used as a spectral
device.

Characteristics of a diffraction grating

Angular dispersion characterizes the degree of stretching
spectrum in the region near a given wavelength
Resolution

Light dispersion

Dependence of the phase velocity of light in a medium on its frequency.
Since v \u003d c / n, then the refractive index of the medium
turns out to be frequency (wavelength) dependent.

The refractive index dispersion indicates how quickly
the refractive index n changes with the wavelength.

Prism as a spectral device

The angle of deflection of the rays by the prism
n is a function of wavelength, so rays of different wavelengths
after passing through the prism will be deflected by
different angles, i.e. the beam of white light behind the prism decomposes
into the spectrum (prismatic spectrum)

Differences in diffraction and prismatic spectra

Diffraction grating
Prism
Decomposes incident light
straight to length
waves, therefore, according to the measured
corners (in directions
maxima) can
calculate the wavelength.
Red beams are deflected
stronger than purple
(red rays have
longer wavelength than
purple.
Breaks down incident light into
indicator values
refraction, so it is necessary
know addiction
refraction of concrete
substances from wavelength
Red beams are deflected
weaker than purple
as for red rays
refractive index
smaller.

Dispersion curves

Dispersion formula (excluding attenuation for
vibrations of one optical electron)

Dispersion formula (without attenuation) for
vibrations of several optical electrons

ABSORPTION (ABSORPTION) OF LIGHT

The phenomenon of a decrease in the energy of a light wave when it
distribution in matter due to transformation
wave energy into other forms of energy.

Bouguer-Lambert law

LIGHT SCATTERING

This is the process of transforming light into matter,
accompanied by a change of direction
propagation of light and the appearance of an improper
luminosity of matter.
Scattering of light in turbid and clean media
Tyndall effect
Molecular scattering

Rayleigh's law

The scattered light intensity is inversely proportional to
the fourth power of the wavelength of the exciting light.
The law describes the Tyndall effect and molecular scattering.
According to Rayleigh's law, the scattered light intensity is inversely
proportional to the fourth power of the wavelength, so blue
and blue rays scatter more than yellow and red,
causing the blue color of the sky. For the same reason, light
passed through a considerable thickness of the atmosphere, it turns out
enriched with longer wavelengths (blue-violet part
spectrum is completely scattered), and therefore at sunset and sunrise
The sun appears red.
Density Fluctuations and Light Scattering Intensity
increase with increasing temperature. Therefore, on a clear summer
day the color of the sky is more saturated compared to this
same winter day.

VAVILOV-CHERENKOV RADIATION

Emission of light by charged particles
when moving in a medium with a constant speed V,
exceeding the phase velocity in this medium as well, i.e., at
condition
(n is the refractive index).
Observed for all transparent
liquids, gases and solids.

Substantiation of the possibility of the existence of Vavilov-Cherenkov radiation

Possibility justification
the existence of the Vavilov radiation
Cherenkov
According to electromagnetic theory, a charged particle
e.g. an electron emits electromagnetic waves
only when moving fast.
Tamm and Frank showed that this is true only up to
as long as the velocity V of the charged particle does not exceed
phase velocity v = c/n of electromagnetic waves in the medium, in
which the particle is moving.
According to Tamm and Frank, if the speed of an electron moving in
transparent medium exceeds the phase velocity of light in
given medium, the electron emits light.
Radiation does not propagate in all directions, but
only for those that form an acute angle with
particle trajectory (along the generators of the cone, the axis
which coincides with the direction of particle velocity).

An electron moves in a medium with a velocity V > v = c/n along
trajectory AE (see figure).
Each point (for example, points A, B, C, D) of the trajectory ABC
charged particle in an optically isotropic medium is
source of a spherical wave propagating with
speed v = c/n.
Any subsequent point is excited with a delay,
therefore, the radii of spherical waves successively
decrease. According to the Huygens principle, as a result
interference these elementary waves
extinguish each other everywhere except
their envelope surface
(wave surface)
with a vertex at point E, where at a given
moment is an electron.

Justification of the directivity of Vavilov-Cherenkov radiation using the Huygens principle

If, for example, an electron traveled the path AE in 1 s, then the light
the wave has traveled the path of AA during this time."
Therefore, the segments AE and AA" are respectively equal to V and v
= c/n.
Triangle AA "E - rectangular with a right angle y
vertices A". Then
The spheres intersect only when
charged particle is moving faster
than light
waves, and then their wave surface
is a cone with a vertex
at the point where it is currently
electron.

Doppler effect for electromagnetic waves in vacuum

0 and - respectively, the frequencies of light waves emitted
source and perceived by the receiver; v - speed
light source relative to the receiver; - angle between
velocity vector v and observation direction,
measured in the reference frame associated with the observer;
c - speed of light propagation in vacuum

Longitudinal Doppler effect

Transverse Doppler effect

Light polarization

The set of phenomena of wave optics, in which
manifests the transverseness of electromagnetic light
waves (according to Maxwell's theory, light waves
transverse: electric strength vectors E
and magnetic H fields of the light wave are mutually
perpendicular and oscillate perpendicular
velocity vector v of wave propagation
(perpendicular to the beam)). Insofar as
for polarization, it suffices to study the behavior
only one of them, namely the vector E, which
is called the light vector.

polarized light
Light in which the direction of oscillation of the light vector
sorted in some way.
natural light
Light with all possible equally probable directions
oscillations of the vector E (and hence H).
Partially polarized light
Light with predominant (but not exclusive!)
the direction of oscillation of the vector E.

Plane polarized (linearly polarized) light
Light in which the vector E (and hence H) oscillates
only in one direction, perpendicular to the beam.
Elliptically polarized light
Light for which the vector E changes with time so that
that its end describes an ellipse lying in a plane,
perpendicular to the beam.
Elliptically polarized light is the most common type
polarized light.

Obtaining plane polarized light

Obtained by passing natural light through polarizers
P, which are media that are anisotropic in
with respect to the oscillations of the vector E (for example, crystals, in
especially tourmaline). Polarizers allow vibrations to pass through
parallel to the main plane of the polarizer, and
completely or partially delay vibrations,
perpendicular to her.

Malus' law

The intensity of light passing through
polarizer and analyzer, proportional to the square
cosine of the angle between their principal planes.

Passage of natural light through two polarizers

Intensity of plane polarized light emitted
from the first polarizer
Intensity of light passing through the second polarizer
Intensity of light passing through two polarizers
Degree of polarization

POLARIZATION OF LIGHT IN REFLECTION AND REFRACTION

The phenomenon of light polarization
Isolation of light waves with specific directions
oscillations of the electric vector - observed at
reflection and refraction of light at the boundary of transparent
isotropic dielectrics.

Reflection and refraction of light at an interface

If the angle of incidence of natural light on the interface,
for example, air and glass, is different from zero, then the reflected
and refracted rays are partially polarized.
In the reflected beam, vibrations predominate,
perpendicular to the plane of incidence (in the figure they
indicated by dots), in the refracted beam - vibrations,
parallel to the plane of incidence
(in the figure, these oscillations
shown by arrows).
Degree of polarization
depends on the angle of incidence.

Brewster's Law

At the angle of incidence of natural light on the boundary
transparent isotropic dielectrics, equal to the angle
Brewster iB defined by the relation
the reflected beam is completely polarized (contains only
vibrations perpendicular to the plane of incidence),
the refracted beam is polarized to the maximum, but not
fully.

Natural light incidence at Brewster's angle

When natural light falls at the Brewster angle iB
reflected and refracted rays mutually
are perpendicular.

POLARIZATION AT BIBREFRONT

Birefringence - the ability of anisotropic
substances to split the incident light beam into two beams,
propagating in different directions with different
phase velocity and polarized in mutually

Uniaxial and biaxial crystals

Anisotropy of substances - dependence of physical properties
substances from direction.
The optical axis of the crystal is the direction in the optical
anisotropic crystal, which propagates
a beam of light without experiencing double refraction.
Uniaxial and biaxial crystals - crystals with one
or two directions along which there is no
double refraction.
The main plane of a uniaxial crystal is a plane,
passing through the direction of the light beam and the optical
crystal axis.

Birefringence in Icelandic spar (uniaxial crystal)

When a narrow light beam falls on a sufficiently thick
the crystal comes out of it two spatially separated
rays parallel to each other - ordinary (o) and
extraordinary (e).

Birefringence in a uniaxial crystal under normal incidence of light

If the primary beam is normally incident on the crystal, then
anyway, the refracted beam is divided into two: one of
them is a continuation of the primary - ordinary
ray (o), and the second is deflected - an extraordinary ray (e). both e-rays are fully polarized in mutually
perpendicular directions.

On the edge of a crystal cut in the form of a plate,
normally incident plane polarized light.
The extraordinary ray (e) in the crystal is deflected and exits
from it parallel to an ordinary ray (o). Both beams on
screen E give light circles o and e (see figure a).
If the crystal is rotated around an axis coinciding with
direction of the o-ray, then the o-circle on the screen will remain
motionless, and the e-circle moves around it along
circle.

Ordinary and extraordinary rays with double refraction

The brightness of both circles changes. If the o-beam reaches
maximum brightness, then the e-ray "disappears", and vice versa.
The sum of the brightnesses of both beams remains constant. So if
e- and o-beams overlap (see figure b), then during rotation
crystal, the brightness of each of the circles changes, and the area
overlap all the time equally bright.

Spherical wave surface

Oscillations of the vector E in any direction
ordinary beam are perpendicular to the optical axis
crystal (its direction is given by the dotted line), so the ray propagates in the crystal in all directions with
the same speed v0 = c/n0.
Let us assume that at point S of the crystal a point source
light emits a light wave, o Ray in the crystal
propagates with the speed v0 = const, so the wave
the surface of an ordinary ray is a sphere.

Ellipsoidal wave surface

For an e-beam, the angle between the direction of oscillation of the vector E and
optical axis is different from the direct one and depends on
beam direction, so the e-ray propagates in
crystal in different directions at different speeds
ve = c/ne. If at point S a point source emits
light wave, then the e-ray in the crystal propagates with
speed ve const, and therefore the wave surface
extraordinary ray - ellipsoid. Along the optical axis
v0 = ve; the greatest discrepancy in speeds - in
direction,
perpendicular
optical axis.

positive crystal

negative crystal

A plane wave is incident normally to a refracting face
positive uniaxial crystal (optical axis OO "
forms an angle with it).
With centers at points A and B, we construct spherical wave
surfaces corresponding to an ordinary ray, and
ellipsoidal - extraordinary ray.
At a point lying on 00, these surfaces are in contact.

Direction of o- and e-rays in a crystal according to the Huygens principle

According to Huygens' principle, the surface tangent to
spheres, will be the front (a-a) of an ordinary wave, and
surface tangent to ellipsoids - front (b-b)
extraordinary wave.
Drawing straight lines to the points of contact, we obtain the directions
distribution of ordinary (o) and extraordinary (e)
rays. As follows from the figure, the o-beam will go along
original direction, and the e-beam deviates from
original direction.

POLARIZERS

Acquisition, detection and analysis devices
polarized light, as well as for research and
measurements based on the phenomenon of polarization. Them
typical representatives are polarizing
prisms and polaroids.
Polarizing prisms are divided into two classes:
giving one plane polarized beam of rays -
single-beam polarizing prisms;
giving two beams of rays polarized in mutually
perpendicular planes, - two-beam
polarizing prisms.

Double Icelandic spar prism glued lengthwise
AB lines with Canadian balsam with n = 1.55.
The optical axis of the OO "prism is with the input face
angle 48°. On the front face of the prism is a natural beam,
parallel to the CB edge, bifurcates into two rays:
ordinary (n0 = 1.66) and extraordinary (ne = 1.51).

Single-beam polarizing prism (Nicol prism, or nicol)

With an appropriate selection of the angle of incidence, equal to or
is greater than the limit, the o-ray experiences total reflection, and
then absorbed by the blackened CB surface. e-beam
leaves the crystal parallel to the incident beam,
slightly offset relative to it (due to
refraction at faces AC and BD).

Double-beam polarizing prism (Icelandic spar and glass prism)

The difference in the refractive indices of the o- and ray rays is used to separate them as far as possible from each other.
An ordinary ray is refracted twice and strongly
is rejected. An extraordinary ray with an appropriate
selection of the refractive index of glass n (n = ne) passes
prism without deflection.

tourmaline crystals

Polarizers whose action is based on the phenomenon
dichroism - selective absorption of light in
depending on the direction of oscillation of the electric
light wave vector.

Polaroids

Films on which, for example, crystals are deposited
herapatitis - a birefringent substance with a strong
pronounced dichroism in the visible region. Apply
to produce plane polarized light.
So, with a thickness of 0.1 mm, such a film is completely
absorbs ordinary rays of the visible region of the spectrum,
being a good polarizer in a thin layer
(analyzer).

A beam of natural light passing through a polarizer
P and becoming plane-polarized at the output, normally
falls on a crystalline plate of thickness d,
cut from a uniaxial negative crystal
parallel to its optical axis OO". Inside the plate, it
divided into ordinary (o) and extraordinary (e)
rays that propagate
in one direction
(perpendicular
optical axis),
but with different
speeds.

Obtaining elliptically polarized light

Oscillations of the vector E in the e-beam occur along the optical
axis of the crystal, and in the o-beam - perpendicular to the optical
axes.
Let the electric vector E of the output from the polarizer
plane polarized beam is with the optical axis
OO" crystal angle a.
Amplitude values ​​of electric vectors in
ordinary (Eo1) and extraordinary (Ee1) rays:

Obtaining elliptically polarized light

The optical path difference of o- and e-beams that have passed through the crystal
plate thickness d.
The phase difference between the oscillations of the o- and e-rays at the output of the plate.
Amplitude values ​​of electric vectors Ee and Eo in e- and o-beams,
passed through the crystalline plate.
The trajectory of the resulting vibration when added mutually
perpendicular oscillations with different amplitudes and phase difference
(t was excluded from the two previous equations)

The passage of plane polarized light through a plate

ANALYSIS OF POLARIZED LIGHT

Plane polarized light
When rotating the analyzer (A) around the beam direction
the light intensity changes, and if at some
position A, the light is completely extinguished, then the light -
plane polarized.

analyzer, the intensity of the transmitted light is not
changes.

Circularly polarized light
In circularly polarized light, the phase difference φ between
any two mutually perpendicular oscillations is equal to
±/2. If a plate "/4" is placed in the path of this light, then
it will introduce an additional phase difference of ±/2. Resultant
phase difference will be 0 or.
Then, at the exit from the plate, the light is plane polarized and
can be extinguished by turning the analyzer.
If the incident light is natural, then during rotation
analyzer at any position of the plate "/4"
intensity does not change. If complete extinction is not achieved, then
incident light - mixture of natural and circular
polarized.

Elliptically polarized light
If in the path of elliptically polarized light we place
plate "/4", the optical axis of which is oriented
parallel to one of the axes of the ellipse, then it will introduce
additional phase difference ± /2. Resultant
phase difference will be 0 or. Then at the exit from the plate
light is plane polarized and can be extinguished
turning the analyzer.
If the incident light is partially polarized, then at
rotation of the analyzer at any position of the plate
intensity varies from
minimum to maximum
but complete extinction is not achieved.

INTERFERENCE OF POLARIZED LIGHT

It has been experimentally proven that coherent rays,
polarized in two mutually perpendicular
planes do not interfere. Interference
observed only when fluctuations in
interacting rays are made along one
directions. So ordinary and extraordinary
rays emerging from the crystal plate, although
are coherent and there is a difference between them
phases, depending on the distance traveled by them in
plate, they cannot interfere, because they
polarized in mutually perpendicular planes.
To observe the interference of polarized
rays, it is necessary to select components from both rays with
the same direction of vibration.

Selection of components with the same vibration directions

A crystalline plate cut from a uniaxial
crystal parallel to the optical axis OO", is placed
between polarizer P and analyzer A. Parallel
the beam of light at the exit from R turns into
plane polarized.
In a crystal plate, o- and e-rays propagate in
direction of fall, but at different speeds.
Analyzer A transmits oscillations polarized in
same plane: electric vectors emerging from
analyzer A o- and e-beams oscillate along
in the same direction, i.e., interference is possible.

ARTIFICIAL OPTICAL ANISOTROPY

The message of optical anisotropy is natural
isotropic substances, if they are subjected to
mechanical stress, are placed in
electric or magnetic field.
As a result, the substance acquires the properties of a uniaxial
crystal, the optical axis of which coincides
according to the directions of deformation,
electric or magnetic fields.

Obtaining Optically Anisotropic Substances

Kerr effect

Optical anisotropy of transparent substances under
exposure to a uniform electric field.
Mechanism of the Kerr effect
Due to the different polarizability of molecules
dielectric in different directions. Electrical
field orients polar molecules along the field and
induces an electric moment in non-polar molecules.]
Therefore, the refractive indices (hence, and
velocity of propagation in the matter of waves,
polarized along and perpendicular] to the vector
electric field strength) become
different k, birefringence occurs.

Kerr cell

Cuvette with liquid containing plates
capacitor, placed between the crossed
polarizer and analyzer.
In the absence of an electric field, light through the system does not
passes. When applied, the environment becomes
anisotropic, and the light leaving the cell is elliptical
polarized and partially passes through the analyzer.

The phase difference φ arising between the ordinary and extraordinary rays

Measured by placing in front of the analyzer
compensator (a device with which the difference
travel between the two beams is reduced to zero).

Rotation of the plane of polarization (or optical activity)

The ability of certain substances (quartz, sugar, water
sugar solution, turpentine, etc.) in the absence of external
influences to rotate the plane of polarization (plane,
passing through the electric vector E and the light beam).
Substances that rotate the plane of polarization are called
optically active.

Observation of the rotation of the plane of polarization

Plane polarized light exiting the polarizer
passes through the sugar solution.
Crossed polarizer and analyzer behind the cuvette with
solution does not completely extinguish the light. If A turn to
angle φ, then complete extinction of light occurs. Hence,
light after passing through the system remains
plane polarized, but the solution rotates the plane
polarization of light by an angle φ.

Angle of rotation of the plane of polarization

Optically active crystals and pure liquids
Optically active solutions
Optical activity is due to both the structure of molecules
substances (their asymmetry), and features
arrangement of particles in the crystal lattice.

Right- and left-hand optically active substances

dextrorotatory substances

towards the beam, turns to the right (clockwise).
Left-handed substances
Substances whose plane of polarization, when viewed
towards the beam, turns to the left (counterclockwise
arrows).

Light interference- the phenomenon of redistribution of the light flux in space when two (or several) coherent light waves are superimposed, as a result of which maxima appear in some places, and intensity minima in others.

coherent called waves, the phase difference of which does not change either in space or in time. The condition for the maximum intensity for the phase difference ; minimum condition

.

To obtain coherent light waves, methods are used to separate a wave emitted by one source into two or more parts, which, after passing through different optical paths, are superimposed on each other.

Let the separation into two coherent waves occur at a certain point O. Before the point M, at which the interference pattern is observed, one wave in a medium with a refractive index n 1 traveled the path S 1 , the second - in a medium with a refractive index n 2 – path S 2 . The phase difference of the oscillations excited by the waves at the point M is equal to

.

The product of the geometric length S of the path of a light wave in a given medium by the exponent n refraction of this medium is called the optical path length L, a  = (L 2 – L 1 ) - the difference in the optical lengths of the paths traversed by the waves - is called the optical path difference. We take into account that /c=2v/c=2/ 0 , where  0 is the wavelength in vacuum.

Interference maximum condition: the optical path difference is equal to an integer number of waves and the oscillations excited at point M by both waves will occur in the same phase  = ± m , where ( m = 0, 1, 2,...).

Interference minimum condition: the optical path difference is equal to a half-integer number of waves and the oscillations excited at the point M by the waves will occur in antiphase
, where ( m = 0, 1, 2,...).

Position of illumination maxima when observing interference from Young slits X max = ±t (l/ d)  , where m is the order of the maximum, d- distance between slots, l – distance to the screen; lows x min = ± (m+1/2)(l/ d)  .

The distance between two adjacent minima, called the width of the interference fringe, is  x = (l/ d)  .

And interferencein thinfilms:

optical path difference

,

G
de n is the relative refractive index of the film, φ is the angle of incidence of light. The term ±/2 is due to the loss of a half-wave when light is reflected from the interface. If a n> n 0 (n 0 is the refractive index of the medium in which the film is located), then the loss of a half-wave will occur upon reflection from the upper surface of the film, and the above term will have a minus sign if n< n 0 , then the loss of the half-wave will occur on the lower surface of the film, and /2 will have a plus sign.

Radii of dark rings in reflected and light Newton's rings in transmitted light
, where m = 1, 2,.. is the ring number, R is the radius of curvature of the lens.

Wave diffraction: a light wave bending around the boundaries of opaque bodies with the formation of an interference redistribution of energy in various directions.

P
Huygens-Fresnel principle: each point of the wave front is a source of waves propagating with a characteristic speed for a given medium. The envelope of these waves gives the position of the wave front at the next moment in time. All points of the wave front oscillate with the same frequency and in the same phase and, therefore, represent a set of coherent sources. Accounting for the amplitudes and phases of the secondary waves makes it possible to find the amplitude of the resulting wave at any point in space.

Fresnel diffraction(from the spherical wave front).

Fresnel zone radii:
, where a is the distance from the source to the screen, b is the distance from the screen with a hole to the diffraction observation screen, m = 1,2,3...

If an even number of Fresnel zones passes through the hole, then a dark spot is observed in the center of the diffraction pattern, if an odd number, then a bright one.

Fraunhofer diffraction(from a flat wave front).

The condition for observing diffraction minima from one slit
(t = 1, 2, 3…).

Diffraction grating is a system of periodically repeating inhomogeneities.

Lattice periodd is the distance between the axes of two adjacent slots.

The condition of the main diffraction maxima from the diffraction grating
, (t= 1, 2, 3…).

Grating angular dispersion
it is equal to

The resolution of the diffraction grating determines the interval δλ, in which two closely spaced wavelengths of the spectrum λ 1 and λ 2 are perceived as separate lines:
, where N is the total number of grating slits that light enters during diffraction.

Polarized light is called light, in which the directions of oscillations of the light vector are somehow ordered. The plane passing through the direction of oscillation of the light vector E plane-polarized wave and the direction of propagation of this wave is called the plane of oscillation, and the plane of oscillation of the vector H called the plane of polarization. Plane polarized light is the limiting case of elliptically polarized light—light for which the vector E (vector H ) changes with time so that its end describes an ellipse lying in a plane perpendicular to the beam. If the polarization ellipse degenerates into a straight line (when the phase difference  is equal to zero or ), then we are dealing with the plane-polarized light considered above, if into a circle (when =±/2 and the amplitudes of the combined waves are equal), then we are dealing with polarized around the world.

The degree of polarization is the quantity
,where I max and I min - maximum and minimum light intensity corresponding to two mutually perpendicular components of the vector E. For natural light I max= I min and R= 0, for plane polarized I min = 0 and R = 1.

LawMalus: I = I 0 cos 2 , where I 0 is the intensity of polarized light incident on the analyzer; α is the angle between the transmission planes of the polarizer and analyzer, I is the intensity of polarized light leaving the analyzer.

When light falls on the dielectric surface at an angle satisfying the relation tgi B = n 21, where n 21 is the refractive index of the second medium relative to the first, the reflected beam is plane polarized (contains only oscillations perpendicular to the plane of incidence). The refracted beam at the angle of incidence i B (Brewster angle) is polarized to the maximum, but not completely.

Brewster's Law: i B + β = π/2 , where β is the angle of refraction.

WAVE OPTICS, a branch of physical optics that studies phenomena associated with the wave nature of light. The wave nature of the propagation of light was established by H. Huygens in the second half of the 17th century. Wave optics received significant development in the studies of T. Young, O. Fresnel, D. Arago, when experiments were carried out that made it possible not only to observe, but also to explain the interference, diffraction and polarization of light, which geometric optics could not explain. Wave optics considers the propagation of light waves in various media, reflection and refraction of light at the boundaries of media (see Fresnel formulas), dispersion and scattering of light in matter, etc. Light waves, which are oscillations of an electromagnetic field, are described by the general equations of classical electrodynamics (see Maxwell equations ). These equations are supplemented by the equations of quantum mechanics, which relate the values ​​of the dielectric and magnetic permeability to the molecular structure and properties of matter. This approach makes it possible to study wave optical phenomena in various media (see Crystal optics, Magneto optics, Molecular optics). Features of the propagation of light waves in moving media (see Electrodynamics of moving media), as well as in strong gravitational fields, are explained in the special and general relativity theory. Wave optics, using the classical description of the light field, is not able to give a consistent explanation of the processes of emission and absorption of light, which requires the introduction of ideas about light quanta - photons (see Quantum optics, Corpuscular-wave dualism). A number of problems in wave optics are also solved with a simpler description of the light field using the wave equation.

Wave optics establishes the limits of applicability of geometric optics and provides a mathematical justification for the relations used in it (the eikonal equation, the Fermat principle, etc.). In the intermediate region, when the wavelength of light is much smaller than the geometrical dimensions of the optical system, but at the same time the diffraction distortions of the beams are significant, methods of quasi-optics are used.

Wave phenomena in nonlinear media are considered in nonlinear optics. The propagation of light waves in randomly inhomogeneous media, including the atmosphere, is studied by the methods of statistical optics. Modern wave optics studies the formation of coherent light beams in the optical resonators of lasers and the transformation of beams by the methods of holography, Fourier optics, and adaptive optics. Rapidly developing areas are also studies of nonlinear optical phenomena in optical fibers (see Fiber optics) and in planar (film) optical systems (see Integrated optics).

Lit. see at st. Optics.