As a result of studying this chapter, the student should: know
- concepts of wave and geometric optics;
- the concept of wave-particle duality;
- four laws of geometric optics;
- the concept of light interference, coherence, train;
- Huygens-Fresnel principle;
- calculation of the interference pattern of two sources;
- interference calculation in thin films;
- principles of clearing optics; be able to
- solve typical applications physical tasks on the laws of geometric optics and the interference of light;
own
- skills to use standard methods and models of mathematics in relation to the laws of geometric optics and interference of light;
- skills to use methods analytical geometry and vector algebra in relation to the laws of geometric optics and light interference;
- skills to conduct physical experiment, as well as processing the experimental results according to the laws of geometric optics and light interference.
Wave and geometric optics. Laws of geometric optics
Wave optics - branch of optics that describes the propagation of light, taking into account its wavelength electromagnetic nature. Within the framework of wave optics, Maxwell's theory made it possible to quite simply explain such optical phenomena, such as interference, diffraction, polarization, etc.
IN late XVII V. Two theories of light took shape: wave(promoted by R. Hooke and H. Huygens) and corpuscular(it was promoted by I. Newton). The wave theory perceives light as wave process, similar to elastic mechanical waves. According to the corpuscular (quantum) theory, light is a stream of particles (corpuscles) described by the laws of mechanics. Thus, the reflection of light can be considered similarly to the reflection of an elastic ball from a plane. For a long time the two theories of light were considered alternative. However, numerous experiments have shown that light in some experiments reveals wave properties, and in others - corpuscular. Therefore, at the beginning of the 20th century. It was recognized that light fundamentally has a dual nature - it has wave-particle duality.
But before presenting the main principles and results of wave optics, let us formulate elementary laws geometric optics.
Geometric optics- a branch of optics that studies the laws of light propagation in transparent media and the rules for constructing images when light passes through optical systems without taking into account its wave properties. In geometric optics the concept is introduced light beam, determining the direction of the flow of radiant energy. It is assumed that the propagation of light does not depend on the transverse dimensions of the light beam. In accordance with the laws of wave optics, this is true if the transverse size of the beam is much greater than the wavelength of light. Geometric optics can be considered as a limiting case of wave optics when the wavelength of light tends to zero. More precisely, the limits of applicability of geometric optics will be determined by studying the diffraction of light.
The basic laws of geometric optics were discovered experimentally long before the discovery physical nature Sveta. Let's formulate four law of geometric optics.
- 1. Law of rectilinear propagation of light:In an optically homogeneous medium, light propagates rectilinearly. This law is confirmed by the sharp shadow cast by a body when illuminated by a point source of light. Another example is when light from a distant source passes through a small hole to produce a narrow, straight beam of light. In this case, it is necessary that the hole size be much larger than the wavelength.
- 2. Law of independence of light beams:The effect produced by a single beam of light is independent of other beams. Thus, the illumination of a surface onto which several beams shine is equal to the sum of the illumination created by the individual beams. The exception is nonlinear optical effects, which can occur at high light intensities.
Rice. 26.1
3.Law of light reflection:incident and reflected rays (as well as perpendicular to the interface between two media, (plane of incidence) along different sides from the perpendicular. Reflection angle at equal to the angle of incidence a(Fig. 26.1):
4. Law of light refraction:incident and refracted rays (as well as perpendicular to the interface between two media, reconstructed at the point of incidence of the beam) lie in the same plane (plane of incidence) on opposite sides of the perpendicular.
The ratio of the sine of the angle of incidence a to the sine of the angle of refraction R there is a quantity, constant for two given environments(Fig. 26.1):
Here n is the refractive index of the second medium relative to the first.
The refractive index of a medium relative to vacuum is called absolute refractive index. Relative refractive index of two media equal to the ratio their absolute refractive indices:
The laws of reflection and refraction have an explanation in wave physics. Refraction is a consequence of changes in the speed of propagation of waves when passing from one medium to another. Physical meaning refractive index - the ratio of the speed of wave propagation in the first medium v ( to the speed of propagation in the second medium v2:
The absolute refractive index is equal to the ratio of the speed of light With in vacuum to the speed of light v in the environment:
A medium with a large absolute refractive index is called optically denser medium. When light passes from an optically denser medium to an optically less dense one, for example from glass to air ( n 2 may take place total reflection phenomenon, i.e. disappearance of the refracted ray. This phenomenon is observed at angles of incidence exceeding a certain critical angle and the one called limiting angle of total internal reflection. For the angle of incidence a = apr, the condition for the disappearance of the refracted ray is
If the second medium is air (p 2 ~ 1), then using formulas (26.2) and (26.3) it is convenient to write the formula for calculating the limiting angle of total internal reflection in the form
Where n = n x > 1 - absolute indicator refraction of the first medium. For the glass-air interface (P= 1.5) critical angle apr = 42°, for the water-air boundary (P= 1.33) and pr = 49°.
Most interesting application total internal reflection is the creation fiber light guides, which are thin (from several micrometers to several millimeters) arbitrarily curved threads made of optically transparent material (glass, quartz, plastic). Light incident on the end of the light guide can travel along it over long distances due to total internal reflection from the side surfaces. The light guide cannot be bent strongly, since with strong bending the condition of total internal reflection (26.7) is violated and the light partially exits the fiber through the side surface.
Note that the first, third and fourth laws of geometric optics can be derived from Fermat's principle(principle of least time): the propagation trajectory of a light beam corresponds to the shortest propagation time. And it's easy to show.
In conclusion, let's look at one of the fun problems in geometric optics - creating an invisibility cap. From an optical point of view, an invisibility cap could be a system for bending light rays around an object.
Making such a system using the law of light refraction is, in principle, not difficult; the main problem is combating the strong attenuation of light in the refractive system. That's why the best option there may be a system of a video recorder of an image behind the object and a television transmitter of this image in front of the object.
Some optical laws were already known before the nature of light was established. The basis of geometric optics is formed by four laws: 1) the law of rectilinear propagation of light; 2) the law of independence of light rays; 3) the law of light reflection; 4) the law of light refraction.
Law of rectilinear propagation of light: light propagates rectilinearly in an optically homogeneous medium. This law is approximate, since when light passes through very small holes, deviations from straightness are observed, the larger the smaller the hole.
Law of independence of light beams: the effect produced by a single beam does not depend on whether the remaining beams act simultaneously or are eliminated. The intersections of the rays do not prevent each of them from propagating independently of each other. By dividing the light beam into separate light beams, it can be shown that the action of the separated light beams is independent. This law is valid only when light intensities are not too high. At intensities achieved with lasers, the independence of the light rays is no longer respected.
Law of Reflection: the ray reflected from the interface between two media lies in the same plane with the incident ray and the perpendicular drawn to the interface at the point of incidence; The angle of reflection is equal to the angle of incidence.
Law of refraction: the incident ray, the refracted ray and the perpendicular drawn to the interface at the point of incidence lie in the same plane; the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant value for given media
sin i 1/sin i 2 = n 12 = n 2 / n 1, obviously sin i 1/sin i 2 = V 1 / V 2 , (1)
where n 12 – relative refractive index the second environment relative to the first. The relative refractive index of two media is equal to the ratio of their absolute refractive indices n 12 = n 2 / n 1.
The absolute refractive index of a medium is called. the value n equal to the ratio of the speed C of electromagnetic waves in a vacuum to their phase speed V in the medium:
A medium with a high optical refractive index is called. optically more dense.
From the symmetry of expression (1) it follows reversibility of light rays, the essence of which is that if you direct a light beam from the second medium to the first at an angle i 2, then the refracted ray in the first medium will exit at an angle i 1 . When light passes from an optically less dense medium to a more dense one, it turns out that sin i 1 > sin i 2, i.e. The angle of refraction is less than the angle of incidence of light, and vice versa. In the latter case, as the angle of incidence increases, the angle of refraction increases to a greater extent, so that at a certain limiting angle of incidence i the refraction angle becomes equal to π/2. Using the law of refraction, you can calculate the value of the limiting angle of incidence:
sin i pr /sin(π/2) = n 2 /n 1, whence i pr = arcsin n 2 /n 1 . (2)
In this limiting case, the refracted beam slides along the interface between the media. At angles of incidence i > i Since light does not penetrate deep into an optically less dense medium, the phenomenon occurs total internal reflection. Corner i called limit angle total internal reflection.
Phenomenon total internal reflection used in total reflection prisms, which are used in optical instruments: binoculars, periscopes, refractometers (devices that allow you to determine optical refractive indices), in light guides, which are thin, bendable threads (fibers) made of optically transparent material. Light incident on the end of the fiber at angles greater than the limiting one undergoes complete internal reflection and propagates only along the light-guide core. With the help of light guides, you can bend the path of the light beam in any way you like. Multicore light guides are used to transmit images. Explain the use of light guides.
To explain the law of refraction and curvature of rays when passing through optically inhomogeneous media, the concept is introduced optical beam path length
L = nS or L = ∫ndS,
respectively for homogeneous and inhomogeneous media.
In 1660, the French mathematician and physicist P. Fermat established extremity principle(Fermat's principle) for the optical path length of a ray propagating in inhomogeneous transparent media: the optical path length of a ray in a medium between two given points minimal, or in other words, light propagates along a path whose optical length is minimal.
Photometric quantities and their units. Photometry is a branch of physics that deals with measuring the intensity of light and its sources. 1.Energy quantities:
Radiation flux F e is a quantity numerically equal to the energy ratio W radiation by time t during which the radiation occurred:
F e = W/t, watt (W).
Energetic luminosity(emissivity) R e – a value equal to the ratio of the radiation flux F e emitted by the surface to the area S of the section through which this flux passes:
R e = F e / S, (W/m2)
those. represents the surface radiation flux density.
Energy luminous intensity (radiant intensity) I e is determined using the concept of a point light source - a source whose dimensions, compared to the distance to the observation site, can be neglected. The energy intensity of light I e is a value equal to the ratio of the radiation flux Ф e of the source to the solid angle ω within which this radiation propagates:
I e = F e /ω, (W/sr) - watt per steradian.
The intensity of light often depends on the direction of the radiation. If it does not depend on the direction of radiation, then such source called isotropic. For an isotropic source, the luminous intensity is
I e = F e /4π.
In the case of an extended source, we can talk about the luminous intensity of the element of its surface dS.
Energy brightness (radiance) IN e is a value equal to the ratio of the luminous energy intensity ΔI e of an element of the emitting surface to the area ΔS of the projection of this element onto a plane perpendicular to the direction of observation:
IN e = ΔI e / ΔS. (W/avg.m 2)
Energy illuminance(irradiance) E e characterizes the degree of illumination of the surface and is equal to the amount of radiation flux incident on a unit of illuminated surface. (W/m2.
2.Light values. In optical measurements, various radiation receivers are used, the spectral characteristics of their sensitivity to light of different wavelengths are different. The relative spectral sensitivity of the human eye V(λ) is shown in Fig. V(λ)
400 555 700 λ, nm
Therefore, light measurements, being subjective, differ from objective, energy ones, and light units are introduced for them, used only for visible light. The basic SI unit of light is luminous intensity - candela(cd), which is equal to the light intensity in a given direction of a source emitting monochromatic radiation with a frequency of 540 10 12 Hz, energetic force of light in this direction is 1/683 W/sr.
The definition of light units is similar to energy units. To measure light values, special instruments are used - photometers.
Light flow. The unit of luminous flux is lumen(lm). It is equal to the luminous flux emitted by an isotropic light source with an intensity of 1 cd within a solid angle of one steradian (with uniformity of the radiation field within the solid angle):
1 lm = 1 cd 1 sr.
It has been experimentally established that a luminous flux of 1 lm generated by radiation with a wavelength of λ = 555 nm corresponds to an energy flux of 0.00146 W. Luminous flux 1 lm generated by radiation with a different λ corresponds to an energy flux
F e = 0.00146/V(λ), W.
1 lm = 0.00146 W.
Illumination E- a value related to the ratio of the luminous flux F incident on a surface to the area S of this surface:
E= F/S, lux (lx).
1 lux is the illumination of a surface on 1 m 2 of which a luminous flux of 1 lm falls (1 lux = 1 lm/m 2).
Brightness R C (luminosity) of a luminous surface in a certain direction φ is a value equal to the ratio of the luminous intensity I in this direction to the area S of the projection of the luminous surface onto a plane perpendicular to this direction:
R C = I/(Scosφ). (cd/m2).
Definition 1
Optics- one of the branches of physics that studies the properties and physical nature of light, as well as its interactions with substances.
This section is divided into three parts below:
- geometric or, as it is also called, ray optics, which is based on the concept of light rays, which is where its name comes from;
- wave optics, studies phenomena in which the wave properties of light are manifested;
- Quantum optics considers such interactions of light with substances in which the corpuscular properties of light make themselves known.
In the current chapter we will consider two subsections of optics. Corpuscular properties lights will be discussed in chapter five.
Long before the understanding of the true physical nature of light arose, humanity already knew the basic laws of geometric optics.
Law of rectilinear propagation of light
Definition 1Law of rectilinear propagation of light states that in an optically homogeneous medium, light propagates in a straight line.
This is confirmed by the sharp shadows that are cast opaque bodies when illuminated using a light source of relatively small size, that is, the so-called “point source”.
Another proof is that famous experiment by the passage of light from a distant source through a small hole, resulting in a narrow beam of light. This experience brings us to the representation of a light ray in the form geometric line, along which light propagates.
Definition 2
It is worth noting the fact that the very concept of a light ray, together with the law of rectilinear propagation of light, loses all its meaning if the light passes through holes whose dimensions are similar to the wavelength.
Based on this, geometric optics, which is based on the definition of light rays, is the limiting case of wave optics at λ → 0, the scope of which will be considered in the section on light diffraction.
At the interface between two transparent media, light can be partially reflected in such a way that some of the light energy will be dissipated after reflection in a new direction, while the other will cross the boundary and continue its propagation in the second medium.
Law of Light Reflection
Definition 3Law of Light Reflection, is based on the fact that the incident and reflected rays, as well as the perpendicular to the interface between the two media, reconstructed at the point of incidence of the ray, are in the same plane (the plane of incidence). In this case, the angles of reflection and incidence, γ and α, respectively, are equal values.
Law of light refraction
Definition 4Law of light refraction, is based on the fact that the incident and refracted rays, as well as the perpendicular to the interface between two media, reconstructed at the point of incidence of the ray, lie in the same plane. sin ratio the angle of incidence α to the sin of the angle of refraction β is a value that is constant for the two given media:
sin α sin β = n .
The scientist W. Snell experimentally established the law of refraction in 1621.
Definition 5
Constant n – is relative indicator refraction of the second medium relative to the first.
Definition 6
The refractive index of a medium relative to vacuum is called - absolute refractive index.
Definition 7
Relative refractive index of two media is the ratio of the absolute refractive indices of these media, i.e.:
The laws of refraction and reflection find their meaning in wave physics. Based on its definitions, refraction is the result of the transformation of the speed of wave propagation during the transition between two media.
Definition 8
Physical meaning of the refractive index is the ratio of the speed of wave propagation in the first medium υ 1 to the speed in the second υ 2:
Definition 9
The absolute refractive index is equivalent to the ratio of the speed of light in a vacuum c to the speed of light v in the medium:
In Figure 3. 1 . 1 illustrates the laws of reflection and refraction of light.
Figure 3. 1 . 1 . Laws of reflection υ refraction: γ = α; n 1 sin α = n 2 sin β.
Definition 10
A medium whose absolute refractive index is smaller is optically less dense.
Definition 11
Under conditions of light transition from one medium inferior to optical density other (n 2< n 1) мы получаем возможность наблюдать явление исчезновения преломленного луча.
This phenomenon can be observed at angles of incidence that exceed a certain critical angle α p r. This angle is called the limiting angle of total internal reflection (see Fig. 3, 1, 2).
For the angle of incidence α = α p sin β = 1 ; sin valueα n r = n 2 n 1< 1 .
Provided that the second medium is air (n 2 ≈ 1), then the equality can be rewritten as: sin α p p = 1 n, where n = n 1 > 1 is the absolute refractive index of the first medium.
Under the conditions of the glass-air interface, where n = 1.5, the critical angle is α p r = 42 °, while for the water-air interface n = 1. 33, and α p p = 48 , 7 ° .
Figure 3. 1 . 2. Total internal reflection of light at the water-air interface; S – point light source.
The phenomenon of total internal reflection is widely used in many optical devices. One such device is a fiber light guide - thin, curved randomly, threads made of optically transparent material, inside of which light entering the end can spread over enormous distances. This invention became possible only thanks to the correct application of the phenomenon of total internal reflection from lateral surfaces (Fig. 3. 1. 3).
Definition 12
Fiber optics- This scientific and technical direction, based on the development and use of optical fibers.
Drawing 3 . 1 . 3 . Propagation of light in a fiber light guide. When the fiber is strongly bent, the law of total internal reflection is violated, and light partially exits the fiber through the side surface.
Drawing 3 . 1 . 4 . Model of reflection and refraction of light.
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The development of almost all optical devices and systems is based on the laws of light propagation. Some of them take into account the dual nature of light, some do not. Most general laws propagations of light that are not related to its nature are considered precisely in geometric optics. You will become familiar with these laws in this lesson.
Subject:Optics
Lesson: Laws of geometric optics
Geometric optics is the most ancient part optics as a science.
Geometric optics- this is a branch of optics in which issues of light propagation in various optical systems (lenses, prisms, etc.) are considered without considering the issue of the nature of light.
One of the basic concepts in optics and, in particular, in geometric optics, is the concept of a ray.
A light ray is a line along which light energy propagates.
Light beam- this is a beam of light, the thickness of which is much less than the distance over which it propagates. This definition is close, for example, to the definition material point, which is given in kinematics.
The first law of geometric optics(Law of rectilinear propagation of light): in a homogeneous transparent medium, light propagates in a straight line.
According to Fermat's theorem: light propagates in a direction in which the propagation time is minimal.
Second law of geometric optics(Laws of reflection):
1. The reflected beam lies in the same plane as the incident beam and perpendicular to the interface between the two media.
2. The angle of incidence is equal to the angle of reflection (see Fig. 1).
|
∟α = ∟β |
Rice. 1. Law of reflection
Third law of geometric optics(Law of refraction) (see Fig. 2)
1. The refracted ray lies in the same plane as the incident ray and the perpendicular restored to the point of incidence.
2. The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant value for these two media, which is called the refractive index ( n).
The intensity of the reflected and refracted beam depends on what the medium is and what the interface is.
Rice. 2. Law of refraction
The physical meaning of the refractive index:
The refractive index is relative since measurements are made relative to two media.
In the event that one of the media is vacuum:
WITH- speed of light in vacuum,
n is the absolute refractive index characterizing the medium relative to vacuum.
If light passes from an optically less dense medium to an optically denser medium, then the speed of light decreases.
An optically denser medium is a medium in which the speed of light is slower.
An optically less dense medium is a medium in which the speed of light is greater.
Exists limit angle refraction - the greatest angle of incidence of the beam at which refraction still occurs when the beam passes into a less dense medium. At angles of incidence greater than the limiting one, total internal reflection occurs (see Fig. 3).
Rice. 3. Law of total internal reflection
The limits of applicability of geometric optics lie in the fact that it is necessary to take into account the size of obstacles to light.
Light has a wavelength of approximately 10 -9 meters
If the obstacles are longer than the wavelength, then the dimensions of geometric optics can be used.
- Physics. 11th grade: Textbook for general education. institutions and schools with depth studying physics: profile level/ A.T. Glazunov, O.F. Kabardin, A.N. Malinin et al. Ed. A.A. Pinsky, O.F. Kabardina. Ross. acad. Sciences, Ross. acad. education. - M.: Education, 2009.
- Kasyanov V.A. Physics. 11th grade: Educational. for general education institutions. - M.: Bustard, 2005.
- Myakishev G.Ya. Physics: Textbook. for 11th grade general education institutions. - M.: Education, 2010.
- St. Petersburg School ().
- AYP.ru ().
- Technical and educational documentation ().
Rymkevich A.P. Physics. Problem book. 10-11 grades - M.: Bustard, 2010. - No. 1023, 1024, 1042, 1054.
- Knowing the speed of light in a vacuum, find the speed of light in a diamond.
- Why, sitting by the fire, do we see objects located opposite us oscillating?
- Comment on the experiment: put a coin on the table and put an empty one on it glass jar(see Fig. 4). Look at the side of the coin through the side of the jar (or have someone look at the coin). Pour a full jar of water and look again from the side at the bottom of the jar. Where did the coin go?
Geometric optics uses the concept of light rays propagating independently of each other, rectilinear in a homogeneous medium, reflected and refracted at the boundaries of media with different optical properties. The energy of light vibrations is transferred along the rays.
Refractive index of the medium. Optical properties transparent media are characterized by a refractive index that determines the speed (more precisely, the phase speed) of light waves:
where c is the speed of light in vacuum. The refractive index of air is close to unity (for water its value is 1.33, and for glass, depending on the type, it can range from 1.5 to 1.95. The refractive index of diamond is especially high - approximately 2.5.
The value of the refractive index, generally speaking, depends on the wavelength R (or on the frequency: This dependence is called the dispersion of light. For example, in crystal (lead glass) the refractive index smoothly changes from 1.87 for red light with a wavelength to 1.95 for blue light with
The refractive index is related to dielectric constant environment (for a given wavelength or frequency) by the relation Environment with great value refractive index is called optically denser.
Laws of geometric optics. The behavior of light rays obeys the basic laws of geometric optics.
1. In a homogeneous medium, light rays are rectilinear (law of rectilinear propagation of light).
2. At the boundary of two media (or at the boundary of a medium with a vacuum), a reflected ray appears, lying in the plane formed by the incident ray and the normal to the boundary, i.e., in the plane of incidence, and the angle of reflection is equal to the angle of incidence (Fig. 224):
(law of reflection, light).
3. The refracted ray lies in the plane of incidence (when light falls on the boundary of an isotropic medium) and forms an angle (angle of refraction) with the normal to the boundary, determined by the relation
(law of light refraction or Snell's law).
When light passes into an optically denser medium, the beam approaches the normal. The ratio is called the relative refractive index of two media (or the refractive index of the second medium relative to the first).
Rice. 224. Reflection and refraction of the sun on a flat boundary of two media
When light falls from a vacuum onto the boundary of a medium with a refractive index, the law of refraction takes the form
For air, the refractive index is close to unity; therefore, when light falls from air onto a certain medium, formula (4) can be used.
When light passes into an optically less dense medium, the angle of incidence cannot exceed the limit value since the angle of refraction cannot exceed (Fig. 225):
If the angle of incidence occurs total reflection, i.e. all the energy of the incident light returns to the first, optically denser, medium. For the glass-air boundary
Rice. 225. Limiting angle of total reflection
Huygens' principle and the laws of geometric optics. The laws of geometric optics were established long before the nature of light was clarified. These laws can be derived from wave theory based on Huygens' principle. Their applicability is limited by diffraction phenomena.
Let us dwell in more detail on the transition from wave concepts of the propagation of light to the concepts of geometric optics. Using Huygens' principle, from a given wave surface of an incident wave, it is possible to construct the wave surfaces of refracted and reflected waves. It should be taken into account that the light rays are perpendicular to the wave surfaces.
Consider a plane light wave incident from medium 1 (with refractive index) onto a flat interface with medium 2 (with refractive index at an angle (Fig. 226). The angle of incidence is the angle between the incident ray and the normal to the interface.
Rice. 226. Huygens' construction for the reflection and refraction of light
At the same time, this is the angle between the interface and the wave surface of the incident wave. Let at some moment this wave surface occupy a position. After some time, it will reach point B of the interface. During the same time, the secondary wave from point A, propagating in the medium X, will expand to a radius. Substituting here we get. Hence it is clear that the wave surface of the reflected wave, which is the envelope of all secondary spherical waves with centers on the segment, is inclined to the interface by an angle that is equal to ( equality of angles and follows from the equality right triangles and having a common hypotenuse and equal legs and Thus, the reflected ray perpendicular to the front of the reflected wave forms an angle with the normal equal to angle falls
Similarly, from this construction of Huygens one can obtain the law of refraction. In medium 2, secondary waves propagate with speed and therefore the spherical wave emerging from point A after time has a radius. Substituting here we find Dividing both sides of this equality by we arrive at the relation
which obviously coincides with the law of refraction (3), since the angle of inclination of the wave surface of the wave in medium 2 is at the same time the angle between the refracted ray and the normal to the interface (angle of refraction, Fig. 226).
Reflection and refraction on a curved surface. A plane wave is characterized by the property that its wave surfaces are unlimited planes, and the direction of its propagation and amplitude are the same everywhere. Often, electromagnetic waves that are not plane can be approximately considered to be plane over a small region of space. To do this, it is necessary that the amplitude and direction of propagation of the wave remain almost unchanged over distances of the order of the wavelength. Then we can also introduce the concept of rays, i.e. lines, the tangent to which at each point coincides with the direction of propagation of the wave. If, in this case, the interface between two media, for example, the surface of a lens, can be considered approximately flat at distances of the order of the wavelength, then the behavior of light rays at such a boundary will be described by the same laws of reflection and refraction.
The study of the laws of propagation of light waves in this case is the subject of geometric optics, since in this approximation the optical laws can be formulated in the language of geometry. Many optical phenomena, such as, for example, the passage of light through optical systems that form an image, can be considered based on the concept of light rays, completely abstracting from the wave nature of light. Therefore, the concepts of geometric optics are valid only to the extent that the phenomena of diffraction of light waves can be neglected. The shorter the wavelength, the weaker the effect of diffraction. This means that geometric optics corresponds to the limiting case of short wavelengths:
Physical model A beam of light rays can be obtained by passing light from a source of negligible size through a small hole in an opaque screen. The light emerging from the hole fills a certain area, and if the wavelength is negligible compared to the size of the hole, then at a short distance from it we can talk about a beam of light rays with a sharp boundary.
Intensity of reflected and refracted light. The laws of reflection and refraction only allow us to determine the direction of the corresponding light rays, but do not say anything about their intensity. Meanwhile, experience shows that the ratio of the intensities of the reflected and refracted rays into which the original beam is split at the interface strongly depends on the angle of incidence. For example, with normal incidence of light on the surface of glass, about 4% of the energy of the incident light beam is reflected, and when incident on the surface of water, only 2% is reflected. But with a grazing incidence, the surfaces of glass and water reflect almost all of the incident radiation. Thanks to this we can admire mirror reflections calm shores clear water mountain lakes.
Rice. 227. In natural singing, oscillations of sector E occur in all possible directions in a plane perpendicular to the beam
Natural light. light wave, like any electromagnetic wave, transverse: vector E lies in a plane perpendicular to the direction of propagation. Light emitted by ordinary sources (for example, hot bodies) is unpolarized. This means that in a light beam, oscillations of the vector E occur in all possible directions in a plane perpendicular to the direction of the beam (Fig. 227). This unpolarized light is called natural light. It can be thought of as an incoherent mixture of two light waves of equal intensity, linearly polarized in two mutually perpendicular directions. These directions can be chosen arbitrarily.
Polarization of light upon reflection. When studying the reflection of unpolarized light from the interface between media, it is convenient to choose one of two independent directions of the vector E in the plane of incidence, and the second - perpendicular to it. The conditions for reflection of these two waves turn out to be different: a wave whose vector E is perpendicular to the plane of incidence (i.e., parallel to the interface) at all angles of incidence (except 0 and 90°) is reflected more strongly. Therefore, the reflected light turns out to be partially polarized, and when reflected at a certain angle (for glass about 56°) it is completely polarized.
This circumstance is used to eliminate glare, for example when photographing a landscape with water surface. By properly selecting the orientation of a polarizing filter that transmits light vibrations of only a certain polarization, you can almost completely eliminate glare in a photograph.
Fermat's principle. The basic laws of geometric optics - the law of rectilinear propagation of light in a homogeneous medium, the laws of reflection and refraction of light at the interface between two media - can be obtained using Fermat's principle. According to this principle, the actual path of propagation of a monochromatic ray of light is the path for which the light requires an extreme (usually minimum) time to travel compared to any other close to it in a conceivable way between the same points.
Rice. 228. To the derivation of the law of light reflection from Fermat’s principle
Let's take for example the law of light reflection. It is immediately clear that it follows directly from Fermat's principle. Let a ray of light emerging from point A be reflected from a mirror at a certain point C and arrive at a given point B (Fig. 228). According to Fermat’s principle, the path traversed by light must be shorter than any other path along a close trajectory, for example. To find the position of the reflection point C, let’s plot an equal segment on the perpendicular from point A to the mirror and connect points A and B with a straight line segment.
The intersection of this segment with the surface of the mirror gives the position of point C. Indeed, it is easy to see that therefore the path of light from point A to point B equal to the segment The path of light from A to B through any other point will be equal to longer, since a straight line is the shortest distance between two points A and B. From Fig. 228 it is immediately clear that precisely this position of point C corresponds to the equality of the angles of incidence and reflection:
Rice. 229. Virtual image of point A in flat mirror
Image in a plane mirror. Point A, located symmetrically to point A relative to the surface of a flat mirror, is an image of point A in this mirror. Indeed, a narrow beam of rays emerging from
A, reflected in the mirror and entering the eye of the observer (Fig. 229), will appear to be coming out of point A. The image created by a flat mirror is called virtual, since at point A it is not the reflected rays themselves that intersect, but their backward extensions. Obviously, the image of an extended object in a flat mirror will be equal in size to the object itself.
What are light rays? How does this concept relate to the concept of a wave surface? What do rays have to do with the direction of propagation of light vibrations?
Under what conditions can the concept of light rays be used?
What is the refractive index of a medium? How is it related to the speed of light?
Formulate the basic laws of geometric optics. What is the plane of incidence? Explain, based on symmetry considerations, why the ray, both during reflection and refraction, does not leave this plane.
Under what conditions will the reflection of light at the interface be complete? What is the limiting angle of total reflection?
Explain how the laws of rectilinear propagation, reflection and refraction can be obtained based on Huygens' principle.
Why can the laws of reflection and refraction of light, formulated for a flat interface, also be applied in the case of curved surfaces (lenses, water drops, etc.)?
Give examples of phenomena you have observed that indicate the dependence of the intensity of reflected light on the angle of incidence.
Why when reflected natural light Does this result in partially polarized light?
State Fermat's principle and show that the law of reflection of light follows from it.
Prove that the image of an object in a plane mirror is equal in size to the object itself.
Fermat's principle and lens formula. The speed of light in a medium with a refractive index is Therefore, Fermat's principle can be formulated as a requirement for the minimum optical length of a ray when light propagates between two given points. The optical beam length is the product of the refractive index and the beam path length. In an inhomogeneous medium, the optical length is the sum of the optical lengths in individual areas. The use of this principle allows us to consider some problems from a slightly different point of view than with the direct application of the laws of reflection and refraction. For example, when considering a focusing optical system, instead of applying the law of refraction, one can simply require that the optical lengths of all rays be equal.
Using Fermat's principle, we obtain the formula thin lens without resorting to the law of refraction. To be specific, we will consider a biconvex lens with spherical refractive surfaces whose radii of curvature are equal (Fig. 230).
It is well known that with the help of a converging lens one can obtain a real image of a point. Let the object, its image. All rays emanating from and passing through the lens are collected at one point. Let it lie on the main optical axis of the lens, then the image also lies on the axis. What does it mean to obtain a lens formula? This means establishing a connection between the distances from the object to the lens and from the lens to the image and the quantities characterizing a given lens: the radii of curvature of its surfaces and the refractive index
From Fermat's principle it follows that the optical lengths of all rays emanating from a source and converging at a point that is its image are the same. Let's consider two of these rays: one going along the optical axis, the second through the edge of the lens (Fig. 230a).
Rice. 230. Towards the conclusion of the formula for a thin lens
Even though the second ray passes longer distance, its path in glass is shorter than that of the first, so the light propagation time is the same for them. Let's express this mathematically. The designations of the values of all segments are indicated in the figure. Let us equate the optical lengths of the first and second rays:
Let's express it using the Pythagorean theorem:
Now let’s use an approximate formula that is valid for, up to terms of order. Considering small compared to up to terms of order, we have
Similarly for we get
We substitute expressions (8) and (9) into the main relation (7) and present similar terms:
In this formula, in the case of a thin lens, we can neglect the values in the denominators on the right side in comparison with and it is obvious that on the left side of the expression should be preserved, since this term is a factor.
With the same accuracy as in formulas (8) and (9), using the Pythagorean theorem can be represented in the form (Fig. 230b)
Now all that remains is to substitute these expressions into left side formula (10) and reduce both sides of the equality by:
This is the desired formula for a thin lens. Introducing the designation
it can be rewritten in the form
Focal length of the lens. From formula (12) it is easy to understand that there is a focal length of a lens: if the source is at infinity (i.e., a parallel beam of rays falls on the lens), its image is in focus. Assuming we get
Aberrations. The resulting property of focusing a parallel beam of monochromatic rays is, as can be seen from the conclusion made, approximate and valid only for narrow beam, i.e. for rays not too far removed from the optical axis. For wide beams of rays, spherical aberration occurs, which manifests itself in the fact that rays far from the optical axis intersect it out of focus (Fig. 231). As a result, the image of an infinitely distant point source, created by a wide beam of rays refracted by the lens, turns out to be somewhat blurred.
In addition to spherical aberration, the lens as an optical device that forms an image has a number of other disadvantages.
For example, even a narrow parallel beam of monochromatic rays that forms a certain angle with the optical axis of the lens does not converge into one point after refraction. When using non-monochromatic light, the lens also exhibits chromatic aberration due to the fact that the refractive index depends on the wavelength. As a result, as can be seen from formula (11), a narrow parallel beam of white light rays intersects after refraction in the lens at more than one point: the rays of each color have their own focus.
When designing optical devices, it is possible to eliminate these disadvantages to a greater or lesser extent by using specially designed complex multi-lens systems. However, it is impossible to eliminate all shortcomings at the same time. Therefore, we have to compromise and, counting optical instruments, intended for specific purpose, seek to eliminate some shortcomings and put up with the presence of others. For example, lenses designed for observing low-brightness objects should transmit possible more light, which forces you to put up with some aberrations that are inevitable when using wide beams of light.
Rice. 231. Spherical aberration of a lens
For telescope lenses where the objects being studied are stars - point sources, located near the optical axis of the device, it is especially important to eliminate spherical and chromatic aberration for wide beams parallel to the optical axis. The easiest way to eliminate chromatic aberration is to use optical system reflection instead of refraction. Since rays of all wavelengths are reflected equally, a reflecting telescope, unlike a refractor, is completely free of chromatic aberration. If you also properly select the shape of the surface of the reflecting mirror, you can completely get rid of spherical aberration for beams parallel to the optical axis. To obtain a point axial image, the mirror must be parabolic.
By squaring both sides and bringing similar terms, we find
This is the equation of a parabola.
Rice. 232. All parallel rays after reflection from a parabolic mirror are collected at a point
Parabolic mirrors are used in all largest telescopes. These telescopes eliminate spherical and chromatic aberration; however, parallel beams, going even at small angles to the optical axis, do not intersect at one point after reflection and give highly distorted off-axis images. Therefore, the field of view suitable for work turns out to be very small, on the order of several tens of arc minutes,
Explain why, in relation to a focusing optical system, Fermat's principle is formulated as the condition for the equality of the optical lengths of all rays from an object point to its image.
Using Fermat's principle, derive the law of refraction of light at the interface between two media.
Formulate approximations that make the thin lens formula valid.
What are the manifestations of spherical and chromatic aberration of a lens?
What advantages and disadvantages does a parabolic mirror have over a spherical one?
Show that an elliptical mirror reflects all rays coming from one focus of the ellipsoid to another focus.