Basics of geometric optics for dummies. Reversibility of light rays

Considering in the previous paragraph the phenomena that occur when light falls on the interface between two media, we assumed that light propagates in a certain direction, indicated in Fig. 180, 181 arrows. Let us now pose the question: what will happen if the light propagates in the opposite direction? For the case of light reflection, this means that the incident beam will not be directed downwards from the left, as in Fig. 182, a, and from the right downwards, as in Fig. 182, b; for the case of refraction, we will consider the passage of light not from the first medium to the second, as in Fig. 182, c, and from the second environment to the first, as in Fig. 182, g,

Accurate measurements show that both in the case of reflection and in the case of refraction, the angles between the rays and the perpendicular to the interface remain unchanged, only the direction of the arrows changes. Thus, if the light beam falls in the direction (Fig. 182, b), then the reflected beam will go in the direction, i.e. it turns out that, compared to the first case, the incident and reflected beams have swapped places. The same is observed during the refraction of a light beam. Let - an incident ray, - a refracted ray (Fig. 182, c). If the light falls in the direction (Fig. 182, d), then the refracted ray goes in the direction, i.e., the incident and refracted rays exchange places.

Rice. 182. Reversibility of light rays during reflection (a, b) and refraction (c, d). If , then

Thus, both during reflection and refraction, light can travel the same path in both directions opposite to each other (Fig. 183). This property of light is called the reversibility of light rays.

The reversibility of light rays means that if the refractive index upon transition from the first medium to the second is equal to , then when passing from the second medium to the first it is equal. Indeed, let light fall at an angle and be refracted at an angle, so that . If, during the reverse course of the rays, light falls at an angle, then it must be refracted at an angle (reversibility). In this case, the refractive index is therefore . For example, when a beam passes from air to glass, and when it passes from glass to air . The property of reversibility of light rays is also preserved during multiple reflections and refractions, which can occur in any sequence. This follows from the fact that with each reflection or refraction the direction of the light ray can be reversed.

Rice. 183. To the reversibility of light rays during refraction

Thus, if, when a light beam emerges from any system of refractive and reflective media, the light beam is forced at the last stage to be reflected exactly back, then it will pass through the entire system in the opposite direction and return to its source.

The reversibility of the direction of light rays can be theoretically proven using the laws of refraction and reflection and without resorting to new experiments. For the case of light reflection, the proof is quite simple (see Exercise 22 at the end of this chapter). A more complex proof for the case of light refraction can be found in optics textbooks.

All laws of geometric optics follow from the law of conservation of energy. All these laws are not independent of each other.

4.3.1. Law of independent propagation of rays

If several rays pass through a point in space, then each ray behaves as if there were no other rays

This is true for linear optics, where the refractive index does not depend on the amplitude and intensity of the transmitted light.

4.3.2. Law of reversibility

The trajectory and path length of the rays do not depend on the direction of propagation.

That is, if a ray that propagates from point to point is launched in reverse (from to), then it will have the same trajectory as in the forward one.

4.3.3. Law of rectilinear propagation

In a homogeneous medium, the rays are straight lines (see paragraph 4.2.1).

4.3.4. Law of refraction and reflection

The law of reflection and refraction is discussed in detail in Chapter 3. Within the framework of geometric optics, the formulations of the laws of refraction and reflection are preserved.

4.3.5. The principle of tautochronism

Let's consider the propagation of light as the propagation of wave fronts (Fig. 4.3.1).

The optical length of any beam between two wavefronts is the same:

Wave fronts are surfaces that are optically parallel to each other. This is also true for the propagation of wave fronts in inhomogeneous media.

4.3.6. Fermat's principle

Let there be two points and , located, possibly, in different environments. These points can be connected to each other by various lines. Among these lines there will be only one, which will be an optical beam that propagates in accordance with the laws of geometric optics (Fig. 4.3.2).

Fig.4.3.2. Fermat's principle.

Fermat's principle:

The optical beam length between two points is minimal compared to all other lines connecting those two points:

There is a more complete formulation:

The optical length of a ray between two points is stationary with respect to the offset of that line.

Ray is the shortest distance between two points. If the line along which we measure the distance between two points differs from the ray by an amount of 1st order of smallness, then the optical length of this line differs from the optical length of the ray by an amount of 2nd order of smallness.

If the optical length of the ray connecting two points is divided by the speed of light, we obtain the time required to cover the distance between two points:

Another formulation of Fermat's principle:

The ray connecting two points follows the path that requires the least time (the fastest path).

From this principle the laws of refraction, reflection, etc. can be derived.

4.3.7 Malus-Dupin law

Normal congruence retains the properties of normal congruence as it passes through various media.

4.3.8 Invariants

Invariants(from the word unchanging) are relationships, expressions that retain their appearance when any conditions change, for example, when light passes through various media or systems.

Integral Lagrange invariant

Let there be some normal congruence (a beam of rays), and two arbitrary points in space and (Fig. 4.3.4). Let's connect these two points with an arbitrary line and find the curvilinear integral.

Fig.4.3.3. Integral Lagrange invariant.

Differential Lagrange invariant

A ray in space is completely described by a radius vector, which contains three linear coordinates, and an optical vector, which contains three angular coordinates. In total, therefore, there are 6 parameters for defining a certain ray in space. However, of these 6 parameters, only 4 are independent, since two equations can be obtained that relate the beam parameters to each other.

The first equation determines the length of the optical vector:

Where is the refractive index of the medium.

The second equation follows from the condition of orthogonality of vectors and:

From expressions (4.3.5) and (4.3.6), using analytical geometry, we can derive the following relationship:

Differential Lagrange invariant:
The quantity retains its value for a given ray when a beam of rays propagates through any set of optical media.

The geometric factor remains invariant when the ray tube propagates through any sequence of different media (Fig. 4.3.5).

The Straubel invariant expresses the law of conservation of energy, since it shows the invariance of the radiant flux.

From the definition of brightness we can obtain the following equality:

(4.3.9) where is the reduced brightness, which is invariant, as already mentioned in Chapter 2.

“Diffraction of light” is a violation of the law of rectilinear wave propagation. Wave optics Diffraction of light. Thus, the wave, after passing through the slit, both expands and deforms. Diffraction by a round hole. Thank you for your attention! Diffraction gratings are used to split electromagnetic radiation into a spectrum.

“Dispersion of light” - The experience described is, in fact, ancient. If you stand facing the rainbow, the Sun will be behind you. Rainbow. The multi-colored stripe is the solar spectrum. Discovery of the phenomenon of dispersion. Ideas about the causes of colors before Newton. Let's consider the refraction of a ray in a prism. Dispersion of light. Rainbow through the eyes of an attentive observer.

“Laws of Light” - Tasks: Mirror. Light laws: Light is visible radiation. Purpose: The presentation was prepared by Gildenbrandt Liliya Viktorovna. Artificial. Light refraction. Law of light reflection. "Information technology in. The work was carried out within the framework of the project.

“Reflection of light” - The first law of geometric optics states that light propagates in a straight line in a homogeneous medium. So, using light rays, you can depict the direction of propagation of light energy. Reflection of light. 5. Laws of reflection. The second law of geometric optics states: the angle of incidence is equal to the angle of reflection, i.e. ?? = ??.

“Diffraction and interference of light” - From the path difference: ?max = 2k. ?/2 – interference maximum?мin = (2k+1) . ?/2 – interference minimum. Addition of wave waves on the surface of a liquid. ?min = (2k+1) . ?/2. ?max = 2k. ?/2. Coherent waves. Observation of interference in thin films. The result of adding waves depends. Interference of light.

“Propagation of light” - D - the distance from the object to the lens. Quantities. Light refraction. Use when solving problems. Rectilinear propagation of light. Test tasks. Astronomical method. Optical instruments. Total reflection. Camera (1837) Projection apparatus Microscope Telescope. Camera. Further. Converging lens (a) Diffusing lens (b).

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 a 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 light guide at angles greater than the limiting one undergoes total internal reflection at the interface between the core and the cladding 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. Multi-core 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 is 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 – 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 luminous intensity in a given direction of a source emitting monochromatic radiation with a frequency of 540·10 12 Hz, the energetic luminous intensity of which 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. A luminous flux of 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).

Topics of the Unified State Examination codifier: the law of light refraction, total internal reflection.

At the interface between two transparent media, along with the reflection of light, it is observed refraction- light, moving to another medium, changes the direction of its propagation.

The refraction of a light ray occurs when it inclined falling on the interface (though not always - read on about total internal reflection). If the ray falls perpendicular to the surface, then there will be no refraction - in the second medium the ray will retain its direction and will also go perpendicular to the surface.

Law of refraction (special case).

We will start with the special case when one of the media is air. This is exactly the situation that occurs in the vast majority of problems. We will discuss the corresponding special case of the law of refraction, and only then we will give its most general formulation.

Suppose that a ray of light traveling in air falls obliquely onto the surface of glass, water or some other transparent medium. When passing into the medium, the beam is refracted, and its further path is shown in Fig. 1 .

At the point of impact, a perpendicular is drawn (or, as they also say, normal) to the surface of the medium. The beam, as before, is called incident ray, and the angle between the incident ray and the normal is angle of incidence. Ray is refracted ray; The angle between the refracted ray and the normal to the surface is called refraction angle.

Any transparent medium is characterized by a quantity called refractive index this environment. The refractive indices of various media can be found in tables. For example, for glass, and for water. In general, in any environment; The refractive index is equal to unity only in a vacuum. In air, therefore, for air we can assume with sufficient accuracy in problems (in optics, air is not very different from vacuum).

Law of refraction (air-medium transition) .

1) The incident ray, the refracted ray and the normal to the surface drawn at the point of incidence lie in the same plane.
2) The ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the refractive index of the medium:

. (1)

Since from relation (1) it follows that , that is, the angle of refraction is less than the angle of incidence. Remember: passing from air to the medium, the ray, after refraction, goes closer to the normal.

The refractive index is directly related to the speed of light propagation in a given medium. This speed is always less than the speed of light in vacuum: . And it turns out that

. (2)

We will understand why this happens when we study wave optics. For now, let's combine the formulas. (1) and (2) :

. (3)

Since the refractive index of air is very close to unity, we can assume that the speed of light in air is approximately equal to the speed of light in a vacuum. Taking this into account and looking at the formula. (3) , we conclude: the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the speed of light in air to the speed of light in the medium.

Reversibility of light rays.

Now let's consider the reverse path of the beam: its refraction when passing from the medium to the air. The following useful principle will help us here.

The principle of reversibility of light rays. The beam path does not depend on whether the beam is propagating in the forward or backward direction. Moving in the opposite direction, the beam will follow exactly the same path as in the forward direction.

According to the principle of reversibility, when transitioning from a medium to air, the beam will follow the same trajectory as during the corresponding transition from air to medium (Fig. 2). The only difference in Fig. 2 from fig. 1 is that the direction of the beam has changed to the opposite.

Since the geometric picture has not changed, formula (1) will remain the same: the ratio of the sine of the angle to the sine of the angle is still equal to the refractive index of the medium. True, now the angles have changed roles: the angle has become the angle of incidence, and the angle has become the angle of refraction.

In any case, no matter how the beam travels - from air to medium or from medium to air - the following simple rule applies. We take two angles - the angle of incidence and the angle of refraction; the ratio of the sine of the larger angle to the sine of the smaller angle is equal to the refractive index of the medium.

We are now fully prepared to discuss the law of refraction in the most general case.

Law of refraction (general case).

Let light pass from medium 1 with a refractive index to medium 2 with a refractive index. A medium with a high refractive index is called optically more dense; accordingly, a medium with a lower refractive index is called optically less dense.

Moving from an optically less dense medium to an optically more dense one, the light beam, after refraction, goes closer to the normal (Fig. 3). In this case, the angle of incidence is greater than the angle of refraction: .

Rice. 3.

On the contrary, moving from an optically denser medium to an optically less dense one, the beam deviates further from the normal (Fig. 4). Here the angle of incidence is less than the angle of refraction:

Rice. 4.

It turns out that both of these cases are covered by one formula - the general law of refraction, valid for any two transparent media.

Law of refraction.
1) The incident ray, the refracted ray and the normal to the interface between the media, drawn at the point of incidence, lie in the same plane.
2) The ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the refractive index of the second medium to the refractive index of the first medium:

. (4)

It is easy to see that the previously formulated law of refraction for the air-medium transition is a special case of this law. In fact, putting in formula (4) we arrive at formula (1).

Let us now remember that the refractive index is the ratio of the speed of light in a vacuum to the speed of light in a given medium: . Substituting this into (4), we get:

. (5)

Formula (5) naturally generalizes formula (3). The ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the speed of light in the first medium to the speed of light in the second medium.

Total internal reflection.

When light rays pass from an optically denser medium to an optically less dense medium, an interesting phenomenon is observed - complete internal reflection. Let's figure out what it is.

For definiteness, we assume that light comes from water into air. Let us assume that in the depths of the reservoir there is a point source of light emitting rays in all directions. We will look at some of these rays (Fig. 5).

The beam hits the water surface at the smallest angle. This ray is partially refracted (ray) and partially reflected back into the water (ray). Thus, part of the energy of the incident beam is transferred to the refracted beam, and the remaining part of the energy is transferred to the reflected beam.

The angle of incidence of the beam is greater. This beam is also divided into two beams - refracted and reflected. But the energy of the original beam is distributed between them differently: the refracted beam will be dimmer than the beam (that is, it will receive a smaller share of energy), and the reflected beam will be correspondingly brighter than the beam (it will receive a larger share of energy).

As the angle of incidence increases, the same pattern is observed: an increasingly larger share of the energy of the incident beam goes to the reflected beam, and an increasingly smaller share to the refracted beam. The refracted beam becomes dimmer and dimmer, and at some point disappears completely!

This disappearance occurs when the angle of incidence corresponding to the angle of refraction is reached. In this situation, the refracted ray would have to go parallel to the surface of the water, but there is nothing left to go - all the energy of the incident ray went entirely to the reflected ray.

With a further increase in the angle of incidence, the refracted beam will even be absent.

The described phenomenon is complete internal reflection. Water does not release rays with angles of incidence equal to or exceeding a certain value - all such rays are completely reflected back into the water. The angle is called limiting angle of total reflection.

The value can be easily found from the law of refraction. We have:

But, therefore

So, for water the limiting angle of total reflection is equal to:

You can easily observe the phenomenon of total internal reflection at home. Pour water into a glass, lift it and look at the surface of the water just below through the wall of the glass. You will see a silvery sheen on the surface - due to total internal reflection, it behaves like a mirror.

The most important technical application of total internal reflection is fiber optics. Light rays launched into a fiber optic cable ( light guide) almost parallel to its axis, fall onto the surface at large angles and are completely reflected back into the cable without loss of energy. Repeatedly reflected, the rays travel further and further, transferring energy over a considerable distance. Fiber optic communications are used, for example, in cable television networks and high-speed Internet access.