Dating back to around 300 BC. e.
Laws of reflection. Fresnel formulas
The law of light reflection - establishes a change in the direction of travel of a light ray as a result of a meeting with a reflecting (mirror) surface: the incident and reflected rays lie in the same plane with the normal to the reflecting surface at the point of incidence, and this normal divides the angle between the rays into two equal parts. The widely used but less precise formulation “angle of incidence equals angle of reflection” does not indicate the exact direction of reflection of the beam. However, it looks like this:
This law is a consequence of the application of Fermat's principle to a reflecting surface and, like all laws of geometric optics, is derived from wave optics. The law is valid not only for perfectly reflective surfaces, but also for the boundary of two media that partially reflects light. In this case, like the law of refraction of light, it does not state anything about the intensity of reflected light.
Reflection mechanism
When an electromagnetic wave hits a conducting surface, a current arises, the electromagnetic field of which tends to compensate for this effect, which leads to almost complete reflection of light.
Types of reflection
The reflection of light can be mirrored(that is, as observed when using mirrors) or diffuse(in this case, upon reflection, the path of the rays from the object is not preserved, but only the energy component of the light flux) depending on the nature of the surface.
Mirror O. s. distinguished by a certain relationship between the positions of the incident and reflected rays: 1) the reflected ray lies in the plane passing through the incident ray and the normal to the reflecting surface; 2) the angle of reflection is equal to the angle of incidence j. The intensity of reflected light (characterized by the reflection coefficient) depends on j and the polarization of the incident beam of rays (see Polarization of Light), as well as on the ratio of the refractive indices n2 and n1 of the 2nd and 1st media. This dependence (for a reflecting medium - a dielectric) is expressed quantitatively by the Fresnel formula. From them, in particular, it follows that when light is incident normal to the surface, the reflection coefficient does not depend on the polarization of the incident beam and is equal to
(n2 - n1)²/(n2 + n1)²
In the very important particular case of a normal fall from air or glass onto their interface (nair " 1.0; nst = 1.5) it is " 4%.
The nature of the polarization of reflected light changes with changes in j and is different for components of incident light polarized parallel (p-component) and perpendicular (s-component) to the plane of incidence. By plane of polarization we mean, as usual, the plane of oscillation of the electric vector of the light wave. At angles j equal to the so-called Brewster angle (see Brewster's law), the reflected light becomes completely polarized perpendicular to the plane of incidence (the p-component of the incident light is completely refracted into the reflecting medium; if this medium strongly absorbs light, then the refracted p-component passes into environment is a very small path). This feature of the mirror O. s. used in a number of polarizing devices. For j larger than the Brewster angle, the reflection coefficient from dielectrics increases with increasing j, tending to 1 in the limit, regardless of the polarization of the incident light. In a specular optical system, as is clear from Fresnel's formulas, the phase of reflected light in the general case changes abruptly. If j = 0 (light falls normally to the interface), then for n2 > n1 the phase of the reflected wave shifts by p, for n2< n1 - остаётся неизменной. Сдвиг фазы при О. с. в случае j ¹ 0 может быть различен для р- и s-составляющих падающего света в зависимости от того, больше или меньше j угла Брюстера, а также от соотношения n2 и n1. О. с. от поверхности оптически менее плотной среды (n2 < n1) при sin j ³ n2 / n1 является полным внутренним отражением, при котором вся энергия падающего пучка лучей возвращается в 1-ю среду. Зеркальное О. с. от поверхностей сильно отражающих сред (например, металлов) описывается формулами, подобными формулам Френеля, с тем (правда, весьма существенным) изменением, что n2 становится комплексной величиной, мнимая часть которой характеризует поглощение падающего света.
Absorption in a reflective medium leads to the absence of a Brewster angle and higher (compared to dielectrics) values of the reflection coefficient - even at normal incidence it can exceed 90% (this explains the widespread use of smooth metal and metallized surfaces in mirrors). The polarization characteristics also differ. light waves reflected from the absorbing medium (due to other phase shifts of the p- and s-components of the incident waves). The nature of the polarization of reflected light is so sensitive to the parameters of the reflecting medium that numerous optical methods for studying metals are based on this phenomenon (see Magneto-optics, Metal-optics).
Diffuse O. s. - its dispersion by the uneven surface of the 2nd medium in all possible directions. The spatial distribution of the reflected radiation flux and its intensity are different in different specific cases and are determined by the relationship between l and the size of the irregularities, the distribution of irregularities over the surface, lighting conditions, and the properties of the reflecting medium. The limiting case of spatial distribution of diffusely reflected light, which is not strictly fulfilled in nature, is described by Lambert’s law. Diffuse O. s. It is also observed from media whose internal structure is inhomogeneous, which leads to the scattering of light in the volume of the medium and the return of part of it to the first medium. Patterns of diffuse O. s. from such media are determined by the nature of the processes of single and multiple light scattering in them. Both absorption and scattering of light can exhibit a strong dependence on l. The result of this is a change in the spectral composition of diffusely reflected light, which (when illuminated with white light) is visually perceived as the color of bodies.
Total internal reflection
As the angle of incidence increases i, the angle of refraction also increases, while the intensity of the reflected beam increases, and the refracted beam decreases (their sum is equal to the intensity of the incident beam). At some value i = i k corner r= π / 2, the intensity of the refracted beam will become equal to zero, all the light will be reflected. With further increase in angle i > i k There will be no refracted ray; the light is completely reflected.
We will find the value of the critical angle of incidence at which total reflection begins, put it in the law of refraction r= π / 2, then sin r= 1 means:
sin i k = n 2 / n 1Diffuse light scattering
θ i = θ r .
The angle of incidence is equal to the angle of reflection
Operating principle of a corner reflector
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The basic optical laws were established a long time ago. Already in the first periods of optical research, four basic laws related to optical phenomena were experimentally discovered:
- law of rectilinear propagation of light;
- law of independence of light beams;
- the law of reflection of light from a mirror surface;
- the law of refraction of light at the boundary of two transparent substances.
The law of reflection is mentioned in the writings of Euclid.
The discovery of the law of reflection is associated with the use of polished metal surfaces (mirrors), which were known in ancient times.
Formulation of the law of light reflection
The incident ray of light, the refracted ray and the perpendicular to the interface between two transparent media lie in the same plane (Fig. 1). In this case, the angle of incidence () and the angle of reflection () are equal:
The phenomenon of total reflection of light
If a light wave propagates from a substance with a high refractive index into a medium with a lower refractive index, then the angle of refraction () will be greater than the angle of incidence.
As the angle of incidence increases, the angle of refraction also increases. This happens until at a certain angle of incidence, which is called the limiting angle (), the angle of refraction becomes equal to 900. If the angle of incidence is greater than the limiting angle (), then all the incident light is reflected from the interface, the phenomenon of refraction does not occur. This phenomenon is called total reflection. The angle of incidence at which total reflection occurs is determined by the condition:
where is the limiting angle of total reflection, is the relative refractive index of the substance in which the refracted light propagates, relative to the medium in which the incident wave of light propagated:
where is the absolute refractive index of the second medium, is the absolute refractive index of the first substance; — phase speed of light propagation in the first medium; — phase speed of light propagation in the second substance.
Limits of application of the law of reflection
If the interface between substances is not flat, then it can be divided into small areas, which individually can be considered flat. Then the course of rays can be sought according to the laws of refraction and reflection. However, the curvature of the surface should not exceed a certain limit, after which diffraction occurs.
Rough surfaces lead to scattered (diffuse) reflection of light. A completely mirror surface becomes invisible. Only the rays reflected from it are visible.
Examples of problem solving
EXAMPLE 1
| Exercise | Two flat mirrors form a dihedral angle (Fig. 2). The incident ray propagates in a plane that is perpendicular to the edge of the dihedral angle. It is reflected from the first, then the second mirror. What will be the angle () by which the beam is deflected as a result of two reflections? |
| Solution | Consider triangle ABD. We see that: From the consideration of triangle ABC it follows that: From the obtained formulas (1.1) and (1.2) we have: |
| Answer |
EXAMPLE 2
| Exercise | What should be the angle of incidence at which the reflected beam makes an angle of 900 relative to the refracted beam? The absolute refractive indices of substances are equal: and . |
| Solution | Let's make a drawing. |
Most of the objects around you: houses, trees, your classmates, etc. are not sources of light. But you see them. The answer to the question “Why is this so?” you will find in this paragraph.
Rice. 11.1. Without a light source, it is impossible to see anything. If there is a light source, we see not only the source itself, but also objects that reflect the light coming from the source
Find out why we see bodies that are not sources of light
You already know that in a homogeneous transparent medium, light travels in a straight line.
What happens if there is some body in the path of the light beam? Some light can pass through a body if it is transparent, some will be absorbed, and some will certainly be reflected from the body. Some reflected rays will hit our eyes, and we will see this body (Fig. 11.1).
Establishing the laws of light reflection
To establish the laws of light reflection, we will use a special device - an optical washer*. Let's fix a mirror in the center of the washer and direct a narrow beam of light at it so that it produces a light stripe on the surface of the washer. We see that a beam of light reflected from the mirror also produces a light stripe on the surface of the washer (see Fig. 11.2).
The direction of the incident light beam is set by the CO ray (Fig. 11.2). This beam is called the incident beam. The direction of the reflected beam of light is set by the OK ray. This ray is called a reflected ray.
From point O of incidence of the beam, draw a perpendicular OB to the surface of the mirror. Let us pay attention to the fact that the incident ray, the reflected ray and the perpendicular lie in the same plane - in the plane of the washer surface.
The angle α between the incident ray and the perpendicular drawn from the point of incidence is called the angle of incidence; The angle β between the reflected ray and a given perpendicular is called the angle of reflection.
By measuring the angles α and β, you can verify that they are equal.
If you move the light source along the edge of the disk, the angle of incidence of the light beam will change and the angle of reflection will change accordingly, and each time the angle of incidence and the angle of reflection of the light will be equal (Fig. 11.3). So, we have established the laws of light reflection:
Rice. 11.3. As the angle of incidence of light changes, the angle of reflection also changes. The angle of reflection is always equal to the angle of incidence
Rice. 11.5. Demonstration of reversibility of light rays: the reflected ray follows the path of the incident ray
rice. 11.6. Approaching the mirror, we see our “double” in it. Of course, there is no “double” there - we see our reflection in the mirror
1. The incident ray, the reflected ray and the perpendicular to the reflection surface drawn from the point of incidence of the ray lie in the same plane.
2. The angle of reflection is equal to the angle of incidence: β = α.
The laws of light reflection were established by the ancient Greek scientist Euclid back in the 3rd century. BC e.
In what direction should the professor turn the mirror so that the “sunbeam” hits the boy (Fig. 11.4)?
Using a mirror on an optical washer, you can also demonstrate the reversibility of light rays: if the incident ray is directed along the path of the reflected one, then the reflected ray will follow the path of the incident one (Fig. 11.5).
Studying the image in a plane mirror
Let's consider how an image is created in a plane mirror (Fig. 11.6).
Let a diverging beam of light fall from a point source of light S onto the surface of a flat mirror. From this beam we select the rays SA, SB and SC. Using the laws of light reflection, we construct the reflected rays LL b BB 1 and CC 1 (Fig. 11.7, a). These rays will travel in a diverging beam. If you extend them in the opposite direction (behind the mirror), they will all intersect at one point - S 1, located behind the mirror.
If some of the rays reflected from the mirror hit your eye, it will seem to you that the reflected rays are coming out of point S 1, although in reality there is no light source at point S 1. Therefore, point S 1 is called the virtual image of point S. A plane mirror always gives a virtual image.
Let's find out how the object and its image are located relative to the mirror. To do this, let's turn to geometry. Consider, for example, a beam SC that falls on a mirror and is reflected from it (Fig. 11.7, b).
From the figure we see that Δ SOC = Δ S 1 OC are right triangles with a common side CO and equal acute angles (since according to the law of light reflection α = β). From the equality of triangles we have that SO = S 1 O, that is, the point S and its image S 1 are symmetrical relative to the surface of a flat mirror.
The same can be said about the image of an extended object: the object and its image are symmetrical relative to the surface of a flat mirror.
So, we have established the general characteristics of images in flat mirrors.
1. A flat mirror gives a virtual image of an object.
2. The image of an object in a flat mirror and the object itself are symmetrical relative to the surface of the mirror, and this means:
1) the image of the object is equal in size to the object itself;
2) the image of the object is located at the same distance from the surface of the mirror as the object itself;
3) the segment connecting a point on the object and the corresponding point on the image is perpendicular to the surface of the mirror.
Distinguish between specular and diffuse reflection of light
In the evening, when the light is on in the room, we can see our image in the window glass. But the image disappears if you close the curtains: we will not see our image on the fabric. And why? The answer to this question is related to at least two physical phenomena.
The first such physical phenomenon is the reflection of light. For an image to appear, light must be reflected specularly from the surface: after specular reflection of light coming from a point source S, the continuations of the reflected rays will intersect at one point S1, which will be the image of point S (Fig. 11.8, a). Such reflection is only possible from very smooth surfaces. They are called mirror surfaces. In addition to an ordinary mirror, examples of mirror surfaces are glass, polished furniture, calm surface of water, etc. (Fig. 11.8, b, c).
If light is reflected from a rough surface, such reflection is called scattered (diffuse) (Fig. 11.9). In this case, the reflected rays propagate in different directions (which is why we see an illuminated object from any direction). It is clear that there are much more surfaces that scatter light than mirror ones.
Look around and name at least ten surfaces that reflect light diffusely.
Rice. 11.8. Specular reflection of light is the reflection of light from a smooth surface
Rice. 11.9. Scattered (diffuse) reflection of light is the reflection of light from a rough surface
The second physical phenomenon that affects the ability to see an image is the absorption of light. After all, light is not only reflected from physical bodies, but also absorbed by them. The best light reflectors are mirrors: they can reflect up to 95% of the incident light. White bodies are good light reflectors, but a black surface absorbs almost all the light falling on it.
When snow falls in the fall, the nights become much lighter. Why? Learning to solve problems
Task. In Fig. 1 schematically shows an object BC and a mirror NM. Find graphically the area from which the image of the object BC is completely visible.
Analysis of a physical problem. To see the image of a certain point of an object in a mirror, it is necessary that at least part of the rays falling from this point onto the mirror is reflected into the observer’s eye. It is clear that if rays emanating from the extreme points of an object are reflected into the eye, then rays emanating from all points of the object will also be reflected into the eye.
Decision, analysis of results
1. Let's construct point B 1 - the image of point B in a flat mirror (Fig. 2, a). The area limited by the surface of the mirror and the rays reflected from the extreme points of the mirror will be the area from which the image B 1 of point B in the mirror is visible.
2. Having similarly constructed the image C 1 of point C, we determine the area of its vision in the mirror (Fig. 2, b).
3. The observer can see the image of the entire object only if the rays that give both images - B 1 and C 1 - enter his eye (Fig. 2, c). This means that the area highlighted in Fig. 2, in orange, is the area from which the image of the object is completely visible.
Analyze the result obtained, look at Fig. again. 2 to the problem and suggest an easier way to find the area of vision of an object in a plane mirror. Test your assumptions by constructing a field of vision for several objects in two ways.
Let's sum it up
All visible bodies reflect light. When light is reflected, two laws of light reflection are satisfied: 1) the incident ray, the reflected ray and the perpendicular to the reflection surface drawn from the point of incidence of the ray lie in the same plane; 2) the angle of reflection is equal to the angle of incidence.
The image of an object in a plane mirror is virtual, equal in size to the object itself and located at the same distance from the mirror as the object itself.
There are specular and diffuse reflections of light. In the case of mirror reflection, we can see a virtual image of an object in a reflective surface; in the case of diffuse reflection, no image appears.
Control questions
1. Why do we see surrounding bodies? 2. What angle is called the angle of incidence? angle of reflection? 3. Formulate the laws of light reflection. 4. Using what device can you verify the validity of the laws of light reflection? 5. What is the property of reversibility of light rays? 6. In what case is an image called virtual? 7. Describe the image of an object in a flat mirror. 8. How does diffuse reflection of light differ from specular reflection?
Exercise No. 11
1. A girl stands at a distance of 1.5 m from a flat mirror. How far is her reflection from the girl? Describe him.
2. The driver of the car, looking in the rearview mirror, saw a passenger sitting in the back seat. Can the passenger at this moment, looking in the same mirror, see the driver?
3. Transfer the rice. 1 in your notebook, for each case construct an incident (or reflected) ray. Label the angles of incidence and reflection.
4. The angle between the incident and reflected rays is 80°. What is the angle of incidence of the beam?
5. The object was at a distance of 30 cm from a flat mirror. Then the object was moved 10 cm from the mirror in a direction perpendicular to the mirror surface, and 15 cm parallel to it. What was the distance between the object and its reflection? What did it become?
6. You are moving towards a mirrored display case at a speed of 4 km/h. At what speed is your reflection approaching you? How much will the distance between you and your reflection decrease when you walk 2 m?
7. The sun's ray is reflected from the surface of the lake. The angle between the incident ray and the horizon is twice as large as the angle between the incident and reflected rays. What is the angle of incidence of the beam?
8. The girl looks into a mirror hanging on the wall at a slight angle (Fig. 2).
1) Construct the girl’s reflection in the mirror.
2) Find graphically which part of her body the girl sees; the area from which the girl sees herself completely.
3) What changes will be observed if the mirror is gradually covered with an opaque screen?
9. At night, in the light of a car’s headlights, a puddle on the asphalt appears to the driver as a dark spot against a lighter background of the road. Why?
10. In Fig. Figure 3 shows the path of rays in a periscope, a device whose operation is based on the rectilinear propagation of light. Explain how this device works. Use additional sources of information and find out where it is used.
LABORATORY WORK No. 3
Subject. Study of light reflection using a plane mirror.
Goal: experimentally test the laws of light reflection.
equipment: light source (candle or electric lamp on a stand), a flat mirror, a screen with a slit, several blank white sheets of paper, a ruler, a protractor, a pencil.
instructions for work
preparation for the experiment
1. Before performing work, remember: 1) safety requirements when working with glass objects; 2) laws of light reflection.
2. Assemble the experimental setup (Fig. 1). For this:
1) place the screen with a slot on a white sheet of paper;
2) by moving the light source, get a strip of light on the paper;
3) place a flat mirror at a certain angle to the strip of light and perpendicular to the sheet of paper so that the reflected beam of light also produces a clearly visible strip on the paper.
Experiment
Strictly follow the safety instructions (see the flyleaf of the textbook).
1. With a well-sharpened pencil, draw a line along the mirror on paper.
2. Place three points on a sheet of paper: the first - in the middle of the incident beam of light, the second - in the middle of the reflected beam of light, the third - in the place where the light beam falls on the mirror (Fig. 2).
3. Repeat the described steps several more times (on different sheets of paper), placing the mirror at different angles to the incident beam of light.
4. By changing the angle between the mirror and the sheet of paper, make sure that in this case you will not see the reflected beam of light.
Processing the experiment results
For each experience:
1) construct a ray incident on the mirror and a reflected ray;
2) through the point of incidence of the beam, draw a perpendicular to a line drawn along the mirror;
3) Label and measure the angle of incidence (α) and angle of reflection (β) of light. Enter the measurement results in the table.
Analysis of the experiment and its results
Analyze the experiment and its results. Draw a conclusion in which you indicate: 1) what relationship you have established between the angle of incidence of the light beam and the angle of its reflection; 2) whether the experimental results turned out to be absolutely accurate, and if not, what were the reasons for the error.
creative task
Using fig. 3, think over and write down an experiment plan to determine the height of a room using a plane mirror; indicate the required equipment.
If possible, conduct an experiment.
Assignment with an asterisk
The reflected and incident rays lie in a plane containing a perpendicular to the reflecting surface at the point of incidence, and the angle of incidence is equal to the angle of reflection.
Imagine shining a thin beam of light onto a reflective surface, like shining a laser pointer at a mirror or polished metal surface. The beam will be reflected from such a surface and will propagate further in a certain direction. The angle between the perpendicular to the surface ( normal) and the original ray is called angle of incidence, and the angle between the normal and the reflected ray is reflection angle. The law of reflection states that the angle of incidence is equal to the angle of reflection. This is completely consistent with what our intuition tells us. A ray incident almost parallel to the surface will only lightly touch it and, having been reflected at an obtuse angle, will continue its path along a low trajectory located close to the surface. A ray falling almost vertically, on the other hand, will be reflected at an acute angle and the direction of the reflected ray will be close to the direction of the incident ray, as required by law.
The law of reflection, like any law of nature, was obtained on the basis of observations and experiments. It can also be derived theoretically - formally, it is a consequence of Fermat’s principle (but this does not negate the significance of its experimental justification).
The key point in this law is that angles are measured from the perpendicular to the surface at the point of impact beam. For a flat surface, for example, a flat mirror, this is not so important, since the perpendicular to it is directed equally at all points. A parallel focused light signal, such as a car headlight or spotlight, can be viewed as a dense beam of parallel rays of light. If such a beam is reflected from a flat surface, all reflected rays in the beam will be reflected at the same angle and remain parallel. This is why a straight mirror does not distort your visual image.
However, there are also distorting mirrors. Different geometric configurations of mirror surfaces change the reflected image in different ways and allow you to achieve various useful effects. The main concave mirror of a reflecting telescope allows light from distant space objects to be focused in the eyepiece. The curved rear view mirror of the car allows you to expand the viewing angle. And the crooked mirrors in the fun room allow you to have a lot of fun looking at the bizarrely distorted reflections of yourself.
Not only light is subject to the law of reflection. Any electromagnetic waves - radio, microwave, x-rays, etc. - behave exactly the same. This is why, for example, both the huge receiving antennas of radio telescopes and satellite television dishes have the shape of a concave mirror - they use the same principle of focusing incoming parallel rays to a point.
The basic laws of geometric optics have been known since ancient times. Thus, Plato (430 BC) established the law of rectilinear propagation of light. Euclid's treatises formulated the law of rectilinear propagation of light and the law of equality of angles of incidence and reflection. Aristotle and Ptolemy studied the refraction of light. But the exact wording of these laws of geometric optics Greek philosophers could not find it.
Geometric optics is the limiting case of wave optics, when the wavelength of light tends to zero.
The simplest optical phenomena, such as the appearance of shadows and the production of images in optical instruments, can be understood within the framework of geometric optics.
The formal construction of geometric optics is based on four laws , established empirically:
· law of rectilinear propagation of light;
· the law of independence of light rays;
· law of reflection;
· law of light refraction.
To analyze these laws, H. Huygens proposed a simple and visual method, later called Huygens' principle .
Each point to which light excitation reaches is ,in its turn, center of secondary waves;the surface that envelops these secondary waves at a certain moment in time indicates the position of the front of the actually propagating wave at that moment.
Based on his method, Huygens explained straightness of light propagation And brought out laws of reflection And refraction .
Law of rectilinear propagation of light :
· light propagates rectilinearly in an optically homogeneous medium.
Proof of this law is the presence of shadows with sharp boundaries from opaque objects when illuminated by small sources.
Careful experiments have shown, however, that this law is violated if light passes through very small holes, and the deviation from straightness of propagation is greater, the smaller the holes.
The shadow cast by an object is determined by straightness of light rays in optically homogeneous media.
Astronomical illustration rectilinear propagation of light and, in particular, the formation of umbra and penumbra can be caused by the shading of some planets by others, for example lunar eclipse , when the Moon falls into the Earth's shadow (Fig. 7.1). Due to the mutual motion of the Moon and the Earth, the shadow of the Earth moves across the surface of the Moon, and the lunar eclipse passes through several partial phases (Fig. 7.2).
Law of independence of light beams :
· the effect produced by an individual beam does not depend on whether,whether other bundles act simultaneously or whether they are eliminated.
By dividing the light flux into separate light beams (for example, using diaphragms), it can be shown that the action of the selected light beams is independent.
Law of Reflection (Fig. 7.3):
· the reflected ray lies in the same plane as the incident ray and the perpendicular,drawn to the interface between two media at the point of impact;
· angle of incidenceα equal to the angle of reflectionγ: α = γ
Rice. 7.3 Fig. 7.4
To derive the law of reflection Let's use Huygens' principle. Let us assume that a plane wave (wave front AB with speed With, falls on the interface between two media (Fig. 7.4). When the wave front AB will reach the reflecting surface at the point A, this point will begin to radiate secondary wave .
For the wave to travel a distance Sun time required Δ t = B.C./ υ . During the same time, the front of the secondary wave will reach the points of the hemisphere, the radius AD which is equal to: υ Δ t= sun. The position of the reflected wave front at this moment in time, in accordance with Huygens’ principle, is given by the plane DC, and the direction of propagation of this wave is ray II. From the equality of triangles ABC And ADC flows out law of reflection: angle of incidenceα equal to the angle of reflection γ .
Law of refraction (Snell's law) (Fig. 7.5):
· 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.
Rice. 7.5 Fig. 7.6
Derivation of the law of refraction. Let us assume that a plane wave (wave front AB), propagating in vacuum along the direction I at speed With, falls on the interface with the medium in which the speed of its propagation is equal to u(Fig. 7.6).
Let the time taken by the wave to travel the path Sun, equal to D t. Then BC = s D t. During the same time, the front of the wave excited by the point A in an environment with speed u, will reach points of the hemisphere whose radius AD = u D t. The position of the refracted wave front at this moment in time, in accordance with Huygens’ principle, is given by the plane DC, and the direction of its propagation - by ray III . From Fig. 7.6 it is clear that
this implies Snell's law :
A slightly different formulation of the law of light propagation was given by the French mathematician and physicist P. Fermat.
Physical research relates mostly to optics, where he established in 1662 the basic principle of geometric optics (Fermat's principle). The analogy between Fermat's principle and the variational principles of mechanics played a significant role in the development of modern dynamics and the theory of optical instruments.
According to Fermat's principle , light propagates between two points along a path that requires least time.
Let us show the application of this principle to solving the same problem of light refraction.
Beam from a light source S located in a vacuum goes to the point IN, located in some medium beyond the interface (Fig. 7.7).
In every environment the shortest path will be straight S.A. And AB. Full stop A characterize by distance x from the perpendicular dropped from the source to the interface. Let's determine the time spent on traveling the path SAB:
.
To find the minimum, we find the first derivative of τ with respect to X and set it equal to zero:
from here we come to the same expression that was obtained based on Huygens’ principle: .
Fermat's principle has retained its significance to this day and served as the basis for the general formulation of the laws of mechanics (including the theory of relativity and quantum mechanics).
Several consequences follow from Fermat's principle.
Reversibility of light rays : if you reverse the beam III (Fig. 7.7), causing it to fall onto the interface at an angleβ, then the refracted ray in the first medium will propagate at an angle α, i.e. it will go in the opposite direction along the beam I .
Another example is a mirage , which is often observed by travelers on hot roads. They see an oasis ahead, but when they get there, there is sand all around. The essence is that in this case we see light passing over the sand. The air is very hot above the road itself, and in the upper layers it is colder. Hot air, expanding, becomes more rarefied and the speed of light in it is greater than in cold air. Therefore, light does not travel in a straight line, but along a trajectory with the shortest time, turning into warm layers of air.
If light comes from high refractive index media (optically more dense) into a medium with a lower refractive index (optically less dense)( > ) , for example, from glass into air, then, according to the law of refraction, the refracted ray moves away from the normal and the angle of refraction β is greater than the angle of incidence α (Fig. 7.8 A).
As the angle of incidence increases, the angle of refraction increases (Fig. 7.8 b, V), until at a certain angle of incidence () the angle of refraction is equal to π/2.
The angle is called limit angle . At angles of incidence α > all incident light is completely reflected (Fig. 7.8 G).
· As the angle of incidence approaches the limiting one, the intensity of the refracted beam decreases, and the intensity of the reflected beam increases.
· If , then the intensity of the refracted beam becomes zero, and the intensity of the reflected beam is equal to the intensity of the incident one (Fig. 7.8 G).
· Thus,at angles of incidence ranging from to π/2,the beam is not refracted,and is fully reflected on the first Wednesday,Moreover, the intensities of the reflected and incident rays are the same. This phenomenon is called complete reflection.
The limit angle is determined from the formula:
;
.
The phenomenon of total reflection is used in total reflection prisms (Fig. 7.9).
The refractive index of glass is n » 1.5, therefore the limiting angle for the glass-air interface = arcsin (1/1.5) = 42°.
When light falls on the glass-air interface at α > 42° will always be a total reflection.
In Fig. 7.9 total reflection prisms are shown, allowing:
a) rotate the beam 90°;
b) rotate the image;
c) wrap the rays.
Total reflection prisms are used in optical instruments (for example, in binoculars, periscopes), as well as in refractometers that make it possible to determine the refractive index of bodies (according to the law of refraction, by measuring , we determine the relative refractive index of two media, as well as the absolute refractive index of one of the media, if the refractive index of the second medium is known).
The phenomenon of total reflection is also used in light guides , which are thin, randomly curved threads (fibers) made of optically transparent material.
Fiber parts use glass fiber, the light-guiding core (core) of which is surrounded by glass - a shell of another glass with a lower refractive index. Light incident on the end of the light guide at angles greater than the limit , undergoes at the core-shell interface total reflection and propagates only along the light-guide core.
Light guides are used to create high-capacity telegraph-telephone cables . The cable consists of hundreds and thousands of optical fibers as thin as human hair. This cable, the thickness of an ordinary pencil, can simultaneously transmit up to eighty thousand telephone conversations.
In addition, light guides are used in fiber optic cathode ray tubes, in electronic counting machines, for information coding, in medicine (for example, gastric diagnostics), and for integrated optics purposes.
