Let us establish the relationship between the displacement x of particles of the medium participating in the wave process and the distance y of these particles from the source of oscillations O for any moment of time. For greater clarity, let us consider a transverse wave, although all subsequent considerations
will also be true for a longitudinal wave. Let the source oscillations be harmonic (see § 27):
where A is the amplitude, circular frequency of oscillations. Then all particles of the medium will also come into harmonic vibration with the same frequency and amplitude, but with different phases. A sinusoidal wave appears in the medium, shown in Fig. 58.
The wave graph (Fig. 58) is superficially similar to the harmonic oscillation graph (Fig. 46), but in essence they are different. The oscillation graph represents the displacement of a given particle as a function of time. The wave graph represents the dependence of the displacement of all particles of the medium on the distance to the source of oscillations at a given moment in time. It is like a snapshot of a wave.
Let us consider a certain particle C located at a distance y from the source of oscillations (particle O). It is obvious that if particle O is already oscillating, then particle C is still oscillating only where is the time of propagation of oscillations from to C, i.e., the time during which the wave traveled the path y. Then the equation of vibration of particle C should be written as follows:
But where is the speed of wave propagation? Then
Relationship (23), which allows us to determine the displacement of any point on the wave at any time, is called the wave equation. By introducing the wavelength X into consideration as the distance between the two closest points of the wave that are in the same phase, for example, between two adjacent wave crests, we can give the wave equation a different form. Obviously, the wavelength is equal to the distance over which the oscillation propagates over a period with a speed
where is the frequency of the wave. Then, substituting into the equation and taking into account that we obtain other forms of the wave equation:
Since the passage of waves is accompanied by vibrations of particles of the medium, the energy of vibrations moves in space along with the wave. The energy transferred by a wave per unit time through a unit area perpendicular to the beam is called wave intensity (or energy flux density). We obtain an expression for the wave intensity
Light waves.
Laws of geometric (ray) optics
Light waves. Light intensity. Light flow. Laws of geometric optics. Total internal reflection
Optics is a branch of physics that studies the nature of light radiation, its propagation and interaction with matter. The branch of optics that studies the wave nature of light is called wave optics. The wave nature of light underlies such phenomena as interference, diffraction, and polarization. The branch of optics that does not take into account the wave properties of light and is based on the concept of a ray is called geometric optics.
§ 1. LIGHT WAVES
According to modern concepts, light is a complex phenomenon: in some cases it behaves like an electromagnetic wave, in others - like a stream of special particles (photons). This property is called particle-wave dualism (corpuscle - particle, dualism - duality). In this part of the lecture course we will consider wave phenomena of light.
A light wave is an electromagnetic wave with a wavelength in a vacuum in the range:
= (0.4¸ 0.76)× 10− 6 m= 0.4¸ 0.76 µm= 400¸ 760 nm= |
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4,000¸ | ||||||||
A – | angstrom is a unit of measurement of length. 1A = 10−10 m. | |||||||
Waves of this range are perceived by the human eye. |
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Radiation with a wavelength less than 400 nm is called ultraviolet, and |
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with greater than 760 nm, – | infrared. | |||||||
Frequency n of the light wave for visible light: | ||||||||
= (0.39¸ 0.75)× 1015 Hz, | ||||||||
c = 3× 108 m/s is the speed of light in vacuum. | ||||||||
Speed | matches | speed | distribution |
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electromagnetic wave. | ||||||||
Refractive index
The speed of propagation of light in a medium, like any electromagnetic wave, is equal to (see (7.3)):
To characterize the optical properties of the medium, the refractive index is introduced. The ratio of the speed of light in a vacuum to the speed of light in a given medium is called absolute refractive index:
Taking into account (7.3) | |||||||
since for most transparent substances μ=1.
Formula (8.2) connects the optical properties of a substance with its electrical properties. For any medium except vacuum, n> 1. For vacuum n = 1, for gases under normal conditions n≈ 1.
The refractive index characterizes optical density of the medium. A medium with a higher refractive index is called optically denser. Let us denote the absolute refractive indices for two media:
n 2 = | |||||||||
Then the relative refractive index is: |
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n 21= | |||||||||
where v 1 and v 2 – | the speed of light in the first and second medium, respectively. |
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dielectric | the permeability of the medium ε depends on the frequency |
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electromagnetic wave, then n = n(ν) orn = n(λ) - the refractive index will depend on the wavelength of light (see lectures No. 16, 17).
The dependence of the refractive index on wavelength (or frequency) is called dispersion.
In a light wave, as in any electromagnetic wave, the vectors E and H oscillate. These vectors are perpendicular to each other and to the direction
vector v. Experience shows that physiological, photochemical, photoelectric and other types of effects are caused by oscillations of the electric vector. Therefore, the light vector is the vector of the electric field strength of a light (electromagnetic) wave.
For a monochromatic light wave, the change in time and space of the projection of the light vector onto the direction along which it
Here k is the wave number; r – distance measured along the direction of wave propagation; E m is the amplitude of the light wave. For a plane wave E m = const, for a spherical wave it decreases as 1/r.
§ 2. LIGHT INTENSITY. LIGHT FLOW
The frequency of light waves is very high, so the light receiver or eye records the time-averaged flux. The intensity of light is the modulus of the time-averaged energy density at a given point in space. For a light wave, as for any electromagnetic wave, the intensity (see (7.8)) is equal to:
For a light wave μ≈ 1, therefore from (7.5) it follows:
μ0 H =ε0 ε E,
whence, taking into account (8.2):
E ~ nE. | |||||||||
Let us substitute formulas (8.4) and (8.5) into (7.8). After averaging we get:
Therefore, the intensity of light is proportional to the square of the amplitude of the light wave and the refractive index. Note that for
vacuum and air n = 1, so I ~ E 2 m (compare with (7.9)).
To characterize the intensity of light, taking into account its ability to cause a visual sensation, the value F, called luminous flux, is introduced. The effect of light on the eye depends greatly on the wavelength. Most
The eye is sensitive to radiation with a wavelength λ з = 555 nm (green).
For other waves, the sensitivity of the eye is lower, and outside the interval (400–760 nm) the sensitivity of the eye is zero.
Luminous flux is the flow of light energy, assessed by visual sensation. The unit of luminous flux is lumen (lm). Accordingly, intensity is measured either in energy units (W/m2) or in light units (lm/m2).
Light intensity characterizes the numerical value of the average energy transferred by a light wave per unit time through a unit area of a site placed perpendicular to the direction of wave propagation. The lines along which light energy travels are called rays. The branch of optics that studies the laws of light propagation
radiation based on ideas about light rays is called geometric or ray optics.
§ 3. BASIC LAWS OF GEOMETRIC OPTICS
Geometric optics is an approximate consideration of the propagation of light under the assumption that light propagates along certain lines - rays (ray optics). In this approximation, the finiteness of the wavelengths of light is neglected, assuming that λ→ 0.
Geometric optics allows in many cases to calculate the optical system quite well. But in a number of cases, real calculations of optical systems require taking into account the wave nature of light.
The first three laws of geometric optics have been known since ancient times. 1. The law of rectilinear propagation of light.
The law of rectilinear propagation of light states that in
In a homogeneous medium, light propagates rectilinearly.
If the medium is inhomogeneous, that is, its refractive index varies from point to point, or n = n(r), then light will not travel in a straight line. At
In the presence of sharp inhomogeneities, such as holes in opaque screens, the boundaries of these screens, deviation of light from rectilinear propagation is observed.
2. The law of independence of light rays states that rays do not disturb each other when crossing. At high intensities, this law is not observed, and light is scattered by light.
3 and 4. The laws of reflection and refraction state that At the interface between two media, reflection and refraction of a light beam occurs. The reflected and refracted rays lie in the same plane as the incident one
ray and perpendicular restored to the interface at the point of incidence
The angle of incidence is equal to the angle of reflection:
for which the indicator
It can vary greatly, and visually we are not able to determine the degree of illumination, since the human eye is endowed with the ability to adapt to different lighting. Meanwhile, lighting intensity is extremely important in a wide variety of areas of activity. For example, you can take the process of filming or video shooting, as well as, say, growing indoor plants.
The human eye perceives light from 380 nm (violet) to 780 nm (red). We best perceive waves with a length that is not the most suitable for plants. Lighting that is bright and pleasing to our eyes may not be suitable for plants in a greenhouse, which may not receive enough waves important for photosynthesis.
Light intensity is measured in lux. On a bright sunny afternoon in our central zone it reaches approximately 100,000 lux, and in the evening it drops to 25,000 lux. In dense shade its value is tenths of these values. Indoors, the intensity of sunlight is much less, because the light is weakened by trees and window glass. The brightest lighting (on the south window in the summer right behind the glass) is at best 3-5 thousand lux, in the middle of the room (2-3 meters from the window) - only 500 lux. This is the minimum lighting required for plant survival. For normal growth, even unpretentious ones require at least 800 lux.
We cannot determine the intensity of light by eye. There is a device for this purpose, the name of which is a lux meter. When purchasing it, it is necessary to clarify the wave range it measures, because The capabilities of the device, although wider than the capabilities of the human eye, are still limited.
Light intensity can also be measured using a camera or photo exposure meter. True, you will have to recalculate the received units into suites. To carry out the measurement, you need to place a white sheet of paper at the measuring location and point the camera at it, the photosensitivity of which is set to 100 and the aperture to 4. Having determined the shutter speed, you should multiply its denominator by 10, the resulting value will approximately correspond to the lighting in lux. For example, with a shutter speed of 1/60 sec. lighting about 600 lux.
If you are interested in growing and caring for flowers, then, of course, you know that light energy is vital for plants to carry out normal photosynthesis. Light affects the growth rate, direction, development of the flower, the size and shape of its leaves. With a decrease in light intensity, all processes in plants slow down proportionally. Its amount depends on how far away the light source is, on the side of the horizon to which the window is facing, on the degree of shading by street trees, on the presence of curtains or blinds. The brighter the room, the more actively the plants grow and the more water, heat and fertilizer they require. If plants grow in the shade, then they require less care.
When shooting a film or television show, lighting is very important. High-quality shooting is possible with illumination of about 1000 lux, achieved in a television studio using special lamps. But acceptable image quality can be obtained with less lighting.
Light intensity in the studio is measured before and during filming using exposure meters or high-quality color monitors that are connected to a video camera. Before starting shooting, it is best to walk around the entire set with a light meter in order to identify darkened or overly lit areas in order to avoid negative phenomena when viewing the footage. In addition, by correctly adjusting the lighting, you can achieve additional expressiveness of the scene being filmed and the necessary directorial effects.
Light plays a huge role not only in the interior, but also in our lives in general. After all, the efficiency of work, as well as our psychological state, depends on the correct lighting of the room. Light gives a person the opportunity not only to see, but also to evaluate the colors and shapes of surrounding objects.
Of course, natural light is most comfortable for human eyes. With this lighting, everything is visible very well and without color distortion. But natural light is not always present; in the dark, for example, you have to make do with artificial light sources.
To prevent your eyes from straining and your vision from deteriorating, it is necessary to create optimal conditions of light and shadow, creating the most comfortable lighting.
The most pleasant lighting for the eyes is natural
Lighting, like many other factors, is assessed according to quantitative and qualitative parameters. Quantitative characteristics are determined by the intensity of light, and qualitative characteristics are determined by its spectral composition and distribution in space.
How and in what terms is light intensity measured?
Light has many characteristics and each has its own unit of measurement:
- Luminous intensity characterizes the amount of light energy that is transferred over a certain time in any direction. It is measured in candelas (cd), 1 cd is approximately equal to the intensity of light emitted by one burning candle;
- Brightness is also measured in candelas; in addition, there are such units of measurement as stilbe, apostilbe and lambert;
- Illumination is the ratio of the luminous flux that falls on a certain area to its surface. It is measured in lux.
It is illumination that is an important indicator for the proper functioning of vision. In order to determine this value, a special measuring device is used. It's called a lux meter.
A lux meter is a device for measuring illumination.
This device consists of a light receiver and a measuring part, it can be of a pointer type or electronic. The light receiver is a photocell that converts the light wave into an electrical signal and sends it to the measuring part. This device is a photometer and has a specified spectral sensitivity. It can be used to measure not only visible light, but also infrared radiation, etc.
This device is used both in industrial premises and in educational institutions, as well as at home. Each type of activity and occupation has its own standards for what the light intensity should be.
Comfortable lighting intensity
Visual comfort depends on many factors. Of course, the most pleasant thing for the human eye is sunlight. But the modern rhythm of life dictates its own rules, and very often you have to work or just be in artificial light.
Manufacturers of lighting fixtures and lamps are trying to create light sources that would meet the characteristics of people’s visual perception and create the most comfortable light intensity.
Light from an incandescent lamp most accurately conveys natural shades
Conventional incandescent lamps use a hot spring as a light source, and therefore this light is most similar to natural light.
Lamps are divided into the following categories based on the type of light they produce:
- warm light with reddish tints, it is well suited for a home environment;
- neutral light, white, used to illuminate workplaces;
- cold light, bluish, intended for places where high precision work is performed or for places with a hot climate.
It is important not only what type of lamps are, but also the design of the lamp or chandelier itself: how many bulbs are screwed in where the light is directed, whether the shades are closed or open - all these features must be taken into account when choosing a lighting device.
Illumination standards are recorded in several documents, the most important are: SNiP (building codes and regulations) and SanPiN (sanitary rules and regulations). There are also MGSN (Moscow city building codes), as well as their own set of rules for each region.
It is on the basis of all these documents that the decision is made on what the lighting intensity should be.
Of course, when thinking about what chandelier to hang in the living room, bedroom or kitchen, no one measures the lighting intensity using a lux meter. However, knowing in general terms which light will be more comfortable for the eyes is very useful.
Table 1 shows lighting standards for residential premises:
Table 1
Table 2 shows lighting standards for offices
At home, without special equipment, it is difficult to measure indoor lighting, and therefore in order to understand which lamp to choose, you should pay attention to the color (cold, neutral or warm) and the number of watts. In recreation rooms it is better to use not too bright ones, and in work rooms - with more intense light.
Since natural light is most pleasant for the eyes, preference in the home environment should be given to lamps that provide warm light. When we come home, our eyes definitely need rest after a busy day at work. Properly selected lamps for chandeliers and lamps in terms of brightness will help create lighting that is suitable in intensity.
Thus, in geometric optics, a light wave can be considered as a beam of rays. The rays, however, themselves determine only the direction of propagation of light at each point; The question remains about the distribution of light intensity in space.
Let us select an infinitesimal element on any of the wave surfaces of the beam under consideration. From differential geometry it is known that every surface has at each point two, generally speaking, different principal radii of curvature.
Let (Fig. 7) be the elements of the main circles of curvature drawn on a given element of the wave surface. Then the rays passing through points a and c will intersect each other at the corresponding center of curvature, and the rays passing through b and d will intersect at another center of curvature.
For given opening angles, the rays emanating from the length of the segments are proportional to the corresponding radii of curvature (i.e., the lengths and); the area of a surface element is proportional to the product of lengths, i.e., proportional. In other words, if we consider an element of a wave surface limited by a certain number of rays, then when moving along them, the area of this element will change proportionally.
On the other hand, intensity, i.e., energy flux density, is inversely proportional to the surface area through which a given amount of light energy passes. Thus, we come to the conclusion that the intensity
This formula should be understood as follows. On each given ray (AB in Fig. 7) there are certain points and , which are the centers of curvature of all wave surfaces intersecting this ray. The distances from point O of the intersection of the wave surface with the ray to the points are the radii of curvature of the wave surface at point O. Thus, formula (54.1) determines the intensity of light at point O on a given ray as a function of the distances to certain points on this ray. We emphasize that this formula is not suitable for comparing intensities at different points of the same wave surface.
Since the intensity is determined by the square of the field modulus, to change the field itself along the ray we can write:
where in the phase factor R can be understood as both and the quantities differ from each other only by a constant (for a given beam) factor, since the difference , the distance between both centers of curvature, is constant.
If both radii of curvature of the wave surface coincide, then (54.1) and (54.2) have the form
This occurs, in particular, always in cases where light is emitted by a point source (the wave surfaces are then concentric spheres, and R is the distance to the light source).
From (54.1) we see that the intensity goes to infinity at points, i.e., at the centers of curvature of the wave surfaces. Applying this to all rays in a beam, we find that the intensity of light in a given beam goes to infinity, generally speaking, on two surfaces - the geometric locus of all centers of curvature of the wave surfaces. These surfaces are called caustics. In the particular case of a beam of rays with spherical wave surfaces, both caustics merge into one point (focus).
Note that, according to the properties of the locus of the centers of curvature of a family of surfaces known from differential geometry, the rays touch the caustics.
It must be borne in mind that (with convex wave surfaces) the centers of curvature of the wave surfaces may turn out to lie not on the rays themselves, but on their extensions beyond the optical system from which they emanate. In such cases we speak of imaginary caustics (or imaginary focuses). In this case, the light intensity does not reach infinity anywhere.
As for turning the intensity to infinity, in reality, of course, the intensity at the points of the caustic becomes large, but remains finite (see the problem in § 59). The formal conversion to infinity means that the geometric optics approximation becomes in any case inapplicable near caustics. The same circumstance is also related to the fact that the change in phase along the ray can be determined by formula (54.2) only in sections of the ray that do not include points of contact with caustics. Below (in § 59) it will be shown that in reality, when passing past a caustic, the field phase decreases by . This means that if in the section of the ray before it touches the first caustic the field is proportional to the multiplier - the coordinate along the ray), then after passing the caustic the field will be proportional. The same will happen near the point of contact of the second caustic, and beyond this point the field will be proportional