Basic laws of geometric optics. Optical path length of a light wave Law of refraction of light
The lengths of light waves perceived by the eye are very small (of the order of ). Therefore, the propagation of visible light can be considered as a first approximation, abstracting from its wave nature and assuming that light propagates along certain lines called rays. In the limiting case, the corresponding laws of optics can be formulated in the language of geometry.
In accordance with this, the branch of optics in which the finiteness of wavelengths is neglected is called geometric optics. Another name for this section is ray optics.
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.
The law of rectilinear propagation states that in a homogeneous medium, light travels in a straight line. This law is approximate: when light passes through very small holes, deviations from straightness are observed, the larger the smaller the hole.
The law of independence of light rays states that harriers do not disturb each other when crossing. The intersections of the rays do not prevent each of them from propagating independently of each other. 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.
The laws of reflection and refraction of light are formulated in § 112 (see formulas (112.7) and (112.8) and the following text).
Geometric optics can be based on the principle established by the French mathematician Fermat in the mid-17th century. From this principle follow the laws of rectilinear propagation, reflection and refraction of light. As formulated by Fermat himself, the principle states that light travels along a path for which it requires the minimum time to travel.
To pass a section of the path (Fig.
115.1) light requires time where v is the speed of light at a given point in the medium.
Replacing v through (see (110.2)), we obtain that Therefore, the time spent by light to travel from point to point 2 is equal to
(115.1)
A quantity having the dimension of length
called optical path length.
In a homogeneous medium, the optical path length is equal to the product of the geometric path length s and the refractive index of the medium:
According to (115.1) and (115.2)
The proportionality of the travel time to the optical path length L makes it possible to formulate Fermat's principle as follows: light propagates along a path whose optical length is minimal. More precisely, the optical path length must be extreme, i.e., either minimum, or maximum, or stationary - the same for all possible paths. In the latter case, all light paths between two points turn out to be tautochronous (requiring the same time to travel).
Fermat's principle implies the reversibility of light rays. Indeed, the optical path, which is minimal in the case of light propagation from point 1 to point 2, will also be minimal in the case of light propagation in the opposite direction.
Consequently, a ray launched towards a ray that has traveled from point 1 to point 2 will follow the same path, but in the opposite direction.
Using Fermat's principle, we obtain the laws of reflection and refraction of light. Let light fall from point A to point B, reflected from the surface (Fig. 115.2; the direct path from A to B is blocked by an opaque screen E). The medium in which the beam passes is homogeneous. Therefore, the minimum optical path length is reduced to the minimum its geometric length. The geometric length of an arbitrary path is equal to (auxiliary point A is a mirror image of point A). It can be seen from the figure that the path of the ray reflected at point O, for which the angle of reflection is equal to the angle of incidence, has the shortest length. Note that as point O moves away from point O, the geometric length of the path increases indefinitely, so in this case there is only one extremum - the minimum.
Now let's find the point at which the beam must refract, propagating from A to B, so that the optical path length is extreme (Fig. 115.3). For an arbitrary beam, the optical path length is equal to
To find the extreme value, differentiate L with respect to x and equate the derivative to zero)
The factors for are equal respectively. Thus, we obtain the relation
expressing the law of refraction (see formula (112.10)).
Let us consider the reflection from the inner surface of an ellipsoid of revolution (Fig. 115.4; - foci of the ellipsoid). According to the definition of an ellipse, paths, etc., are the same in length.
Therefore, all rays that leave the focus and arrive at the focus after reflection are tautochronous. In this case, the optical path length is stationary. If we replace the ellipsoid surface with a MM surface, which has less curvature and is oriented so that the ray emerging from the point after reflection from the MM hits the point, then the path will be minimal. For a surface that has a curvature greater than that of the ellipsoid, the path will be maximum.
Stationarity of optical paths also occurs when rays pass through a lens (Fig. 115.5). The beam has the shortest path in air (where the refractive index is almost equal to unity) and the longest path in glass ( The beam has a longer path in air, but a shorter path in glass. As a result, the optical path lengths for all rays are the same. Therefore the rays are tautochronous and the optical path length is stationary.
Let us consider a wave propagating in an inhomogeneous isotropic medium along rays 1, 2, 3, etc. (Fig. 115.6). We will consider the inhomogeneity to be small enough so that the refractive index can be considered constant on segments of rays of length X.
Optical path length
Optical path length between points A and B of a transparent medium is the distance over which light (Optical radiation) would propagate in a vacuum during its passage from A to B. The optical path length in a homogeneous medium is the product of the distance traveled by light in a medium with refractive index n by refractive index:
For an inhomogeneous medium, it is necessary to divide the geometric length into such small intervals that the refractive index could be considered constant over this interval:
The total optical path length is found by integration:
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See what “Optical path length” is in other dictionaries: The product of the path length of a light beam and the refractive index of the medium (the path that light would travel in the same time, propagating in a vacuum) ...
Big Encyclopedic Dictionary Between points A and B of a transparent medium, the distance over which light (optical radiation) would spread in a vacuum in the same time as it takes to travel from A to B in the medium. Since the speed of light in any medium is less than its speed in vacuum, O. d ...
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optical path length- (s) The sum of the products of the distances traveled by monochromatic radiation in various media and the corresponding refractive indices of these media. [GOST 7601 78] Topics: optics, optical instruments and measurements General optical terms... ... Technical Translator's Guide
The product of the path length of a light beam and the refractive index of the medium (the path that light would travel in the same time, propagating in a vacuum). * * * OPTICAL PATH LENGTH OPTICAL PATH LENGTH, the product of the path length of a light beam by... ... encyclopedic Dictionary
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Optical path, between points A and B of the transparent medium; the distance over which light (Optical radiation) would spread in a vacuum during its passage from A to B. Since the speed of light in any medium is less than its speed in ... ... Great Soviet Encyclopedia
The product of the path length of a light beam and the refractive index of the medium (the path that light would travel in the same time, propagating in a vacuum) ... Natural science. encyclopedic Dictionary
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PATH LENGTH between points A and B of a transparent medium is the distance over which light (optical radiation) would spread in a vacuum in the same time it takes to travel from A to B in the medium. Since the speed of light in any medium is less than its speed in a vacuum... Between points A and B of a transparent medium, the distance over which light (optical radiation) would spread in a vacuum in the same time as it takes to travel from A to B in the medium. Since the speed of light in any medium is less than its speed in vacuum, O. d ...
From (4) it follows that the result of the addition of two coherent light rays depends on both the path difference and the light wavelength. The wavelength in vacuum is determined by the quantity , where With=310 8 m/s is the speed of light in vacuum, and 
– frequency of light vibrations. The speed of light v in any optically transparent medium is always less than the speed of light in vacuum and the ratio 
called optical density environment. This value is numerically equal to the absolute refractive index of the medium.
The frequency of light vibrations determines color light wave. When moving from one environment to another, the color does not change. This means that the frequency of light vibrations in all media is the same. But then, when light passes, for example, from a vacuum into a medium with a refractive index n the wavelength must change 
, which can be converted like this:
,
where 0 is the wavelength in vacuum. That is, when light passes from a vacuum into an optically denser medium, the wavelength of the light is decreases V n once. On the geometric path 
in an environment with optical density n will fit
waves (5)
Magnitude 
called optical path length light in matter:
Optical path length 
light in a substance is called the product of its geometric path length in this medium and the optical density of the medium:
.
In other words (see relation (5)):
The optical path length of light in a substance is numerically equal to the path length in a vacuum, on which the same number of light waves fits as on the geometric length in the substance.
Because the result of interference depends on phase shift between interfering light waves, then it is necessary to evaluate the result of interference optical path difference between two rays
,
which contains the same number of waves regardless on the optical density of the medium.
2.1.3.Interference in thin films
The division of light beams into “halves” and the appearance of an interference pattern is also possible under natural conditions. A natural “device” for dividing light beams into “halves” are, for example, thin films. Figure 5 shows a thin transparent film with a thickness 
, to which at an angle 
A beam of parallel light rays falls (a plane electromagnetic wave). Beam 1 is partially reflected from the upper surface of the film (beam 1), and partially refracted into the film
ki at the angle of refraction 
. The refracted beam is partially reflected from the lower surface and exits the film parallel to beam 1 (beam 2). If these rays are directed at a collecting lens L, then on the screen E (in the focal plane of the lens) they will interfere. The result of interference will depend on optical the difference in the path of these rays from the “division” point 
to the meeting point 
. From the figure it is clear that geometric the difference in the path of these rays is equal to the difference geom . =ABC–AD.
The speed of light in air is almost equal to the speed of light in vacuum. Therefore, the optical density of air can be taken as unity. If the optical density of the film material n, then the optical path length of the refracted ray in the film ABCn. In addition, when beam 1 is reflected from an optically denser medium, the phase of the wave changes to the opposite, that is, half a wave is lost (or vice versa, gained). Thus, the optical path difference of these rays should be written in the form
wholesale . = ABCn – AD / . (6)
From the figure it is clear that ABC = 2d/cos r, A
AD = ACsin i = 2dtg rsin i.
If we put the optical density of air n V=1, then known from the school course Snell's law gives for the refractive index (optical density of the film) the dependence
. (6a)
Substituting all this into (6), after transformations we obtain the following relation for the optical path difference of the interfering rays:
Because when beam 1 is reflected from the film, the phase of the wave changes to the opposite, then conditions (4) for the maximum and minimum interference are reversed:
- condition max
- condition min. (8)
It can be shown that when passing light through a thin film also produces an interference pattern. In this case, there will be no loss of half a wave, and conditions (4) are met.
Thus, the conditions max And min upon interference of rays reflected from a thin film, are determined by relation (7) between four parameters - 
It follows that:
1) in “complex” (non-monochromatic) light, the film will be painted with the color whose wavelength 
satisfies the condition max;
2) changing the inclination of the rays ( 
), you can change the conditions max, making the film either dark or light, and when illuminating the film with a diverging beam of light rays, you can get stripes« equal slope", corresponding to the condition max by angle of incidence 
;
3) if the film has different thicknesses in different places ( 
), then it will be visible strips of equal thickness, on which the conditions are met max by thickness 
;
4) under certain conditions (conditions min when the rays are incident vertically on the film), the light reflected from the surfaces of the film will cancel each other out, and reflections there won't be any from the film.
OPTICAL PATH LENGTH is the product of the path length of a light beam and the refractive index of the medium (the path that light would travel during the same time, propagating in a vacuum).
Calculation of the interference pattern from two sources.
Calculation of the interference pattern from two coherent sources.
Let's consider two coherent light waves emanating from sources u (Fig. 1.11.).
The screen for observing the interference pattern (alternating light and dark stripes) will be placed parallel to both slits at the same distance. Let us denote x as the distance from the center of the interference pattern to the point P under study on the screen.
Let us denote the distance between the sources as d. The sources are located symmetrically relative to the center of the interference pattern. From the figure it is clear that
Hence
and the optical path difference is equal to
The path difference is several wavelengths and is always significantly smaller, so we can assume that
Then the expression for the optical path difference will have the following form:
Since the distance from the sources to the screen is many times greater than the distance from the center of the interference pattern to the observation point, we can assume that.
, (1.96)
e. Substituting value (1.95) into condition (1.92) and expressing x, we obtain that intensity maxima will be observed at values where is the wavelength in the medium, and m max is the order of interference, and
X
, (1.97)
- coordinates of intensity maxima.
Substituting (1.95) into condition (1.93), we obtain the coordinates of the intensity minima
An interference pattern will be visible on the screen, which looks like alternating light and dark stripes. The color of the light stripes is determined by the filter used in the installation.
The distance between adjacent minima (or maxima) is called the interference fringe width. From (1.96) and (1.97) it follows that these distances have the same value. To calculate the width of the interference fringe, you need to subtract the coordinate of the adjacent maximum from the coordinate value of one maximum
For these purposes, you can also use the coordinate values of any two adjacent minima.
Coordinates of intensity minima and maxima.
Optical length of ray paths. Conditions for obtaining interference maxima and minima.
In a vacuum, the speed of light is equal to , in a medium with a refractive index n the speed of light v becomes less and is determined by the relation (1.52)
Let two point coherent light sources emit monochromatic light (Fig. 1.11). For them, the coherence conditions must be satisfied:
To point P, the first ray travels in a medium with a refractive index - a path, the second ray passes in a medium with a refractive index - a path.
The distances from the sources to the observed point are called the geometric lengths of the ray paths. The product of the refractive index of the medium and the geometric path length is called the optical path length L=ns. L 1 = and L 1 = are the optical lengths of the first and second paths, respectively.
, (1.87)
Let u be the phase velocities of the waves.
, (1.88)
The first ray will excite an oscillation at point P:
, (1.89)
and the second ray is vibration
The phase difference between the oscillations excited by the rays at point P will be equal to:
The multiplier is equal to (- wavelength in vacuum), and the expression for the phase difference can be given the form
there is a quantity called the optical path difference. When calculating interference patterns, it is the optical difference in the path of the rays that should be taken into account, i.e., the refractive indices of the media in which the rays propagate. Substituting value (1.95) into condition (1.92) and expressing x, we obtain that intensity maxima will be observed at values From formula (1.90) it is clear that if the optical path difference is equal to an integer number of wavelengths in vacuum
then the phase difference and oscillations will occur with the same phase. Number
, (1.93)
is called the order of interference. Consequently, condition (1.92) is the condition of the interference maximum. 
If equal to half an integer number of wavelengths in vacuum,
That
, so that the oscillations at point P are in antiphase. Condition (1.93) is the condition of the interference minimum.
So, if at a length equal to the optical path difference of the rays, an even number of half-wavelengths fits, then a maximum intensity is observed at a given point on the screen. If there is an odd number of half-wavelengths along the length of the optical ray path difference, then a minimum of illumination is observed at a given point on the screen. Recall that if two ray paths are optically equivalent, they are called tautochronous. Optical systems - lenses, mirrors - satisfy the condition of tautochronism. 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. 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 experimentally: · the law of rectilinear propagation of light; · the law of independence of light rays; · the law of reflection; · the law of refraction of light. 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 bends around 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. Fig 7.1 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 movement 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
To derive the law of reflection Let's use Huygens' principle. Let us assume that a plane wave (wave front AB 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.
Derivation of the law of refraction. Let us assume that a plane wave (wave front AB), propagating in vacuum along direction I with 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, i.e. .This implies Snell's law : A slightly different formulation of the law of propagation of light 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. Ray 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 it takes to travel the path S.A.B.:
.To find the minimum, we find the first derivative of τ with respect to m and equate it 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). From Fermat's principle has several consequences. 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 ray decreases, and the reflected ray increases. · If , then the intensity of the refracted ray becomes zero, and the intensity of the reflected ray 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 boundary at α > 42° there will always be total reflection. In Fig. Figure 7.9 shows total reflection prisms that allow: a) to rotate the beam by 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. Fig. 7.10 In fiber parts, glass fiber is used, the light-guiding core (core) of which is surrounded by glass - a shell made 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. Through such a cable, the thickness of an ordinary pencil, up to eighty thousand telephone conversations can be simultaneously transmitted. In addition, light guides are used in fiber-optic cathode ray tubes, in electronic counting machines, for encoding information, in medicine (for example, stomach diagnostics), for purposes of integrated optics.
