Fresnel zones - are areas into which the surface of sound or light waves for computing the results of diffraction of sound or light.This method was first applied in 1815 O.Frenel.
Background
Augustin-Jean Fresnel (10.06.1788-14.07.1827) - French physicist.He devoted his life to the study of the properties of physical optics.He had in 1811 under the influence of E. Malus began to study physics on their own, soon became interested in experimental research in the field of optics.In 1814, "rediscovered" the principle of interference, and in 1816 added the well-known principle of Huygens, which introduced the idea of coherence and interference of elementary waves.In 1818, building on the work done, he developed the theory of diffraction of light.He introduced the practice of considering the diffraction from the edge, as well as a round hole.I conduct experiments, now classics, with double prism and bizerkalami of light interference.In 1821, he proved the fact of the transverse nature of light waves, in 1823 opened a circular and elliptical polarization.He explained on the basis of wave concepts chromatic polarization, as well as the rotation of the polarization of light and birefringence.In 1823 he established the laws of refraction and reflection of light on a fixed flat surface between the two media.Along with Jung considered the creator of wave optics.He is the inventor of a series of interference devices, such as a mirror or a Fresnel biprism Fresnel.It is considered the founder of a fundamentally new way of lighthouse illumination.
little theory
Identify Fresnel diffraction can be both with a hole of any shape, and none at all.But in terms of practical expediency is best seen at the opening of its round shape.In this light source and the observation point must be on a line that is perpendicular to the plane of the screen and passes through the center of the hole.In fact, in the Fresnel zone can break any surface through which the light waves.For example, the surfaces of equal phase.However, in this case it will be easier to break a hole in the flat area.For this we consider the elementary optical problems, which will allow us to determine not only the radius of the first Fresnel zone, but also follow-up with arbitrary numbers.
task of sizing rings
To begin to imagine what the surface of a planar holes located between the light source (point C) and the observer (point H).It is perpendicular to the line of HF.CH segment passes through the center of the round hole (point A).Since our goal is the axis of symmetry, the Fresnel zone will be in the form of rings.A decision will be confined to the determination of the radius of the circle with an arbitrary number (m).The maximum value is called the radius of the zone.To solve the problem it is necessary to make an additional construction, namely: select an arbitrary point (A) in the plane of the opening and connect its straight line segments with the observation point and the light source.The result is a triangle SAN.Then you can make it so that the light waves coming to the observer along the path SAN pass a longer path than the one that will go on the path CH.This implies that the path difference CA + AN-CH determines the difference between the wave phase, which took place on secondary sources (A and D) to the observation point.From this value depends on the resultant interference of waves from the position of the observer, and therefore the light intensity at this point.
Calculate first radius
turns out that if the path difference is equal to half the wavelength of light (λ / 2), then the light will come to the observer in the opposition.It can be concluded that, if the path difference is less than λ / 2, then the light will arrive in the same phase.This condition CA + AN-SN≤ λ / 2 is by definition the condition that the point A is in the first ring, that is, it is the first Fresnel zone.In this case, the boundary of the circle path difference is equal to half the wavelength of light.So this equation to determine the radius of the first zone, which we denote by P1.If the path difference corresponding to λ / 2, it will be equal to the segment OA.In that case, if the distance SO far exceed the diameter of the hole (it is usually considered such options), from geometrical considerations radius of the first zone is determined by the following formula: P1 = √ (λ * SB * OH) / (CO + OH).
Calculation of Fresnel zone radius
formula for determining future values of the radii of the rings identical discussed above, only the numerator is added to the number of the desired zone.In this case, the equality of the path difference will be: CA + AN-SN≤ m * λ / 2 or CA + AN-CO-ON≤ m * λ / 2.It follows that the radius of the desired area with the number "m" to the following formula: PM = √ (m * λ * SB * OH) / (CO + OH) = R1√m
Summarizing the interim results
may be noted that the breakingin the area - a division of the secondary light source to the sources having the same area as Pm = π * π * Rm2- PM-12 = π * P12 = P1.Light from neighboring zones will in opposite phase as path difference neighboring ring by definition be equal to half the wavelength of light.Generalizing this result, we conclude that the breaking of the holes on the circles (such that the light from the neighboring comes to the observer with a fixed phase difference) would mean breaking the ring at the same area.This assertion is easily proved with the help of the task.
Fresnel Zone for a plane wave
Consider the breakdown square holes on a thin ring with equal area.These circles are the secondary light sources.The amplitude of the light wave that came out of each ring to the observer about the same.In addition, the phase difference between the adjacent range at the point H is also the same.In this case, the complex amplitudes at the point in adding an observer on a single complex plane form part of the circle - arc.The total amplitude of the same - a chord.Now consider how the changing pattern of summing complex amplitudes in the case of change of the opening while maintaining the other parameters of the problem.In that case, if the hole opens to the viewer only one zone, the picture will be presented to the addition part of the circumference.The amplitude of the last ring is rotated an angle π relative to the central part, ie. K. The path difference of the first zone, by definition, equal to λ / 2.This angle π means that the amplitude will be half of the circle.In this case, the sum of these values at the observation point is zero - zero chord length.If you will open three rings, the picture will present a half circle, and so on.The amplitude at the observer for an even number of rings is zero.And in the case where an odd number of wheels is used, it will be equal to the maximum value of the length and diameter of the complex plane adding amplitudes.The above objectives are fully disclose the method of Fresnel zones.
Brief about particular cases
Consider rare conditions.Sometimes the task of states that used a fractional number of Fresnel zones.In this case, a half ring understand quarter circle pattern, which will correspond to half the area of the first zone.Similarly calculated any other fractional value.Sometimes the condition suggests that certain fractional number of rings is closed, and so much open.In this case, the total amplitude of the field is as a vector difference between the amplitudes of two tasks.When all zones are open, then there is no obstacle in the path of the light waves, the picture will be in the form of a spiral.It turns out, because when you open a large number of rings to consider the dependence of the emitted secondary light source to the point of the observer and the direction of the secondary source.We find that the light from the area with a large number has a small amplitude.Center coil is received in the middle of the circumference of the first and second rings.Therefore, the field amplitude in the case where all the visible area is less than half than the first one when the open circle, and the intensity differs by four times.
Fresnel diffraction of light
Let's look at what is meant by the term.Fresnel diffraction condition is called when a hole is opened through several zones.If you will open a lot of rings, then this option can be ignored, that is exerted in the approach to geometrical optics.In the case where the through hole is opened for the observer substantially less than one zone, this condition is referred to as Fraunhofer diffraction.He is considered to be satisfied if the light source and the point of the observer is at a sufficient distance from the hole.
Compare and zone plate lenses
If you close all odd or all the even Fresnel zone, while at the observer will light wave with greater amplitude.Each ring gives the complex plane half circle.So if left open the odd zones, then the total will be only half the spiral of circles that contribute to the overall amplitude of the "bottom-up".The obstacle in the path of the light wave, in which only one type of open rings, called the zone plate.The light intensity at the observer outweigh the light intensity at the plate.This is due to the fact that the light wave of each open ring misses the viewer in the same phase.
similar situation is observed with focusing light with a lens.It, unlike the plate, no rings are not closed, and moves the light in phase by π * (2 + π * m) from the circles that closed zone plate.As a result, the amplitude of the light wave is doubled.Moreover, the lens eliminates so-called reciprocal phase shifts which are within a single ring.It expands on the complex plane half circle for each zone in a line segment.As a result, the amplitude increases by π times, and the whole complex plane spiral lens will unfold in a straight line.
