TALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS 17. FRESNEL DIFFRACTION ON A ROUND APERTURE

Transcription

1 7. FRESNEL DIFFRACTION ON A ROUND APERTURE. Objective Exaining diffraction pattern on a round aperture, deterining wavelength of light source.. Equipent needed Optical workbench, light source, color filters, screens with different apertures, ocular (eyeglass). 3. Theory Diffraction is a process where light (waves in general) bends in hoogenous and isotropic edia behind obstacles on its way. Light turns away fro straight propagation path due to eeting a non-transparent obstacle, not because of bending, reflection or scattering. Laws of geoetrical optics do not describe this feature. The closer is obstacles (or apertures) diaeter to lights wavelength, the ore light diffracts. Phenoena of diffraction can be explained by a clever atheatical construction Huygens- Fresnel principle. According to it every eleent ( point ) of the wave front is a source of coherent eleentary waves. New light in every point of space in front of our wave front is fored as a result of interference of eleentary waves. Eleentary waves are also called secondary waves since they are generated fro front of original (priary) wave. Secondary waves originating fro the part uncovered by obstacle propagating in every direction although having different intensities, foring a sharp angle to wave front eleents norals are inspected in case of Fresnel diffraction. As a result of interference of all secondary waves intensity of light is redistributed diffraction pattern occurs behind the aperture (i.e. in place where norally a shadow should be observed). Two cases of diffraction are traditionally entioned: Fresnel and Fraunhofer diffractions. We have Fresnel diffraction if a spherical wave reaches an obstacle (or aperture in it) and diffraction pattern is observed relatively close to the obstacle. In case of a Fraunhofer diffraction obstacle (or aperture in it) is lit by a flat wave (parallel beas) and diffraction pattern is observed relatively far away fro the obstacle. Suing secondary waves can be considered planar beas are practically parallel. It is said to be diffraction in parallel beas (beas ust always intersect in case of diffraction but in this case eeting angle is relatively sall). In case of Fresnel diffraction beas can not be considered being parallel. Suing waves are spherical. It is said to be diffraction in converging beas. There is no principle difference between the Fresnel and Fraunhofer diffractions. In both cases is light observed as a wave. Distance of light source and detection point fro the obstacle and easures of the obstacle ake the difference. Resulting diffraction patterns are different aswell. Iportant paraeters are wavelength of light source λ, characteristic easure R of aperture, distance between aperture and detection screen b and distance of light source fro aperture a (characteristic easure of round aperture is it's diaeter, in case of a slit it's width). A special cobination of entioned 4 paraeters is used to describe lights distribution behind the obstacle:

2 where ρ F p =, R λab ρ F =. If paraeter p >>, we have a case of so-called far field and a Fraunhofer diffraction. If p, we are in near-field and observe Fresnel diffraction. If p <<, diffraction cannot be observed and laws of geoetrical optics are held. Different kinds of diffraction can be distinguished by the paraeter p, which is called the Fresnel constant. Fraunhofer diffraction pattern has always a bright axiu of intensity in the centre and pattern does not alter when paraeter b changes. In case of Fresnel diffraction things are different. There is no sharp line between both odes of diffraction. A Huygens-Fresnel principle is applied to explain the Fresnel diffraction. Let us consider propagation of light waves fro point source S (see figure ) and deterine aplitude of waves in soe arbitrary chosen point P. In case of isotropic and hoogenous edia a spherical wave front is fored around source at distance a = OS = c t ( c speed of light) during tie t. According to Huygens-Fresnel principle oscillations (secondary waves) propagate fro every point of the sphere in arbitrary directions (generally in direction of point P) and in soe tie reach point P where interference takes place. Aplitude of fored suary wave can be estiated by Fresnel s ethod described below.

3 Figure 7. Wave front is divided into annular s so that distance between s edges and point P λ differs by half wavelength. Hence distance of 's outer edge b and point P (see figure 7.) equals to: λ b = b +, () where b is distance between wave front s axiu curvature point O and point P. Such division of wave front let's one ake a conclusion that waves originating fro corresponding points of any two neighbouring s will reach point P in opposite phase and aplitude A of resulting oscillations in this point can be characterised with following series of ebers with alternating signs: A A A + A A +... A..., () = where A, A, A 3, are suary waves aplitudes in point P coposed by waves originated fro s.,., 3.,... correspondingly. Value of A depends on the surface of, on angle α between s (part of wave front) surface noral N r and light bea MP heading to detection point and also on distance between selected and detecting point (see figure ). α is called a diffraction angle. In order to find aplitude A we first exaine how aplitude A changes with increase of nuber. For this we deonstrate that surfaces of all Fresnel s are approxiately equal. We derive a forula for calculating Fresnel s surface for this purpose. Relying on figure we can write: fro which λ r = a + ( a h ) = b + ( b h ), (3) λ bλ + h = + ( a b). (4) We can disiss the second eber in nuerator since it has very sall value due to wave length λ. So: h bλ =. + (5) ( a b) Using forula for calculating surface of a spherical segent: Thus surface of Fresnel nuber equals: S π ab = π ah = λ. (6) 3

4 S = S S π abλ =. (7) Since this forula does not depend of nuber, surfaces of Fresnel s are approxiately equal and the effect of their irradiations in detection point is deterined by difference of diffraction angle and distance only. Diffraction angle increases together with the nuber which leads to decreasing aplitude in detection point. Aplitude in detection point decreases also due to increase of path of secondary waves. Angle and path are increased onotonously. This leads to a conclusion that suary aplitude of secondary waves in detection point also decreases onotonously together with increasing of Fresnel nuber: A > A > A3 >... > A >.... Accordingly resulting aplitude of secondary waves originated fro soe in detection point can be estiated by taking ean value of resulting aplitudes of waves originated fro neighbouring s: A A+ A =. (8) Let us apply last relation to forula (). Firstly let us write it in the following way: + A A A3 A3 A5 A = + A + + A (9) It concludes fro the forula (8) that expressions in brackets equal to zero, so: A A =. (0) Hence resulting aplitude in point P equals to half of aplitude caused by wave fro the first Fresnel. Approxiate radius r of outer edge of Fresnel can be expressed (using forulas 3 and 5): ab r = λ. () Let us explore Fresnel diffraction in case of a round aperture. Non-transparent screen with a round aperture is inserted between point light source S and detection point P (see figure 7.). Screen blanks all Fresnel s in the wave front except the one with nuber n located in aperture with diaeter D (on figure 7. n = ). Resulting aplitude of oscillations in point P is following: A = A A + A3 A ± A n, () where aplitude A n has positive sign in case of s with odd nubers n. Applying soe conversions analogous to those used in forula (9) and accounting condition (8), one can write: A An A = ±. (3) 4

5 Figure 7. Aplitude A n does not differ uch of A if nuber fro Fresnel s in aperture is not big. In this case according to forula (3) resulting aplitude is A = A when nubers n are odd and A = 0 in case of even nubers n. In first case diffraction axiu (bright spot) can be observed in point P, second case leads to diffraction iniu (dark dot). Nuber of Fresnel s in aperture can be calculated as follows: ( ) D n =. (4) 4abλ Diffraction inius or axius can be observed on detection screen in point P only in case of integer values of n. Investigating variations of axius and inius in the centre of diffraction pattern caused by change of distance b and keeping distance a fixed, one can deterine nuber of Fresnel s n in aperture and calculate wavelength of used light source as follows: ( ) D λ = (5) 4abn Diagra of experient stand is displayed in figure 7.3. Light fro source passes color filter and heads to a sall round aperture in screen 3. Aperture can be considered being a point light source. A spherical wave originating fro it falls to a screen 4 with round aperture. Diffraction pattern is observed via ocular 5. All entioned coponents are fixed to optical bench. 5

6 Figure Experiental procedure. Check that location of all optical coponents atches the one given on figure 3. Choose distance a between screens 3 and 4 on optical bench within the range 80 0 c. Switch on light source.. Mount ocular 5 so that diffraction pattern will be in the centre of oculars field. 3. Move oculars lowly away fro aperture and look for change of axius and inius in patterns centre. This corresponds to change of nuber n of Fresnel s fro odd to even and vice versa whereas n itself decreases. A bright unfocused spot will be seen through ocular when there is one left in aperture. 4. Found position corresponds to axial distance. Move ocular slowly closer to aperture and look for changes in diffraction pattern. Nuber n = when you have reached first diffraction iniu and a dark dot appears in ocular. Coing closer leads to a bright dot in ocular surrounded by a dark ring, corresponding to n=3. Then another iniu appears (dark dot, n=4) etc. 5. Measure distance b for every iniu and axiu of diffraction pattern using scale on optical bench. Also write down corresponding nuber of Fresnel s n. You will find diaeter D arked on screen. 6. Use different values of n for the experient. Write results in table. Table Deterining light sources wavelength with Fresnel diffraction Experient nr n b λ a = D = λ = Calculate wavelength λ for each pair of b and n using forula (5). Calculate arithetic ean of acquired results and its A-type extended uncertainty U A ( λ ). Estiate reality of your result (you know color of light source). 6

7 5. Questions and tasks. What is a spherical wave and how is it obtained in present experient?. What is wave front and what is wave surface? 3. What is diffraction of light, what's its cause and when can it be observed? 4. Postulate Huygens-Fresnel principle. 5. Write the analytic forula for Huygens-Fresnel principle. 6. What is Fresnel diffraction, how is it used? 7. Describe the principle of constructing Fresnel s. 8. What does nuber of Fresnel s depend on? 9. Why can a dark dot be in soe cases observed in the centre of diffraction pattern? 0. Why a color filter is used in this experient? What pattern would be produced in case of white light?. What distance easured in experient is affected by wavelength?. What distance is affected by oculars focal length? 3. What kind of diffraction pattern could be observed behind a round obstacle? 4. What is a Poisson dot? 5. Will results of experient depend on aperture's diaeter? 6. Literature. Halliday, D., Resnick, R., Walker, J. Fundaentals of Physics. 6th ed. New York, John Wiley & Sons, Inc., 00, 36-3, 37-,