Fresnel Diffraction. monchromatic light source

Transcription

1 Fesnel Diffaction Equipment Helium-Neon lase (632.8 nm) on 2 axis tanslation stage, Concave lens (focal length 3.80 cm) mounted on slide holde, iis mounted on slide holde, m optical bench, micoscope slide with seveal small shapes glued to it (ball beaing, staight edge pin), calipes, tape measue. upose To undestand Fesnel diffaction. To test the Huygens-Fesnel wave desciption of light by examining Fesnel diffaction pattens geneated by a cicula apetue with vaiable diamete and seveal othe objects. To measue the wavelength of lase light. Theoy Suppose, as in Figue, that light fom some monochomatic souce is shone upon a small cicula apetue. The light passing though the apetue is then allowed to fall on a sceen positioned at some distance, ', fom the slit, with its plane pependicula to the path of the light's popagation. The esulting diffaction patten displayed upon the sceen can be classified as aising fom one of two types of diffaction. If the system is aanged so that the light falling on the apetue is plana and the ays of light aiving at the sceen ae essentially paallel, then the patten displayed on the sceen is a consequence of Faunhoeffe diffaction. If, howeve, one o both of these constaints is not satisfied then the esulting patten is a consequence of Fesnel diffaction. This moe geneal second type of diffaction shall be the focus of the discussion pesented hee. monchomatic light souce diffaction patten apetue Figue ' sceen To poceed it will be necessay to make use of the Huygens-Fesnel pinciple which, in its unadulteated fom, claims that any point on a wave font can itself be egaded as a point souce emitting cicula wave fonts. Thee is a fundamental poblem with this desciption of wave behavio, made evident by the consideation of the popagation of plane paallel light, emitted, fo example, fom a lase. Figue 2 shows seveal wave fonts of plane paallel light, with the Figue 2 Diection of wave opagation. diection of popagation indicated by a ay. In addition, at seveal points,, 2, 3, and 4, the cicula wave fonts, pedicted by the Huygens-Fesnel pinciple, have been depicted. Since point souces emit waves in a cicula patten with no pefeed diection, 2, 3, and 4 not only contibute to the fowad popagation of the oiginal wave but also, in theoy, contibute to a wave popagating in the evese diection. This pedicted wave, taveling towads the oiginal souce of the light, is not, howeve, obseved expeimentally. It follows that thee must be a pefeed diection in which the seconday point souces on any given wave font emit. The cicula wave function that descibes wave popagation away fom a point souce must be modified by multiplying by an appopiate facto that will take this diectionality into account. Such a facto is called the obliquity o inclination facto, K, and it has the fom, deived fom the Kichoff desciption of wave popagation,

2 K( θ) = ( cos θ), 2 () θ whee θ is the angle between the diection of oiginal wave popagation and a ay adiating fom the Huygens-Fesnel point Diection of wave souces on any given wave font. Figue 3 shows the opagation. seconday point souce and the emitted wave fonts modified by the obliquity facto. Figue 3 Befoe consideing the effect of a cicula apetue on the passage of light it is convenient to fist conside the unobstucted popagation of light fom a monochomatic point souce. Such a consideation equies the intoduction of the wave function, E, which descibes mathematically the spheical wave emitted by a point souce. In spheical coodinates, with the point souce at the oigin, such a wave function can be witten explicitly in the fom E = 0 ωt kρ ρ cos( ), whee ω and k ae the angula fequency and wave numbe of the wave espectively, ρ is the adius, t is the time, and 0 is the amplitude of the wave at the souce, ρ = 0, when t = 0. hysically, Equation 2 can be intepeted as descibing the electic field component of light as a function of both time and distance fom the cental souce. Suppose, as in Figue 4, that the pimay wave font geneated by a point souce, S, has taveled a distance. Accoding to the Huygens-Fesnel pinciple, all points on the pimay wave font, a sphee of adius, can be consideed seconday point souces, emitting spheical wave fonts modified by the obliquity facto. (2) ' 3 λ/2 ' λ ' λ/2 S 2 3 ' Figue 4 If the Huygens-Fesnel pinciple is coect it should be possible to sum the contibutions of each seconday souce at a point, a distance ' fom S, to obtain the initial pimay wave. It follows fom Equation 2 that the wave function at the point should be given by 0 E = ωt k ρ ' cos( ( ' ' )). (3) As illustated in Figue 4, the spheical wave font is divided into annula egions, called Fesnel o half-peiod zones, centeed on an axis taced fom S to. In each egion the uppe bounday (the bounday futhest fom the S- axis) is half a wavelength longe than the path length fom any point on the lowe bounday (the bounday closest to the S- axis). In zone 2, fo example, the path lengths fom the uppe and lowe bounday to the point ae ' λ/2 and ' λ espectively. It follows that fo any point in a zone thee is a coesponding point in an adjacent zone that is futhe than by an amount λ/2. Note that the Fesnel zones, although small, ae finite in extent. Conside the yth Fesnel zone, shown in Figue 5, having uppe and lowe bounday path lengths to point of ' yλ/2 and ' (y - )λ/2 espectively. Let da be an infinitesimal ing-shaped aea within the yth zone. The assumption shall be made that the point souces within the ing ae coheent, each having the

3 same phase and amplitude. In addition, we shall assume that each of the point souces on the wave font at adius adiate in phase with the pimay wave. Accoding to these assumptions, the seconday wavelets emitted fom the infinitesimal ing all aive at, afte having taveled a distance, with the same phase, ωt - k( ). Since thee ae an infinite numbe points within the infinitesimal aea da it is convenient to conside the electic field at geneated pe unit dφ S φ θ ρ sin φ da d ' yth zone aea, σ, athe than summing the electic fields geneated by each individual Huygens-Fesnel point souce. In analogy to the wave function fo the pimay wave, given by Equation 2, we expect σ to have the fom σ = K A cos[ ωt k( )], (4) whee A is the souce stength pe unit aea of the seconday emittes on da. Note that the obliquity facto, K, has been included in Equation 4 to take into account the diectionality of the seconday point souces on da. It follows fom Equation 4 that the contibution, de, to the wave at due to the aea da is given by the equation Figue 5 de = σ da A = K cos[ ωt k( )] da. (5) In ode to integate Equation 5 and obtain the total contibution of the yth zone, E y, to the wave at, an expession fo da is equied. Applying the law of cosines to the geomety depicted in Figue 5 esults in = ( ') 2( ')cos φ, (6) o, afte diffeentiating and eaanging, sin φ φ d d = ( '). (7) Equation 7 makes use of the fact that both and ' ae constant. Again fom the geomety pesented in Figue 5 we find which, with the help of Equation 7 becomes da = dφ2 π( sinφ), da = 2π d. ( ') (9) Equation 9, in addition to facilitating the integation suggested by Equation 5, allows us to detemine the aea of the Fesnel zones. Integating Equation 9 fom ' (y - )λ/2 to ' yλ/2 and assuming small values of y it is found that (8)

4 A = = ' yλ / 2 2π d ( ') = ' ( y ) λ/ 2 = π' λ. ' Equation 0 shows that the aea of the Fesnel zones, fo small values of y, is a constant. Substituting Equation 9 into Equation 0 and integating fom ' (y - )λ/2 to ' yλ/2 esults in E = 2πK ( ') y y A = ( ) = ' yλ / 2 = ' ( y ) λ/ 2 cos[ ωt k( )] d 2K λ sin[ ωt k( ' )], ( ') y y A Whee the assumption has been made that the obliquity facto K y emains constant ove the entie zone. Accoding to Equation, the amplitude of E y altenates between positive and negative values depending on whethe y is odd o even. This means that adjacent Fesnel zones ae of opposite phase and will tend to cancel one anothe. Also note that as y inceases the value of θ will incease esulting in a deceased obliquity facto, and, in tun, a deceased contibution to the oveall wave function. It follows that zones of lage y, and thus small K, will contibute little to the wave obseved at the point. If the sphee of adius has been divided into m Fesnel zones then the wave function at the point should be given by the Equation E = E E2 E3 L E m. Altenatively, if the sign change given by Equation is taken into account this seies can be ecast in the fom E = E E2 E3 L ± E m. (3) If m is odd, Equation 3 can be ewitten in two diffeent ways o, altenatively, E 3 3 E = ( E2 ) ( E Em 2 m m L ( Em ), E2 E2 E4 E4 E6 E = E ( E E 3 ) ( 5 ) Em 3 m m L ( Em 2 ) Em Note that the yth zone may have a value of E y that is eithe geate o less than the aithmetic mean of its two neighbos, ( E y- E y )/2. In the fist case, with E y geate than the aithmetic means of its neighbos, each backeted tem appeaing in Equations 4 and 5 is negative. Consequently the two inequalities and 5 ) (0) () (2) (4) (5) E < m 2 2 (6) E2 Em E > E Em 2 2 (7)

5 can be constucted fom Equations 4 and 5. Since the obliquity facto vaies slowly fom zone to zone the appoximation E y E y± can be made so that Equation 7 becomes It follows fom Equations 6 and 8 that E > m 2 2. (8) E m 2 2. (9) Note that the last Fesnel zone, the mth zone, is expected to have the lagest possible value of θ, which is π. Substituting into Equation we find that the obliquity facto fo this zone becomes 0 and, consequently, E m, given by Equation also becomes 0. Hence, Equation 9 can be ewitten as E E 2. (20) The same esult is obtained if it is assumed that E y is less than the aithmetic mean of its two neighbos. Similaly, if the numbe of tems, m, in Equation 3 is even, the same esult follows by an analogous pocedue. Substituting Equation into 20, noting that the obliquity facto fo the fist zone has a value of, gives an expession fo the total wave function at a point ', A λ E ωt k ρ ( ') sin[ ( ' ' )]. (2) A compaison of Equations 3 and 2 suggests that 0 = A λ. Although the esult geneated by the Huygens- Fesnel pinciple, Equation 2, is quite simila to Equation 3 thee is a phase diffeence between the two equations that suggests a poblem with the pinciple. The moe igoous Kichoff fomulation of wave popagation coects this deficiency of the Huygens-Fesnel desciption. Having now consideed the popagation of an unobstucted wave in some detail we ae now in a position to conside the effect of placing a cicula apetue between the souce of the light and the point of obsevation. It is at this point that the pactical advantage of defining Fesnel zones becomes evident. Figue 6 depicts the effect of a cicula apetue, placed a distance fom the souce, on the passage of light fom seveal Fesnel zones to the point, lying on the symmety axis of the apetue. As can be seen fom the diagam, the light fom only a small numbe of zones, all located close to the symmety axis, passes unaffected though the apetue. The est of the zones ae effectively blocked. It follows that an apetue essentially tuncates the seies given by Equation. Suppose the diamete of the apetue was deceased so that light fom only the fist zone could each the point. In this situation a bight spot would be visible at. If, howeve, the apetue was widened so that light fom the fist two zones eached, a dak spot would be visible, a consequence of the afoementioned cancellation of adjacent zones. Thus, if the apetue was opened S zone zone 2 zone 3 zone 4 zone 5 zone 6 slowly, the point would altenately become a dak and bight spot. It should be noted that fo lage sizes of the apetue the obliquity facto comes into play and, as a consequence, the vaiation in bightness of the point is less ponounced. Figue 6 apetue '

6 If is moved away fom the symmety axis the bightness of the point will vay depending upon the numbe of zones (and factions of those zones) that emit light unobstucted by the apetue. The esult is a diffaction patten that consists of a seies of concentic ings which ae altenately dak and bight. The total numbe of ings, both dak and bight, coesponds to the numbe of zones "visible" to a point located on the symmety axis of the system. The use of Fesnel zones can also be used to explain othe diffaction zone phenomena. Of paticula inteests is its zone 2 use in explaining (and pedicting) the zone 3 zone 4 existence of the oisson bight spot, the zone 5 obsevation of which initially helped confim zone 6 the wave-like popeties of light. Suppose the apetue in Figue 6 is eplaced by a S small cicula object. As seen in Figue 7 the effect of the object is to pevent light emitted by the cental zones fom eaching the point, lying on the symmety axis. The pesence of the object does not, howeve, pevent light fom the oute zones Figue 7 fom eaching. The esult is a cental bight spot, the oisson bight spot, suounded by a seies of concentic ings. cicula object Figue 8 shows the expeimental aangement that will be used to study Fesnel diffaction pattens. The lase, mounted on the optics bench by means of a two-axis tanslation stage, shines on the lens, mounted on a slide holde. The lens speads the light, effectively alteing the beam of the lase so that it appeas to oiginate fom a point souce lying at the focal point of the lens. Having passed though the lens the light then falls upon the iis o micoscope slide mounted in anothe slide holde. The esulting diffaction patten is displayed on a wall. two-axis tanslation stage y-adjust x-adjust lase concave lens slide holde optics bench iis o micoscope slide optical axis ' Wall Figue 8 Fo a numbe of diffeent iis positions on the optics bench the aea of the Fesnel zones is detemined by measuing the diamete of the iis apetue, calculating its aea, and dividing the esult by the numbe of ings displayed on the wall. The assumption is being made that the Fesnel zones involved in ceating the obseved diffaction pattens lie essentially in a plane paallel to the wall. This assumption is only justified fo Fesnel zones with small values of y. Equation 0 can be used to detemine the wavelength of the lase fom the zone aeas along with thei associated iis positions. If the iis is eplaced by the micoscope slide the diffaction patten geneated by seveal objects can be examined. Expeimental ocedue. Aange the equipment so that the optical bench is diected towads a wall that is appoximately 5.30 m away fom the closest end of the bench. The lase should be mounted at the opposite end of the bench so that it is as fa away as possible fom the wall. Mount the lens diectly in font of the lase and the iis appoximately cm fom the lens. Thoughout this expeiment emembe that looking diectly into the lase can cause etinal damage.

7 2. Align the optics of the system so that the beam of the lase, having passed though the lens, uns paallel to the optical bench. 3. Decease the iis apetue to its smallest possible size. Open the iis slowly and obseve the changes in the diffaction patten. You should see the cental point incease and decease in its intensity. Adjust the apetue so that thee ae 8-0 dak ings aound a cental dak point. When this has been accomplished the apetue allows the light fom an exact intege numbe of unobstucted zones to each the cental pat of the diffaction patten. The numbe of Fesnel zones "visible" to the cental point will then coespond to the total numbe of ings, bight and dak, including the cental dak point. In theoy it should be possible to adjust the iis so that a bight spot is visible, counting the ings again to detemine the numbe of unobstucted zones. In pactice, howeve, it is easie to identify the dak point minima athe than the bight spot maxima. 4. Measue the iis - wall distance with the tape measue and the iis - lens focal point distance using the optical bench scale. In addition, use the calipes to measue the diamete of the iis apetue. Recod all values including the numbe of unobstucted Fesnel zones. 5. Repeat steps 3-4 at least 6 times moving the iis 5.0 cm away fom the lens each time. 6. Replace the iis with the micoscope slide. Examine and descibe the diffaction patten of each object on the micoscope slide. Use the tanslation stage to adjust the beam position of the lase. 7. efom measuements fo the optional components of the expeiment as necessay. Eo Analysis The eo in all measued distances is half the smallest division of the scale fom which they ae ead. Assume that the focal length of the lens is exact. opagate all eos accoding to standad methods of eo popagation. To be handed in to you laboatoy instucto at the time he o she specifies. ( mak) Fill in the steps of the integation that ae missing in Equation 0, ecalling that the assumption of small y has been made. 2. (2 maks) Fill in the steps of the integation that ae missing in Equation. 3. (2 mak) Table of values listing lens focal point to iis distance,, iis - wall distance, ', numbe of unobstucted Fesnel zones, y, and the diamete of the iis apetue. Include all uncetainties and have pope headings and units fo the columns. 4. (2 maks) Table of values listing the quantity '/( ') and the coesponding value of Fesnel zone aea. Include all uncetainties an have pope headings and units fo the columns. 5. (2 maks) A gaph of the Fesnel zone aea as a function of '/( ') with uncetainty bas if these ae lage enough to show up. 6. (2 maks) Obtain the egession equation fo the gaph in pat 5. Fom the slope of the line detemine a value fo the wavelength of the lase and compae it to the actual value of nm. 7. (2 maks) A desciption and diagams, whee appopiate, of the diffaction patten geneated by the vaious objects on the micoscope slide.