O9e Fringes of Equal Thickness

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Eugene Pearson
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1 Fakutät für Physik und Geowissenschaften Physikaisches Grundpraktikum O9e Fringes of Equa Thickness Tasks 1 Determine the radius of a convex ens y measuring Newton s rings using ight of a given waveength. Using the experimenta arrangement of task 1 determine the waveength of ight passing a fiter. 3 Determine the thickness of meta fois or wires using the interference at wedge-shaped ayers. Perform at east one of these additiona experiments: Z1 Determine the thickness or the diameter of an oject chosen y yoursef, e.g. a strand of hair. Z Measure Newton s rings with water as medium etween ens and gass pate. Z3 Determine Poisson s numer μ from the principa radii of curvature of a curved pate using fringes of equa thickness. Literature Physics, P. A. Tiper, 3rd Edition, Vo., Chap. 33-1, 33- University Physics, H. Benson, Chap. 37.1, 37.5, 36.3 Physikaisches Praktikum, 13. Aufage, Hrsg. W. Schenk, F. Kremer, Optik,.0.1,.0.,.1 Accessories Ae comparator for measuring the interference pattern, spectroscopic amp, fiters, enses, gass pates, fois, wires, pate with ending device Keywords for preparation - Interference, coherence, coherence ength - Superposition of waves - Interference (destructive, constructive), phase reationships, optica path difference - Fringes of equa thickness, fringes of equa incination, Newton s rings - Principe of ight fiters, e.g. interference fiters 1

2 emarks At the eginning of the experiment the demonstrator wi introduce you to the use of the Ae comparator for the measurement of the interference pattern. This optica precision instrument consists of two microscopes to oserve the oject (interference pattern) as we as a precision ength scae with a resoution of 0. µm. In order to minimize errors arising from the movement of the comparator tae in different directions, the measurements shoud e performed whie moving the tae in one direction ony. The anaysis of the data sets is done y inear regression. For each task aout 10 measurements shoud e performed. Newton s ings Newton s rings occur when monochromatic ight interferes in the thin intermediate gap etween a convex ens and a pane gass pate, see Fig. 1. In the Ae comparator monochromatic ight comes in horizontay, strikes the gass pate and is refected downwards toward ens and pane gass pate. The ight is partiay refected at each air/gass interface. Here we are interested in the refections at the convex side of the ens and the upper side of the pane gass pate, since these waves interfere to generate Newton s rings. The contact etween ens and gass pate must not e idea. This is modeed y the introduction of a contact distance d 0 ; d 0 might e either positive, when e.g. dust partices are ocated etween ens and gass pate, or negative, when the gravity pressure sighty indents the pane pate. ay 1 refected at the top of the air fim interferes with ray refected at the ottom of the air fim, see Fig. 1. At a distance r k from the contact point the air fim has a thickness d + d 0, where d denotes the idea thickness due to the convex curvature. The refractive index of air is taken to e unity. ay is refected at an opticay denser medium and acquires an additiona phase shift of π. Fig. 1. Schematica setup for the measurement of Newton s rings showing the convex ens on the pane gass pate as we as the gass pate (G) used for defection of the incoming ight (L).

3 The optica path difference etween ray 1 and ray is Δ x= ( d+ d0) + λ / (1a) and the corresponding phase shift is Δ x 4π δ = π = ( d+ d0) + π. (1) λ λ The reationship etween the radius of the kth interference ring r k and the radius of the ens foows from Fig. : d( d) = rk. () Fig.. Using the third Pythagorean theorem to reate d and r k. Using Eqs. (1) and () as we as the condition for constructive interference (right rings) one otains in the imit d << 1 rk = k λ d0. (3) Derive the equation for destructive interference (dark rings). For the anaysis of the data in tasks 1 and pot r k vs. k and determine resp. λ from the sope. In order to estimate the uncertainty use r λ = k r k k1 k 1, (4) where k 1 and k denote the first and highest measured diffraction order. Compare this uncertainty with that determined from the sope. Interference in wedge-shaped ayers A wedge-shaped air gap etween two fat gass pates can e created y pacing a fim, meta foi, thin wire, strand of hair etc. at one end of the gass pates as shown in Fig. 3. As in the case of Newton s rings monochromatic ight is incident normay on the gass pates and the interference fringes are oserved in refection. 3

4 Fig. 3. Interference at wedge-shaped air fims. Let x k e the horizonta position of the kth right fringe, corresponding to a pate separation d k as shown in the Fig. 3. Then the phase shift is δ = π Δ x / λ= 4 πd / λ+ π and with the condition for k k k constructive interference and the reation tanα = d k /x k = D/ one otains x k λ λ = k. (5) D 4D Derive the equation for dark fringes. For adjacent equidistant dark or right fringes separated y Δ one has λ =. (6) D Measure 10 different vaues x k in dependence on k and pot x k vs. k. Determine D from the sope, the measured vaue and the given waveength λ. Additiona task: Newton s rings with a water ayer Fig. 4. Newton rings with a water ayer etween ens and gass pate. The anaysis foows the reasoning aove for the case of Newton s rings at an air gap. Here the optica path difference is modified y the refractive index n that differs significanty from unity λ Δ= nd ( + d0) +. (7) 4

5 This eads to a modified equation for the right fringes: λ rk = ( k 1) d0. (8) n Derive the corresponding equation for the dark fringes. Additiona task: Determination of Poisson s numer Consider a thin pate (ength, width and thickness h) suspended on two ades S, S, see Fig. 5. Via two additiona ades S 1, S 1, see Fig. 5, a force aong the z-direction is exerted on the pate y some fastening device. Under this oad the pate ends in the yz-pane (primary ending). In this ending the upper ayers of the pate are stretched, experiencing at the same time a atera contraction, whereas the ower ayers are compressed and simutaneousy ateray expanded. The atera contraction and expansion eads aso to a ending in the xz-pane (secondary ending). Fig. 5. Transparent pate under pressure in a fastening device. The surface profie of the ent pate is shown, aeit rather exaggerated, in Fig. 6. Both in the yz- and the xz-pane the surface profie can e characterized in a good approximation y its radius of curvature, yieding a sadde-point profie. Fig. 6. Sadde-point surface-profie of the ent pate with strongy exaggerated curvature. 5

6 Fig. 7. Cross-section through the ent pate in the yz- (eft) and xz-pane (right). From Fig. 7 one reads off the reative eongation 1 βh h = =, (y-direction) (9) and the reative contraction 1 β h h = =. (x-direction) (10) Therefore, for Poisson s numer µ one otains in good approximation μ =. (11) The sadde-point surface-profie shown in Fig. 6 is descried y x y z = +. (1) Experimentay, x and y truncated after the terms ( x / ) and ( / ), such that the Tayor expansion of the square roots can e y to yied x y z =. (13) Sections through the sadde-point profie parae to the xy-pane (z = const) yied hyperoas. For z = 0 one otains the pair of ines, see Fig. 8, =+ =. (14) y x and y x 6

7 The smaer of the two anges etween these ines e denoted y α, see Fig. 8. Then one otains for Poisson s ratio μ = = tan α. (15) Fig. 8. Interference pattern. The fastening device is positioned under the microscope of the Ae comparator. On the ent transparent pate a fat gass pate is paced; the whoe setup is iuminated y monochromatic ight as descried aove and an interference pattern of the type shown in Fig. 8 is oserved. For the distances x k and y k of the apexes of the hyeroas from the origin one otains x =± z and k k yk =± zk, where the z k now denote optica path differences. Measure the engths x k and y k for severa orders of the interference pattern, pot () xk = f1 k and y () k = f k and determine the sopes S and S, respectivey. Then Poisson s ratio is otained from S S μ = =. (16) 7

Measurements with the 'Ae-Komparator' Fig. 9.

measuring microscope (enarged image of the oject), 5 microscope

001 mm), 6 gass scae, 7 kno for the fineadjustment of the movae

reading = 5 mm + 0.3 mm +0.04 mm + 0.0008 mm = 5.

8 Measurements with the 'Ae-Komparator' Fig. 9. Ae-Komparator 1 tae, fixation screw for movae tae, 3 oject, 4 measuring microscope (enarged image of the oject), 5 microscope to measure the enarged gass scae (1 mm, 0.1 mm, 0.01 mm, mm), 6 gass scae, 7 kno for the fineadjustment of the movae tae Fig. 10. Gass scae as seen in microscope (1). reading = 5 mm mm mm mm = mm with an uncertainty of mm Fig. 11. Newton s rings. The fringes are not equay spaced. Measure from the eft highest order to the right highest order or vice versa. Cacuate the diameter from the difference of the readings: e ri = r e ri 5 5 8

Multiple Beam Interference

Multiple Beam Interference MutipeBeamInterference.nb James C. Wyant 1 Mutipe Beam Interference 1. Airy's Formua We wi first derive Airy's formua for the case of no absorption. ü 1.1 Basic refectance and transmittance Refected ight