7/27/2019 Measurement of Refractive Index Using a Michelson Interferometer

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Measurement of refractive index using a Michelson interferometer

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1982 Phys. Educ. 17 209

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T l r rorMeasurement ofrefractive indexus ing a Michelsonin terferome-ferJ J Fend l e yThe Michelsonnterferometer hasong been apopular piece of equipment in the physics under-gradu ate labora tory. On e possible application is themeasurement of the refractivende x of ahinparallel-side p late of transp are nt ma teria l of kno wnthickness.Alternatively, i f the refractivendex iskno wn, the thickness of the plate may be determined.)This is acco mp lished by using white ight' fringessuperimposednmon ochrom atic fringes an d isdescribed in stand ard abora torymanuals Whit t leand Yarwood 1973). In thisarticle anoth er me thodfor measuringhe refractivendex is described.Monochromatic fringes are used and plates rangingin thicknessrom a few micro me treso severalmillimetres m ay be measured .Consider heplate in one arm of the Michelsoninterfero me ter. Let the ray w hich is no rm al o hemirror M make anangle of incidence ' p i with the plate(figure 1). The change in phase of the ray as it passesthrough he plate may be determinedfrom consid-ering figure 2. The phase change of the ray in goingfrom P to Q is

2nnnnd "p1. i. coscp,where i is the wavelength of the monochromatic ight.

n is th e refractive index, d is the thickness. cp is th eangle of refraction and sin cpi = n sin cp,.

2la te

Monochromatlc

Mi r ror

vEy e

Figure 1 Transparent plate. at angle , to the beam , n onearm of the Michelson interferometerThe equivalent ray in the other arm of the inter-ferometer will pass throughorresponding irthickness an d suffer aphase hange

Thus the phase difference between thewo raysintroduced by the plate isA = 27- 1nndnd cos(cpi - cp,)1. co s cp * i coscpThe factor 2 is required ecause the ray passesthroug h the plate twice.When the plate is normal to the ray ( c p j = cp r = O),

Figure 2 Details of the light path through the plate

Jack Fendley has heenanassociateprofesso r in thecol lege of engineering,KingAhdulAzizC'niwrsit!*,Jedduh, Suudi Arabia s ince 1977. A,fter obtaining hisB S c and PI1 D ut Exet er Unir wsi t ) . , he work ed us areseardlof f icer , for E R A Technolog). Ltd and thenlectured in th e pll.vsics depurtnwnt of Chelsea Col lege,London C'niwsi t ) , .isrev iouseseardl IWSconcerned nit11 th e electricalroperties of semi-conductors an d riielrctric,.~ hut he i s ( ,urrentlJ. )! ,orkingon solar energ) ' ,partic,ulurl), pltotovoltuic cells and th eoptic,ul propertieso f sur faces .0031 91 2C 82 050203+33S2 00 1 9 8 2 The nstl!.!te c' P P \ j i s 209

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the phase difference between the two rays introducedby the plate isAo=2(F-Y)

Inurningheplate romhenormal positionthroughhe angle cp i , the change in thehasedifference between he two rays is (A - Ao ) . If thiscorresponds to m fringes, then (A - Ao ) = 2nm an d

if n is known, d can be found romequation (1 )immediately. If d is known and n unknown, it is notpossible to rearrange equation (1) to give a solutionfor n since cp which itself depend s on n, is unknown.However, n can be found by using a com puter andstepping n until

is a minimum.An approximate value of n may be foundimme diately for small values of qi for which sincp Xcp, cos cp z (1 - cp2:'2) an d c p i = ncp,. Equation (1)then becomesn = cpc p i - i./d (2)This gives astart ing value for n to be used in the

computer program.Experimental procedureThe experim ental arrang eme nt is shown in figure 3.The plate was a sta nd ard glass slide of dime nsion s76 mm x 25 m m x 1 m m . A holder for theplate,which allowed the plate tobe rotated, was mad e froman inexpensive protracto r. A spindle cem ented to theprotractor was located in a hole drilled into the baseof the interferometer.

It is imp ortan t hat he plate be mad e initiallynormalo the beam. A check with a et-squa reshowed that the plate was perpend icular to the baseof the interferometer to within *is. Using equation(2) and he values of n, i. an d d given in the nextsection, this gives anerror in m of k0.04 which isnegligible. To ensure that the platewas initially set atc p i = 0, the plate was rotate d in its holder 3' to 4' offnormal. A s the plate was movedback toward s henormal position. the fringes moved one way, stoppedand then moved n the reverse direction. By notingthe position of the pr otr ac tor for thre e fringes eitherside of the normal, the p i = 0 position could be foundand a reference m ark mad e on the base of the inter-ferometer at th e zero of the pro tracto r.With a plate thickness of 1 mm ther e was a shift ofapproximately half a fringe/degree of rotat ion t

I / \ I/SpindleP ln/ G lasss l l d e', /

UFigure 3 Holder for rotating t he glass slidenorm al incidence rising to approximately 15 fringes1degree at cp = 30'. By at taching n rm oheprotractor , some 2Ocm long , it was possible to ro tatethe slide slowly enough ocoun t he fringes at hehigher angles of incidence.The interferometer was set up for circular fringes.Wh en he glass slide was placed in on e arm at henormalosition,he fringes rema inircular. Atheore tical analysis show s hat he fringes becomeslightly elliptical as the slide is rota ted , bu t with thepresent arrang em ent he effect was too small to benoticed with the unaided eye.ResultsA mercury discharge lampwith a green filter was usedas the light source o give mo noc hro ma tic light ofwavelength 546.2 nm . T he thickness of the glass slidewas measured using a dial gauge and had a value ofd = 1.025i .002 mm . The angles c p i from the normalthro ug h which the slide was rota ted for variou s fringeshifts m are given in table 1. The values of therefractivendex n calculatedrom eachair ofvariables using a com puter progra m are also given.Theave rag e value of the refractive ndex was n =1.481+0.008. This should be compa red with a valueof 1.482 +0.00 4 found romheapparent epthmethod' (Nelkon and Parker 1970).Table 1 show s that the estimate d recision in m andc p i was k i fringe an d respectively. Th e relativeerror in m an d cpi-and hence in n-will be large forsmall values of m and cp i an d a minimum value of m= 20 corresponding to c p i = 10.4' was chosen as the

210

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starting point. In calcu lating the averag e value of ngiven in table 1. no attem pt was mad e to weight theindividual values of n since uch a detailed erroranalysis would be inconsistent with theunknownaccuracy of the protractor.DiscussionThe values of n given in table1appear oshowasystematic increase with increasing v,. Possiblereasons for this include a n erro r in setting the normalposition of the plate an d/or a systematic error in theangular graduations of the protractor . T he systematicchange in n is comparable with the random error andthe precision of th eexperimen t must be improvedbefore it can be decided if the effect is real or no t. Th eprecision could be increased by using aproperlymachined holder with an angu lar vernier scale. Sucha holder should be provided with a screw arrang e-ment to allow the slide to be turned slowly. With sucha device, readings of cp to within + S minutes of arccould be obtained.Inderivingequation ( l ) , the multiple eflectionswithin thelate have not been considered. Ananalysis of this effect, assu min g a non com plex valueof n (a perfect dielectric) sho ws tha t m is increased by( p - Po)/nwhere p is the addit ional phase chan ge atangle of incidence cp, an d p,, that at norm al ncidence,an dand

t a n b = r2 sin26l-r2 co s 26t a n P o = rn2 sin26,I-r, cos26,with 6 = 2nndcosrp,/ i , 6, = 2nnd!i ,ro = ( n- ) /( n + 1) and r is the reflection coefficient at the anglerp, an d is depe nden t on the polarisa tion of the light.Fo r light polarised parallel a nd perpend icular to theplane of incidence.

n coscp, - coscp,n coscp, + coscp,=

an d - coscp, - n cosrp,rl c o s @ ;+ n cos@., .F or a full discussion of the reflection of light, seeKlein (1970). For the present exp erimental conditionsthe ma ximu m value of I(B - po)/nI