The condition for maximum intensity by the transmitted light in a plane parallel air film is. For an air film, μ = 1. (2-1)
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1 hapte Two Faby--Peot ntefeomete A Faby-Peot intefeomete consists of two plane paallel glass plates A and B, sepaated by a distance d. The inne sufaces of these plates ae optically plane and thinly silveed so that they can eflect 70% of the incident light. The oute faces of these plates ae also paallel to each othe but inclined to thei espective inne sufaces to avoid intefeence effect due to multiple eflection and efaction. With this intefeomete, finges of constant inclination ae obtained by tansmitted light afte multiple eflections between the glass plates. [Fig. 1] Fig. 1 Light fom an extended souce S 1 of monochomatic souce is endeed paallel by the collimating lens L 1. Each of the paallel ays suffes multiple eflections in the ai film enclosed between the plates A and B and emeges fom the plate B. A ay oiginating fom the point souce S on S 1 afte suffeing intenal eflection issues fom B in the fom of a paallel beam which intefee and ae bought to focus at P by the lens L. The finges fomed in the focal plane of L ae cicula. The condition fo imum intensity by the tansmitted light in a plane paallel ai film is Fo an ai film, μ 1. μdcosφ nλ dcosφ nλ (-1) whee n 0, 1,, 3, etc. is the intege giving the ode of intefeence fo a paticula finge. This condition is satisfied fo all the points on a cicle though P with cente at O. The value of the path diffeence depends upon the angle φ, so that we have a seies of
2 - - hapte Two concentic cicles with O as the cente on the sceen o eyepiece. Finges of constant inclination ae called Haidinge Finges. n this intefeomete, the two plates ae invaiably kept paallel. One of the plates is fixed and the othe plate is moved with a ack and pinion aangement. f the distance between the two plates is deceased, the value of φ deceases fo a given λ and n. t means with a decease in d, the ings shink and disappea at the cente. Wheneve t is deceased by λ, one ing disappeas at the cente. Also, the ode of the ings deceases fom cente outwads. The ing patten of a Faby-Peot intefeomete is extemely shap in compaison to the ing patten obtained with a Michelson intefeomete. Hence, Faby-Peot intefeomete is vey useful fo esolving vey small wavelength diffeences. ntensity of the tansmitted beam n a Faby-Peot intefeomete, the shapness of the finges depends on the eflection coefficient of the silveed sufaces. Let the Faby-Peot plates be sepaated by t. 6 Suppose the tansmitted amplitudes ae,,,,... etc. is the eflection X A B Fig. coefficient and is the poduct of the oiginal amplitude and the tansmission coefficient at the two plates [Fig. ]. The tansmitted beams have a constant phase diffeence whee Y π λ ( t cos φ) Applying the law of polygon of vectos, the intensity is popotional to the squae of the esultant amplitude. 6 p.d φ φ...jj...raoo { t cosφ t (Path diffeence p.d) t cosφ. p.d t cosφ. On the othe side, p.d t cosφ. Total p.d
3 Faby-Peot ntefeomete ( + cos + cos +...) + ( ) 1 3 X component 1 3 Y component [ ] O, ( cos + cos +...) + ( ) (-) Mathematically, i i 6 i e + e + e +... i e whee i 1. i Multiply the numeato and the denominato of the ight hand side by 1 e. e i + e i + e 6 i e i i i ( e + e ) + (-3) Also, e ± i cos ± i. Substituting its value in Eq. (-3), ( cos + i ) + ( cos + i ) Sepaating the eal and imaginay tems, +... ( cos i ) [ ] cos + i + cos i cos + i ( 1 cos + cos +...) + i( ) + (-) cos + Equating the eal and imaginay tems, and cos ( 1 cos + cos +...) + + (-5) cos + ( ) (-6) cos + As < 1, < 1. Substituting the values fom (-5) and (- 6) in (-), ( cos ) ( cos + ) ( ) ( ) cos JJ...Raoo
4 - - ( ) ( ) O, ( ) cos + cos + hapte Two i.e. cos + cos + ( ) cos + i.e. cos + ( ) cos + O, cos + i.e. ( ) + O, k ( ) + (-7) Special ases (i) When 0, π, π,..., 0, π, π,... and 0. n this case, the denominato in Eq. (-7) will have a minimum value. Hence, the intensity of the finges will be a imum. O, (-8) k ( ) ( ) π 3π 5π (ii) When π, 3π, 5π,...,,,,... and 1. n this case, the denominato in Eq. (-7) has a imum value. Hence, intensity of the finge will be a minimum....jj...raoo min ( ) +
5 Faby-Peot ntefeomete O, min ( ) (-9) O, min k ( ) Visibility of finges The visibility of finges is Substituting the values of and min, V V min (-10) k + min ( ) ( ) k + k k ( ) ( ) ( ) ( ) i.e. V ( ) + ( ) O, V (-11) Eq. (-11) shows that the visibility of the finges is a function of the eflection coefficient only. The visibility of the finges inceases with incease in the value of. Shapness of the finges The esultant intensity at the point of inteface in the focal plane is k ( ) ( ) + (-7) and the imum intensity is...jj...raoo k ( ) (-8) Following Faby, the intensity distibution within the finges may be witten as follows:
6 - 6 - k ( ) ( ) ( ) [ ] hapte Two o, F (-1) ( ) whee F is the coefficient of finesse. ( ) Half-width A measue of shapness of finges, i.e. how apidly the intensity diminishes on eithe side of the imum, is the finge half width. The finge half width is the total width of a finge at points whee the intensity falls to half the intensity imum. Fo a given value of, if a gaph is plotted between and, the half-width is quite small [Fig. 3]. ntensity of the peak at half width is. Fom Eq. (-7) and (-8), ( ) Half Width Fig.3 k k + ( ) ( ) o, ( ) + o, (-13) ( )...JJ...Raoo
7 Faby-Peot ntefeomete At half-width, and ( ) 1 O, { phase diffeence (-1) Fom Eq. (-1), when 1, 0. That is, the visibility is imum. at half-width deceases with incease in the value of. Accoding to Eq. (-1), 1 F Fo values of geate than ½, 1 F is so small that the appoximation adian can be made without any seious eo. Then, F. But, the phase diffeence between two successive ima is π. The total half width, measued in ode numbes is, theefoe, W (-15) π π F π The half width and theefoe, the shapness of the intefeence finges depends upon the coefficient of finesse F. Fo Michelson intefeomete, the intensity of the finges is given by cos At half-width,....jj...raoo
8 - 8 - hapte Two cos 1 1 π 1.57 adian. Fo a Faby-Peot intefeomete with 0. 9 at half-width, 0.11adian [fom Eq. (-7)]. This shows that the finges obtained with a Faby-Peot intefeomete ae compaatively much shape than those obtained with a Michelson intefeomete. Resolving powe of a Faby Peot intefeomete Altenative fom of Rayleigh s citeion of esolution Rayleigh s citeion can be expessed in an altenative fom, if the spectal lines have equal intensities (i.e. thee ae no seconday ima and minima), which will be applicable to the esolution of Faby Peot finges. The intensity distibution in the finges of gating specta is given by α Nβ R o (-16) α β whee N is the numbe of lines on the gating suface and R o is the esultant amplitude, α is the phase diffeence between the exteme ays fom the fist slit and β is the phase A B 1 λ D λ+δλ n n n + 1N Fig. : Gating finges at the limit of esolution diffeence between the exteme ays fom the second slit. The fist facto R ( α α ) o gives the intensity distibution in the diffaction by a gle slit. The second facto Nβ β gives the intefeence patten fo N slits. Neglecting the diffaction tem in equation (-16), the intensity distibution in the finges of gating specta is given by Nβ β (-17) Pincipal imum occu when β ± n π whee n is an intege and N. The fist minimum on eithe side of the pincipal imum occus when N β nnπ ± π....jj...raoo
9 Faby-Peot ntefeomete As shown in Fig, the point, the intesection of intensity cuves of two spectal lines when just esolved, is midway between thei centes. Hence at (half width), 1 N β nnπ ± π o, β nπ ± π N (-18) The intensity at,, fo each line is given by ( nnπ ± π ) 1 N ( nπ ± π N) ( ± π N) π π 0.05 (-19) Hence, ( ) and 0.81 (-0) D Thus, Rayleigh s citeion of esolution implies that the two spectal lines of equal intensity may be said to be just esolved if midway between thei ima the esultant intensity due to ovelap is 0.81 of the imum in the combined intensity patten, i.e (o 8π ) of the height of the two peaks in the esultant intensity patten. Resolving powe of Faby Peot intefeomete: Definition At any point in the patten due to ai spaced Faby Peot instument, the ode of intefeence is expessed as n λ e cos θ (-1) whee n is not necessaily an intege. Fo small angles of incidence, equation (-1) becomes n λ e (-) The change in the ode coesponding to the change in the wavelength is obtained by diffeentiating equation (-). n Δλ + λ Δn 0 Hence, λ n R.P (-3) Δλ Δn ntepetation of equation (-3) n measuing the shift of one spectal line λ Δλ in tems of the ode numbe n of the othe line λ, thee should be a shift of at least Δ n n R. P to obtain the limit of esolution. The negative sign indicates that a positive incease in λ coesponds to a negative change (decease) in n....jj...raoo
10 An expession fo esolving powe λ λ+δλ At half width hapte Two 0 Suppose, that two wavelengths λ and λ + Δλ ae emitted by a souce with equal intensities. The bight finges of each wavelength will be fomed alongside one anothe as shown in Fig 5. Accoding to Aiy s fomula, the intensity is given by (-1) ( ) whee is the phase diffeence between the tansmitted (emegent) beams. Fo just esolution, Fig (this occus at half width. Ode fo λ ). Theefoe, 0.05 ( ) o, ( ) 0.05( ) i.e ( ) i.e. 1 1 ( 0.05)( ) JJ...Raoo
11 Faby-Peot ntefeomete ( ) o, ( ) ( ) (-) hange in ode by 1 coesponds to a phase change of π. hange in ode by Δn coesponds to a phase change of π ( Δn). Now, accoding to Taylo s citeion, thee is a just detectable dop in intensity between two finges so that they can be esolved, if the sepaate intensity cuves intesect at, i.e. if the sepaation of the ima is equal to the half value width. Theefoe, phase change between two ima. Fom equations (-) and (-5), ( Δn) π π Δn (-5) o, ( ) Δn π π ( ) (-6) n view of equation (-6), equation (-3) becomes λ Δλ Δn n 1 πn [ ( ) ] o, λ Δλ πn [ ( )]...JJ...Raoo πn.595n R.P (-7) ( ) ( ) Theefoe, in a Faby Peot intefeomete, the esolving powe inceases with an incease in the eflecting powe.
12 - 1 - Application of Faby-Peot ntefeomete hapte Two To detemine the diffeence between two closely spaced wavelengths f a souce of light consisting of two wavelengths λ 1 and λ which diffe slightly is used, then two sets of finges coesponding to the two wavelengths λ 1 and λ ae poduced. These finges on supeimposition give ise to positions of imum and minimum intensity. A position of imum intensity is poduced when a bight finge of one wavelength coincides with the bight finge of the othe wavelength (consonance). A position of minimum intensity is poduced when the bight finge of one wavelength coincides with the dak finge of the othe wavelength (dissonance). The distance between the two plates is adjusted so that a imum intensity position is obtained. The eading of the micomete scew is noted. The movable plate is futhe moved till again a imum position is obtained. The eading of the micomete scew is again noted. Let λ 1 > λ. f l is the distance though which the plate has been moved, then ( n + ) l nλ λ 1 1 l l o, 1 λ λ λ1λ i.e. λ 1 λ l Taking λ as the mean wavelength, the diffeence in wavelengths λ is Questions 1 λ λ (-8) l 1. Discuss the constuction and woking of a Faby-Peot intefeomete.. Descibe the fomation of finges by Faby-Peot intefeomete and discuss the intensity distibution. 3. Define visibility V of finges. alculate the same fo finges obtained by Faby-Peot intefeomete.. Show that the finges obtained with a Faby-Peot intefeomete ae compaatively much shape than those obtained with a Michelson intefeomete. 5. Deive an expession fo the esolving powe of a Faby-Peot intefeomete. 6. State the altenative fom of Rayleigh s citeion of esolution....jj...raoo
13 Faby-Peot ntefeomete Explain the use of Faby-Peot intefeomete to detemine the diffeence between two closely spaced wavelengths. Poblems 1. White light is incident nomally on a Faby Peot intefeomete with a plate sepaation of x 10-6 m. alculate the wavelengths fo which thee ae intefeence ima tansmitted beam in the ange of 000 to 5000 Å?. White light is incident nomally on a Faby Peot intefeomete with a plate sepaation of x 10 - cm. alculate the wavelengths fo which thee ae intefeence ima in the tansmitted beam in the ange 000 to 500 Å. 3. n a spectomete, one half of the slit is illuminated with mecuy light and the second half is illuminated with white light though a Faby Peot intefeomete. n the wavelength ange of 000 to 5000Å, 0 finges ae obseved in the continuous spectum. alculate the plate sepaation of the Faby Peot intefeomete.. Fo a Faby-Peot intefeomete, the eflection coefficient is 0.9 and What is the esolving powe of the instument at λ 500 Å? What is the minimum sepaation in wavelength that can be seen with the instument? Sepaation of the plates is 3 mm. 5. Fo a Faby-Peot intefeomete, find the half width of the finges, given the coefficient of eflection Two Faby-Peot intefeometes have equal plate sepaation and thei coefficients of eflection ae 0.8 and 0.9. Find the atio of half-width of ima in the two cases....jj...raoo
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