DIFFRACTION OF LIGHT

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1 DIFFRACTION OF LIGHT The phenomenon of bending of light round the edges of obstcles or nrrow slits nd hence its encrochment into the region of geometricl shdow is known s diffrction. P Frunhofer versus Fresnel Diffrction S In Fresnel type of diffrction the incident wvefronts re sphericl or cylindricl. i.e. the source of light is t finite distnce from the diffrcting perture. The screen on which the diffrction pttern is displyed is lso t finite distnce from the diffrcting perture. In Frunhofer type of diffrction both the incident nd emergent wvefronts re plne (the rys re prllel) i.e. both the source nd the screen re effectively t infinite distnces. In lbortory Frunhofer diffrction is relized by using converging lenses for conversion of sphericl wvefront into plne wvefront nd vice vers. L P L 1 S f f 1

2 /19 MAHE-MIT-BE-ENGINEERING P HYSICS DIFFRACTION OF LIGHT FRESNEL DIFFRACTION : Hlf period zones In Fresnel diffrction the given wvefront is divided into number of zones (Fresnel s hlf period zones) to study the intensity of light t given point. W PLANE WAVEFRONT M m M b + m / r m M 1 O b + / b + / b P r r 1 Trces of concentric spheres on the wvefront (circles) m th hlf period zone W Consider plne wvefront WW (prllel bem) of light of wvelength pproching point p t which the intensity of light is to be determined. Consider stright line PO perpendiculr to the wvefront. Distnce PO b. With P s centre drw concentric sphere with rdii 3 m b + b + b +... b +. They cut the wvefront into circles of rdii r 1 OM 1 r OM r 3 OM 3. r m OM m. First hlf-period zone is the re enclosed by the first circle. nd hlf-period zone is the re of the nnulr strip enclosed between the nd ring nd the first ring. m th hlf-period zone is the re of the nnulr strip enclosed between the rings of rdii r m nd r m-1. Wvelets from the two consecutive zones rech p with time difference of hlf period. (They re 18 out of phse they hve pth difference of ).

3 MAHE-MIT-BE-ENGINEERING P HYSICS DIFFRACTION OF LIGHT 3/19 W M m r m b + m / 1 st hlf period zone M M 1 O M 1 M b + / b b + / P Are of the hlf period zones: In the right ngled ΔOM m P. OP + OM m M m P M m W b + r m b + m r m b m bm b Neglect since smll r m b m r m m. Are: A m π r m π r m-1 πbm πb(m 1) πb constnt (independent of m). Intensity of wvelets t P due to ech zone depends on the re A m distnce (b + m ) nd obliquity ( OM m P). A m is constnt but distnce nd obliquity increse s m increses. Due to this reson the mplitudes m of wvelets t P from the hlf period zones 1 3. m re in decresing order of mgnitude nd lternte light vectors re opposite in direction ( 1 > > 3 >.. > m ). Resultnt mplitude: m m zero zero neglect since smll for lrge m 1 Amplitude due to the wvelets from hlf the re of the first hlf-period zone.

4 4/19 MAHE-MIT-BE-ENGINEERING P HYSICS DIFFRACTION OF LIGHT Resultnt intensity I I I 1 where I1 1 4 I 1 intensity t P due to the first hlf-period zone. If the wvelets from the lternte zones re blocked (s in zone plte) then the negtive terms ( 4.) in the epression for mplitude re eliminted nd the mplitude (nd hence the intensity) becomes mimum t P. ( ). If screen contins circulr perture of rdius r m the intensity t P due to the wvelets from the perture depends on the vlue of m. For m 1 P is drk. For m P is somewht bright. 3 ZONE PLATE: It is trnsprent sheet with circulr concentrtive zones (nnulr strips) of equl re in which odd numbered zones re trnsprent nd even numbered zones re opque. It converges both prllel bem of light nd lso diverging bem of light incident on it due to diffrction of light. Consider diverging wvefront of wvelength from point source S incident on the zone plte. Let the light wves from odd numbered zones reinforce t I on the opposite side of the zone plte. This is possible only if the pth difference between the light wves from the successive trnsprent zones is. i.e. the points S nd I re on the is of he zone plte such tht

5 MAHE-MIT-BE-ENGINEERING P HYSICS DIFFRACTION OF LIGHT 5/19 Section of zone plce O centre of zone plte M m M 4 M 3 M M 1 M Opque M 3 M 4 portions M 5 M 6 etc M 1 S object u S O object distnce M 1 O v O I Imge distnce I Imge M M 3 M 4 M m SM 1 + M 1 I SO + OI + u + v +. (1) SM + M I u + v +.. SM m + M m I u + v + m From Δ SOM 1 SM 1 SO + OM 1 u + r 1 SM 1 1 r + 1 u 1 u r + 1 r + 1 u 1 u 1 since r 1 << u u u SM 1 u r1 +. Similrly M 1 I u v r1 +. v r1 1 1 SM 1 + M 1 I u + v + + u v.. ()

6 6/19 MAHE-MIT-BE-ENGINEERING P HYSICS DIFFRACTION OF LIGHT Compre (1) nd (). r 1 1 u 1 + v (3) u v r 1 Compre (3) with the lens eqution 1 u v f Thus the ( primry ) focl length of the zone plte f. Bright imge of S is formed t I. Hence the zone plte behves like converging lens. There re lso finter imges corresponding to focl lengths f f f... etc becuse t these distnces the pth differences between light wves from successive odd numbered zones of the zone plte re etc respectively. r 1 Comprison of zone plte with conve lens: 1. Both the zone plte nd the conve lens produce rel imge of n object on the side other thn the object. But in the cse of conve lens the wves rrive in phse in the cse of zone plte wves rrive with incresing phse differences of π from the successive trnsprent zones.. For given wvelength of light conve lens hs number of foci of diminishing intensity. 3. Chromtic berrtion eists in both. In the cse of conve lens red light is focused t lrger distnce thn violet light where s in the cse of zone plte red light is focused t shorter distnce thn violet light.

7 MAHE-MIT-BE-ENGINEERING P HYSICS DIFFRACTION OF LIGHT 7/19 FRAUNHOFER DIFFRACTION AT A SINGLE SLIT P III II SECONDARY MAXIMA I O s ds Δ P o PRINCIPAL MAXIMUM I II MINIMA PARALLEL BEAM LENS f III CROSS SECTION OF SLIT SCREEN DIFFRACTION PATTERN A prllel bem of monochromtic light of wvelength is incident normlly on rectngulr slit of width. The light wves from the vrious points in the slit get diffrcted in ll directions. These wves re converged on screen by mens of conve lens of focl length f. O is the centre of the slit. ds is n element of the light-wvefront in the plne of the slit t distnce s from the centre O. Diffrcted wves trveling norml to the plne of the slit will be focused t P o. Diffrcted wves trveling t n ngle to the slit-norml will rech P distnt from O (distnt + Δ from ds). The infinitesiml electric field (de) t P produced by the diffrcted wve from ds is given by E de s sin[ t - k( + Δ )] de s ds ω E mplitude of the incident wve E ds sin sin [ ωt k k s ] Δ s sin.

8 8/19 MAHE-MIT-BE-ENGINEERING P HYSICS DIFFRACTION OF LIGHT Adding contributions from the pir of elements (ds) t +s nd s one gets de de s + de +s E ds. [ sin( ωt k ks sin ) + sin( ωt k + ks sin ) ] Using the identity sin α + sin β α β cos one gets E ds de [ cos ( ks sin) sin( ωt k) ]. sin α + β Integrting de form s to s one gets the totl electric field (E ) t P produced by the wves diffrcted from the slit: E sin( ωt k) [ cos( ks sin ) ] E E sin( t k) ( ) sin ks sin ω k sin ds E E ( k sin ) sin ½ ½ k sin sin ( ω t k) sin α o A o α E A sin( ωt k) E π α ½ k sin ½ sin π sin α is hlf the phse difference between the contributions coming from the opposite edges of the slit. sin α E A sin (ωt k) A A o α mplitude. The intensity on the screen is given by I A A o sin α α sin α I m. α

9 MAHE-MIT-BE-ENGINEERING P HYSICS DIFFRACTION OF LIGHT 9/19 INTENSITY VALUES I A o PRINCIPAL MAXIMUM A 3 π o A 5 π o MINIMA SECONDARY MAXIMA III II I I II α 3π π π π π 3π sin 3 α π π sin π sin sin 1 1 sin width of the slit

10 1/19 MAHE-MIT-BE-ENGINEERING P HYSICS DIFFRACTION OF LIGHT FRAUNHOFER DIFFRACTION AT A DOUBLE-SLIT Intensity distribution: c o d P P o 4 A o sin α INTENSITY α PARALLEL BEAM LENS CROSS-SECTION OF THE SLITS SCREEN DIFFRACTION PATTERN The infinitesiml electric field (de) t P produced by the diffrcted wves from the pir of symmetric elements (ds) t +s nd s from O is de E ds [ cos (ks sin ) sin( ωt k) ] E mplitude of the incident wve OP k π ω πν ν frequency ngle of diffrction. Integrting from d d s to s + one gets totl electric field E t P produced by the wves diffrcted from the two slits.

11 4AMAHE-MIT-BE-ENGINEERING P HYSICS DIFFRACTION OF LIGHT 11/19 d s + E E sin( ωt k) [ cos( k s sin ) ] ds d E s E sin( t k) E k sin s + ( ) sin k s sin ω k sin. k sin d d s k ( d + ) sin sin ( d ) sin sin( ωt - k). α + β α β Using sin α sin β cos sin k with α ( d + ) sin β ( d ) sin k from which α + β k d sin α β k sin E E k sin k d cos sin k sin sin sin(wt - k) E k sin cos k sin sin sin sin( ω t k) α E sin α α + c π π d α sin β ( sin) E cos β sin( ωt - k) k A sinα α E A cosβ sin( ωt k). Intensity (mplitude) AMPLITUDE I sinα α c( os)βsingle-slit DIFFRACTION PATTERN YOUNG S DOUBLE-SLIT INTERFERENCE PATTERN

12 ()1/19 MAHE-MIT-BE-ENGINEERING P HYSICS DIFFRACTION OF LIGHT FRAUNHOFER DIFFRACTION AT MULTIPLE SLITS (DIFFRACTION GRATING) Intensity distribution: se Aα If there re N slits ech contributing n mplitude nd the α wves due to the djcent slits differing in phse by δ the resultnt comple mplitude is A e i E + E e 1δ + E e i δ E e i(n-1)δ inδ 1 e E iδ 1 e The resultnt intensity due to diffrction t multiple slits I A (Ae i ) (Ae i ) A inδ inδ 1 e 1 e E E iδ iδ 1 e 1 e in E 1 1 ( δ) cos N cos δ 1 - cos δ δ sin sin sin E AI A ( Nδ ) ( δ ) δ π d sin β diffrction ngle d distnce between djcent slits. sin α sin (N β) π α sin width of the slit. α sin β A si nαα sin s. Nββin SINGLE N SLIT DIFFRACTION PATTERN INTERFERENCE TERM FOR N SLITS MAXIMA: Since lim β mπ sin(n β) sin β lim β mπ N ( β) cos N cos β ± N sin (N β) sin(n β) N sin β sin β for β ± m π Where m integer order of diffrction.

13 MAHE-MIT-BE-ENGINEERING P HYSICS DIFFRACTION OF LIGHT 13/19 A) I is mimum when β ± m π or π d sin ± m π or d sin ± m These re principl mim with intensity I A sinα N( α.minima: I is minimum ( zero) when sin (N β) sin β i.e. when N β p π p integer ecept when p N N.. etc which correspond to principl mim. p π the condition for minimum is β ecept when p m N N m integer order of diffrction. π d sin p π i.e. the condition for minimum is N N 3 (N 1) (N + 1) or d sin etc N N N N N N in which... etc re omitted. N N Secondry Mim: A secondry mimum occurs in between two successive minim. The secondry mim re much smller in intensity thn the principl mim. There re N secondry mim between the two principl mim of successive orders. Effect of incresing the number of slits (N) in Frunhofer diffrction (Diffrction grting): By incresing the vlue of N the principl mim become much more intense nd much nrrower. The subsidiry (secondry) mim become less intense (of negligible intensity). This is the principle of formtion of spectr by diffrction grting. The principl mim re the spectrl lines. d sin m is the grting eqution for norml incidence of light. m order of the spectrum. centrl mimum. Principl mimum of zeroth order (m ) is the

14 ΔΔdcos14/19 MAHE-MIT-BE-ENGINEERING P HYSICS DIFFRACTION OF LIGHT PARALLEL BEAM d 1 CROSS SECTION OF DIFFRACTION GRATIONG (N-SLITS) LENS CENTRAL MAXIMUM ( 1 + ) I SINGLE SLIT DIFFRACTION PATTERN 1 sin α α A o m 1 m m 1 m 5 m 3 m m m 3 m 5 Dispersive power of diffrction grting: The ngulr dispersion of diffrction grting D Δ. ΔΔ ngulr seprtion between two spectrl lines of wvelength 1 nd. Δ 1 GRATING SPACE ORDER OF DIFFRACTION Grting eqution: d sin m. Difference with respect to : d (cos d) md. Substitute differences (Δ) in plce of infinitesimls (d) d (cos Δ) m Δ D m

15 MAHE-MIT-BE-ENGINEERING P HYSICS DIFFRACTION OF LIGHT 15/19 Width of the principl mim: Minim re formed by diffrction grting when N β p π π d sin β NUMBER OF SLITS INTEGER IN THE GRATING N π d sin pπ or d sin p N For the m th principl mimum t by grting: d sin m. For the first minimum (p1) t + Δ fter the m th principl mimum dsinδm+ + ( ) N d sin cos 13 Δ + cos sin 13 Δ 1 Δ d1 sin 3 + d cos Δ m + ( ) N m + N FIRST MINIMUM (p 1 ) AT + } m + ( d cos) Δ m + N Δ N d cos ANGULAR HALF-WIDTH OF m TH PRINCIPAL MAXIMUM AT m th PRINCIPAL MAXIMUM AT Resolving power of diffrction grting: For two close spectrl lines of wvelength 1 nd just resolved by the grting the chromtic resolving power is defined s R where Δ 1 Δ 1 + nd. Rleigh s criterion for opticl resolution: According to this two close spectrl lines re just resolved when the principl mimum of one of the lines flls on the first minimum of the other line nd vice vers.

16 16/19 MAHE-MIT-BE-ENGINEERING P HYSICS DIFFRACTION OF LIGHT RESULTANT RESULTANT RESULTANT WELL RESOLVED JUST RESOLVED UNRESOLVED For just resolution of 1 nd Δ is given by the ngulr hlf-width of the principl mimum: Δ. N d cos The wvelength seprtion (Δ) for this ngulr seprtion (Δ) is given by the ngulr dispersion: Δ m D. Δ d cos Substituting the vlue of Δ in this eqution N d cos Δ m d cos or R Δ N m N totl number of slits in the grting m order of the grting spectrum.

17 MAHE-MIT-BE-ENGINEERING P HYSICS DIFFRACTION OF LIGHT 17/19 Diffrction Grting: Norml incidence of light: d sin Angle of diffrction ( Angle of devition) d d Grting Spce SECTION OF A GRATING Consider prllel bem of light of wvelength incident normlly on grting. The diffrcted prllel bem of light is t n ngle with respect to the grting-norml. Then the pth difference of the diffrcted rys from the two corresponding points in the djcent slits is d sin. The condition for getting the principl mimum of m th order is d sin m GRATING EQUATION FOR NORMAL INCIDENCE. To determine the wvelength () of the diffrcted light grting with known vlue of the grting spce (d) nd spectrometer cn be used.

18 18/19 MAHE-MIT-BE-ENGINEERING P HYSICS DIFFRACTION OF LIGHT Diffrction Grting: Oblique incidence nd minimum devition (Determintion of wvelength of spectrl line using spectrometer): d d sin i i i d sin i D i + i Angle of Incidence Angle of diffrction D Angle of Devition d Grting spce i Consider prllel bem of light of wvelength incident t n ngle i with respect to the grting norml. The diffrcted prllel bem of light is t n ngle with respect to the grting norml nd is devited through n ngle D ( i + ) from the incident bem. Then the pth difference of the diffrcted rys from the two corresponding points in the djcent slits is d sin i + d sin. The condition for getting the principl mimum of m th order is d sin i + d sin m D i. d [sin i + sin (D i)] m. For minimum devition vrition of D should be zero nd hence d di [ d{ sin i sin(d i) }] d di SECTION OF A GRATING + d constnt. [ sin i sin(d i) ] + or cos i cos (D i) Cos i cos (D i) or i D i D i D or i D nd D i D D.

19 MAHE-MIT-BE-ENGINEERING P HYSICS DIFFRACTION OF LIGHT 19/19 Hence t the minimum devition position of the grting the diffrction condition is d [sin i + sin ] m d sin D D + sin m d sin D m D ngle of minimum devition. To determine the wvelength () of the diffrcted light grting with known vlue of the grting spce (d) nd spectrometer cn be used. The ngle D is the ngle between the direct imge nd the m th order spectrl line t the minimum devition. Comprison between prism spectr nd grting spectr Prism Spectr 1. Only one spectrum. Intensity of the spectrl lines is more becuse in this cse the entire incident light is used to form only one spectrum. 3. Angle of devition decreses s wvelength increses. Devition for violet light is more thn tht for red. 4. Since the dispersive power of the prism depends on the wvelength the prism spectrum is irrtionl spectrum. 5. The resolving power is limited to the refrctive inde of the mteril of the prism. Grting Spectr mny spectr (spectr of severl orders) Intensity is less becuse the incident light is distributed in primry mimum nd mny spectr. Angle of devition increses s wvelength increses. Devition for violet light is less thn tht for red. Since the dispersive power of the grting is constnt the grting spectrum is rtionl spectrum. Resolving power depends on the grting spce nd cn be mde lrge

Diffraction. Diffraction and Polarization. Diffraction. Diffraction. Diffraction I. Angular Spread: θ ~ λ/a

$Diffraction. Diffraction and Polarization. Diffraction. Diffraction. Diffraction I. Angular Spread: θ ~ λ/a$ 1 nd Polriztion Chpter 38 Rleigh s Criterion Polriztion n geometricl optics, we modeled rs like this! n fct wht hppens is this... A sphericl wve propgtes out from the perture. All wves do this.. For double