FRAUNHOFFER DIFFRACTION AT DOUBLE SLIT While dealing with

FRAUNHOFFER DIFFRACTION AT DOUBLE SLIT ØWhile dealing with interference, we considered the interference of the secondary waves originating from the two slits. Each slit ØSo, the resultant pattern at any pattern on the screen will also produces a diffraction pattern of its own. consists of: 1) Interference pattern due to two slits 2) Two diffraction patterns, one due to each slit

FRAUNHOFFER DIFFRACTION AT DOUBLE SLIT (contd. ) ØConsider two rectangular slits AB and CD parallel to each other and perpendicular to the plane of paper. The width of each slit is ‘a’ and are separated from each other by opaque portion whose width is ‘b’. Lens L is placed between slits and the screen, focuses the interference-cum-diffraction pattern on the screen.

FRAUNHOFFER DIFFRACTION AT DOUBLE SLIT (contd. ) ØLet a plane wave front be incident on the surface XY. All the secondary waves travelling in a direction parallel to OP come to focus at P. Therefore point P corresponds to the position of the central bright maximum. INTERFERENCE MAXIMA AND MINIMA: ØLet us consider the secondary waves travelling in a direction inclined at an angle θ with initial direction. From the triangle CAN

INTERFERENCE MAXIMA AND MINIMA (contd. ): ØIf the path difference is equal to odd multiple of the direction of minima due to interference pattern of the secondary waves from the two slits. , θ gives ØHence ØPutting n = 1, 2, 3, …. . the values of θ 1, θ 2, θ 3, θ 4 etc. corresponding to the directions of minima can be obtained.

INTERFERENCE MAXIMA AND MINIMA (contd. ): ØFrom equation (I) Ø On the other hand if the secondary waves travel in a direction such that the path difference is even multiple of , gives the direction of the maxima due to interference of light waves emanating from the two slits.

INTERFERENCE MAXIMA AND MINIMA (contd. ): ØPutting n = 1, 2, 3, …. . the values of etc. corresponding to the directions of the maxima can be obtained. From equation (II) we get and

INTERFERENCE MAXIMA AND MINIMA (contd. ): or ØAs θ is small ØThus, the angular separation between any two consecutive minima (or maxima) is equal to The angular separation is inversely proportional to (a+b), the distance between two slits.

FRAUNHOFFER DIFFRACTION AT DOUBLE SLIT (contd. ) DIFFRACTION MAXIMA AND MINIMA : ØLet us consider the secondary waves travelling in a direction inclined at an angle φ with the initial direction of the incident light. The path difference then will be given by ØIf the path difference BM is equal to λ the wavelength of light used, then we can divide the slit into two equal halves each of width a/2, corresponding to each point in the upper half of the slit there will be a point in lower half of the slit, such that the path difference between the waves originating from them

FRAUNHOFFER DIFFRACTION AT DOUBLE SLIT (contd. ) DIFFRACTION MAXIMA AND MINIMA (CONTD. ): ØSo the angle φ in this case will give the direction of minimum. If φn be the angle of diffraction for nth diffraction minimum, then ØPutting n = 1, 2, 3, …. . the values of etc. corresponding to the direction of diffraction minima can be obtained.

FRAUNHOFFER DIFFRACTION AT DOUBLE SLIT (contd. )

FRAUNHOFFER DIFFRACTION AT DOUBLE SLIT (contd. )

FRAUNHOFFER DIFFRACTION AT DOUBLE SLIT (contd. ) MISSING ORDERS IN A DOUBLE SLIT DIFFRACTION PATTERN: ØIn the diffraction pattern due to a double slit, the slit width is taken as ‘a’ and the separation between the slits as ‘b’. If the slit width ‘a’ is kept constant, the diffraction pattern remains the same. ØKeeping ‘a’ constant, if the spacing ‘b’ is altered the spacing between the interference maxima changes. Depending on the relative values of ‘a’ and ‘b’ certain orders of interference maxima will be absent in the resultant pattern.

MISSING ORDERS IN A DOUBLE SLIT DIFFRACTION PATTERN (CONTD. ): ØThe directions of interference maxima are given by ØThe directions of diffraction minima are given by ØIn equations ‘(A) and (B) ‘n’ and ‘p’ are integers. If the values of ‘a’ and ‘b’ are such that the equation are satisfies simultaneously for the same value of θ, then the position of interference maxima corresponds to the diffraction minima at the same position on the screen

MISSING ORDERS IN A DOUBLE SLIT DIFFRACTION PATTERN (CONTD. ): ØLet Then a=b and ØIf p = 1, 2, 3, 4 etc. then n = 2, 4, 6 etc. Thus orders 2, 4, 6 etc. of the interference maxima will be missing in the diffraction pattern. There will be three interference maxima in the central diffraction maxima.

MISSING ORDERS IN A DOUBLE SLIT DIFFRACTION PATTERN (CONTD. ): ØIf 2 a = b Then and If p = 1, 2, 3, 4 etc. then n = 3, 6, 9 etc. Thus orders 3, 6, 9 etc. of the interference maxima will be missing in the diffraction pattern. On the both sides of central maximum, the number of interference maxima is 2 and hence there will be five interference maxima in the central diffraction maximum. The position of the third interference maximum will corresponds to the first diffraction minimum.

MISSING ORDERS IN A DOUBLE SLIT DIFFRACTION PATTERN (CONTD. ): ØIf The two slits join and all the orders of interference maxima will be missing. The diffraction pattern observed on the screen is similar to that due to a single slit of width equal to 2 a.

DIFFERENCE BETWEEN SINGLE SLIT AND DOUBLE SLIT DIFFRACTION PATTERN: ØThe main difference between the two diffraction patterns is that double slit pattern is the superposition of single slit diffraction pattern and a double slit interference pattern. ØThe principal maximum of double slit pattern, envelops a number of interference maxima and minima. ØThe intensity of the principal maximum of double slit diffraction pattern is about four times as compared to that of single slit

DIFFERENCE BETWEEN SINGLE SLIT AND DOUBLE SLIT DIFFRACTION PATTERN (CONTD. ): The intensity of principal maximum of single slit diffraction pattern decreases gradually to zero but that of double slit diffraction pattern varies between certain maximum and minimum values, before attaining zero value. This happens because of superposition of interference maxima and minima on principle maximum. DIFFRACTION GRATING

PLANE DIFFRACTION GRATING

PLANE DIFFRACTION GRATING ØA diffraction grating is an extremely useful device and in one of its forms it consists of a very large number of narrow slits side by side. The slits are spaces. light is ØWhen a wave front is separated incident onby a opaque grating surface, transmitted through the slits and obstructed by the opaque portion. Such a grating is called Transmission grating. ØThe secondary wave front from the position of the slits interfere with each other similar to the interference of waves in Young’s double slit experiment.

PLANE DIFFRACTION GRATING ØFrauhoffer used the first grating which consisted of large numb of parallel fine wires stretched on a frame. Now a days, grating are prepared by ruling equidistant lines on a glass surface. T lines are drawn with a fine diamond point. The space between any two lines is transparent to light and the line portion is o to light. Such surface acts as transmission grating. ØOn the other hand , lines are drawn on a silvered surface (plane or concave) then light is reflected from the position of the mirror in between any two lines and such surface acts as Reflection grating.

PLANE DIFFRACTION GRATING ØIf the spacing between the lines is of the order of the wavelength of light, then an appreciable deviation of light is produced. Grating used to study visible region of spectrum contains 10000 lines per cm. ØGratings, with originally ruled surface are only few. For practical purpose replicas of the original grating are prepared. On the original grating surface a thin layer of collodion solution is poured and the solution is allowed to harden. Then the film of collodion is removed from the grating surface and fixed between two glass plates. This arrangement serves as Transmission grating.

PLANE DIFFRACTION GRATING THEORY OF PLANE TRANSMISSION GRATING: ØLet XY be the grating surface and MN be the screen, both are perpendicular to the plane of paper. Here AB is the slit and BC is an opaque portion. The width of slit is ‘a’ and opaque spacing between two slits is ‘b’.

THEORY OF PLANE TRANSMISSION GRATING(contd. ): Ølet a plane wave front is incident on the grating surface. All the secondary wave fronts traveling in the same direction as that of incident light will come to focus at point P on the screen. The point P where all the secondary waves reinforce one another corresponds to the position of the central bright maximum.

THEORY OF PLANE TRANSMISSION GRATING(contd. ): ØConsider the secondary waves traveling in a direction inclined at an angle θ with the direction of incident light. The collecting lens also suitably rotated such that the axis of the lens is ØThe secondary waves come to focus waves at P 1. The intensity parallel to the direction of secondary. at P 1 will depend on path difference between the secondary waves originating from the corresponding points A and C of two neighboring slits.

THEORY OF PLANE TRANSMISSION GRATING(contd. ): PRINCIPAL MAXIMA: ØAs AB = a and BC = b. Path difference between the secondary waves starting A and C is equal to AC Sin θ. But AC = AB + BC = a + b So path difference= δ = AC Sin θ = (a+b) Sin θ If δ = nλ Then (a+b) is called Grating element/constant. δ = (a+b) Sin θn= nλ For a grating with 15, 000 lines per inch, The value of grating constant is (A) (a+b) =2. 54/15, 000 cm where n = 1, 2, 3, 4, ……… gives P 1 as a point of maximum intensity.

THEORY OF PLANE TRANSMISSION GRATING(contd. ): PRINCIPAL MAXIMA: ØFor such points, where relation (A) is satisfied the waves from all slits are exactly in phase. Hence they are points of maximum possible intensity and are termed as Principal maxima. δ = (a+b) Sin θn= nλ (A) ØEquation(A) gives the condition for nth order Principal maximum. The position of Principal maxima is independent of the number of slits. Since amplitude of disturbance from each slit is independent of θ, so all principal maxima have almost same intensity. Ø Between the two principal maxima, there exist a number of

THEORY OF PLANE TRANSMISSION GRATING(contd. ): CONDITION FOR SECONDARY MININA: ØIf the angle of diffraction changes from θn to θn+dθ, there will be corresponding change in the path difference between the ØIf the path difference so. Aintroduced , (where N is waves originating from and C. be the total number of lines on the grating surface) then the total path difference between the rays originating from the corresponding points of the first and the last strips will be:

THEORY OF PLANE TRANSMISSION GRATING(contd. ): CONDITION FOR SECONDARY MININA: ØIf the diffraction grating be divided into two halves, then the corresponding points in the upper and lower halves will have ØThe superposition a path difference ofof the. waves from the corresponding points in the upper and lower halves will lead to destructive interference and the direction θn+dθ will correspond to first minimum after the nth Principal Maximum ØBetween nth and (n+1)th principal maxima, there will be (N-1) secondary minima corresponding to a path difference:

THEORY OF PLANE TRANSMISSION GRATING(contd. ): CONDITION FOR SECONDARY MININA: ØThe number of secondary maxima is (N-2). Thus the condition for n’th secondary minimum after the nth principal maximum is

THEORY OF PLANE TRANSMISSION GRATING(contd. ): WIDTH OF CENTRAL MAXIMA: ØThe direction of nth principal maximum is given by: ØLet and give the directions of the first seconda minima on the two sides of the nth order primary maxima, then Where N is the total number of lines on the grating surface.

THEORY OF PLANE TRANSMISSION GRATING(contd. ): WIDTH OF CENTRAL MAXIMA: ØDividing (B) by (A) we get:

THEORY OF PLANE TRANSMISSION GRATING(contd. ): WIDTH OF CENTRAL MAXIMA: ØExpanding the equation, we get ØFor small value of dθ ;

WIDTH OF CENTRAL MAXIMA (conclusions): ØIn equation (C), dθ refers to half the angular width of the principal maximum. The half width dθ is Inversely proportional to N, the total number of lines and ØThe value of is more for higher orders because the increase in the value of is less than the half width of the principal maximum increase in the order n is less for higher orders. Øthe larger the number of lines on the grating surface the smaller is the value of Ø is higher for longer wavelengths and hence spectral lines are sharper towards violet than the red end of the spectrum.

OBLIQUE INCIDENCE Ø Let a parallel beam of light be incident obliquely on the grating surface at an angle of incidence i. Then the path difference between the secondary waves passing through the points A and C = FC + CE ØFrom triangle AFC it is seen that Above equation holds good if the beam is diffracted upwards.

OBLIQUE INCIDENCE(contd. ) ØIf the beam is diffracted downwards then the path difference becomes: ØFor nth primary maximum

OBLIQUE INCIDENCE(contd. ) ØThe deviation of the diffraction beam = ØFor deviation to be minimum, must be This is possible if the value of is maximum: minim Thus, the deviation produced in the diffracted beam is a minimum when the angle of incidence is equal to the angle of diffraction. ØLet Dm be the angle of minimum deviation. Then NEXT SLIDE

OBLIQUE INCIDENCE(contd. ) Condition of the principal maximum of the nth order for a wavelength λ.

ABSENT SPECTRA WITH A DIFFRACTION GRATING ØIf Then not possible Hence the first order spectrum will be absent. ØSimilarly second, third etc. orders will be absent if ØIn general if nth order spectrum will be absent

ABSENT SPECTRA WITH A DIFFRACTION GRATING(contd. . ) ØThe condition for absent spectra can be obtained from following considerations. For the nth order principal maximum ØFurther if the value of a and are such that: ØThen the effect of the wave front from any slit will be zero. Considering each slit to be made of two halves, the path difference between the secondary waves from the corresponding points will be they cancel one another’s effect

ABSENT SPECTRA WITH A DIFFRACTION GRATING(contd. . ) ØIf two conditions given by equations (A) and (B) are satisfied simultaneously then The values of n = 1, 2, 3, 4……. Refers to the orders of principal maxima that are absent in the diffraction pattern

HIGHEST POSSIBLE ORDER OF PRINCIPAL MAXUMUM ØThe maximum value for ØSubstituting this value in equation (A) Here n is the highest order for principal maximum visible. If n is not an integer, then the highest possible order is given by the integer lower than n.

INTENSITY DISTRIBUTION OF DIFFRACTION GRAT