Diffraction Grating Date : 9. Diffraction Grating - IITKGP

:Diffraction DateGrating9. Diffraction GratingBackgroundFraunhofer diffractionFresnel diffractionAngular dispersionResolving powerSpectral linesAim of the experimentTo determine the wavelengths of the prominent lines of mercury by aplane transmission diffraction grating, hence to find (a) the chromaticresolving power of the plane transmission diffraction grating and (b) thedispersive power of the grating.Apparatus requiredSpectrometerPlane transmission diffraction gratingMercury-lampSpirit levelTheoryIf a parallel beam of monochromatic light is incident normally on theface of a plane transmission diffraction grating, bright diffraction maxima areobserved on the other side of the grating. These diffraction maxima satisfythe grating condition : a b sin n n , (1)where(a b) the grating element ( 2.54/N, N being the number ofrulings per inch of the grating), n the angle of diffraction of the nth maximumn the order of spectrum which can take values 0, 1, 2, 3 . the wavelength of the incident lightClearly, the diffraction is symmetrical about 0 0. If the incidentbeam contains different colours of light, there will be different ncorresponding to different in the same order n. By measuring n andknowing N, can be calculated.Chromatic resolving power of a grating is defined as its power ofdistinguishing two nearby spectral lines and is defined asChromatic R.P 94 (2)

Diffraction GratingWhere is the separation of two wavelengths which the grating canresolve; the smaller the value of , the larger the resolving power.Employing Rayleigh’s criterion for the limit of resolution, one can show inthe case of a gratingR.P nN. (3)The angular dispersion or dispersive power of a grating is defined asthe rate of change of angle of diffraction with the change of wavelength in aparticular order of the spectrum. Differentiating eqn. (1) with respect to ,we getd n. (4) d a b cos Eqn.(4) shows that for a given small wavelength difference the angularseparation is directly proportional to the order n. When is small (lessthan 60), cos is constant and hence is proportional to . Such aspectrum is called a normal spectrum.Procedure(a)1.2.3.(b)4.5.Adjustment of the Collimator and the Telescope :Level the prism table, telescope and collimator with spirit level suchthat telescope axis and collimator, axis intersect the principal verticalaxis of the spectrometer. A prism may be used for this purpose.Focus the eye-piece of the telescope on the cross-wire by drawing itin or out of the telescope tube until the cross-wire is seen clearly.Use Schuster’s method for focusing telescope and collimator forparallel rays [see Page no. 117, topic (v)]Adjustment of the Grating :The grating is to be adjusted on the prism table such that light fromthe collimator falls normally on it. For achieving this :First the collimator and the telescope are brought in one line and theimage of the slit is focused on the vertical cross-wire. Thecorresponding reading on both the verniers is noted.The telescope is rotated through 900.95

Diffraction Grating6.7.8.(c)12.Mount the grating on the prism table and rotate the prism table so thatthe reflected image is seen on the vertical cross-wire in the telescope.Take the vernier readings.Turn the prism table from this position through 450 or 1350, so that‘writing’ on the grating is away from the collimator. In this position,the grating is normal to the incident beam (see Fig.1).The slit is rotated in its place till the spectral lines are very sharp andbright. This brings the slit parallel to the lines of grating.Measuring the Diffraction Angles :The spectrum is shown in Fig. 2.9. Rotate the telescope to the leftside of the direct image and adjust iton different spectral lines (startingwith first order blue line andfinishing with second order yellowline) turn by turn. It should be takencare that the movement of telescopeis in one direction.10.Note the vernier readings V1and V2.11.Now rotate the telescope tothe right side of the direct image andrepeat steps 9 and 10.Thedifference of corresponding vernierreadings will give twice the angle of diffraction.Find the angles of diffraction for prominent lines in the first and thesecond order spectra.ObservationsVernier constant of the spectrometer (Least Count) :Number of lines per inch of the grating (N) :2.54Grating element (a b) cmNTable 1To set the unruled surface of the grating for normal incidenceDirect reading of the telescope withoutgratingMainVernierTotalScale (M)(V)(T M V)Telescopeis rotatedthrough900 andset atangle96Reading of the prism table when theangle of indicence is 450MainVernierTotalScale (M)(V)(T M V)Prism table isrotated through450 or 1350 andset atangle

Diffraction GratingTable 2Vernier No.Readings for the diffracted imageswith the telescope at theMainscale(M)LeftVernier(V)Total(T M V)Mainscale (M)RightVernier(V)Total(T M V)Differencebetween theleft andrightreadings ofvernier (2 )1stColour of the lineOrder No.(n)Determination of the angles of diffraction for the lines of different colour and order( a1)( a2)(a1 a2)( a)( b1)( b2)(b1 b2)( b)( a1)( a2)(a1 a2)( a)( b2)(b1 b2)( b)1stBlue2nd12nd2( b1)97Mean(2 )a b()2Angle ofDiffraction( )

Vernier No.Readings for the diffracted imageswith the telescope at theMainscale(M)LeftVernier(V)Total(T M V)RightVernier(V)Mainscale (M)Total(T M V)Differencebetween theleft andrightreadings ofvernier (2 )1stColour of the lineOrder No.(n)Diffraction Grating( a1)( a2)(a1 a2)( a)( b1)( b2)(b1 b2)( b)( a1)( a2)(a1 a2)( a)( b2)(b1 b2)( b)1stGreen2nd12nd2( b1)98Mean(2 )a b()2Angle ofDiffraction( )

Vernier No.Readings for the diffracted imageswith the telescope at theMainscale(M)LeftVernier(V)Total(T M V)RightVernier(V)Mainscale (M)Total(T M V)Differencebetween theleft andrightreadings ofvernier (2 )1stColour of the lineOrder No.(n)Diffraction Grating( a1)( a2)(a1 a2)( a)( b1)( b2)(b1 b2)( b)( a1)( a2)(a1 a2)( a)( b2)(b1 b2)( b)1stYellow2nd12nd2( b1)99Mean(2 )a b()2Angle ofDiffraction( )

Diffraction GratingCalculation and ResultsTable 3Determination of wavelength of unknown linesNo. of linesper cm ofthe gratingsurface (N)(given)Colour ofthe lineOrderno.(n)Angle ofdiffraction ( )(From Table 2)Wavelength ofthe spectralline( )(Å)Mean (Å)1Blue21Green21Yellow2Table 4Determination of Resolving power and dispersive power of the gratingColourofthe lineOrderNo.(n)Angle ofdiffraction (from Table 2)No. of gratinglines illuminatedby thecollimator (N)100Resolvingpowerof thegrating nNAngular dispersionof the grating nN2.54 cos

Diffraction GratingError calculationThe wavelength of unknown spectral line is determined from the relation:sinθλ nNTherefore, the maximum proportional error in the determination of isδλ cos δ λsinθδλ δθ .(A) λ tanθ2 measured from the difference between two readings corresponding to two positions ofthe telescope. Hence is equal to the value of one vernier constant (in radian).Substituting the measured values of and the value of in eqn. (A) and multiplying by100, the maximum percentage error in can be calculated.101

Diffraction GratingDiscussion(i)When mounting the grating on the prism table, if the ruled surface of the grating istowards the collimator, two images are viewed in the telescope placed with its axis normalto that of the collimator. The two images are formed by reflection at the front and backsurfaces of the grating. In this case, work is to be done with the front surface image. Todistinguish between the front and back surface images, an electrical lamp is to be placedbehind the sodium flame. Both the monochromatic sodium light and the white light areincident on the grating. The image formed by reflection of the white light from the backsurface of the grating will be coloured. This image is ignored and the adjustments formaking the plane of the grating vertical are to be done with the other image.(ii)When the ruled surface of the grating is in the side of the collimator, the prism tableis to be rotated through 450 in the proper direction to make the unruled surface of gratingnormal to the rays from the collimator. Also, it should be placed on the prism table so as toget the maximum area of the surface exposed to the incident light.(iii) The slit should be made very narrow to increase the brightness of the higher orderdiffracted images.(iv)The source position should be so adjusted as to make the diffracted images on bothsides of the central one equally bright.(v)If necessary, the slit illumination can be increased by forming an image of the sourceon the slit by inserting a convex lens of short focal length between the slit and the source.(vi)While rotating the telescope, it should be moved always in the same direction so asto avoid any back-lash error.Questions1.2.3.4.5.6.7.8.9.10.11.12.13.14.In this experiment, how does diffraction occur?What is a plane transmission diffraction grating?What is a reflection grating?How are commercial gratings made?What type of grating do you use for your experiment?Define grating element and corresponding points.What is the effect of increasing the number of lines per cm on the grating?What do you understand by the angular dispersive power of the grating?How does the angular dispersive power of the grating vary with (i) the ordernumber n of the spectrum, (ii) the grating element or the number of lines per cmin the grating, and (iii) the wavelength ?Distinguish between a grating spectrum and a prismatic spectrum.What will happen if the slit is illuminated with white light?What will happen if the rulings of the grating are not parallel and the distancebetween two consecutive rulings is not constant?What is the SI unit of wavelength?What happens if the ruled surface of the grating faces the collimator?102

Diffraction Grating15.16.17.What do you mean by the resolving power of a grating?How can you experimentally verify that the incident rays are normal to thegrating surface?What are the uses of a diffraction grating?References1. Fundamental of Optics by F. Jenkins and H. White 535 JEN/F2. Optics by A.Ghatak 535 GHA/O3. Optics by E. Hecht535 HEC/O103

Plane transmission diffraction grating Mercury-lamp Spirit level Theory If a parallel beam of monochromatic light is incident normally on the face of a plane transmission diffraction grating, bright diffraction maxima are observed on the other side of the grating. These diffraction maxima satisfy the grating condition : a b sin n n , (1)