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Diffraction of light
A Phenomenon of bending of light or spreading of light by an obstacle is known as diffraction. The resulting intensity distribution of light if observed on the screen is called the diffraction pattern. An Italian scientist Grimaldi observed this pattern first time.
A’
A
S ( Source )^ B
B’ Slit Screen To study the phenomena, consider a source of monochromatic light S. AB is the small aperture slit. A’B’ is the illuminated portion on the screen, above A’ and below the B’ are the region of geometrical shadow. This geometrical shadow is called the diffraction. Difference between interference and diffraction
Type of diffraction: On the basis of distance between source of light and screen, the diffraction phenomenon is divided into two parts:
- Fraunhofer Diffraction
- Fresnel Diffraction
Fraunhofer’s diffraction at a single slit: X
P
θ
O S N
Monochromatic source Y Slit Convex lens Screen
Let a parallel beam of monochromatic light of wavelength λ be incident upon a narrow slit AB of the width e. According to the Huygens principle, each point ( A & B ) sends out secondary wavelets in all possible direction. O is the point on screen will have the maximum intensity, because it is placed equidistant from point A and B. But slit is very narrow (slit width is the order of wavelength of light, λ). So that, the diffraction effect becomes prominent. Let us consider the beam diffracted at angle θ. The ray proceeding in the same direction as that of the incident ray are focused at point O , while those diffracted through and angle θ are focused at P. Let us find out the resultant intensity at point P. Draw a perpendicular AN. The path difference between the two diffracted rays from A and B , The path difference (Δ) = BN.
So that the path difference = e sin θ
We know that: phase difference = Then the phase difference =
Therefore,
This equation gives the direction of first, second, third, etc. minima by putting m = 1, 2, 3, ----------------. It is to be noted that m = 0 not admissible because it gives, which corresponds to principle maxima. To find the direction of maximum intensity, let is differentiate equation (1) with respect to α and equate to zero. i. e.
Put the value of I from equation (1), we have
or
or
This equation is solved graphically by plotting the curves and
y
O
Fig: Plot of and versus
Fraunhofer Diffraction at N -slit (Diffraction grating): A diffraction grating is a device consists of a large number of parallel slits of equal width separated from each other by equal space. These slits can be obtained by resulting equidistant lines either on a transparent glass surface or on a silvered surface with the help of fine diamond point. These lines are drawn on a plane transparent surface so it is called plane transmission grating. If these lines are drawn on a silvered surface then the grating is called the reflection grating. The number of ruled lines varies 15000 to 30000 per inch on the surface.
X
e A P d S 1 S 2 K S 3 S 4 S 5 O
B Y
Let e is the width of each slit and d is the space between two slits. Then ( e + d ) is called the grating element. The point of two consecutive slit is separated by the distance ( e + d ) or it is the distance between corresponding points. Let parallel beam of monochromatic light of wavelength λ be incident normally on the grating. According to the Huygen’s principle all points in each slit become the source of secondary disturbances in all directions or in other words- all points in a slit send out the secondary wavelets in all possible directions.
2 Nβ
2 β 6 β β 4 β P 2 2 β M N P 1
Draw a perpendicular on MP (^) 1 from the centre. MP 1 = 2 MN or Therefore, Also, MP 2 = 2 MP 1
We have, or So, (1) Similarly, (2)
From above two equations:
or Resultant amplitude(R) A o or amplitude of each vibration
Then the resultant intensity ( I ) is:
The first order gives the diffraction pattern due to single slit; while the second factor gives the intensity distribution pattern due to interference pattern for N slits.
Let us consider the intensity distribution due to the second factor. Principal Maxima: When i. e. where n = 0, 1, 2, 3, ----------- Then and thus indeterminate, Let us find the value of usual method of differentiating the numerator and the denominator. By using ‘ L ’ Hospital rule:
So that the intensity from equation (1), we have, (2) This is the maximum intensity. There are most intense and called principal maxima, which is obtained in the direction given by
where n = 0, 1, 2, 3, ---------- For n = 0, we get the zero order maxima. For n = ±1, ±2, ±3, -------------, we obtain the first, second, third order principal maxima respectively. ± sign shows that there are two principal maxima for each order lying both side from zero the order maxima.
For Minima: When but
Hence, from equation (1) we get I = 0 This is minimum intensity. These minima are obtained in the direction given by or because (3) Where m takes all integral values except 0, N , 2 N , 3 N , -------- nN , because these value of m make, which gives the principal maxima. It is clear from above that m = 0 gives a principal maxima and m = 1, 2, 3, -------- ( N -1) give minima. Then m = N gives again a principal maxima. Thus, there are ( N -1) minima between two consecutive principal maxima.
According to Pythagoras theorem:
Then So that
This equation shows that the intensity of the secondary maxima is proportional to , where as the intensity of the principal maxima is proportional to , therefore
Hence, greater value of N (Number of slits), the weaker are secondary maxima, In actual grating, N is very large. Hence these secondary maxima are not visible in the grating spectrum.
Formation of Multiple Spectra by the grating: When a beam of wavelength fall normally on the grating, the principal maxima are formed in the direction given by n = 0, 1, 2, 3, ------------------- This equation shows that for a given order n , the angle of diffraction varies with wavelength. Hence, longer wavelength will have the greater angle of diffraction, if the incident light be white, then each order will certain principal maxima of different wavelengths in the different direction. II nd^ order
I st^ order
Zero order I st^ order
II nd^ order
The principal maxima of all wavelengths corresponding to n = 1 will form the first order spectra and so on ………., the principal maxima of all wavelength corresponding to n = 1 means Hence the zero order maxima (principal maxima) will be white, having both side first order, second order and so on………...
Condition for absent spectra: Sometimes it so happen that the certain order of spectrum is absent. One order (first order) spectrum is clearly visible but second order spectrum may not be visible and third order spectrum my again be visible and so on. These conditions will depends upon the ratio of e and d. Where e is width of the slit or transparent space/ transparency and d is the space between two slits or opaque space/ opacity. We have calculated that The intensity of light due to diffraction grating in a direction making an angle θ with the normal to the grating is given by-
where and is the factor gives the diffraction pattern due to single slit, while the second factor gives the intensity distribution pattern due to interference pattern due to N slits (or say combined effect of all N slits). The principal maxima in the grating spectrum are obtained in the directions is given by (1) where n is the order of maximum
or Hence the spectra of second order will be seen.
Dispersive power of a grating: The dispersive power of a grating is defined as “the rate of change of the angle of diffraction with the wavelength of light”. It is expressed as. The angle of diffraction, for the principal maxima is related to the corresponding wavelength by grating equation
Differentiating the above equation with respect to, we get
This is the expression for the dispersive power of grating. The above equation shows that the dispersive power is-
- directly proportional to the order n.
- inversely proportional to the grating element ,
- inversely proportional to. It means larger value of, smaller value of then the value of dispersive power is higher.
Resolving Power of a Diffraction grating: The resolving power of grating is defined as its ability to show two spectral lines of wavelengths very close as separated. It can also be defined as the ratio of the wavelength of any spectral line to the smallest wavelength difference between two neighboring lines. Mathematically, it can be expressed as , where is the wavelength of any spectral line and is the difference in the wavelengths of this line and a neighboring line. To drive the expression for the resolving power of a plane transmission grating we shall apply the Rayleigh limit. Let the parallel beam of light of wavelength and be incident normally on the grating.
We have already derived the expression for the n th^ principal maxima due to the
wavelength are formed in the direction.
(1) Where N is the total number of lines on grating surface and is the grating element. Let the first minima adjacent to the n th^ order maxima be formed in the direction then the
grating equation for minima is (2) According to Rayleigh criterion, two spectral lines of wavelengths and will appear just resolved. When n th^ maxima of the wavelength () fall on the first minima of the wavelength, which is adjacent to the n th^ maxima of (), we have
or (3) Equating equation (2) and (3), we have
or This is the expression for resolving power.
Relation between Resolving Power and Dispersive Power: The resolving power of a grating is equal to the product of the order of the spectrum and total number of lines on the grating. (5)
We know that the dispersive power of the grating is so that from the above equation, we have is the total aperture width. Where is the total width of ruled space on the grating.
