INTERFERENCE 1.1 NATURE OF LIGHT

Size: px

Start display at page:

Download "INTERFERENCE 1.1 NATURE OF LIGHT"

Phyllis Palmer
6 years ago
Views:

1 1 INTERFERENCE 1.1 NATURE OF LIGHT In the year 1678, Christian Huygens proposed the wave theory of light. According to this, a Luminous body is a source of disturbance in hypothetical medium called ether and these disturbances are propagated in the form of longitudinal waves through space. This theory could satisfactorily explain reflection, refraction and double refraction but could not explain the phenomena of polarization and rectilinear propagation of light. The concept of longitudinal nature of light was corrected by Fresnel and Young. They suggested that light waves are transverse and not longitudinal as suggested by Huygens. On the basis of this assumption, they could explain the phenomena of interference,diffraction and polarization but could not explain the propagation of transverse waves through vacuum. In the year 1873, Maxwell proposed Electromagnetic theory of light. According to this theory light consists of electric and magnetic fields mutually perpendicular to each other and at right angles to the direction of propagation of the wave. As electric and magnetic fields can be produced in vacuum also, hence no medium like luminiferous ether is required for the propagation of light waves. Christian Huygens (April 14, 169 July 8, 1695) was a Dutch mathematician, astronomer and physicist. He proposed wave theory of light, and in particular demonstrated how waves might interfere to form a wavefront, propagating in a straight line. In 1887, Michelson and Morley proved conclusively that there was no ether surrounding the earth or elsewhere. An electromagnetic wave having wavelength and frequency ν is represented by the instantaneous value of electric field vector of the wave (E) at any point in space, E=E sin (k.r ùt) (1.1) or E=E cos (k.r ùt) (1.) where r is the position vector of the point. E is the amplitude of electric field vector of light wave, ð k is the propagation vector of the wave Magnitude k = and ω = πv is the angular frequency. ë

2 1. Engineering Physics-I π The representations (1.1) and (1.) differ only in initial phase as cosθ = sin + θ. Similarly the = sin. ω E = E sin ωt kr. also differ in intial phase only as representations E E ( kr t) and ( ) sin( π + θ) = sin( θ). 1. COHERENT SOURCES The atom or molecule of any substance in general remains in its lowest energy state, i.e., ground state. If by any interaction it absorbs some energy, it gets excited to the higher energy state. In this excited 8 state it remains for an interval of the order of 1 sec. after which it returns to the lower energy state thereby emitting energy. This process is known as spontaneous emission. This emission of energy is totally random. There are billions of atoms in a light source which randomly keep on emitting light. Therefore the phase difference between light waves obtained from two different sources never remains constant, rather it keeps on changing randomly. Hence the sources of light, the waves emitted by which do not have a definite phase correlationship, i.e., the phase difference between the waves keeps on changing with respect to time, are defined as Incoherent sources of light. The light waves from incoherent sources reaching any point in space cannot produce sustained interference pattern. The binding condition for constructive or destructive interference at any point changes at very high rate, i.e., 8 1 times per second. Such rapid changes cannot be detected by humman eye. Hence in such cases only average intensity of light is observed. If the phase difference between the waves from two sources reaching any point in space does not change with time rather it remains constant then these sources of light are known as coherent sources of light. This property of the waves is known as coherence. In practice it is impossible to obtain such two sources of light. Hence for all practical purposes one source of light is diveded in two parts by partial reflection or partial refraction, which act as coherent sources of light. 1.3 PRINCIPLE OF SUPERPOSITION When two or more disturbances (displacement of the particle of medium in case of sound waves and electric field vector in case of light waves) superpose over each other in a region, the resultant disturbance is equal to the vector sum of the individual disturbances. If E1, E, E 3,... etc. are the individual disturbances, then resultant disturbance is given by: E = E1 + E + E E n (1.3) If the two disturbances meet at a point of the region in the same phase i.e. the phase difference between them is an integral multiple of π, they support each other. In other words the resultant disturbance is equal to the sum of two disturbances, i.e., E = E1 + E (1.4) If the two disturbances meet in the opposite phase i.e. the phase difference between them is an odd multiple of π, they oppose each other. In other words the resultant disturbance is equal to the difference of two disturbances, i.e. E = E1 E (1.5)

3 Interference INTERFERENCE OF LIGHT When two coherent light waves (electromagnetic waves) of equal frequency, travelling in the same direction, superimpose over each other, the intensity of light is redistributed in space. This phenomenon is defined as interference of light. There are two types of interference. (i) Constructive interference When the two waves superpose in the same phase (phase difference δ = nπ, n=,1,,3,... n is an integer), the amplitude and intensity both are maximum. This phenomenon is called constructive interference of light. (ii) Destructive interference When the two waves superpose in the opposite phase (phase difference δ = (n 1) π, n= 1,,3,.. ), the amplitude and intensity both are minimum. This phenomenon is called destructive interference of light. 1.5 YOUNG S DOUBLE SLIT EXPERIMENT In the year 181, Thomas Young experimentally demonstrated the phenomenon of interference of light waves. Fig 1.1 shows the experimental arrangement of Young s double slit experiment. S is a fine hole in a window shutter, placed in front of a monochromatic source of light. A fine beam of light (wavefront) emerges from this hole and falls on two narrow, parallel slits S 1 and S in a piece of cardboard as illustrated in the figure. A screen is placed to the right of the slits. A pattern of alternate bright and dark regions (bands) is observed on the screen. These bright and dark bands are called interference fringes.* Central Ray S 1 S Shutter θ L Sunligh t d Fig 1.1: Young s double slit experiment In the above experiment two slits S 1 and S act as two coherent light sources that send out identical waves. Brightness occurs on the screen at the fringe labelled because the waves travelling to this position meet in the same phase ( = ). Similarly the fringes labelled 1,,... etc. are bright * The interference fringes are hyperbolic. The eccentricities of the hyperbolae are given by e = d/. In the optical experiments, the path difference is of the order of 1-8 cm and d of the order of 1 - cm. The eccentricity is therefore very large of the order of 1 6. As a consequence, hyperbolae are practically straight lines as shown in fringe pattern of Young s double slit experiment.

4 1.4 Engineering Physics-I ( = ë, ë,...). At the fringes labelled 1, 3,... etc. the waves meet in the opposite phase 3 =,,... and hence nullify each other s effect. Hence, brightness at these fringes is minimum (dark fringes) Fig 1.: Fringe pattern of Young s double slit experiment 1.6 THEORY OF TWO COHERENT BEAM INTERFERENCE Let S 1 and S are two coherent light sources emitting light waves of same frequency and nearly same amplitudes. Let the waves originating from these sources superpose over a common point P distant r 1 and r from S 1 and S respectively. If the vibrations of electric field vector produced by two sources are represented by sinusoidal funcions, their magnitudes at point P will be: r 1 P S 1 r O S Fig. 1.3: Two beam interference E = E sin( ωt kr ) (1.6) E = E sin( ωt kr ) (1.7) where E1 and E are the amplitudes of waves originating from S 1 and S, ω is the angular frequency of the waves, and π K = is the propagation constant. According to the principle of superposition, the resultant electric field at point P is given by

5 Interference 1.5 E = E1 + E (1.8) Substituting the values from Eqns. (1.6) and (1.7) E = E sin( ωt kr) + E sin( ωt kr ) 1 1 E E1 sinωtcos kr1 E1 cosωtsin kr1 = + E sinωtcos kr E cosωtsin kr ( ) E E cos kr E cos kr sin t ( E sinkr + E sin kr )cosωt (1.9) = ω 1 1 Let us put E1 cos kr1 + E cos kr = E cosθ (1.1) and 1 1 E sin kr + E sin kr = E sinθ (1.11) Putting these values in Eqn. (1.9), we get E = E (sin ωt cos θ cos ωt sin θ ) (1.1) E = E sin( ωt θ) This is the equation of the resultant wave. To determine the amplitude E of the resultant wave, square and add Eqns. (1.1) and (1.11) E (sin θ + cos θ) = E (sin kr + cos kr ) + E (sin kr + cos kr ) E E (coskr coskr + sinkr sin kr ) E = E1 + E + E1Ecos k( r1 r) (1.13) π E = E1 + E + E1Ecos ( r r 1) (1.14) where ( r r1) is the path difference between the waves from S 1 and S at point P. 1 = ( r r) (1.15) As path difference of ë corresponds to a phase difference of π π phase difference = path difference π δ = ( r r ) (1.16) 1 π δ = (1.17) where δ is the phase difference between two waves meeting at point P. Eqn. (1.14) can be rewritten as

6 1.6 Engineering Physics-I E = E + E + E E cosδ (1.18) 1 1 As intensity of a wave is proportional to the square of its amplitude, hence I E I = k. E (1.19) The intensity of first wave is I = k. E 1 1 and intensity of second wave is (1.) I = k. E Substituting these values in Eqn. (1.18), we get intensity of the resultant wave at point P. I = I1+ I+ I1I cosδ State I: Constructive interference: (1.1) (1.) For constructive interference the resultant amplitude and intensity at point P must be maximum. π π cosδ = cos cos ( r r1) 1 = = + (1.3) i.e., the path difference between two waves must be = r r = n = n (1.4) ( 1) ( ) = even multhiple of or the phase difference between two waves must be δ = nπ (1.5) = even multiple of π The maximum amplitude of the resultant wave will be E = E + E + E E max 1 1 E = E + E (1.6) max 1 The maximum intensity of the resultant waves will be I = I + I + I I max 1 1 ( ) I = I + I (1.7) max 1

7 Interference 1.7 State II: Destructive interference: For destructive interference the resultant amplitude and intensity at point P must be minimum π π cosδ cos cos ( r r1) 1 (1.8) i.e., the path difference between the two waves must be = ( r r) = (n 1) (1.9) 1 = odd multiple of or the phase difference between the two waves must be δ = (n 1) π = odd multiple of π (1.3) The minimum amplitude of the resultant wave will be E = E + E E E min 1 1 E = E E (1.31) min 1 The minmum intensity of the resultant wave will be I = I + I I I min 1 1 I ( I I ) = (1.3) min 1 The ratio of maximum and minimum intensities will be I I max 1 + I E1+ E Emax = = = I I I E E E min 1 1 min (1.33) State III: Waves with equal amplitudes: If the amplitudes of two waves are equal, i.e. E1= E then I1 = I = I (say) The resultant amplitude is given as E = E + E + E E cosδ = E (1+ cos δ) δ 4 cos E = E1 δ E = E1cos (1.34)

8 1.8 Engineering Physics-I Maximum amplitude E = E (1.35) max 1 Minimum amplitude E min = (1.36) The resultant intensity is given by ä I = 4Icos ( as I E ) (1.37) Maximum intensity I = 4I (1.38) max Minimum intensity Visibility of Fringes I min = (Perfect darkness) (1.39) The quality of the fringes produced by interferometric system can be discribed quantitatively using the visibility (V) which is given by V I = I max max I + I min min (1.4) It was first formulated by Michelson. Here I max and I min are the intensities corresponding to the maximum and adjacent minimum in the fringe system. There will be best contrast in fringes or visibility when the difference between I max and I min maximum. As we know that I = I + I + I I and Imin = I1 + I I1I max 1 1 II V = I + I 1 1 (1.41) 1.7 INTENSITY DISTRIBUTION IN INTERFERENCE In interference phenomenon energy is neither created nor destroyed, rather there occurs redistribution of energy. Energy is transferred from the points of minimum intensity to the points of maximum intensity. Thus interference fringes are formed in accordance with the law of conservation of energy. Fig. 1.4 shows the intensity pattern for double slit interference. The horizontal solid line I describes the uniform intensity pattern on the screen if one of the slits is covered up. If the two sources were incoherent, the intensity would be uniform over the screen and would be I. It is indicated by the horizontal dashed line in the above figure. For coherent sources energy is merely redistributed over the screen as indicated by wave like pattern in the figure.

9 Interference CONDITIONS FOR INTERFERENCE Fig. 1.4: Intensity distribution in interference In a well defined interference pattern, the intensity at regions corresponding to destructive interference must be zero while at regions corresponding to constructive interference, it must remain maximum for all values of time. To accomplish this it is essential that the following conditions must be fulfilled. 1. The two beams of light which interfere must originate from the same source of light i.e., the two sources must be coherent.. The waves must have the same frequency or wavelength. 3. The amplitudes of two waves must be equal or nearly equal. 4. The original source must emit light of single wavelength or it must be very nearly monochromatic. On the other hand if the light source is heterogeneous, the optical path difference between the two interfering beams must be very small. 5. The two interfering waves must be propagated in almost the same directions or the two interfering wave-fronts must intersect at a very small angle. 6. If the interference phenomenon is produced by polarized light, the two interfering waves must be in the same state of polarization. 1.9 EXPRESSION FOR FRINGE WIDTH As shown in Fig. 1.5, S is a monochromatic source of light which emits waves of wavelength. S 1 and S are two narrow slits, equidistant from source S. Hence slits will behave like two coherent sources. Let the distance between these sources is d. YY is a screen placed at a distance D from the sources. The central point O of the screen is equidistant from the sources S 1 and S, at which the path difference between the waves from sources will be zero. Therefore, the intensity at this point is always maximum. Let us now consider a point P on the screen distant y (say) from the center O of the screen. Whether a maximum or a minimum is obtained at point P, depends upon the path difference between the waves reaching this point, which is determined as follows: Now = SP SP (1.4) 1 d AP = y and d BP = y +

10 1.1 Engineering Physics-I Y P S 1 A y S d O S D B Y Fig. 1.5: Production of interference fringes By Pythagoras theorem, from triangle PA S 1 d ( SP 1 ) = D + y which can be rearranged as 1 d y SP 1 = D 1+ D On expanding the right side of the above relation by Binomial theorem and neglecting the d y higher powers of D (as D>>d and y) we get d 1 y SP 1 = D 1+ D Similarly from triangle PB S ( ) d = + + SP D y hence d 1 y + SP = D1+ D 1 d d yd ( SP SP 1 ) = = y+ y = D D (1.43)

INTERFERENCE OF LIGHT

INTERFERENCE OF LIGHT Physics Without Fear Want to be good in Physics? Remember: PHYSICS IS AN ACTIVE SPORT Physics is like a sport. To be good in a sport, you must practice. Likewise, to be good in Physics