Wave Optics Physics Notes

Wavefront A wavefront is defined as the continuous locus of all such particles of the medium which are vibrating in the same phase at any instant. An arrow drawn perpendicular to a wavefront in the direction of propagation of a wave is called a ray. Huygens' Principle According to Huygens' Princ...

Wavefront A wavefront is defined as the continuous locus of all such particles of the medium which are vibrating in the same phase at any instant. An arrow drawn perpendicular to a wavefront in the direction of propagation of a wave is called a ray. Huygens' Principle According to Huygens' Principle, each point on a wavefront is a source of secondary waves, which add up to give a wavefront at any late time. This principle is based on the following assumptions: 1. Each point on a wavefront acts a fresh source of new disturbance, called secondary waves or wavelets. 2. The secondary wavelets spread out in all directions with the speed of light in the given medium. 3. The new wavefront at any later time is given by the forward envelope (tangential surface in the forward direction) of the secondary wavelets at that time. Laws of reflection on the basis of Huygens' wave theory Consider a plane wavefront AB incident on the plane reflecting surface XY, both the wavefront and the reflecting surface being perpendicular to the plane of paper. First the wavefront touches the reflecting surface at B and then at the successive points towards C. In accordance with Huygens' Principle, from each point on BC, secondary wavelets start growing with the speed c. During the time the disturbance from A reaches the point C, the secondary wavelets from B must have spread over a hemisphere of radius BD = AC = ct, where t is the time taken by the disturbance to travel from A to C. The tangent plane CD drawn from the point C over this hemisphere of radius ct will be the new reflected wavefront. Let angles of incidence and reflection be i and r respectively. In ΔABC and ΔDCB, we have \[Each is 90°\] BC = BC \[Common\] AC = BD \[Each is equal to ct\] Hence or i = r i.e., the angle of incidence is equal to the angle of reflection. This proves the first law of reflection. Since the incident ray SB, the normal BN and the reflected ray BD are respectively perpendicular to the incident wavefront AB, the reflecting surface XY and the reflected wavefront CD (all of which are perpendicular to the plane of the paper), therefore they all lie in the plane of the paper, i.e., in the same plane. This proves the second law of reflection. Laws of refraction on the basis of Huygens' wave theory Consider a plane wavefront AB incident on a plane surface XY, separating two media 1 and 2. Let ν~1~ and ν~2~ be the velocities of light in the media, with ν~2~ \< ν~1~. The wavefront first strikes at point A and then at the successive points towards C. According to Huygens' Principle, from each point on AC, the dray wavelets start growing in the second medium with speed ν~2~. Let the disturbance take time t to travel disturbance from B to C, then BC = ν~1~t. During the time the disturbance from point A must have spread over a hemisphere of radius AD = ν~2~t in the second medium. The tangent plane CD drawn from point C over this hemisphere of radius ν~2~t will be the new refracted wavefront. Let the angles of incidence and refraction be i and r respectively. From right ΔABC, we have From right ΔADC, we have or (a constant) This proves Snell's law of refraction. The constant ^1^μ~2~ is called the refraction index of the second medium with respect to first medium. Further, since the incident ray SA, the normal AN and the refracted ray AD are respectively perpendicular to the incident wavefront AB, the diving surface XY and the refracted wavefront CD (all perpendicular to the plane of the paper), therefore, plane. This proves another law of refraction. 1. The frequency ν remains the same as light travels from one medium to another. 2. The wavelength in a medium is directly proportional to the phase speed (or wave speed) and inversely proportional to its refractive index. 3. The speed of light in an optically rarer medium is greater than that in an optically denser medium. Behaviour of a prism Behaviour of a convex lens Behaviour of a concave mirror  C:\\Documents and Settings\\Administrator\\My Documents\\My Pictures\\Picture\\picuture 026.jpg  Principle of superposition of waves When a number of waves travelling through a medium superpose on each other, the resultant displacement at any point at a given instant is equal to the vector sum of the displacements due to the individual waves at that point. If are the displacements due to the different waves acting separately, then according to the principle of superposition, the resultant displacement when all the waves act together is given by the vector sum: Interference of light When two light waves of the same frequency and having zero or constant phase difference travelling in the same direction superpose each other, the intensity in the region of superposition gets redistributed, becoming maximum at some points and minimum at others. This phenomenon is called interference of light. Conditions for constructive and destructive interference Expression for intensity at any point interference pattern Suppose the displacements of two light waves from two coherent sources S~1~ and S~2~ at point P on the observation screen at any time t are given by and where a~1~ and a~2~ are the amplitudes of the two waves, φ is the constant phase difference between the two waves. By the superposition principle, the resultant displacement at point P is or Put......... (1) and......... (2) Then or Thus, the resultant wave is also a harmonic wave of phase angle Ѳ. To determine A, squaring and adding equations (1) and (2), we get or or......... (3) But intensity of a wave proportional to (amplitude)^2^ We write and where k is proportionality constant. The equation (3) can be written as or......... (4) The total intensity also contains a third term. It is called interference term. Constructive interference The resultant intensity at the point P will be maximum when cos φ = 1 or φ = 0, 2π, 4π,... Since a phase difference of 2π corresponds to a path difference of λ, therefore, if p is the path difference between the two superposing waves, then or p = 0, λ, 2λ, 3λ,... = nλ where n = 0,1,2,3.... Hence the resultant intensity at a point is maximum when the phase difference between the two superposing waves is an even multiple of π or path difference is an integral multiple of wavelength λ. This is the condition of constructive interference. Destructive interference The resultant intensity at the point P will be minimum when cos φ = -1 or φ = π, 3π, 5π,... or or where n = 1,2,3,4.... Hence the resultant intensity at a point is minimum when the phase difference between the two superposing waves is an odd multiple of π or the path difference is an odd multiple of λ/2. This is the condition of destruction interference. Coherent and incoherent sources Two sources of light which continuously emit light waves of same frequency (or wavelength) with a zero or constant phase difference between them are called coherent sources. Two sources of light which do not emit light waves with a constant phase difference are called incoherent sources. Theory of interference fringes: fringe width Suppose a narrow slit S is illuminated by monochromatic light of wavelength λ. S~1~ and S­~2~ are two narrow slits at equal distance from S. Being derived from the same parent source S, the slits S~1~ and S­~2~ act as two coherent sources, separated by a small distance d. Interference fringes are obtained on a screen placed at distance D from the sources S~1~ and S­~2~. Consider a point P on the screen at distance x from the center O. The nature of the interference at the point P depends on path difference, p = S~2~P -- S~1~P From right-angle ΔS~2~BP and ΔS~1~AP, or or In practice, the point P lies very close to O, therefore S~2~P = S~1~P = D. Hence or Positions of bright fringes For constructive interference, or where n = 0, 1, 2, 3,... Clearly, the positions of various bright fringes are as follows: For n = 0, x~0~ = 0 Central bright fringe For n = 1, First bright fringe For n = 2, Second bright fringe For n = n, n^th^ bright fringe Positions of dark fringes For destructive interference, or where n = 1, 2, 3,... Clearly, the positions of various dark fringes are as follows: For n = 1, First dark fringe For n = 2, Second dark fringe For n = n, n^th^ dark fringe Since the central point O is equidistant from S~1~ and S­~2~, the path difference p for it is zero. There will be a bright fringe at the centre O. But as we move from O upwards or downwards, alternate dark and bright fringes are formed. Fringe width. It is the separation between two successive bright or dark fringes, Width of a dark fringe = Separation between two consecutive bright fringes Width of a bright fringe = Separation between two consecutive dark fringes, Clearly, both the bright and dark fringes are of equal width. Hence the expression for the fringe width in Young's double slit experiment can be written as β = Dλ/d As β is independent of n (the order of fringe), therefore all the fringes are of equal width. In the case of light, λ is extremely small, D should be much larger than d, so that the fringe width β may be appreciable and hence observable. Conservation of energy in interference In an interference pattern, the intensities at the points of maxima and minima are such that I~max~ = ∝ (a~1~ + a~2~)^2^ and I~min~ = ∝ (a~1~ - a~2~)^2^ or If there is no interference between the light waves from the two sources, then intensity at every point would be same. That is which is same as I~av~ in the interference pattern. So there is no violation of the law of conservation of energy in interference. Whatever disappears from a dark fringe, an equal energy appears in a bright fringe. Comparison of intensities at maxima and minima Let a~1~ and a~2~ be the amplitudes and I~1~ and I~2~ be the intensities of light waves from two different sources. As intensity ∝ Amplitude^2^ Amplitude at a maximum in interference pattern = a~1~ + a~2~ Amplitude at a minimum in interference pattern = a~1~ - a~2~ Therefore, the ratio of intensities at maxima and minima is or where amplitude ratio of the two waves. Diffraction of light The phenomenon of bending of light around the corners of small obstacles or apertures and its consequent spreading into the regions of geometrical shadow is called diffraction of light. Diffraction at a single slit A source S of monochromatic light is placed at the focus of a convex lens L~1~. A parallel beam of light and hence a plane wavefront WW gets incident on a narrow rectangular slit AB of width d. According to Huygens theory, all parts of the slit AB will became source of secondary wavelets, which all start in the same phase. These wavelets spread out in all directions, thus causing diffraction of light after it emerges through slit AB. Suppose the diffraction pattern is focused by a convex lens L~2~ on a screen placed in its focal plane. Central maximum The wavelets from any two corresponding points of the two halves of the slit reach the central point O in the same phase, they add constructively to produce a central bright fringe. Calculation of path difference Suppose the secondary wavelets diffracted at an angle θ are focussed at point P. then the path difference between the wavelets from A and B will be p = BP -- AP = BN = AB sin *θ* = d sin *θ*. Positions of minima Let the point P be so located on the screen that the path difference, p = λ and angle θ = θ~1~. Then from the above equation, we get d sin *θ*~1~ = λ we can divide the slit AB into two halves AC and CB. Then the path difference between the wavelets from A and C will be λ/2. Similarly, corresponding to every point in the upper half AC, there is a point in the lower half CB for which the difference is λ/2. Hence the wavelets from the two halves reach the point P always in opposite phases. They interfere destructively so as to produce a minimum. Thus the condition for first dark fringe is d sin *θ*~1~ = λ Similarly, the condition for second dark fringe will be d sin *θ*~2~ = 2λ Hence the condition for n^th^ dark fringe can be written as d sin *θ*~n~ = nλ, n = 1, 2, 3,... The directions of various minima are given by *θ*~n~ ≈ sin *θ*~n~ = nλ/d \[as λ \