Wave Optics Lecture Notes

Wave Optics Dr. Sharad B. Patil Assistant Professor Department of Applied Physics, School of Engineering and Management, D Y Patil Education Society (Deemed to be University) Kasaba Bawada, Kolhapur  Contents Introduction:...

Wave Optics Dr. Sharad B. Patil Assistant Professor Department of Applied Physics, School of Engineering and Management, D Y Patil Education Society (Deemed to be University) Kasaba Bawada, Kolhapur  Contents Introduction: interference, diffraction, review of the geometric path, optical path. Theory of plane diffraction grating and grating equation. Resolving power of plane diffraction grating. Newton’s Ring: Experimental arrangement. Diameter of Bright and Dark ring. Determination of wavelength of monochromatic light using Newton’s ring. Applications of interference in anti-reflecting coatings. Wave? Types? General transverse sinusoidal wave equation => 2 k=  2 = 𝑇 T is the time-period of oscillation Principle of Superposition of Waves => Let y1(x, t) and y2(x, t) be the displacements that the string would experience if each wave traveled alone. The displacement of the string when the waves overlap is then the algebraic sum y’(x, t) = y1(x, t) + y2(x, t) => Overlapping waves algebraically add to produce a resultant wave (or net wave). Two waves traveling in opposite directions along a stretched string. The superposition => Overlapping waves do not, in any way, alter the travel of each other. principle applies as the pulses move through each other.  Interference What will be the result? Prediction 1 Prediction 2 Interference “The phenomenon in which two or more waves superimpose to form a resultant wave of greater, lower or the same amplitude.” Interference Constructive Interference Destructive Interference If the crest of a wave meets the crest of another wave of the same frequency at the same point, then the resultant amplitude is the sum of individual amplitudes, known as constructive interference. When a wave’s crest meets another wave’s trough, then resultant amplitude is equal to the difference in the individual amplitudes, known as destructive interference. Interference “The phenomenon in which two or more waves superpose to form a resultant wave of greater, lower or the same amplitude.” y1(x, t) = ym sin (kx − t) wave-1 y2(x, t) = ym sin (kx − t + ) wave-2 (wave-2 is shifted from wave-1 by ) From the principle of superposition, the resultant wave is given by y’(x, t) = y1(x, t) + y2(x, t) = ym sin (kx − t) + ym sin (kx − t + ) Using y’(x, t) = [2ym cos  /2] sin (kx − t + /2) Phase difference The phase difference between two waves can change if the waves travel through different materials having different refractive index The refractive index () of the medium is given by the ratio of the speed of light in a vacuum to the speed of light in the medium Path difference We call this extra distance the path difference. Geometric Path and Optical Path Geometric path length is the actual distance light travels in an optical system. The optical path length is the geometric path length multiplied by the refractive index of the media the light travels in. For example, light travelling through 10 𝑐𝑚 of water then it has The geometric path length (GP) = 10 cm. And The optical path length (OP), 𝑂𝑃 = 𝐺𝑃 × 𝜂 10cm Where, 𝜂 = Refractive index of water. OP = 10 × 1.333 OP = 13.3𝑐𝑚. Diffraction “The bending or spreading of light waves as they pass through an aperture or around the sharp edges of objects is called diffraction” We can observe single-slit diffraction when light passes through a single slit whose width (w) is on the order of the wavelength of the light. The diffraction pattern on the screen will be at a distance L >> w away from the slit. We can observe the bending phenomenon of light or diffraction that causes light from a coherent source to interfere and produce a distinctive pattern on the screen called the diffraction pattern. Diffraction is evident when the sources are small enough that they are relatively the size of the wavelength of light. For large slits, the spreading out is small and generally unnoticeable. Diffraction pattern Plane Diffraction Grating: Theory An arrangement consisting of a large number of parallel slits of the same width and separated by equal opaque spaces is known as a diffraction grating. Fraunhofer (1821) used the first grating, which consisted of a large number of parallel wires placed very closely side by side at regular intervals. Now gratings are constructed by ruling equidistant parallel lines on a transparent material such as glass with a fine diamond point. The ruled lines are opaque to light while the space between any two lines is transparent to light and acts as a slit. This is known as Plane diffraction grating. When the spacing between the lines is of the order of the wavelength of light, then an appreciable deviation of light is produced. Let, ‘a’ be the width of the line and ‘b’ be the width of the slit. a+b Then (a+b) is known as the grating element. If ‘N’ is the number of lines per inch (LPI) on the grating, then N (a+b) =1”=2.54 cm 1 this implies; N = a+b a+b = 2.54 𝑐𝑚 a 𝑁 b When light falls on the grating, the light gets diffracted through each slit. As a result, both diffracted light interfere in forming a diffraction pattern.  Plane Diffraction Grating: Theory The figure represents the section of a plane diffraction grating placed perpendicular to the plane of the paper. Let, ‘a’ be the width of opaque line and ‘b’ be the width of transparent slit. Then, (a+b) is the grating element. XY is the screen placed perpendicular to the plane of a paper. Suppose a parallel beam of monochromatic light of wavelength 𝜆 is incident normally on the grating. By Huygens's principle, each slit sends secondary wavelets in all directions. The secondary wavelets travelling in the same direction of incident light will come to a focus at point P 0 of the screen as the screen is placed at the focal plane of the convex lens. The point P0 will be the central maximum. Now, consider the secondary waves travelling in a direction inclined at an angle 𝛳 with the direction of the incident light. These waves reach point P1 on passing through the convex lens in different phases. As a result, dark and bright bands on both sides of the central maximum are obtained. Grating Equation P Q P0 From, ∆AMC, CM 𝑠𝑖𝑛𝜃 = AC 𝐶𝑀 = 𝐴𝐶 𝑠𝑖𝑛𝜃 (1) But, From Fig., 𝐴𝐶 = 𝑎 + 𝑏 which is Grating Element  1 But, 𝑎+𝑏 = 𝑁 𝑆𝑖𝑛𝜃 Therefore, 𝐶𝑀 = (3) 𝑁 As CM is the path difference between light beams AP and CQ which is always integral multiple of 𝜆 i.e. 𝑛𝜆 to get the constructive interference at P1 on the screen, therefore, 𝐶𝑀 = 𝑛𝜆 (4) Put eq. (4) in eq. (3) we get, 𝑆𝑖𝑛𝜃 𝑛𝜆 = 𝑁 𝑠𝑖𝑛𝜃 = 𝑁𝑛𝜆 (5) This equation is called grating equation. The equation (5) is also referred as the condition for the existence of a principal maximum. Case-I: Monochromatic wave When incident light consists of the monochromatic wave, only a single value of , then the grating equation is 𝑠𝑖𝑛𝜃 = 𝑁𝑛𝜆 Case-II: Polychromatic wave If incident light consists of a number of wavelengths, i.e.  and +d, which diffract at an angle  and +d, then the grating equation becomes; sin( + 𝑑) = 𝑁𝑛(𝜆 + 𝑑) Condition for minima From Fraunhofer diffraction theory, the condition for minima is 𝑁 a + b 𝑠𝑖𝑛𝜃 = ±𝑚𝜆 where m can have all integral values except 0, N, 2N, 3N…., nN, because these values of m give rise to different principal maxima. It is clear from above equation that m = 0 gives principal maxima of zero order. m = 1, 2, 3, … (N-1) give minima and m = N gives again principal maxima of first order. Thus there are N-1 equispaced minima between two consecutive principal maxima. Conclusions from grating equation: 1. For a particular wavelength λ, the angle of diffraction Θ is different for the principal maxima of different orders. 2. From Fraunhofer diffraction theory, it has found that, as the number of lines in the grating increases, the intensity of diffracted waves increases. These diffracted waves form maxima which appear as sharp, bright parallel lines and are termed as spectral lines. 3. For white light and a particular order of n, the light of different wavelengths will be diffracted in different directions. 4. At the center, Θ=0 gives the maxima of all wavelengths which coincides to form the central image of the same colour as that of the light source. This forms zero order diffraction. 5. The principal maxima of all wavelengths form the first, second,... order spectra for n=1,2,... 6. The longer the wavelength, the greater the angle of diffraction. Thus, the spectrum consists of violet being in the innermost position and red being in the outermost position. 7. Most of the intensity goes to zero order and the rest is distributed among other orders. 8. Spectra of different orders are situated symmetrically on both sides of zero order. 𝒂+𝒃 9. The maximum number of orders available with the grating is 𝒏𝒎𝒂𝒙 = 𝝀 Fig. Grating Spectrum  Resolving power of Plane Diffraction Grating The resolving power of an analyzing instrument is its ability to just separate two close spectral lines in their diffraction patterns. The ability of an optical instrument to produce distinctly separate images of two objects located very close to each other is called its resolving power. Rayleigh criterion of resolution According to Rayleigh two equally bright point sources could be just resolved by any optical system if the distance between them is such that the central maximum in the diffraction pattern due to the one source coincides exactly with the first minimum in the diffraction pattern due to the other. This is known as Rayleigh criteria of resolution. Unresolved Just resolved Resolving power of Plane Diffraction Grating Consider a parallel beam of light of wavelengths  and  + d incident normally on the plane transmission grating having grating element (a + b) and total number of rulings N. Then the resolving power of the grating is defined as the ratio of wavelength () to the difference d of the wavelength i.e., /d. The separate diffraction pattern for  and  + d is shown in the Figure. According to Rayleigh criterion, these spectral lines are just resolved as the principal maxima of one line lies just on the first minima of the other. Now the direction of nth principal maximum for a wavelength  is given as 𝑎 + 𝑏 𝑠𝑖𝑛𝜃 = 𝑛𝜆 (1) The direction of nth principal maximum for a wavelength 𝜆 + 𝑑𝜆 is given by 𝑎 + 𝑏 𝑠𝑖𝑛 𝜃 + 𝑑𝜃 = 𝑛 𝜆 + 𝑑𝜆 , (2) The equation of minima for wavelength 𝜆 is N 𝑎 + 𝑏 𝑠𝑖𝑛𝜃 = 𝑚𝜆 (3) Here m has all the integral values except 0, N, 2N, … nN, because for these values of m the condition for maxima is satisfied. Thus, first minimum adjacent to nth principal maximum in the direction ( + d) can be obtained by substituting the values of m as (nN + 1) in Eq. (3). Therefore, first minima in the direction ( + d) is given by N 𝑎 + 𝑏 sin(𝜃 + 𝑑) = (𝑛𝑁 + 1)𝜆 (3) (𝑛𝑁+1)𝜆 𝑎 + 𝑏 sin(𝜃 + 𝑑) = 𝑁 𝜆 𝑎 + 𝑏 sin 𝜃 + 𝑑 = 𝑛 + 𝑁 (4) From eq. (2) and eq. (4) 𝜆 𝑛( + 𝑑) = 𝑛 + 𝑁 𝜆 n𝑑 = 𝑁 𝜆 = 𝑛𝑁 𝑑 This is the required expression for the resolving power of the plane diffraction grating. This says that the number of lines per cm of a grating should be larger in order to increase its resolving power.  Newton’s Ring If a plano-convex lens is placed such that its curved surface lies on a glass plate, then an air film of gradually increasing thickness is formed between the two surfaces. When a beam of monochromatic light is allowed to fall normally on this film and viewed as shown in Figure, an alternating dark and bright circular fringes are observed. These circular fringes are formed because of the interference between the reflected waves from the top and the bottom surfaces of the air film. These fringes are circular since the air film has a circular symmetry and the thickness of the film corresponding to each fringe is same throughout the circle. The interference fringes so formed were first investigated by Newton and hence known as Newton’s rings. Newton’s Ring: Experimental Arrangement When a plano-convex lens with its convex surface is placed on a plane glass plate, an air film of gradually increasing thickness is formed between the lens and the glass plate. The thickness of the air film is almost zero at the point of contact O and gradually increases as one proceeds toward the periphery of the lens. If monochromatic light is allowed to fall normally on the lens, and the film is viewed in reflected light, alternately bright and dark concentric rings are seen around the point of contact. If it is viewed with the white light then colored fringes are obtained. The experimental arrangement of Newton’s Ring apparatus is shown in the figure. Consider a parallel beam of monochromatic light incidents normally on the upper surface of the air film. The beam gets partly reflected and partly refracted. The refracted beam in the air film is also reflected partly at the lower surface of the film. The two reflected rays, i.e. produced at the upper and lower surface of the film, are coherent and interfere constructively or destructively. When the light reflected upwards is observed through microscope M which is focused on the glass plate, a pattern of dark and bright concentric rings are observed from the point of contact O. These concentric rings are known as Newton's Rings  K Consider two plane surfaces OM and OM’ inclined at an angle  enclosing a wedge shaped air film of increasing thickness, as shown in figure. L E A beam of monochromatic light is incident on the upper surface of the film and the M’ interference occurs between the rays reflected at its upper and lower surfaces. i A D i r i r+ t The interference occurs between the reflected rays BK and DL, both of which are B r r+ obtained from the same incident ray of light AB.  O C H M The path difference between the two reflected rays  r+  = [BC + CD]in film – [BE]in air I’ I  = (BC + CD) – BE (1) Fig.: Inference in wedge shaped film Since CD = CI  = (BC + CI) – BE (2) From right angled triangle BED 𝐵𝐸 sin i = 𝐵𝐷 (3)  K From right angled triangle BCD L 𝐵𝐶 E sin r = (4) 𝐵𝐷 M’ i Dividing equation (3) by equation (4), we get; A D i r i r+ t sin 𝑖 𝐵𝐸/𝐵𝐷 𝐵𝐸 B = == r r+ sin 𝑟 𝐵𝐶/𝐵𝐷 𝐵𝐶  O C H M BE = BC (5)  r+ Using equation (5), equation (2) becomes I’ I  = (BC + CI) – BC Fig.: Inference in wedge shaped film  = CI (6) From right angled triangle DCI 𝐶𝐼 cos (r + ) = 𝐷𝐼 CI = DI cos (r + ) = 2t cos (r + ) (7) (Since DI = DH + HI = t + t = 2t) Using equation (7), equation (6) becomes K L  = 2t cos (r + ) (8) E M’ As the light gets reflected from top surface of air film (rare medium), there is no i A D phase change. However, when light gets reflected from lower surface of air film i r i (from denser medium), there is a phase shift of  (Stokes phase change). r+ t B Therefore, the total path difference between the reflected rays, r r+  O C H M   = 2t cos (r + ) + (9) 2  r+ Equation (9) shows that the path difference  depends on the thickness t. I’ I However, t is not uniform and it is different at different positions. Fig.: Inference in wedge shaped film At t = 0, equation (9) becomes  = (10) 2 which is the condition for darkness. Therefore, the centre of the rings appears to be dark. This is called zero-order band. For normal incidence, i = 0 and r = 0. Then, the path difference is   = 2t cos  + (11) 2 Since, θ is very small, cos θ = 1. Hence, equation (11) reduces to 𝜆 ∆ = 2t + (12) 2 At the point of contact of the lens and the glass plate (O), the thickness of the film is effectively zero i.e. t = 0 𝝀 ∆= (13) 𝟐 This is the condition for minimum intensity. Hence, the center of Newton's rings generally appears dark. Condition for maxima The constructive interference takes place when  = n where, n = 0, 1, 2, 3,.... Therefore equation (12) becomes  𝑛 = 2𝑡 + 2  2t = (2n − 1) 𝟐 (14) where, n = 0, 1, 2, 3,.... Equation (14) is the condition for maxima in Newton’s rings experiment. Condition for minima In order to get destructive interference, the path difference is 1  = (n + 2 )  Therefore equation (12) becomes 1  (n + 2 )  = 2t + 2. 2t = n (15) where, n = 0, 1, 2, 3,.... Equation (15) is the condition for minima in Newton’s rings experiment. Diameter of Newton’s Ring Diameter of Bright Rings Consider a ring of radius ‘r’ due to the thickness ‘t’ of air film as shown in the figure. According to the property of the circle, the product of intercepts of the intersecting chord is equal to the product of sections of the diameter. Plano Convex lens 𝐷𝐵 × 𝐵𝐸 = 𝐴𝐵 × 𝐵𝐶 (1) 𝑟 × 𝑟 = 𝑡(2𝑅 − 𝑡) (2) t 𝑟 2 = 2𝑅𝑡 − 𝑡 2 (3) Since t is very small, 𝑡 2 is also be a negligible one, thus, 𝑟 2 = 2𝑅𝑡 (4) 1 𝑟2 r 𝑡= (5) 2 2𝑅 nth where R is radius of curvature of plano convex lens. bright ring We know the condition for maxima is 𝜆 2 𝜇𝑡 = (2𝑛 − 1) where 𝑛 =1, 2, 3..... (6) 2 Using equation (5), the equation (6) becomes, 𝑟2 𝜆 2 𝜇( ) = (2𝑛 − 1) (7) 2𝑅 2 Thus the radius of the nth bright ring becomes, 𝜆𝑅 𝑟𝑛2 = (2𝑛 − 1) (8) 2𝜇 Thus the diameter of the nth bright ring is, 𝐷 𝜆𝑅 ( )2 = (2𝑛 − 1) 2 2𝜇 𝜆𝑅 𝐷2 = 2(2𝑛 − 1) 𝜇 𝜆𝑅 (9) 𝐷= 2(2𝑛 − 1) 𝜇  For air medium 𝜇 = 1 and eq. (9) simplifies to 𝐷= 2(2𝑛 − 1)𝜆𝑅 (10) 𝐷∝ (2𝑛 − 1) (11) where 𝑛 =1, 2, 3..... Thus, the diameter of the bright rings is proportional to the square root of the odd natural number. Diameter of Dark Rings: We know that the condition for minima is 2 𝜇𝑡 = 𝑛𝜆 where 𝑛 = 1, 2, 3 …. (12) Using equation (5) in equation (12), we get 𝑟2 2𝜇 = 𝑛𝜆 (13) 2𝑅 The radius of the nth dark ring becomes, 𝑛𝜆𝑅 𝑟𝑛2 = (14) 𝜇 Thus, the diameter of the nth dark ring is, 𝐷𝑛 2 𝑛𝜆𝑅 ( ) = 2 𝜇 4𝑛𝜆𝑅 𝐷2𝑛 = 𝜇 4𝑛𝜆𝑅 where 𝑛 = 1, 2, 3 …. (15) 𝐷𝑛 = 𝜇 For air medium 𝜇 = 1 and equation (15) simplifies to 𝐷𝑛 = 4𝑛𝜆𝑅 where 𝑛 = 1, 2, 3 …. (16) 𝐷𝑛 ∝ 4𝑛𝜆𝑅 Thus diameter of the dark rings is proportional to the square root of the natural numbers. Determination of wavelength of monochromatic light using Newton’s ring experiment The diameter of the nth dark ring is given by, 𝐷𝑛2 = 4𝑛𝜆𝑅 (1) Similarly, the diameter of the (n + p)th dark ring is given by, 2 𝐷𝑛+𝑝 = 4(𝑛 + 𝑝)𝜆𝑅 (2) Subtracting equation (1) from equation (2), we get 2 𝐷𝑛+𝑝 − 𝐷𝑛2 = 4 𝑛 + 𝑝 𝑅 − 4𝑛𝑅 2 𝐷𝑛+𝑝 − 𝐷𝑛2 = 4𝑝𝑅 This implies 𝑫𝟐𝒏+𝒑 − 𝑫𝟐𝒏 𝝀= (3) 𝟒𝒑𝑹 Therefore, the measurement of diameters of the nth and (n + p)th dark fringes together with the radius of curvature of the lens gives us the wavelength of monochromatic light with the help of above formula. Determination of radius of curvature of plano convex lens using Newton’s ring experiment The diameter of the nth dark ring is given by, 𝐷𝑛2 = 4𝑛𝜆𝑅 (1) Similarly, the diameter of the (n + p)th dark ring is given by, 2 𝐷𝑛+𝑝 = 4(𝑛 + 𝑝)𝜆𝑅 (2) Subtracting equation (1) from equation (2), we get 2 𝐷𝑛+𝑝 − 𝐷𝑛2 = 4 𝑛 + 𝑝 𝑅 − 4𝑛𝑅 2 𝐷𝑛+𝑝 − 𝐷𝑛2 = 4𝑝𝑅 This implies 𝑫𝟐𝒏+𝒑 − 𝑫𝟐𝒏 𝑹= (3) 𝟒𝒑𝝀 Therefore, the measurement of diameters of the nth and (n + p)th dark fringes together with the wavelength of monochromatic light gives us the radius of curvature of given plano convex lens with the help of above formula.