Optics and Semiconductor Physics 20PYC01 Past Paper PDF

OPTICS AND SEMICONDUCTOR PHYSICS (20PYC01) COURSE MATERIAL B.E Common to CSE, CSE (AI&ML, IOT & CS, BC Technology), IT AND IT (AIDS) DEPARTMENT OF PHYSICS CHAITANYA BHARATHI INSTITUTE OF TECHNOLOGY (A) GANDIPET, HYDERABAD...

OPTICS AND SEMICONDUCTOR PHYSICS (20PYC01) COURSE MATERIAL B.E Common to CSE, CSE (AI&ML, IOT & CS, BC Technology), IT AND IT (AIDS) DEPARTMENT OF PHYSICS CHAITANYA BHARATHI INSTITUTE OF TECHNOLOGY (A) GANDIPET, HYDERABAD TELANGANA STATE  Unit-I Wave Optics Interference: Huygens’ Principle –Superposition of waves –Interference of light by wave front splitting and amplitude splitting–Fresnel’s biprism – Interference in thin films in reflected light– Newton’s rings Diffraction: Fraunhofer diffraction from a single slit –Double slit diffraction – Rayleigh criterion for limit of resolution–Concept of N-slits–Diffraction grating and its resolving power. Interference Interference of light: The redistribution of energy in accordance with the superposition of two or more coherent light waves. (OR) In simple words, if two or more light waves of same frequency and having constant phase difference travelling along the same direction are allowed to superimpose over each other, there is a modification in the intensity of light in the region of super position is known as interference of light. If sources are said to be coherent, they emit light waves of the same frequency, amplitude and always in phase with each other. This is possible only if two sources are produced from the same part of the source. However, if phase difference changes with time then the source is known as incoherent source. Huygen’s Principle: According to Huygen’s principle of wave theory, each point on the on primary wave front acts as a source for the spherical secondary wavelets and these wavelets move in forward direction with the speed of light. The envelope of these wavelets (i.e. the curve connecting the common tangent of the spherical wave fronts) gives the shape of a new wave front in forward direction. The direction of propagation of rays is called rays and these are normal to the wave front in homogeneous medium. Superposition of waves: The spherical wave fronts from monochromatic source starting from slit 𝑆 falls on slits S1 and S2. The crests and troughs of waves are represented by complete arcs and dotted arcs. The super position of two spherical wave fronts produced interference pattern in the form of alternate bright and dark fringes of equal width on the screen 𝑋𝑌. At points where a crest or trough due to one wave falls on the crest or trough due to another wave the resultant amplitude is the sum of the amplitudes of the individual waves in the absence of the other. As the intensity proportional to square of the amplitude, the resultant intensity at these points increases. This is known as constructive interference. At points where a crest due to one wave falls on the trough due to another wave the resultant amplitude is the difference of the amplitudes of the individual waves. And hence the resultant intensity at these points decreases. This is known as destructive interference. Thus on the screen a number of alternate bright and dark fringes of equal width called interference fringes are observed. An interference phenomenon can be obtained by division of wave front and division of amplitude. Division of wave front: The incident wave front is divided into two parts by utilizing the phenomenon of reflection or diffraction. These two parts of wave front travel unequal distances and reunite to produce interference pattern. Ex. Fresnel biprism and Lloyd’s mirror. Division of amplitude: The amplitude of incoming beam is divided into two parts by utilizing the parallel reflection or refraction. These two parts of wave front travel unequal distances and reunite to produce interference pattern. Ex. Newton’s rings and Michelson Interferometer. The conditions for interference of light are that the two sources should be monochromatic, the two sources derived from the same part of the source, the two sources emitting light waves should be coherent and the two sources must emit light waves of same wavelength and frequency. The light must be continuously emitted. Fresnel Biprism A freshnel biprism is a thin double prism placed base to base and have very small refracting angle ( ). This is equivalent to a single prism with one of its angle nearly 179° and other two of each. Here the interference is observed by the division of wave front. Monochromatic light through a narrow slit S falls on biprism ABC , which divides it into two components. One of these component is refracted from portion AC of biprism and appears to come from S1 where the other one refracted through portion BC appears to come from S2. Thus S1 and S2 act as two virtual coherent sources formed from the original source. Light waves arising from S1 and S2 interfere in the shaded region and interference fringes are formed which can be observed by placing a screen MN. If d is the separation between the virtual sources S1and S2, Z1 is separation between source S and biprism and Z2 is the separation between biprism and the screen then Z1 + Z2 = D and fringe width is The separation ‘d' between the source can be measured experimently and is described in the following section. Fresnel biprism can be used to determine the wavelength of a light source (monochromatic), thickness of a thin transparent sheet/ thin film, refractive index of medium etc. The method of determination is described below. The experiment can easily be done in the laboratory. Interference phenomenon in thin parallel film due to reflected light Consider a thin parallel film of thickness t and of refractive index µ. Let a monochromatic light ray (AB) of wavelength λ strikes the upper surface of the film at point B. Due to division of amplitude, a component of light ray is reflected from the upper surface as BC while the refracted component travels through the films twice (BD and DE) and later emerges as ray EF. The two light rays BC and EF are coherent and the effective path difference between them is obtained to be, Path difference (Δ) = Path travelled in the film – the path travelled in air Δ = µ(BD+DE) – BQ (1) A normal is drawn at B on the surface of the film, and extended backwards as NOP. Similarly the ray DE is extended backwards as EDP. The lines EP and BP will meet at P. A normal is drawn at E to BC and at B to DE. From the figure, ∠ABN = i and ∠NBC = i. So ∠QBE = 90°− i. Hence, ∠BEQ = i [because angle at Q in triangle BQE is 90°] ∠PBD = r and ∠BDE = 2r ; ∠DBE = 90°− r In triangle DBE, ∠BED = 90°− r. Hence in triangle MBE, ∠MBE = r Also in triangle PBE, ∠BPE = r In triangle BPD, the angles at B and P are equal, so OB = OP. The refractive index ‘μ’ in Equation (1) is given as μ = sini/sinr, In triangle BQE, sini = BQ/BE and in triangle BME, sin r = ME/BE and hence, μ = BQ/BE/(ME/BE) = BQ/ME or BQ = μME (2) Substituting Equation (2) in (1), we have Δ = (BD + DE) μ − μ ME = (BD + DE − ME)μ = (PE − ME)μ [since BD = PD] = (PM)μ= μPB cos r =[since cos r = PM / PB] = μ.2t.cos r = 2μt cos r (3) The incident ray AB is reflected on the surface of a denser medium. Hence, it experiences an additional phase change π or path difference λ/2. Δ = 2μtcosr + λ/2 If the path difference is equal to nλ (n = 1, 2, 3 and so on) then the constructive interference occurs and the film appears bright i.e. Δ = 2μtcosr + λ/2 = nλ 2μtcosr = (2n-1)λ/2 If the path difference is equal to (n+1/2)λ (n = 1, 2, 3 and so on) then the constructive interference occurs and the film appears dark i.e. Δ = 2μtcosr + λ/2 = (n+1/2)λ 2μtcosr = nλ Newton’s Rings Experimental arrangement Consider a plano-convex lens of radius of curvature R placed on an optically plane glass plate (P). A thin air film is now enclosed between them in such a way that the thickness of the film gradually increases from the point of contact of lens and glass plate (O) to the edges. Let monochromatic light rays of wavelength λ are incident on the lens normally by using another glass plate inclined at an angle of 45o with the source of light ‘S’. The light rays pass through the lens without refraction and are reflected from the upper and lower surfaces of the enclosed air film of each thickness, producing coherent sources. Due to the interference of these coherent sources, the air film appears in the form of alternate dark and bright concentric rings having center at point ‘O’ which can be observed using a microscope M placed above the lens. Theory The path difference between two reflected rays is, Δ = 2μtcosr +λ/2 (1) For air film µ = 1 and for normal incidence r = 0, then Eq. (1) reduces to, 2t +λ/2 (2) At point of contact t = 0 and the effective path difference is λ/2 which is condition for minimum intensity. Hence the centre of the ring appears dark. Condition for bright ring 2t +λ/2 = nλ or 2t = (2n-1)λ/2 (3) Condition for dark ring 2t +λ/2 = (n+1/2)λ or 2t = nλ (4) Diameter of the Newton’s rings Let R be the radius of curvature and r be the radius of dark ring corresponding to the air film of the thickness t as shown in the figure. From the theorem of intersecting chords of the circle, BA.AP = OA.AD which leads to the condition, r.r = t.(2R-t) → r2 = 2Rt-t2. Since, t ≪ R hence t2 may be neglected in comparison to R and r2 = 2Rt t = r2/(2R) (5) For bright ring: 2t = 2[r2/(2R)] = (2n-1)λ/2 r2 = R(2n-1)(λ/2) D2 = 4R(2n-1) (λ/2) = 2R(2n-1)λ (where r = D/2) ∴ 𝐷𝑛 = 2 √𝑅 𝜆 √2𝑛 − 1 OR 𝐷𝑛 ∝ √2𝑛 − 1 (6) The diameter of bright rings is proportional to square root of odd natural number. For dark ring: 2t = 2[r2/(2R)] = nλ r2 = Rnλ D = 4Rnλ = 2Rnλ (where r = D/2) 2 2 ∴ 𝐷𝑛 = 4𝑅𝑛𝜆 OR 𝐷𝑛 ∝ √𝑛 (7) Diameter of the dark rings is proportional to square root of natural numbers. Applications of Newton’s Rings Experiment i It is used to determine the refractive index of given liquid. ii It is used to find wavelength of monochromatic light. iii It is used to find the radius of curvature of convex lens. iv It is used for the quality control of optical flatness of the surfaces. Determination of wave length of given source of light From the relation for the diameters of dark rings, 2 𝐷𝑚 = 4𝑅𝑚λ (mth order dark) (8) 𝐷𝑛2 = 4𝑅𝑛𝜆 (nth order dark ring) (9) 𝐷𝑛2 − 𝐷𝑚 2 = 4𝑅(𝑛 − 𝑚)𝜆 (10) 𝐷2 −𝐷2 𝑛 𝑚 ∴ 𝜆 = 4𝑅(𝑛−𝑚) (11) Determination of the refractive index of a given liquid Consider a liquid film of refractive index μ placed between the lens and the glass plate. Let 𝐷𝑛′ and 𝐷𝑚 ′ diameters of the 𝑛 th and mth dark rings respectively. 4𝑅𝑛𝜆 4𝑅𝑚𝜆 For the nth and mth order dark rings, 𝐷𝑛′2 = ′2 and 𝐷𝑚 = 𝜇 𝜇 4𝑅(𝑛−𝑚)𝜆 Now, 𝐷𝑛′2 − 𝐷𝑚 ′2 = (12) 𝜇 Using Eq.’s (10) and (12), we can obtain a relation for the refractive index as 𝐷2𝑛 −𝐷2𝑚 𝜇= (13) 𝐷′2 ′2 𝑛 − 𝐷𝑚 The schematic diagram of Newton’s Rings  Diffraction Diffraction refers to bending of the wave when it hits an obstacle whose size is comparable to that of the wavelength of the light incident on it and produces diffraction fringes. Fresnel applied Huygen’s principle of secondary wavelets in conjunction with the principle of interference and calculated the position of fringes. Types of Diffraction Diffraction phenomenon can be classified into the following 1. Fraunhoffer’s Diffraction: In this class of diffraction,the source and the screen or telescope are placed at infinity or effectively at infinity. The wave front incident on the aperture/obstacle is plane. 2. Fresnel’s Diffraction: In this class of diffraction, the source and the screen are placed at finite distances. The incident wave fronts are either spherical or cylindrical. Diffraction due to single slit: Let a plane wave front of monochromatic light of wavelength ⎣ propagating normally to the slit be incident on it. Let the diffracted light be focussed by means of a convex lens on a screen placed in the focal plane of the lens. According to Huygens –Fresnel, every point of the wave front in the plane of the slit is a source of secondary spherical wavelets, which spread out to the right in all directions. The secondary wavelets travelling normally to the slit are brought to the focus at the centre by the lens. The secondary wavelets making an angle with the normal are focussed at any other point other than the central spot. Fig 1: Single-slit diffraction pattern. (a) Monochromatic light passing through a single slit has a central maximum and many smaller and dimmer maxima on either side. The central maximum is six times higher than shown. (b) The diagram shows the bright central maximum, and the dimmer and thinner maxima on either side. ANALYTICAL EXPLANATION Light from the source S is incident as a plane wave front on the slit AB. According to Huygen’s wave theory , every point in AB sends out secondary waves in all directions. The undeviated rays from AB are focused at C on the screen by the lens L2 while the rays diffracted through an angle θ are focused at a point P on the screen. The rays from the ends A and B reach at C in same phase and hence the intensity is maximum. To find the intensity at P let us draw normal AN on BN. Since the path beyond AN is the same therefore the path difference between the extreme rays is =BN = ABsinθ = esinθ [ AB = e ,width of the slit] 2𝜋 or the phase difference = esinθ ------(1) 𝜆 Let AB is divided into a large number n of equal parts then there may be infinitely large number of point sources of secondary wavelets between A and B. The phase difference between any two consecutive parts is therefore 1 2𝜋 esinθ = δ (say) -------(2) 𝑛 𝜆 Thus we have to find the resultant amplitude of n such vibrations each of amplitude a and phase difference δ between successive vibrations. According to the theory of composition of n vibrations each of amplitude a and common phase difference δ between successive vibrations the resultant at P is given by  𝑛δ sin ( ) 2 R=a δ 𝑠𝑖 𝑛 2 𝜋𝑒𝑠𝑖𝑛θ sin ( ) 𝜆 R=a πesinθ -------(3) 𝑠𝑖 𝑛 𝑛 𝜋𝑒𝑠𝑖𝑛θ Let us substitute = α ,then 𝜆 sin (α) R=a α 𝑠𝑖 𝑛 𝑛 α α Since for large value of n ; is very small, therefore sin( ) may be replaced by 𝑛 𝑛 The above equation then takes the form sin (α) sin (α) R=R=a α = na α 𝑛 sin (α) R =A -------(4) α Where A = na The resultant intensity at P is proportional to the square of the amplitude. Taking proportionality constant as unity, the resultant intensity at P is given by sin (α) 2 I = A2 ( ) -----------(5) α Conditions for Maximum and Minima The resultant amplitude as given by equation (4) may be expanded as sin (α) 𝐴 α3 α5 α7 α2 α4 α6 R=A = [α − + − - + ⋯] = A [1 − + − - + ⋯] α α 3! 5! 7! 3! 5! 7! R will be maximum if the negative terms in the above equation vanish. This is possible only when α = 0 or 𝜋𝑒𝑠𝑖𝑛θ = 0 or θ = 0. 𝜆 The resultant intensity at P will be maximum for θ = 0 and called the principal Maxima. Hence the intensity of the principal Maxima = A2. Position of Minima It is clear from equation(5) that the intensity will be minimum when sinα = 0 but α ≠ 𝟎 i.e α = ±mπ where m = 1,2,3…. 𝜋𝑒𝑠𝑖𝑛θ = ±mπ 𝜆 esinθ = ±m𝜆 The values of m =1,2,3…. Gives the directions of first, second, third,……minima Secondary Maxima In diffraction pattern there are secondary maxima in addition to principal maxima. The Condition of secondary maxima may be obtained by differentiating equation(5) with respect to α and equating it to zero. Hence  𝑑𝐼 𝑑 sin(α) 2 sin (α) αcosα−sinα = {A2 ( ) } = A22 [ ]=0 𝑑α 𝑑α α α α2 sin (α) Obviously either = 0 or αcosα − sinα = 0 α α = tanα sin(α) The condition = 0 gives the position of principal maxima except α = 0. α The position of secondary maxima is given by α = tanα ---------(7) The equation (7) can be solved graphically by plotting the curves Y = α and Y = tan α The curve is shown in fig. The equation Y = α gives a straight line passing through the origin and making an angle 45o with X-axis. The equation Y = tan α gives a discontinuous curve. The point of intersection of these two curves gives the value of α satisfying the equation α = tanα. These points correspond to the value of 3π 5π 7π α = 0 ,± ,± , ± …… 2 2 2 The first value α = 0 gives the position of principal maxima while the value of 3π 5π 7π α=± ,± , ± ……gives the position of first secondary maximum, second 2 2 2 secondary maxima, third secondary maxima and so on respectively. The intensity of first secondary maxima is given by 3𝜋 2 sin 2 𝐴2 𝐼𝑂 4 I1 = A ( 3π )2 = = = A2 22 22 9𝜋2 2 where I0 is the intensity of principal maxima. Similarly the intensity of second secondary maxima is given by 5𝜋 2 sin 2 𝐴2 𝐼𝑂 4 I2 = A ( 5π )2 = = = A2 62 62 25𝜋2 2 Thus it is obvious from the values of IO, I1, I2…..etc, that the relative intensities of successive maxima are nearly  4 4 4 1: : : …… 9𝜋2 25𝜋2 49𝜋2 Fig 3:A graph of single-slit diffraction intensity showing the central maximum to be wider and much more intense than those to the sides. Double Slit Diffraction: The diffraction pattern of two slits of width a that are separated by a distance d is the interference pattern of two point sources separated by d multiplied by the diffraction pattern of a slit of width a. In other words, the locations of the interference fringes are given by the equation dsinθ=nλ, the same as when we considered the slits to be point sources, but the intensities of the fringes are now reduced by diffraction effects. Interference and diffraction effects operate simultaneously and generally produce minima at different angles. This gives rise to a complicated pattern on the screen, in which some of the maxima of interference from the two slits are missing if the maximum of the interference is in the same direction as the minimum of the diffraction which are referred to as missing orders. Diffraction at double slit: The double slits have been represented as A1B1 and A2B2 in Fig.. The slits are narrow and rectangular in shape. The plane of the slits is perpendicular to plane of the paper. Let the width of both the slits be equal and it is ‘e’ and they are separated by opaque length ‘d’. A monochromatic plane wave front of wave length ‘λ’ is incident normally on both the slits. Light is made incident on arrangement of double slit. The secondary wavelets travelling in the direction of OP0 are brought to focus at P0 on the screen SS′ by using a converging lens L. P0 corresponds to the position of the central bright maximum. The intensity distribution on the screen is the combined effect of interference of diffracted secondary waves from the slits. The diffracted intensity on the screen is very large along the direction of incident beam [i.e along OP0]. Hence it is maximum at P0. This is known as principal maximum of zero order. The intensity at point P1 on the screen is obtained by applying the Fraunhofer diffraction theory at single slit and interference of diffracted waves from the two slits. The diffracted wave amplitude due to single slit at an angle θ with respect to incident beam is , where 2α is the phase difference between the secondary wavelets arising at the end points of a slit. This phase difference can be estimated as follows: Draw a normal from A1 to B1Q. Now, B1C is the path difference between the diffracted waves at an angle ‘θ’ at the slit A1B1. From the triangle A1B1C The corresponding phase difference..1 The diffracted wave amplitudes at the two slits combine to produce interference. The path difference between the rays coming from corresponding points in the slits A1B1 and A2B2 can be found by drawing a normal from A1 to A2R. A2D is the path difference between the waves from corresponding points of the slits. In the triangle or the path difference A2D = A1A2 sin θ = (e + b) sin θ The corresponding phase difference Applying the theory of interference on the wave amplitudes at the two slits gives the resultant wave amplitude (R). …2 The intensity at P1 is..3 Eqn 3 represents the intensity distribution on the screen. The intensity at any point on the screen depends on α and β. The intensity of central maximum is 4Io. The intensity distribution at different points on the screen can be explained in terms of path difference between the incident and diffracted rays as follows. In eqn (3) the term cos2β corresponds to interference and corresponds to diffraction. Conditions for interference and diffraction maxima and minima: Interference maxima and minima: If the path difference A2D = (e + b) sinθn = ± nλ where n = 1, 2, 3… then ‘θn’ gives the directions of the maxima due to interference of light waves coming from the two slits. The ± sign indicates maxima on both sides with respect to the central maximum. On the other hand if the path difference is odd multiples of λ/2 i.e., then θn gives the directions of minima due to interference of the secondary waves from the two slits on both sides with respect to central maximum. Diffraction maxima and minima: If the path difference B1C = e sinθn = ± nλ, where n = 1, 2, 3… then θn gives the directions of diffraction minima. The ± sign indicates minima on both sides with respect to central maximum. For diffraction maxima is the condition. The ± sign indicates maxima on both sides with respect to central maximum. The intensity distribution on the screen due to double slit diffraction is shown in the following Figure. Fig. (a) represents the graph for interference term, Fig. (b) shows the graph for diffraction term and Fig (c) represents the resultant distribution. Based on the relative values of e and b certain orders of interference maxima are missing in the resultant pattern. The direction of interference maxima are given as (e + b) sin θn = nλ where n = 1, 2, 3, ….. and the directions of diffraction minima are given as e sin θm = mλ where m = 1, 2, 3, … Intensity distribution due to diffraction at double slit. For some values of θn, the values of e and b are satisfied such that at these positions the interference maxima and the diffraction minima are formed. The combined effect results in missing of certain orders of interference maxima. Now we see certain values of e and b for which interference maxima are missing. (i) Let e = b Then, 2 e sinθn = nλ and e sinθm = mλ If m = 1, 2, 3 … then n = 2, 4, 6… i.e., the interference orders 2, 4, 6 … missed in the diffraction pattern (ii) If 2e = b Then 3e sin θm = nλ and e sin θm = mλ if m = 1, 2, 3… Then n = 3, 6, 9… i.e the interference orders 3, 6, 9… are missed in the diffraction pattern (iii) if e + b = e i.e b = 0 the two slits are joined. So, the diffraction pattern is due to a single slit of width 2e.  Unit-II LASERS AND FIBRE OPTICS Lasers: Characteristics of lasers – Einstein’s coefficients – Amplification of light by population inversion - Ruby, Nd-YAG, He-Ne, CO2 and semiconductor laser – Applications of lasers. Fiber Optics: Introduction – Construction – Principle – Propagation of light through an optical fiber – Numerical aperture and acceptance angle – Step-index and graded-index fibers – Pulse dispersion – Fiber losses – Fiber optic communication system – Applications ------------------------------------------------------------------------------------------------------------------------------- -------- Introduction LASER stands for light amplification by stimulated emission of radiation. It is different from conventional light (such as tube light or electric bulb), there is no coordination among different atoms emitting radiation. Laser is a device that emits light (electromagnetic radiation) through a process is called stimulated emission. Characteristics of Laser Light (i). Coherence: Coherence is one of the unique properties of laser light. It arises from the stimulated emission process. Since a common stimulus triggers the emission events which provide the amplified light, the emitted photons are in step and have a definite phase relation to each other. This coherence is described interms of temporal and spatial coherence. (ii). Monochromaticity: A laser beam is more or less in single wave length. I.e. the line width of laser beams is extremely narrow. The wavelengths spread of conventional light sources is usually 1 in 106, where as in case of laser light it will be 1 in 105.I.e. if the frequency of radiation is 1015Hz., then the width of line will be 1 Hz. So, laser radiation is said to be highly monochromatic. The degree of non-monochromaticity has been expressed as ξ =(dλ/λ) =dν/ν, where dλ or dν is the variation in wavelength or variation in frequency of radiation. (iii) Directionality: Laser beam is highly directional because laser emits light only in one direction. It can travel very long distances without divergence. The directionality of a laser beam has been expressed interms of divergence. Suppose r1 and r2 are the radii of laser beam at distances D1 and D2 from a laser, and then we have. Divergence, ∆θ= (r1 - r2)/ D2-D1  The divergence for a laser beam is 0.01mille radian where as incase of search light it is 0.5 radian. (iv) High intensity: In a laser beam lot of energy is concentrated in a small region. This concentration of energy exists both spatially and spectrally, hence there is enormous intensity for laser beam. The power range of laser is about 10-13w for gas laser and is about 109 w for pulsed solid state laser and the diameter of the laser beam is about 1 mm. then the number of photons coming out from a laser per second per unit area is given by Nl=P/ hνπr2≈1022to1034photons/m-2-sec By assuming hν=10-19 Joule, Power P=10-3to109watt r=0.5×10-3meter Based on Planck’s black body radiation, the number of photons emitted per second per unit area by a body with temperature T is given by Nth= (2hπC/ λ4)(1/e(hν/kT)-1) dλ≈1016photons/m2.sec By assuming T=1000k, λ=6000A0 This comparison shows that laser is a highly intensive beam. Spontaneous and stimulated emission In lasers, the interaction between matter and light is of three different types. They are: absorption, spontaneous emission and stimulates emission.Let E1 and E2 be ground and excited states of an atom. The dot represents an atom. Transition between these states involves absorption and emission of a photon of energy E2-E1=hν12. Where ‘h’ is Planck’s constant. Absorption: As shown in fig8.1(a), if a photon of energy hν12(E2-E1) collides with an atom present in the ground state of energy E1 then the atom completely absorbs the incident photon and makes transition to excited state E2. Spontaneous emission:As shown in fig8. 1. (b), an atom initially present in the excited state makes transition voluntarily on its own. Without any aid of external stimulus or an agency to the ground. State and emits a photon of energy hν12(=E2-E1).this is called spontaneous emission. These are incoherent. Stimulated emission:As shown in fig8.1.(c), a photon having energy hν12(E2-E1)impinges on an atom present in the excited state and the atom is stimulated to make transition to the ground state and gives off a photon of energy hν12. The emitted photon is in phase with the incident photon. These are coherent. This type of emission is known as stimulated emission. 2  (a) Absorption ;(b) Spontaneous emission;(c) Stimulated emission Differences between Spontaneous emission and stimulated emission of radiation Spontaneous emission Stimulated emission 1. Polychromatic radiation 1. Monochromatic radiation 2. Less intensity 2. High intensity 3. Less directionality, more angular spread 3. High directionality, so less angular spread during propagation during propagation. 4. Spatially and temporally in coherent 4. Specially and temporally coherent radiation radiation. 5. Spontaneous emission takes place when 5. Stimulated emission takes place when a excited atoms make a transition to lower photon of energy equal to h ν12(=E2- energy level voluntarily without any E1)stimulates an excited atom to make external stimulation. transition to lower energy level. Population inversion Usually in a system the number of atoms (N1) present in the ground state (E1) is larger than the number of atoms(N2) present in the higher energy state. The process of making N2>N1 called population inversion. Conditions for population inversion are: 3 a) The system should posses at least a pair of energy levels(E2>E1), separated by an energy of equal to the energy of a photon (hν). b) There should be a continuous supply of energy to the system such that the atoms must be raised continuously to the excited state. Population inversion can be achieved by a number of ways. Some of them are (i) optical pumping (ii) electrical discharge (iii) inelastic collision of atoms (iv) chemical reaction and (v) direct conversion Metastable state: The excited states (particularly electronic states in laser gain media) which have a relatively long lifetime due to slow radiative and non-radiative decay are called Metastable states. Pumping: For maintaining a state of population inversion atoms have to be raised continuously to excited state. It requires energy to be supplied to the system. The process of supplying energy to the medium with a view to transfer it into state of population inversion is known as pumping. Commonly used pumping types are: — 1. Optical pumping 2. Electric discharge 3. Atom-Atom collision 4. Direct conversion 5. Chemical reactions 6. Injection current Lasing action: After completion of lifetime of electrons in the Meta stable state, they fall back to the lower energy state or ground state by releasing energy in the form of photons. This process of emission of photons is called spontaneous emission. When this emitted photon interacts with the electron in the Meta stable state, it forces that electron to fall back to the ground state. As a result, two photons are emitted. This process of emission of photons is called stimulated emission. 4 When these photons again interacted with the electrons in the Meta stable state, they forces two Meta stable state electrons to fall back to the ground state. As a result, four photons are emitted. Likewise, a large number of photons are emitted. As a result, millions of photons are emitted by using small number of photons. Einstein’s Coefficients We know that, when light is absorbed by the atoms or molecules, then it goes from the lower energy level (E1) to the higher energy level (E2) and during the transition from higher energy level (E2) to lower energy level (E1) the light is emitted from the atoms or molecules. Let us consider an atom exposed to light photons of energy E2 -E1= hv , three distinct processes take place. a. Absorption b. Spontaneous emission c. Stimulated Emission Absorption: An atom in the lower energy level or ground state energy level E1 absorbs the incident photon radiation of energy hv and goes to the higher energy level or excited level E2 as shown in figure. This process is called absorption If there are many numbers of atoms in the ground state then each atom will absorb the energy from the incident photon and goes to the excited state. The probability that the number of atoms in state m absorb a photon and rise to state n in unit time is given by P m→n = B m→n u(ν) Where Nm - Number of atoms in the lower state m P m→n - Probability of absorption u(ν) - Energy density B m→n - Proportionality constant - Einstein’s coefficient of absorption Therefore, the number of atoms in state m absorb a photon and rise to state n in unit time is given by Nm P m→n = Nm B m→n u(ν) Normally, the atoms in the excited state will not stay there for a long period of time , rather it comes to ground state by emitting a photon of energy E=hν Such an emission takes place by one of the following two methods. 5 Spontaneous Emission: The atom in the excited state returns to the ground state by emitting a photon of energy E=(E2-E1) = hν Spontaneously without any external triggering as shown in the figure. This process is known as spontaneous emission. Such an emission is random and is independent of incident radiation. Stimulated Emission: The atom in the excited state can also return to the ground state by external triggering or inducement of photon thereby emitting a photon of energy equal to the energy of the incident photon, known as stimulated emission. Thus, results in two photons of same energy, phase difference and of same directionality as shown. The probability of emission is sum of two parts, one which is independent of the radiation ' density and the other proportional to it. The probability that the number of atoms in state n that drop to m, either spontaneously or under stimulation, emitting a photon per unit time is P n→m = A n→m + B n→m u(ν) Where A n→m - Einstein’s coefficient of spontaneous emission B n→m -Einstein’s coefficient of induced emission The number of atoms in state n that drop to m, either spontaneously or under stimulation, emitting a photon per unit time is Nn P n→m = Nn [ A n→m + B n→m u(ν) ] In thermal equilibrium, emission and absorption must balance Nm B m→n u(ν) = Nn [ A n→m + B n→m u(ν) ] Nm B m→n u(ν) - Nn B n→m u(ν) = Nn A n→m u(ν) (Nm B m→n - Nn B n→m ) = Nn A n→m u(ν) = (Nn A n→m) / (Nm B m→n - Nn B n→m ) on dividing with Nn B n→m u(ν) = (A n→m / B n→m) / { (Nm/Nn) (B m→n / B n→m ) – 1 } But, It was proved by Einstein that B m→n = B n→m u(ν) = (A n→m / B n→m) / { (Nm/Nn) – 1 } According to the Boltzmann’s law Nm / Nn = exp[(En – Em) / kT ] = exp(hν/kT) 6  u(ν) = [A n→m / B n→m] / [exp(hν/kT) – 1] But, According to Planck’s radiation formula u(ν) = 8 Π h ν3 / c3 [exp(hν/kT) – 1] A n→m / B n→m = 8 Π h ν3 / c3 This is the formula for the ratio between the spontaneous emission and induced emission coefficients. The ratio is proportional to ν3. This shows that the probability of spontaneous emission increases rapidly with the energy difference between two states. Ruby Laser Ruby Laser is a solid state pulsed, three level lasers. It consists of a cylindrical shaped ruby crystal rod of length varying from 2 to 20cms and diameter varying 0.1 to 2cms. This end faces of the rod are highly flat and parallel. One of the faces is highly silvered and the other face is partially silvered so that it transmits 10 to 25% of incident light and reflects the rest so as to make the rod-resonant cavity. Basically, ruby crystal is aluminum oxide [Al 2O3] doped with 0.05 to 0.5% of chromium atom. These chromium atoms serve as activators. Due to presence of 0.05% of chromium, the ruby crystal appears in pink color. The ruby crystal is placed along the axis of a helical xenon or krypton flash lamp of high intensity. Ruby laser 7  Energy level diagram of chromium ions in a ruby crystal Construction: Ruby (Al2O3+Cr2O3) is a crystal of Aluminum oxide in which some of Al+3 ions are replaced by Cr +3 ions.When the doping concentration of Cr+3 is about 0.05%, the color of the rod becomes pink. The active medium in ruby rod is Cr+3ions. In ruby laser a rod of 4cm long and 5mm diameter is used and the ends of the rod are highly polished. Both ends are silvered such that one end is fully reflecting and the other end is partially reflecting. The ruby rod is surrounded by helical xenon flash lamp tube which provides the optical pumping to raise the Chromium ions to upper energy level (rather energy band). The xenon flash lamp tube which emits intense pulses lasts only few milliseconds and the tube consumes several thousands of joules of energy. Only a part of this energy is used in pumping Chromium ions while the rest goes as heat to the apparatus which should be cooled with cooling arrangements as shown in fig.2.5. The energy level diagram of ruby laser is shown in fig.2.6 Working: Ruby crystal is made up of aluminum oxide as host lattice with small percentage of Chromium ions replacing aluminum ions in the crystal chromium acts as do pant. A do pant actually produces lasing action while the host material sustains this action. The pumping source for ruby material is xenon flash lamp which will be operated by some external power supply. Chromium ions will respond to this flash light having wavelength of 5600A0. When the Cr +3ions are excited to energy level E3 from E1 the population in E3 increases. Chromium ions stay here for a very short time of the order of 10-8 seconds then they drop to the level E2 which is mat stable state of life time 10-3s. Here the level E3is rather a band, which helps the pumping to be more effective. The transitions from E3toE2 are non-radioactive in nature. During this process heat is given to crystal lattice. Hence cooling the rod is an essential feature in this method. The life time in mete stable state is 10 5times greater than the lifetime in E3. As the life of the state E2 is much longer, the number of ions in this state goes on increasing while ions. In this state goes on increasing while in the ground state (E1)goes on decreasing. By this process population inversion is achieved between the exited Meta stable state E2 and the ground state E1. When an 8 excited ion passes spontaneously from the metastable state E2 to the ground state E1, it emits a photon of wave length 6943A0. This photon travels through the rod and if it is moving parallel to the axis of the crystal, is reflected back and forth by the silvered ends until it stimulates an excited ion in E2 and causes it to emit fresh photon in phase with the earlier photon. This stimulated transition triggers the laser transition. This process is repeated again and again because the photons repeatedly move along the crystal being reflected from its ends. The photons thus get multiplied. When thephoton beam becomes sufficiently intense, such that part of it emerges through the partially silvered end of the crystal. Drawbacks of ruby laser: 1. The laser requires high pumping power to achieve population inversion. 2. It is a pulsed laser.. Output pulses with time. Nd: YAG laser It is another solid state laser made up of yttrium alluminium garnet. Y3Al5O12. A few of Y+3 Ions of replaced with Nd+3 ions. Hence the active medium is Neodium ions. Yttrium alluminium garnet Is the supporting medium. It is a four level laser. 9  In the setup to produce laser the medium is taken in the form of rod equal to integral multiples wavelength of laser. The pumping sources is a flash tube. Again the ends of the rod are property polished. The energy level diagram is is as shown in the figure. When Nd+3 iron are excited they jump into to upper energy bands E4 or E5 either with the absorption of 0.73 micro m or 0.8 micro m. there onwards the jump to to lower metastable state E3, then to to another lower level with the emission of laser. the another transition is to the ground state. it is a radiation less transition. Laser emission takes place with a wavelength 1.06 micro m, which falls in IR region. it emits both pulsed and continuous waves. Helium-Neon gas laser Helium-Neon gas laser is a continuous four level gas laser. It consists of a long, narrow cylindrical tube made up of fused quartz. The diameter of the tube will vary from 2 to 8 mm and length will vary from 10 to 100 cm. The tube is filled with helium and neon gases in the ratio of 10:1. The partial pressure of helium gas is 1mm of Hg and neon gas is 0.1mm of Hg so that the pressure of the mixture of gases inside the tube is nearly 1 mm of Hg. Laser action is due to the neon atoms.Helium is used for selective pumping of neon atoms to upper energy levels. Two electrodes are fixed near the ends of the tube to pass electric discharge through the gas. Two optically plane mirrors are fixed at the two ends of the tube at Brewster angle normal to its axis. One of the mirrors is fully silvered so that nearly 10 100%reflection takes place and the other is partially silvered so that 1%of the light incident on it will be transmitted. Optical resources column is formed between these mirrors.. Helium-Neon gas laser Working When a discharge is passed through the gaseous mixture, electrons are accelerated down the tube. These accelerated electrons collide with the helium atoms and excite them to higher energy levels. The different energy levels of Helium atoms and Neon atoms is shown in fig. The helium atoms are excited to the levels F2 and F3 these levels happen to be metastable energy states. Energy levels and hence Helium atoms exited levels spend sufficiently large amount of time before getting de excited. As shown in the fig, some of the excited states of neon can correspond approximately to the same energy of excited levels F2 and F3. Thus, when Helium atoms in level F2 and F3 collide with Neon atoms in the ground level E1, an energy exchange takes place. This results in the excitation of Neon atoms to the levels E4 and E6and de excitation of Helium atoms to the ground level (F1). Because of long life times of the atoms in levels F2 and F3, this process of energy transfer has a high probability. Thus the discharge through the gas mixture continuously populates the neon atoms in the excited energy levels E4 and E6. This helps to create a state of population inversion between the levels E4 (E6) to the lower energy level (E3 and E5). The various transitions E6→E5, E4→E3, E6→E3 leads to the emission of wave lengths 3.39mm, 1.15 um and 6328 A0. Specific frequency selection may be obtained by employing mirrors The excited Neon atoms drop down from the level E3 to the E2 by spontaneously emitting a photon around wavelength 6000A0. The pressures of the two gases in the mixture are so chosen that there is an effective transfer of energy from the Helium to the Neon atoms. Since the level E2 is a meta stable state, there is a finite probability of the excitation of Neon, atoms from E2 to E3 leading to population inversion, when a narrow tube is used, the neon atoms in the level E2 collide with the walls of the tube and get excited to the level E1. The transition from E5 to E3 may be non radioactive. The typical power outputs of He-Ne laser lie between 1 and 50 mw of continuous wave for inputs of 5-10W. 11  Energy level diagram of He-Ne atoms. Carbon dioxide laser it is a gas laser and four level laser. it also emits laser in IR region 10.6 micro m. Just like Helium neon gas laser in this laser also there is mixture of carbon dioxide helium and Nitrogen gas. the real active medium is carbon dioxide and the other two gases work as supporting media. the Mixture is in the ratio 1:1:8 for Carbon dioxide nitrogen and Helium gases. This laser works based on transition of molecular energy levels. They are both rotational and vibrational energy levels. carbon dioxide molecule has three types of Vibrational modes. it is a linear molecule as shown in the figure. The three modes are symmetrical, asymmetrical and bending modes. 12 During the symmetric mode vibration of oxygen atoms takes place symmetrically towards are away from carbon atom at a time where am I in asymmetric Mode vibration off oxygen atoms take place in the same direction. it means one atom moves towards carbon atom and the other one moves away from the carbon atom. Bending takes place at some angle. All these vibrations occur at different frequency. Lasing action takes place when the carbon dioxide molecule makes transition from higher asymmetric mode to the other two modes of vibrations. in this laser also nitrogen molecules transfer energy to carbon dioxide molecules so as to maintain the metastable state in asymmetric mode as shown in the figure. From the asymmetric mode based on the transition there will be emission of laser in the region with wavelength 9.4 and 10.6 micrometre. The Other downward transitions are non -radiative. 13 As in the case of helium neon laser here also the mixture of gases will be taken inside a glass tube. the ends of the glass tube are fixed with reflecting and semi reflecting mirrors. Again when active centres of carbon dioxide are excited by means of electric discharge through the mixture then laser emission takes place through semi silvered glass plate. Semiconductor laser It is specifically fabricated p-n junction diode. This diode emits laser light when it is forward biased. When a p-n junction diode is forward biased, the electrons from n – region and the holes from the p- region cross the junction and recombine with each other. During the recombination process, the light radiation (photons) is released from a certain specified direct band gap semiconductors like Ga-As. This light radiation is known as recombination radiation. The photon emitted during recombination stimulates other electrons and holes to recombine. As a result, stimulated emission takes place which produces laser. 14 Construction Figure shows the basic construction of semiconductor laser. The active medium is a p-n junction diode made from the single crystal of gallium arsenide. This crystal is cut in the form of a platter having thickness of 0.5μmm. The platelet consists of two parts having an electron conductivity (n- type) and hole conductivity (p-type). The photon emission is stimulated in a very thin layer of PN junction (in order of few microns). The electrical voltage is applied to the crystal through the electrode fixed on the upper surface. The end faces of the junction diode are well polished and parallel to each other. They act as an optical resonator through which the emitted light comes out. Working Figure shows the energy level diagram of semiconductor laser. When the PN junction is forward biased with large applied voltage, the electrons and holes are injected into junction region in considerable concentration The region around the junction contains a large amount of electrons in the conduction band and a large amount of holes in the valence band. If the population density is high, a condition of population inversion is achieved. The electrons and holes recombine with each other and this recombination’s produce radiation in the form of light. When the forward – biased voltage is increased, more and more light photons are emitted and the light production instantly becomes stronger. These photons will trigger a chain of stimulated recombination resulting in the release of photons in phase. The photons moving at the plane of the junction travels back and forth by reflection between two sides placed parallel and opposite to each other and grow in strength. 15 After gaining enough strength, it gives out the laser beam of wavelength 8400o A. The wavelength of laser light is given by Where Eg is the band gap energy in joule. Type: It is a solid state semiconductor laser. Active medium: A PN junction diode made from single crystal of gallium arsenide is used as an active medium. Pumping method: The direct conversion method is used for pumping action Power output: The power output from this laser is 1mW. Nature of output: The nature of output is continuous wave or pulsed output. Wavelength of Output: gallium arsenide laser gives infrared radiation in the wavelength 8300 to 8500o A. It is very small in dimension. The arrangement is simple and compact. It exhibits high efficiency. The laser output can be easily increased by controlling the junction current. It is operated with lesser power than ruby and CO2 laser. It requires very little auxiliary equipment. It can have a continuous wave output or pulsed output. It is difficult to control the mode pattern and mode structure of laser. The output is usually from 5 degree to 15 degree i.e., laser beam has large divergence. The purity and monochromacity are power than other types of laser. Threshold current density is very large (400A/mm2). It has poor coherence and poor stability. Holography In the normal photography two dimensional details of plain will be recorded. in the normal photography only intensity variations will be recorded whereas in the holography it is a three dimensional photography in which phase of the scattered waves by the object also will be recorded. Holography means the whole writing according to Latin. In the holography there are two steps 1. Recording of hologram 2. Reconstruction of image from the hologram 16 Recording of the hologram Recording of the Hologram explains by means of laser. the laser beam is split into two parts. 1 is allowed to strike the photographic plate directly and the other reaches the photographic plate by the reflection of object for which Hologram has to be developed as shown in the figure. Reconstruction of image To reconstruct the image from the hologram The same blazer use it for recording will be used for the purpose. when the laser is allowed to pass through the hologram on the other side of the hologram you can see the formation of real image. virtual image also appears when observed through the hologram on the other side as shown in the figure. 17  Fiber optics Introduction 1. An optical fiber (or fiber) is a glass or plastic fiber that carries light along its length. 2. Fiber optics is the overlap of applied science and engineering concerned with the design and application of optical fibers. 3. Optical fibers are widely used in fiber-optic communications, which permits transmission over long distances and at higher band widths (data rates) than other forms of communications. 4. Specially designed fibers are used for a variety of other applications, including sensors and fiber lasers. Fiber optics, though used extensively in the modern world, is a fairly simple and old technology. Construction of Optical Fiber Optical fiber is a cylinder of transparent dielectric medium and designed to guide visible and infrared light over long distances. Optical fibers work on the principle of total internalreflection. Optical fiber is very thin and flexible medium having a cylindrical shape consisting of three sections 1) The core material 2) The cladding material 3) The outer jacket The structure of an optical is shown in figure. The fiber has a core surrounded by a cladding material whose reflective index is slightly less than that of the core material to satisfy the condition for total internal reflection. To protect the fiber material and also to give mechanical support there is a protective cover called outer jacket. In order to avoid damages there will be some cushion between cladding protective cover. Structure of an optical fiber 18 Principle of Optical Fiber When a ray of light passes from an optically denser medium into an optically rarer medium, the refracted ray bends away from the normal. When the angle of incidence is increased angle of refraction also increases and a stage is reached when the refracted ray just grazes the surface of separation of core and cladding. At this position the angle of refraction is 90 degrees. This angle of incidence in the denser medium is called the critical angle (θc) of the denser medium with respect to the rarer medium and is shown in the fig. If the angle of incidence is further increased then light is totally reflected. This is called total internal reflection. Let the reflective indices of core and cladding materials be n1 and n2 respectively. Total internal reflection. When a light ray travelling from an optically denser medium into an optically rarer medium, is incident at angle greater than the critical angle for the two media, then the ray is totally reflected back into the medium by obeying the loss of reflection. This phenomenon is known as totally internal reflection. According to law of refraction, n1 sinθ1= n 2 sinθ2 Here θ1=θc, θ2=90 n 1 sinθc=n2 sin 90 𝑛 Sinθc = 𝑛2 1 𝑛 θc = sinˉ¹(𝑛2 )→ (1) 1 Equation (1) is the expression for condition for total internal reflection. In case of total internal reflection, there is absolutely no absorption of light energy at the reflecting surface. Since the entire incident light energy is returned along the reflected light it is called total internal 19  reflection. As there is no loss of light energy during reflection, hence optical fibers are designed to guide light wave over very long distances. Acceptance Angle and Acceptance Cone Acceptance angle:Itis the angle at which we have to launch the beam at its end to enable the entire light to propagate through the core. Fig.8.12 shows longitudinal cross section of the launch of a fiber with a ray entering it. The light is entered from a medium of refractive index n0 (for air n0=1) into the core of refractive index n1. The ray (OA) enters with an angle of incidence to the fiber end face i.e. the incident ray makes angle with the fiber axis which is nothing but the normal to the end face of the core. Let a right ray OA enters the fiber at an angle to the axis of the fiber. The end at which light enter the fiber is called the launching pad. Path of atypical light ray launched into fiber. Let the refractive index of the core be n1 and the refractive index of cladding be n2. Here n1>n2. The light ray reflects at an angle and strikes the core cladding interface at angle θ. If the angleθ is greater than its critical angle θc, the light ray undergoes total internal reflection at the interface. According to Snell’s law n0sinαi=n1sinαr → (2) From the right angled triangle ABC αr+θ=900 αr=900 –θ → (3) Substituting (3) in (2), we get n0sinαi =n1sin (900 –θ) = n1cos θ 𝑛 sinαi=(𝑛1 ) cos θ →(4) 0 When θ= θc, αi= αm=maximum α value 𝑛 sinαm=(𝑛1 ) cos θc →(5) 0 20  𝑛 From equation (1) Sinθc= 𝑛2 1 𝑛 √𝑛1 2 −𝑛2 2 cos θc=√1 − 𝑠𝑖𝑛2 θc =√1 − (𝑛2 )2 = →(6) 1 𝑛1 Substitute equation (6) in equation (5) 𝑛 √𝑛1 2 −𝑛2 2 √𝑛1 2 −𝑛2 2 sinαm=(𝑛1 ) = →(7) 0 𝑛1 𝑛0 If the medium surrounding fiber is air, then n0=1 sinαm=√𝑛1 2 − 𝑛2 2→(8) This maximum angle is called the acceptance angle or the acceptance cone half angle of the fiber. The acceptance angle may be defined as the maximum angle that a light ray can have with the axis of the fiber and propagate through the fiber. Rotating the acceptance angle about the fiber axis (fig.) describes the acceptance cone of the fiber. Light launched at the fiber end within this acceptance cone alone will be accepted and propagated to the other end of the fiber by total internal reflection. Larger acceptance angles make launching easier. Light gathering capacity of the fiber is expressed in terms of maximum acceptance angle and is termed as “Numerical Aperture”. Acceptance cone Numerical Aperture Numerical Aperture of a fiber is measure of its light gathering power. The numerical aperture (NA) is defined as the sign of the maximum acceptance angle. Numerical aperture (NA)= sinαm=√𝑛1 2 − 𝑛2 2 →(9) = √(𝑛1 − 𝑛2 ) (𝑛1 + 𝑛2 ) = √((𝑛1 + 𝑛2 ) 𝑛1 ∆) → (10) 21  (𝑛1 −𝑛2 ) Where ∆= called as fractional differences in refractive indices 𝑛 and 𝑛2 are the 𝑛1 refractive indices of core and cladding material respectively. As n1≈ n2, we can take n1+ n2= 2n1 Then numerical aperture= (2n12∆) 1/2= n1 (2∆) 1/2→ (11) Numerical aperture is a measure of amount of light that can be accepted by a fiber. From equation (9) it is seen that numerical aperture depends only on the refractive indices of core and cladding materials and it is independent on the fiber dimensions. Its value ranges from 0.1 to 0.5. A large NA means that the fiber will accept large amount of light from the source. Classification optical fibers Step index fibers and graded index fibers Based on the variation of refractive index of core, optical fibers are divided into: (1) step index and (2) graded index fibers. Again based on the mode of propagation, all these fibers are divided into: (1) single mode and (2) multimode fibers. In all optical fibers, the refractive index of cladding material is uniform. Now, we will see the construction, refractive index of core and cladding with radial distance of fiber, ray propagation and applications of above optical fibers. i.Step index fiber: The refractive index is uniform throughout the core of this fiber. As we go radially in this fiber, the refractive index undergoes a step change at the core-cladding interface. Based on the mode of propagation of light rays, step index fibers are of 2 types: a) single mode step index fiber & b) multimode step index fibers. Mode means, the number of paths available for light propagation of fiber. Single mode step index fiber: The core diameter of this fiber is about 8 to 10µm and outer diameter of cladding is 60 to 70 µm. There is only one path for ray propagation. So, it is called single mode fiber. The cross sectional view, refractive index profile and ray propagation are shown in fig. (i). In this fiber, the transmission of light is by successive total internal reflections i.e. it is a reflective type fiber. Nearly 80% of the fibers manufactured today in the world are single mode fibers. So, they are extensively used. Single mode step index fiber ;( a) Cross sectional view and refractive index profile ;( b) Ray propagation 22 Multimode step index fiber: The construction of multimode step index fiber is similar to single mode step index fiber except that its core and cladding diameters are much larger to have many paths for light propagation. The core diameter of this fiber varies from 50 to 200 µm and the outer diameter of cladding varies from 100 to 250 µm. The cross- sectional view, refractive index profile and ray propagations are shown in fig. Light propagation in this fiber is by multiple total internal reflections i.e it is a reflective type fiber. Transmission of signal in step index fiber: Generally the signal is transmitted through the fiber in digital form i.e. in the form of 1’s and 0’s. The propagation of pulses through the multimode fiber is shown in fig. The pulse which travels along path 1(straight) will reach first at the other end of fiber. Next the pulse that travels along with path 2(zig-zag) reaches the other end. Hence, the pulsed signal received at the other end is broadened. This is known as intermodal dispersion. This imposes limitation on the separation between pulses and reduces the transmission rate and capacity. To overcome this problem, graded index fibers are used. 2) Graded index fiber: In this fiber, the refractive index decreases continuously from center radially to the surface of the core. The refractive index is maximum at the center and minimum at the surface of core. This fiber can be single mode or multimode fiber. The cross sectional view, refractive index profile and ray propagation of multimode graded index fiber are shown in fig. The diameter of core varies from 50 to 200µm and outer diameter of cladding varies from 100 to 250 µm. The refractive index profile is circularly symmetric. As refractive index changes continuously radially in core, light rays suffer continuous refraction in core. The propagation of light ray is not due to total internal reflection but by refraction as shown in fig. In graded index fiber, light rays travel at different speed in different paths of the fiber. Near the surface of the core, the refractive index is lower, so rays near the outer surface travel faster than the rays travel at the center. Because of this, all the rays arrive at the receiving end of the fiber approximately at the same time. This fiber is costly. 23 Transmission of signal graded index fiber: In multimode graded index fiber, large number of paths is available for light ray propagation. To discuss about inter modal dispersion, we consider ray path 1 along the axis of fiber. Fig. (ii).Multimode step index fibre(a)Cross sectional view and refractive index profle(b)Ray propagation As shown in fig. (ii)(b) and another ray path 2. Along with the axis of fiber, the refractive index of core is maximum, so the speed of ray along path 1 is less. Path 2 is sinusoidal and it is longer, along this path refractive index varies. The ray mostly travels in low refractive region, so the ray 2 moves slightly faster. Hence, the pulses of signals that travel along path 1 and path 2 reach other end of fiber simultaneously. Thus, the problem of intermodal dispersion can be reduced to a large extent using graded index fibers. 24 Pulse Dispersion Consider a ray of light OA be incident at an angle Ѳi on the entrance aperture of the fibre as shown in fig. The ray is refracted into the core along AB and makes an angle Ѳr with the axis of the core. Now, the ray strikes at the upper-core cladding at B. After this the ray is totally internally reflected back inside the core. Further it strikes at point C of lower cladding and after reflection it again strikes the upper interface at D. Let t be the time taken by the light ray to cover the distance B to C and then from C to D with the velocity v. Then 𝐵𝐶+𝐶𝐷 𝑡= (1) 𝑣 If n1 be the refractive index of core and c is the speed of light in vacuum, then 25  𝑐 𝑛1 = 𝑣 𝑐 𝑣= (2) 𝑛1 From the ΔBCG (figure), 𝐵𝐺 𝐶𝑜𝑠 𝜃𝑟 = 𝐵𝐶 𝐵𝐺 𝐵𝐶 = 𝐶𝑜𝑠 𝜃𝑟 Similarly 𝐺𝐷 𝐶𝐷 = 𝐶𝑜𝑠 𝜃𝑟 𝐵𝐺+𝐺𝐷 𝑙 𝐵𝐶 + 𝐶𝐷 = = (3) 𝐶𝑜𝑠 𝜃𝑟 𝐶𝑜𝑠 𝜃𝑟 Substituting 2 and 3 in equation 1, 𝑙 𝑛1 𝑡= x 𝐶𝑜𝑠 𝜃𝑟 𝑐 The time taken by the ray in traversing an axial length “l” of the fibre after a series of total internal reflections is 𝑛1 𝑙 𝜏= 𝑐 𝑐𝑜𝑠 𝜃𝑟 All the light rays lying between angle 0 and critical angle 𝜃𝑐 are present. The time taking by the rays making 0 angle with the axis of the fibre will be minimum and is given by 𝑛1 𝑙 𝜏𝑚𝑖𝑛 = 𝑐 The maximum time is given by 𝑛1 𝑙 𝜏𝑚𝑎𝑥 = 𝑐 𝑐𝑜𝑠 𝜃𝑐 But cos 𝜃𝑐 = 𝑛2 /𝑛1 𝑛12 𝑙 𝜏𝑚𝑎𝑥 = 𝑐 𝑛2 𝑛12 𝑙 𝑛1 𝑙 ∆𝜏 = 𝜏𝑚𝑎𝑥 − 𝜏𝑚𝑖𝑛 = − 𝑐 𝑛2 𝑐 26  𝑛1 𝑙 𝑛1 ∆𝜏 = ( − 1) 𝑐 𝑛2 𝑛1 𝑙 ∆𝜏 = ∆ 𝑐 𝑛1 − 𝑛2 𝑎𝑠 𝑡ℎ𝑒 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑟𝑒𝑓𝑟𝑎𝑐𝑡𝑖𝑣𝑒 𝑖𝑛𝑑𝑒𝑥 𝑖𝑠 ∆ = ( ) 𝑛2 But the numerical aperture is given by 𝑁. 𝐴. = 𝑛1 √2 ∆ 𝑙 (𝑁𝐴)2 So in terms of numerical aperture the delay difference ∆𝜏 = 2 𝑛1 𝑐 Fibre losses Fiber optic transmission has various advantages over other transmission methods like copper or radio transmission. Fiber optic cable which is lighter, smaller and more flexible than copper can transmit signals with faster speed over longer distance. However, many factors can influence the performance of fiber optic. To ensure the nice and stable performance of the fiber optic, many issues are to be considered. Fiber optic loss is a negligible issue among them, and it has been a top priority for many engineers to consider during selecting and handling fiber optic. When a beam of light carrying signals travels through the core of fiber optic, the strength of the light will become lower. Thus, the signal strength becomes weaker. This loss of light power is generally called fiber optic loss or attenuation. This decrease in power level is described in dB. During the transmission, something happened and causes the fiber optic loss. To transmit optical signals smoothly and safely, fiber optic loss must be decreased. The cause of fiber optic loss located on two aspects: internal reasons and external causes of fiber optic, which are also known as intrinsic fiber core attenuation and extrinsic fiber attenuation. Intrinsic Fiber Core Attenuation Internal reasons of fiber optic loss caused by the fiber optic itself, which is also usually called intrinsic attenuation. There are two main causes of intrinsic attenuation. One is light absorption and the other one is scattering. Light absorption is a major cause of losses in optical fiber during optical transmission. The light is absorbed in the fiber by the materials of fiber optic. Thus light absorption in optical fiber is also known as material absorption. Actually the light power is absorbed and transferred into other forms of energy like heat, due to molecular resonance and wavelength impurities. Atomic structure is in any pure material and they absorb selective wavelengths of radiation. It is impossible to manufacture materials that are total pure. Thus, fiber optic manufacturers choose to dope germanium and other materials with pure silica to optimize the fiber optic core performance. 27 Scattering is another major cause for losses in optical fiber. It refers to the scattering of light caused by molecular level irregularities in the glass structure. When the scattering happens, the light energy is scattered in all direction. Some of them is keeping traveling in the forward direction. And the light not scattered in the forward direction will be lost in the fiber optic link as shown in the following picture. Thus, to reduce fiber optic loss caused by scattering, the imperfections of the fiber optic core should be removed, and the fiber optic coating and extrusion should be carefully controlled. Extrinsic Fiber Attenuation Intrinsic fiber core attenuation including light absorption and scattering is just one aspect of the cause in fiber optic loss. Extrinsic fiber attenuation is also very important, which are usually caused by improper handling of fiber optic. There are two main types of extrinsic fiber attenuation: bend loss and splicing loss. Bend loss is the common problems that can cause fiber optic loss generated by improper fiber optic handling. Literally, it is caused by fiber optic bend. There are two basic types. One is micro bending, and the other one is macro bending (shown in the above picture). Macro bending refers to a large bend in the fiber (with more than a 2 mm radius). To reduce fiber optic loss, the following causes of bend loss should be noted: Fiber core deviate from the axis; Defects of manufacturing; Mechanical constraints during the fiber laying process; Environmental variations like the change of temperature, humidity or pressure. Fiber optic splicing is another main causes of extrinsic fiber attenuation. It is inevitable to connect one fiber optic to another in fiber optic network. The fiber optic loss caused by splicing cannot be avoided, but it can be reduced to minimum with proper handling. Using fiber optic connectors of high quality and fusion splicing can help to reduce the fiber optic loss effectively. The above picture shows the main causes of losses in optical fiber, which come in different types. To reduce the intrinsic fiber core attenuation, selecting the proper fiber optic and optical components is necessary. To decrease extrinsic fiber attenuation to minimum, the proper handling and skills should be applied. Fiber optic communication system Fiber oppticsessentially deals of with the communication(including voice signals,video signals or digital data) by transmission of light through optical fibers. Optical fiber communication system essentially consists of three parts:(a)transmitter (b) optical fiber and (c) receiver.The 28  transmitter includes modulator, encoder, light source, drive circuits and couplers. The receiver includes amplifier and decoder,binary electrical signal and light decoder. Block diagram represents optical fibre communication system 1. Encoder: It is an electric circuit where in the information is encoded into binary sequences of zeros and one. 2. Transmitter: An electric signal is applied to the optical transmitter. The optical transmitter consists of driver circuit and the light source. Driver circuit drives the light source. Light source converts electrical signal to optical signal. 3. Optical fiber: The optical fiber acts as a wave guide and transmits the optical pulses towards the receiver, by the principle of total internal reflection. 4. Receiver: The light detector receives the optical pulses and converts them into electrical pulses. These signals are amplified by the amplifier. 5. Decoder: The amplified signals are decoded by the decoder. 5.16 Applications of optical fibers: Optical Fibres uses in Medical industry 29 Because of the extremely thin and flexible nature, it used in various instruments to view internal body parts by inserting into hollow spaces in the body. It is used as lasers during surgeries, endoscopy, microscopy and biomedical research. Optical Fibres used in Communication In the communication system, telecommunication has major uses of optical fibre cables for transmitting and receiving purposes. It is used in various networking fields and even increases the speed and accuracy of the transmission data. Compared to copper wires, fibre optics cables are lighter, more flexible and carry more data. Optical Fibres used in Defense Purpose Fibre optics are used for data transmission in high-level data security fields of military and aerospace applications. These are used in wirings in aircraft, hydrophones for SONARs and Seismics applications. Optical Fibres are used in Industries These fibres are used for imaging in hard to reach places such as they are used for safety measures and lighting purposes in automobiles both in the interior and exterior. They transmit information in lightning speed and are used in airbags and traction control. They are also used for research and testing purposes in industries. Optical Fibres used for Broadcasting These cables are used to transmit high definition television signals which have greater bandwidth and speed. Optical Fibre is cheaper compared to the same quantity of copper wires. Broadcasting companies use optical fibres for wiring HDTV, CATV, video-on-demand and many applications. Uses of Optical Fibre for Lightening and Decorations By now, we got a fair idea of what is optical fibre and it also gives an attractive, economical and easy way to illuminate the area and that is why it is widely used in decorations and Christmas trees. Optical Fibres used in Mechanical Inspections On-site inspection engineers use optical fibres to detect damages and faults which are at hard to reach places. Even plumbers use optical fibres for inspection of pipes. Before the discovery of super conductivity, it was though that the electrical resistance is zero 30  Unit-III Principles of Quantum Mechanics Introduction –Wave nature of particles – de-Broglie hypothesis –Physical significance of ψ –Time-dependent and time-independent Schrodinger equations – Born interpretation – Probability current –Wave packets –Uncertainty principle –Particle in infinite square well potential –Scattering from potential step – Potential barrier and tunneling. -------------------------------------------------------------------------------------------------------------------------------- ------- Introduction Quantum mechanics came from Planck’s Quantum theory in 1900, and formulated as Quantum mechanics in 1925. The Quantum mechanics proved to be very much evident in solving problems that were unable to be solved by the so called classical mechanics. Classical Mechanics aims to provide a precise and consistent description of the dynamics of particles and the working of system of particles. Quantum physics explain the behaviour of matter and radiation at the atomic level. To say simply, for understanding macroscopic particles Classical mechanics is used and for microscopic particles Quantum mechanics is used. WAVES & PARTICLES Particle: The concept of a particle is easy to grasp. It has mass, it is located at some definite point, and it can move from place to another, it gives energy when slowed down or stopped. The particle is specified by (i) mass m (ii) velocity v (iii) momentum P (iv) energy E. The motion of particle is given by formulae like 𝐝𝐩 𝐝𝐯 F= ma (or) F= 𝐝𝐭 = F= m 𝐝𝐭 Wave: The concept of wave is a bit more difficult than that of a particle. A wave is spread out over a relatively large region of space, it cannot be said to be located just here and there. Actually, a wave is nothing but rather a spread out of disturbance. A wave is specified by its (i) frequency (ii) wavelength (iii) phase of wave velocity (iv) amplitude (v) intensity , the displacement regarding wave is y= A sinωt. Debroglie hypothesis or Debroglie’s Matter Waves The energy or radiation behaves both like wave and particle. In 1924, Louis Debroglie put a bold suggestion that the correspondence between wave and particle should not be confined only to electromagnetic radiation, but it should also be valid for material particles i.e., like radiation matter also has a dual (particle like & wave like) characteristics.  A moving particle has always got a wave associated with it and the particle is controlled by the wave in a manner similar to that in which a photon is controlled by waves. According to Debroglie. Nature loves symmetry, since energy or radiation exhibits wave particle duality, matter must also possess this dual character according to Einstein mass energy relation. Close parallelism between optics & mechanics. Fermat’s least action principle of light ray is similar to Hamiltonian least action principle of a body in mechanics. The periodicity of electron motion in Bohr’s atom model provides a clue that a material particle might behave like a wave. Derivation of Debroglie- Wavelength According to Debroglie, a moving particle is associated with a wave which is known as Debroglie wave or matter wave. According to Planck’s theory the energy E of a photon of frequency ν is given by hc E= hν = ---------- (1) λ Where ‘λ’ is wavelength of photon ‘C’ is speed or velocity of light. If photon is treated as a particle, its energy as given by Einstein’s mass-energy relation E=mc2 --------- (2) where ‘P’ is momentum of photon P=mc From (1) & (2) hc = mc2 λ λ=h/mc ------- (3) For a particle of mass ‘m’ moving with velocity ‘v’, the wavelength of matter waves associated with particle is given by equation. λ=h/mv where mv=p, momentum λ=h/p----- (4) If E is the kinetic energy of material particle, then E= ½ mv2 = ½ m2v2 => E= P2/2m => P = √2mE h Therefore, Debroglie wavelength is λ= √2mE When a charged particle carrying a charge ‘q’ is accelerated by a potential difference ‘V’ volts then its kinetic energy is given by E = qV h Therefore λ= √2mqV h 12.26 If the charged particle is an electron then q=e then λ= = A0. √2meV √V There is a comparison between Debroglie equation and electronic orbits in the atom. According to Bohr orbital angular momentum is given by  nh mvr= where ‘r’ is radius of permitted circular orbit, 2π ‘n’ is integer. From above equation and ‘λ’ we have nh 2πr = mv = nλ (2πr) is circumference of electronic orbit. Thus, the permitted orbits correspond to integral multiples of Debroglie wavelength. Matter waves According to Debroglie, a material particle like electron or neutron is pictured as a group of waves. These waves are called matter waves or Debroglie waves. Every moving particle is associated with a wave called as Debroglie wave. Properties of matter waves (i) Wavelength: The wavelength of matter is λ=h/mv or λ=h/p. so, wavelength is inversely proportional to velocity. So, if v=0, λ=∞, and if λ is small, v is greater. This shows that matter waves are generated by the motion of particles. (ii) Group velocity: A group of waves each wave having wavelength given above, is associated with the particle. This group as a whole must travel with the particle velocity v Hence group velocity of matter waves= vgr = v. Since particle is guided by the group of matter waves, these waves are sometimes called pilot waves. (iii) Phase velocity: Each wave of the group of matter waves travel with a velocity known as phase velocity of the wave. It is given by ω vph= k { but ω=2πν; k=2π/λ} 2πν hν Therefore, vph = 2π/λ = λν = mv but we know that E=hν= mc2  vph = mc2/mv = c2/v Obviously particle velocity v < c hence vph >c So this is an unexpected result. From vgr and vph values we can write vgr. vph= c2. Differences between Matter wave & EM wave MATTER WAVE EM WAVE Matter wave is associated with a particle. Oscillating charged particle gives rise to electromagnetic wave. Wavelength depends on the mass of the Wavelength depends on the energy of the particle and its velocity λ= h/mv photon λ= hC/E  Can travel with a velocity greater than the Travels with velocity of light. velocity of light. Matter wave is not electromagnetic wave. Electric field and magnetic field oscillate perpendicular to each other. Physical significance of Ψ & Max- Born Interpretation The actual physical significance was not clear. Max Born’s interpretation of ‘Ψ’, given in 1926, is generally accepted at present. As ‘Ψ’ is a complex function Ψ*Ψ = |Ψ|2 is a real value. |Ψ|2 at a point is proportional to the probability of finding the particle at the point at any given instant. The probability density at any point is represented by |Ψ|2, the probability P of finding the particle within any element of volume dxdydz is given by P=Ψ*Ψ dxdydz Since the total probability of finding the particle somewhere is unity, Ψ is such a function that satisfies condition ∭ |𝛹|2 dxdydz = 1 Ψ satisfying above equation is called a normalized function. Besides this Ψ is a single valued continuous function. Schrodinger time independent wave equation Schodinger, in 1926 developed wave equation for moving particles. Consider a particle of mass ‘m’ moving with a velocity ‘v’ which is associated with a group of waves. ❖ Let ‘Ψ’ be the wave function of the particle. ❖ Consider a simple form of progressing wave associating with the particle. ❖ So, wave function Ψ(x,t)=Ψ0 exp{− 𝑖(𝜔𝑡 − 𝑘𝑥)}--------------> (1) Where Ψ=Ψ0 is amplitude , k=2π/λ, propagation constant. 2𝜋 ω = 2πν = , angular frequency 𝑇 ❖ Now, differentiating (1) with respect to ‘x’ we get 𝜕𝛹 = ikΨ0 exp{−i(𝜔𝑡 − 𝑘𝑥)} 𝜕𝑥 𝜕2𝛹 = i2k2 Ψ0 exp{−i(𝜔𝑡 − 𝐾𝑥)} 𝜕𝑥 2 𝜕2𝛹 = -k2 Ψ [since from (1) 𝜕𝑥 2 𝜕2𝛹 +k2 Ψ = 0 ----------------- (2) 𝜕𝑥 2 OR 𝜕2𝛹 4𝜋 2 + Ψ = 0 --------------- (3) {since k=2π/λ} 𝜕𝑥 2 𝜆2 ❖ Equation (2) or (3) is the differential form of classical wave equation. ❖ Now, we know that Debroglie wavelength λ= h/mv  𝜕2𝛹 4𝜋 2 𝑚2 𝑣 2 + 𝝍 = 0 ------------------ (4) 𝜕𝑥 2 ℎ2 ❖ Now, we know that the total energy E of the particle is sum of its kinetic energy K and potential energy V Therefore, E = K + V and K= ½ mv2 m2v2 = 2m (E-V) --------- (5) From (4) & (5) 𝜕2𝛹 8𝜋 2 𝑚(𝐸−𝑉) + 𝛹= 0 𝜕𝑥 2 ℎ2 ❖ The value of h/2π is considered as ℏ Therefore, ℏ=h/2π 𝜕2𝛹 2𝑚(𝐸−𝑉) + 𝛹= 0 ---------- (6) 𝜕𝑥 2 ℏ2 This equation is a one dimensional equation. ❖ In three dimensional it can be written as 𝜕2𝛹 𝜕2𝛹 𝜕2𝛹 2𝑚(𝐸−𝑉) + 𝜕𝑦 2 + 𝜕𝑧 2 + 𝛹= 0 ---------- (7) 𝜕𝑥 2 ℏ2 Where Ψ= Ψ(x, y, z) 𝜕2 𝜕2 𝜕2 ❖ Using laplacian operator 𝛻2 = 𝜕𝑥 2 +𝜕𝑦 2 +𝜕𝑧 2 2𝑚(𝐸−𝑉) We have ∇2 Ψ+ 𝛹= 0 ----------- (8) ℏ2 ❖ This is the Schrodinger wave equation, since time factor doesn’t appear (8) is called time independent Schrodinger’s wave equation. Schrodinger Time dependent wave equation ❖ Let ‘Ψ’ be the wave function of the particle. ❖ Consider a simple form of progressing wave associating with the particle. So, we know the wave function Ψ(x,t)=Ψ0 exp{− 𝑖(𝜔𝑡 − 𝑘𝑥)} Where Ψ=Ψ0 is amplitude , k=2π/λ, propagation constant. 2𝜋 ω = 2πν = , angular frequency 𝑇 It is expressed as Ψ=Ψ0 exp{− 𝑖(2πν𝑡 − 2π/λ𝑥)} 2π Ψ=Ψ0 exp{− 𝑖(ℎν𝑡 − h/λ𝑥)} ℎ 𝑖 Ψ=Ψ0 exp{− ℏ(ℎν𝑡 − h/λ𝑥)} since ℏ = h/2π 𝑖 Ψ=Ψ0 exp{− (𝐸𝑡 − p𝑥)} ----(1) since E=hν and p = h/λ ℏ Differentiating equation(1) w.r.t ‘x’ 𝜕𝛹 𝑖 𝑖 = p Ψ0 exp{− ℏ(𝐸𝑡 − p𝑥)} 𝜕𝑥 ℏ Once again differentiating w.r.t ‘x’ 𝜕2𝛹 𝑖 𝑖 = (ℏ p)2Ψ0 exp{− ℏ(𝐸𝑡 − p𝑥)} 𝜕𝑥 2 𝜕2𝛹 𝑝2 = - ℏ2 Ψ since from equ(1) 𝜕𝑥 2  𝜕2 𝛹 p2 Ψ = −ℏ2 ------(2) 𝜕𝑥 2 Differentiating equ(1) w.r.t ‘t’ once 𝜕𝛹 𝑖 𝑖 = - ℏ E Ψ0 exp{− ℏ(𝐸𝑡 − p𝑥)} 𝜕𝑡 𝜕𝛹 𝑖 = - ℏE Ψ since from equ(1) 𝜕𝑡 ℏ 𝜕𝛹 EΨ=- 𝑖 𝜕𝑡 𝜕𝛹 E Ψ = iℏ -------(3) 𝜕𝑡 The total energy is E =KE+PE E=½ mv2 + V(x,t) 𝑝2 E= 2𝑚 + 𝑉(𝑥, 𝑡) 𝑝2 EΨ= Ψ + 𝑉(𝑥, 𝑡)Ψ 2𝑚 Substituting equ(2) and equ(3) in the above 𝜕𝛹 ℏ2 𝜕 2 𝛹 iℏ = - 2𝑚 + V(x,t)Ψ 𝜕𝑡 𝜕𝑥 2 In three dimensions the above equations transfers to 𝜕𝛹