MECH 6491 Engineering Metrology and Measurement Systems. Lecture 4 Cont d. Instructor: N R Sivakumar

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1 MECH 6491 Engineering Metrology and Measurement Systems Lecture 4 Cont d Instructor: N R Sivakumar 1

Here, we are talking about separating out different parts of light when

2 Light Polarization In 1669, Huygens studied light through a calcite crystal observed two rays (birefringence). Here, we are talking about separating out different parts of light when we discuss polarization. Specifically, we are interested in the electric field of the electromagnetic wave.

3 Light Polarization Net electric field is zero Unpolarized light! Electric field only going up and down say it is linearly polarized. Light can have other types of polarizations such as circularly polarized or elliptically polarized. We will only look at linearly polarized light.

4 Plane-polarized light Vertical E y A sin( x/ t) Horizontal E z A sin( x/ t)

5 Circularly polarized light E y Right circular Ax sin( / t 90 ) E z A sin( x/ t) E y Left circular Ax sin( / t 90 ) E z A sin( x/ t)

with electric fields perpendicular to the axis.

6 Light Polarization I. Polarizers- Polarizers are made of long chained molecules which absorb light with electric fields perpendicular to the axis. Because I I E I o 2 cos 2 Malus s Law I 0 is the initial intensity, and θ i is the angle between the light's initial polarization direction and the axis of the p

7 Light Polarization II. Scattering -

8 Light Polarization III. Reflection - n sin n n n n sin n p p o p tan p 2 sin 2 cos p Brewster s Law

Spherical Mirrors A spherical mirror is a mirror which has the shape of a piece cut out of a spherical surface. There are two types concave, and convex mirrors.

9 Spherical Mirrors A spherical mirror is a mirror which has the shape of a piece cut out of a spherical surface. There are two types concave, and convex mirrors. Concave mirrors magnify objects placed close to them shaving mirrors and makeup mirrors. Convex mirrors have wider fields of view passenger-side wing mirrors of cars but objects which appear in them generally look smaller (and, therefore, farther away) than they actually are.

Spherical Mirrors - Concave The normal to the mirror centre is called principal axis. V, at which the principal axis touches the mirror surface is called the vertex.

10 Spherical Mirrors - Concave The normal to the mirror centre is called principal axis. V, at which the principal axis touches the mirror surface is called the vertex. The point C, on the principal axis, equidistant from all points on the reflecting surface of the mirror is called the centre of curvature. C to V is called the radius of curvature of the mirror R Rays parallel principal axis striking a concave mirror, are reflected by the mirror at F (between C and V) is focal point. The distance along the principal axis from the focus to the vertex is called the focal length of the mirror, and is denoted f.

11 Spherical Mirrors - Convex The definitions of the principal axis, C, R, V of a convex mirror are same as that of concave mirror. When parallel light-rays strike a convex mirror they are reflected such that they appear to emanate from a single point F located behind the mirror. This point is called the virtual focus of the mirror. The focal length f of the mirror is simply the distance between V and F. f is half of R

Image Formation - Concave Graphical method - just four simple rules: An incident ray // to principal axis is reflected through the focus F of the mirror An incident ray which passes through the F of

12 Image Formation - Concave Graphical method - just four simple rules: An incident ray // to principal axis is reflected through the focus F of the mirror An incident ray which passes through the F of the mirror is reflected // to the principal axis An incident ray which passes through the C of the mirror is reflected back along its own path (since it is normally incident on the mirror) An incident ray which strikes the mirror at V is reflected such that its angle of incidence wrt the principal axis is equal to its angle of reflection. ST is the object at distance p from the mirror (P > f). Consider 4 light rays from tip T to strike the mirror 1 is // to axis so reflects through point F. 2 is through point F so reflects // to the axis. 3 is through C so it traces its path back. 4 is towards V so it has same angles of incidence and reflectance. The point at which all these meet is where the object will be S T at distace Q (if seen further from q, the image is seen) This is a real image

Image Formation - Concave We only need 2 rays (minimum) to get the image position: In this case, the object ST is located within the focal length of the mirror f Consider 2 lines from tip T of the

13 Image Formation - Concave We only need 2 rays (minimum) to get the image position: In this case, the object ST is located within the focal length of the mirror f Consider 2 lines from tip T of the object Line 2, passes through F and reflects // to the principal axis Line 3 passes through C and is reflected along its path If these two lines are connected, the image is formed on the other side of the lens Here the image is magnified, but not inverted like the previous case There are no real light-rays behind the mirror Image cannot be viewed by projecting onto a screen This type of image is termed a virtual image The difference between a real and virtual image is, immediately after reflection from the mirror, light-rays from object converge on a real image, but diverge from a virtual image.

Image Formation - Convex Graphical method - just four simple rules: Incident ray // to principal axis is reflected as if it is from virtual focus F of mirror.

14 Image Formation - Convex Graphical method - just four simple rules: Incident ray // to principal axis is reflected as if it is from virtual focus F of mirror. Incident ray directed towards the virtual focus F of the mirror is reflected // to principal axis. Incident ray directed towards C of the mirror is reflected back along its own path (since it is normally incident on the mirror). Incident ray which strikes the mirror at V is reflected such that its angle of incidence wrt the principal axis is equal to its angle of reflection.. ST is the object. Consider 2 light rays from tip T to strike the mirror 1 is // to axis so appears to reflect through point F. 2 is through point C so so it traces its path back. The point of intersection of these two lines is where the image will be Vitural or Real???? Inverted and Magnified or otherwise?????

Image Formation - Concave If we do this by analytical method, we will get the following formula Magnification M is given by the object and image distances It is negative if the image is inverted and

15 Image Formation - Concave If we do this by analytical method, we will get the following formula Magnification M is given by the object and image distances It is negative if the image is inverted and positive if it is not For expression that relates the object and image distances to radius of curvature If object is far away (p = ) all the lines are // and focus on the focal point F The focal length is R/2 which can be combine to give

16 Image Formation plane mirror For plane mirrors the radius of curvature R = If R =, then f = ±R/2 = because 1/f = 0 1/p + 1/q = 1/f = 0 q = -p (which means that it is a virtual image far behind the mirror as much the object is in front) Magnification is q/p = 1 So plane mirror does not magnify or invert the image

17 Image Formation plane mirror Sign conventions may vary based on different text books. So follow consistently which ever method is used

18 Image Formation Lenses For thin lenses this distance is taken as 0

Image Formation Lenses A lens is a transparent medium bounded by two curved surfaces (spherical or cylindrical) Line passing normally through both bounding surfaces of a lens is called the optic axis.

19 Image Formation Lenses A lens is a transparent medium bounded by two curved surfaces (spherical or cylindrical) Line passing normally through both bounding surfaces of a lens is called the optic axis. The point O on the optic axis midway between the two bounding surfaces is called the optic centre. There are 2 basic kinds: converging, diverging Converging lens - brings all incident light-rays parallel to its optic axis together at a point F, behind the lens, called the focal point, or focus. Diverging lens spreads out all incident light-rays parallel to its optic axis so that they appear to diverge from a virtual focal point F in front of the lens. Front side is conventionally to be the side from which the light is incident.

20 Image Formation Lenses Relationship between object and image distances to focal length is given by Magnification of the lens is given by Example (Object outside Focal Point) Object distance S = 200mm Object height h = 1mm Focal length of the lens f = 50mm Find image distance S and Magnification m

21 Image Formation Lenses Relationship between object and image distances to focal length is given by Magnification of the lens is given by Example (Object inside Focal Point) Object distance S = 30mm Object height h = 1mm Focal length of the lens f = 50mm Find image distance S and Magnification m

22 Image Formation Lenses Relationship between object and image distances to focal length is given by Magnification of the lens is given by Example (Object at Focal Point) Object distance S = 30mm Object height h = 1mm Focal length of the lens f = -50mm (diverging lens) Find image distance S and Magnification m

23 F-Number and NA The calculations used to determine lens dia are based on the concepts of focal ratio (f-number) and numerical aperture (NA). The f-number is the ratio of the lens focal length of the to its clear aperture (effective diameter ). The f-number defines the angle of the cone of light leaving the lens which ultimately forms the image. The other term used commonly in defining this cone angle is numerical aperture NA. NA is the sine of the angle made by the marginal ray with the optical axis. By using simple trigonometry, it can be seen that

24 Different Lenses

Spherical Aberration Spherical aberration comes from the spherical surface of a lens The further away the rays from the lens center, the bigger

$TSA = LSA X tan u Spherical aberration is dependent on lens shape, orientation and index of refraction of the lens Aspherical lenses offer best$

25 Spherical Aberration Spherical aberration comes from the spherical surface of a lens The further away the rays from the lens center, the bigger the error is Common in single lenses. The distance along the optical axis between the closest and farthest focal points is called (LSA) The height at which these rays is called (TSA) TSA = LSA X tan u Spherical aberration is dependent on lens shape, orientation and index of refraction of the lens Aspherical lenses offer best solution, but difficult to manufacture So cemented doublets (+ve and ve) are used to eliminate this aberration

Astigmatism When an off-axis object is focused by a spherical lens, the natural asymmetry leads to astigmatism. The system appears to have two different focal lengths.

26 Astigmatism When an off-axis object is focused by a spherical lens, the natural asymmetry leads to astigmatism. The system appears to have two different focal lengths. Saggital and tangential planes Between these conjugates, the image is either an elliptical or a circular blur. Astigmatism is defined as the separation of these conjugates. The amount of astigmatism depends on lens shape 26

27 Astigmatism 27

28 Chromatic Aberration Material usually have different refractive indices for different wavelengths n blue >n red This is dispersion blue refracts more than the red, blue has a closer focus

29 Achromatic Doublets As in the case of spherical aberration, positive and negative elements have opposite signs of chromatic aberration. By combining elements of nearly opposite aberration to form a doublet, chromatic aberration can be partially corrected It is necessary to use two glasses with different dispersion characteristics, so that the weaker negative element can balance the aberration of the stronger, positive element.

30 Achromatic Doublets R 1 R 2 R 3 Lens maker s formula n 1 n 2 Achromatic doublet (achormat) is often used to compensate for the chromatic aberration the focuses for red and blue is the same if ( n b nr1)( ) ( nb 2 nr 2)( ) R R R R

31 MECH 691T Engineering Metrology and Measurement Systems Lecture 5 Instructor: N R Sivakumar 31

32 Outline Introduction General Description Coherence Interference between 2 plane waves Laser Doppler velocimetry Interference between spherical waves Interferometry Wavefront Division Amplitude Division Heterodyne Interferometry 32

33 Light as Waves Waves have a wavelength Waves have a frequency 33

34 Frequency Thousand (10 3 ) oscillations/second - kilohertz (khz) Million (10 6 ) oscillations/second - megahertz (MHz) Billion) (10 9 ) oscillations/second - gigahertz (GHz) Thousand billion (10 12 ) oscillations per second - terahertz (THz) Million billion) (10 15 ) oscillations per second - petahertz (PHz) 34

35 Introduction The superposition principle for electromagnetic waves implies that, two overlapping fields Ul and U2 add to give Ul + U2. This is the basis for interference. 35

36 Interference in water waves 36

37 Overlapping Semicircles 37

38 Superposition +1 Constructive Interference t t In Phase +2 t -2 38

39 Superposition +1 Destructive Interference t t Out of Phase 180 degrees +2-2 t 39

40 Superposition Different f ) Constructive 2) Destructive 3) Neither 40

41 Superposition Different f ) Constructive 2) Destructive 3) Neither 41

42 Interference Requirements Need two (or more) waves Must have same frequency Must be coherent (i.e. waves must have definite phase relation) 42

43 General Description Interference can occur when two or more waves overlap each other in space. Assume that two waves described by and overlap The electromagnetic wave theory tells us that the resulting field simply becomes the sum The observable quantity is intensity (irradiance) I which is Where e i = (cos + isin ) and 43

44 General Description Resulting intensity is not just (I 1 + I 2 ). When 2 waves interfere is called the interference term We also see that when then and I reaches minima (cos 180 ) which means destructive interference Similarly when then and I reaches maxima (cos 0 ) constructive interference When 2 waves have equal intensity I 1 = I 2 = I 0 44

45 Coherence Detection of light is an averaging process in space and time We assume that ul and u2 to have the same single frequency Light wave with a single frequency must have an infinite length However sources emitting light of a single frequency do not exist 45

46 Coherence Here we see two successive wave trains of the partial waves The two wave trains have equal amplitude and length Lc, with an abrupt, arbitrary phase difference a) shows the situation when the two waves have traveled equal paths. We see that although the phase of the original wave fluctuates randomly, the phase difference remains constant in time 46

47 Coherence In c) wave 2 has traveled Lc longer than wave 1. The head of the wave trains in wave 2 coincide with tail of the corresponding wave trains in partial wave 1. Now the phase difference fluctuates randomly as the successive wave trains pass by Here cos varies randomly between +1 and -1 and for multiple trains it becomes 0 (no interference) I = I 1 + I 2 47

48 Coherence In b) wave 2 has traveled l longer than wave 1 where 0<l<L c. For many wavetrains the phase difference varies in time proportional to we still can observe an interference pattern but with a reduced contrast is the coherence length and is the coherence time For white light, the coherence length is 1 micron 48

49 Plane Wave Interference When two plane wave interfere the resultant fringe spacing is given by 49

50 Laser Doppler Velocimetry (LDV) Method for measuring the velocity of moving objects Based on the Doppler effect light changes its frequency (wavelength) when detected by a stationary observer after being scattered from a moving object example - when the whistle from a train changes from a high to a low tone as the train passes by 50

51 Laser Doppler Velocimetry (LDV) Particle is moving in a test volume where two plane waves interfere at an angle These two waves will form interference planes which are parallel to the bisector of a and separated by a distance As the particle moves through test volume, it will scatter light when it passes a bright fringe and scatter no light while passing a dark fringe The resulting light pulses can be recorded by a detector The time lapse between pulses is t d and frequency is f d = 1/t d 51

52 Laser Doppler Velocimetry (LDV) 52

53 Laser Doppler Velocimetry (LDV) If there are many particles of different Vs many different frequencies can be recorded on a frequency analyzer and the resulting spectrum will tell how the particles are distributed among the different velocities You know, the and the f can be recorded. V can be calculated This method does not distinguish between particles moving in opposite directions LDV can be applied for measurement of the velocity of moving surfaces, turbulence in liquids and gases (where the liquid or gas seeded with particles). Examples - of stream velocities around ship propellers, velocity distributions of oil drops in IC engines 53

54 Interference between other Waves Figure shows the fringe pattern in xzplane when spherical waves from two point sources P1 and P2 on the z-axis interfere. Fringe density increases as distance between PI and P2 increases 54

55 Interference between other Waves The intensity distribution in XY plane is Where This called circular zone pattern 55

56 Interference between other Waves By measuring the distance between interference fringes over selected planes in space, quantities such as the angle and distance can be found. One further step would be to apply for a wave reflected from a rough surface By observing the interference - can determine the surface topography For smoother surfaces, however, such as optical components (lenses, mirrors, etc.) where tolerances of the order of fractions of a wavelength are to be measured, that kind of interferometry is quite common. 56

57 Interferometry Light waves interfere only if they are from the same source (why???) Most interferometers have the following elements light source element for splitting the light into two (or more) partial waves different propagation paths where the partial waves undergo different phase contributions element for superposing the partial waves detector for observation of the interference 57

58 Interferometry Depending on how the light is split, interferometers are commonly classified Wavefront division interferometers Amplitude division interferometers 58

59 Wavefront Division Example of a wavefront dividing interferometer, (Thomas Young) The incident wavefront is divided by passing through two small holes at S I and S 2 in a screen 1. The emerging spherical wavefronts from S I and S 2 will interfere, and the pattern is observed on screen 2. The path length differences of the light reaching an arbitrary point x on S 2 is found from Figure When the distance D between screens is much greater than the distance d between S 1 and S 2, we have a good approximation 59

60 Wavefront Division y m=2 m=1 m=0 m=1 m=2 D 60

61 Wavefront Division dsin m m = 0,1,2,3... Maximum tan y m D or y m Dtan Dsin y m m D d Maxima m y m +/ D /d 2D /d 3D /d m y m +/ Minima D /2d 3D /2d 5D /2d 7D /2d dsin (m 1 ) m = 0,1,2,3... Minimum 2 dsin y m (m 1/2) D d 61

62 Wavefront Division 13E Suppose that Young s experiment is performed with blue-green light of 500 nm. The slits are 1.2mm apart, and the viewing screen is 5.4 m from the slits. How far apart the bright fringes? From the table on the previous slide we see that the separation between bright fringes is D /d D /d (5.4m)( m)/0.0012m m 2.25mm 62

63 Wavefront Division A) Fresnel Biprism B) Lloyds Mirror C) Michelsons Stellar Interferometer 63

64 Amplitude Division Example of a amplitude dividing interferometer, (Michelson) Amplitude is divided by beamsplitter BS which partly reflects and partly transmits These divided light go to two mirrors M 1 and M 2 where they are reflected back. The reflected lights recombine to form interference on the detector D The path length can be varied by moving one of the mirrors or by mounting that on movable object (movement of x give path difference of 2x) and phase difference 64

65 Amplitude Division As M 2 moves the displacement is measured by counting the number of light maxima registered by D By counting the number of maxima per unit time will give the velocity of the object. The intensity distribution is given by Ezekiel, Shaoul. RES Video Demonstrations in Lasers and Optics, Spring (Massachusetts Institute of Technology: MIT OpenCourseWare), (Accessed 15 May, 2012). License: Creative Commons BY-NC-SA 65

66 Michelson Interferometer Split a beam with a Half Mirror, the use mirrors to recombine the two beams. Mirror Light sourc e Half Mirror Screen Mirror 66

67 Michelson Interferometer If the red beam goes the same length as the blue beam, then the two beams will constructively interfere and a bright spot will appear on screen. Mirror Light sourc e Half Mirror Screen Mirror 67

68 Michelson Interferometer If the blue beam goes a little extra distance, s, the the screen will show a different interference pattern. Mirror Light sourc e Half Mirror Screen s Mirror 68

69 Michelson Interferometer If s = /4, then the interference pattern changes from bright to dark. Mirror Light sourc e Half Mirror Screen s Mirror 69

70 Michelson Interferometer If s = /2, then the interference pattern changes from bright to dark back to bright (a fringe shift). Mirror Light sourc e Half Mirror Screen s Mirror 70

71 Michelson Interferometer By counting the number of fringe shifts, we can determine how far s is! Mirror Light sourc e Half Mirror Screen s Mirror 71

72 Michelson Interferometer If we use the red laser ( =632 nm), then each fringe shift corresponds to a distance the mirror moves of 316 nm (about 1/3 of a micron)! Mirror Light sourc e Half Mirror Screen s Mirror 72

73 Amplitude Division Twyman Green Interferometer Mach Zehnder Interferometer 73

74 Dual Frequency Interferometer We stated that two waves of different frequencies do not produce observable interference. By combining two plane waves The resultant intensity becomes If the frequency difference V I - V 2 is very small and constant, this variation in I with time can be detected This is utilized in the dual-frequency Michelson interferometer for length measurement Also called as Heterodyne interferometer 74

75 Dual Frequency Interferometer 75

Chapter Ray and Wave Optics

109 Chapter Ray and Wave Optics 1. An astronomical telescope has a large aperture to [2002] reduce spherical aberration have high resolution increase span of observation have low dispersion. 2. If two