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Lehigh University Lehigh Preserve Theses and Dissertations 2006 A Michelson interferometric technique for measuring refractive index of sodium zinc tellurite glasses Deepak N. Iyer Lehigh University Follow this and additional works at: http://preserve.lehigh.edu/etd Recommended Citation Iyer, Deepak N., "A Michelson interferometric technique for measuring refractive index of sodium zinc tellurite glasses" (2006). Theses and Dissertations. Paper 954. This Thesis is brought to you for free and open access by Lehigh Preserve. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of Lehigh Preserve. For more information, please contact [email protected]. Iyer, Deepak N. A Michelson Interferometric Technique for Measuring Refractive Index... January 2007 A MICHELSON INTERFEROMETRIC TECHNIQUE FOR MEASURING REFRACTIVE INDEX OF SODIUM ZINC TELLURITE GLASSES. By Deepak. N. Iyer A Thesis Presented to the Graduate and Research Committee of Lehigh University in Candidacy for the Degree of Master of Science In Electrical Engineering Lehigh Universitv'" .... December 2006 Dedicated to my parents 111 ACKNOWLEDGEMENTS I would like to take this opportunity to thank my advisor, Dr. Jean Toulouse for providing me an opportunity to pursue research under his supervision. His advanced and profound knowledge in optics has been a constant source of invaluable guidance, without which this work would never have been accomplished. I am indebted to Dr. Radha Pattnaik for his immense support throughout this work. His technical input has been an invaluable asset and he has been a source of inspiration for me at every step of my work. This will be incomplete without thanking my parents, Padma Narayanan and A.Narayanan Iyer for their constant support, love and encouragement, without which I would not have come this far. I also thank my sister for her continuous support and motivation. Thank you for being there for me! 1\' TABLE OF CONTENTS CERTIFICATE OF APPROVAL 11 ACKNOWLEDGEMENTS IV TABLE OF CONTENTS V LIST OF FIGURES Vlll LIST OF TABLES x ABSTRACT CHAPTER 1 MICHELSON INTERFEROMETER THEORY 2 1.1 Introduction 2 1.2 Michelson Interferometer 3 1.3 Interference of waves with single frequency 5 1.3.1 Condition for interference- monochromatic light 7 Interference of two waves with different frequencies. 8 1.4 CHAPTER 2 EXPERIMENTAL SETUP FOR REFRACTIVE INDEX MEASUREMENTS 2.1 J 10 Measuring refractive index by counting the number of fringes 10 2.1.1 Procedure for alignment 13 2.1.2 Theory behind rotating the sample 17 2.1.3 Derivation of the relationship between the refractive index, number of fringes collapsing and the angle of rotation of the sample 2.2 Measuring the refractive index from the intensity of the fringe pattern 2.2.1 Procedure for alignment CHAPTER 3 DERIVATIONS AND RESULTS 3.1 Measuring the refractive index of BK7 Windows 19 21 22 23 23 3.1.1. Plotting angle of rotation with respect to the number of fringes 23 3.1.2. Difference Angle derivation 26 3.1.3. Plotting difference in angle of rotation with respect to the number of fringes 28 3.1.4. Derivation of the expression for intensity in tenus of refractive index. thickness of the sample and the angle of rotation of the sample 35 3.1.5. Analysis of the stored \\'avefonn 36 3.1.6. Measuring the refractive index of BK7 Windows by plotting the intensity of the fringe pattern with respect 40 to the angle of rotation 3.2 Measuring refractive indices of core and cladding sodium zinc tellurite glass samples 3.2.1 57 Intensity plots for measuring refractive index of core glass sample 3.2.2 60 Intensity plots for measuring refractive index of cladding glass sample 69 CHAPTER 4 CONCLUSION 77 CHAPTER 5 FUTURE WORK 78 REFERENCES 79 VITA 82 \"11 LIST OF FIGURES Figure 1: Schematic illustration of a Michelson interferometer 4 Figure 2: Formation of circles on interference 6 Figure 3: Beat Signal from two input frequencies into a Michelson Interferometer 9 Figure 4: Michelson Interferometric setup for measuring refractive index of BK7 sample by manually counting the number of fringes collapsing or appearing 10 Figure 5: Successive fields of view in interferometer alignment 15 Figure 6: Demonstrating the change in path length 18 Figure 7: Analysis of the sample being rotated 19 Figure 8: Michelson Interferometric setup for measuring refractive index of BK7 sample by measuring the intensity of the interference fringe pattern 21 Figure 9(a-d): Plotting cosine of the angle of rotation with respect to the number of fringes collapsing or appearing 24 Figure 10: Simulation for ideal valucs ofrefractivc index for BK7 sample 28 Figure II (a-h): Plotting cosine of thc difference in angle of rotation with respcct to the number of fringes collapsing or appearing 30 Figure 12: Refractive Index bar graph for Figs. II (a-h) 34 ~lotion 37 Figure 13: Trapezoidal Profile for the controller Figure 14: Intensity Fringe Pattern Simulation for mcasuring rcfractivc index VIII 40 Figure 15(a-I): Plotting intensity with respect to the angle of rotation for the BK7 glass sample 41 Figure 16: Refractive Index bar graph for Figs. 15(a-u) 57 Figure 17(a-I): Plotting intensity with respect to the angle of rotation for the core glass sample 60 Figure 18: Refractive Index bar graph for Figs. 17(a-l) 69 Figure 19(a): Plotting intensity with respect to the angle of rotation for the cladding glass sample. 70 Figure 20: Refractive Index bar graph for Figs. 19 (a-g) 76 IX LIST OF TABLES Table 1: The measured values of refractive indices presented in Figs. 9(a-d) along with the mean and standard deviations 26 Table 2: Density, Hardness and Refractive Index of Zinc Tellurite Glasses 58 Table 3: Refractive Indices at 532 and 1064 nm and d 33 values of poled Na20-ZnO-Te02 glasses 59 ABSTRACT The Michelson interferometer is an extremely versatile instrument that can be used to make an accurate comparison of wavelengths, measure the refractive index of gases and transparent solids, and determine small changes in lengths quite precisely .The main purpose of our research is to develop a Michelson Interferometric setup to measure the refractive index of Sodium Zinc Tellurite glasses of known thickness. This is accomplished by rotating the test sample by means of an automated rotation stage connected to a controller. The intensity of the fringe pattern is measured with the help of an oscilloscope and non-linear curve fitting is done with the appropriate formulae to accurately obtain the value of refractive index. It is successfully demonstrated that refractive index can be obtained to two digits of precision and samples ranging from a few microns in thickness to a few millimeters can be measured. CHAPTER 1: MICHELSON INTERFEROMETER-THEORY 1.1 Introduction There are, in general, a number of types of optical instruments that produce optical interference. These instruments are grouped under the generic name of interferometers. The Michelson interferometer is perhaps the best known and most basic in a family of interferometers which includes the Fabry-Perot interferometer, the Twyman-Green interferometer and the Mach-Zehnder interferometer. The Michelson Interferometer, first developed by Albert Michelson in 1881, has proved of vital importance in the development of modem physics and is an optical instrument of high precision and versatility. This versatile instrument was used to establish experimental evidence for the development for the validity of the special theory of relativity, to detect and measure hyperfine structure in line spectra, to measure the tidal effect of the moon on the earth and to provide a substitute standard for the meter in temlS of wavelengths of light 1.2. Michelson himself pioneered much of this work. It is generally uscd in invcstigations that involvc small changes in optical path lengths and is ablc to dctcct very small movements (on the order of nanometers). With Michelson intcrferomcter. circular and straight-line fringes of both monochromatic light and white light can bc produced. These fringes can be uscd to make an accurate comparison of wavelengths, measure the refractive index of gases and transparent solids, and determine small changes in length quite precisely. The instrument can be used as a stable mode selecting resonator element in laser cavities as well. Interferometry has been used for thickness and index measurements of thin films 3 ,4 and optical fibers 5. In principle, these measurements can be made to great precision, typically on the order of nanometers, since interferometric technique uses the interference of nanometer wavelengths. We propose a Michelson Interferometric technique to measure the refractive index of Sodium Zinc Tellurite bulk glasses. This is achieved by rotating the test sample in one arm of the interferometer thereby creating a path length difference between the two optical paths of the interferometer. Non-linear curve fitting is performed on the measured waveform to relate its intensity to the refractive index of the sample thereby obtaining values of refractive index to two digits of accuracy. 1.2 Michelson Interferometer The Michelson interferometer causes interference by splitting a beam of light into two parts. Each part is made to travel a different path and brought back together where they interfere according to their path length difference 1.2. A diagram of the apparatus is shO\m in Fig. 1 ., j Screen M 1 -. .- Light source I -. I I I_ C J -.jd ~ Figure 1: Schematic illustration of a Michelson interferometer. The Michelson interferometer operates on the principle of division of amplitude rather than on division of wavefront. Light from a light source strikes the beam splitter (designated by S) and is split into two parts. The beam splitter allows 50% of the radiation to be transmitted to the translatable mirror MI. The other 50% of the radiation is reflected back to the fixed mirror M2. Both these mirrors, M I and M2, are highly silvered on their front surfaces to avoid multiple internal reflections. The compensator plate C is introduced along this path to have the same optical path length when M I and M2 are of same distance from the beam splitter. After returning from MI , 50% of the light is reflected toward the frosted glass screen. Likewise. 50% ofthc light returning from tvb is transmitted to the glass screen. Thc two beams arc superposed and one can obser.e the interference fringe pattern on the screen. The character of the fringes is directly related to the differcnt optical path lengths travcled by the two bcams and. thcrcforc. is relatcd to whatevcr causcs a diffcrcncc in thc optical path lcngths. 4 1.3 Interference of Waves With a Single Frequency If two waves simultaneously propagate through the same regIOn of space, the resultant electric field at any point in that region is the vector sum of the electric field of each wave. This is the principle of superposition. (it is assumed that all waves have the same polarization). If two beams emanate from a common source, but travel over two different paths to a detector, the field at the detector will be determined by the optical path difference, which we will denote by So if two waves of the same frequency, OJ, /jx = x2 - XI but of different amplitude and different phase impinge on one point they are superimposed, or interfere, so that: (I) where 01 and 02 are the amplitudes of both the waves and a l and a 2 are the phase angles at any time I. The resulting wave can be described as y = A sin(OJI- a) (2) with A being the resultant amplitude and a the resultant phase. (3) where !1¢ is the phase difference which is givcn by (4) whcre i. is thc wavelcngth of the light source uscd. 5 In a Michelson interferometer, light is split into two beams by the beamsplitter (amplitude splitting), reflected by two mirrors, and passed again through the glass plate to produce interference phenomena behind it. A lens is inserted between the light beam and the beamsplitter so that the light source lies at the focal point, since only enlarged light spots can exhibit interference rings. If the actual mirror M z is replaced by its virtual image Mz', which is formed by reflection at the glass plate, a point P of the real light source is formed as the points P' and P" of the virtual light sources L( and Lz as shown in Figure 2. p' 2d p-...". -::::J= ... 2dCOS9~ 8 L, ... 1 L2 ...., /~ ( M I ....~ 1 Ip'I(~1 Mz l/t\,11, \.11 rT1----------I \1 8r"lI-~ff::::E_ 1\ 11:1\ 1;'11 II -H-Htlf-1~f-----";L...-+--H--+-===------ff----+--H- 1\1' I )1.1 I ... I \.t l';d--r-- ... ) d Figure 2: Formation of circles on intcrfercnce So based on the different light paths, the phase difference, using the symbols of Figure 2, is: 2ii 2,7 i1cp = -tlX = -2d cosO I. ;. (5) The intensity distribution for a1 = a 2 = a according to (3) is: , , , DqJ I : : : AL = 4a Lcos L- ') (6) ~ 6 1.3.1. Condition for interference-Monochromatic Light Constructive interference (Maxima) occurs when I1rp = 2nm, 111 = 0, ±l, ±2, ±3 (7) So we have from equation (5), 2d cosO = l1lA; 111 = 0, ±I, ±2, ±3 (8) that is circles are produced for a fixed value of 111 and d since 0 remains constant (see Fig. 2). So if both the optical path lengths are the same or if these two paths differ by an integral number of wavelengths A ,the condition for constructive interference is met. Thus, bright fringes will be formed for that wavelength. If, on the other hand, the two optical paths differ by an odd integral number of half wavelengths ~ A , where 111 2 = ±I, ±3, ±5, and so on, the condition for destructive interference is met and dark fringes will be formed. So destructive interference occurs when I1rp = ±(2111 + I), 111 = 0, 1,2.3 (9) This might appear at first sight to violate conservation of energy. However energy is conserved. because there is a re-distribution of energy at the detector in which the energy at the destructive sites arc re-distributed to the constructive sites. The effect of the interference is to alter the share of the reflected light which heads for the detector 7 and the remainder which heads back in the direction of the source. If the position of the movable mirror M1 is changed so that d for example decreases then, according to Eq. (8), the diameter of the ring will also decrease since 111 is fixed for this ring. A ring thus disappears each time d is reduced by half. The ring pattern disappears if d = O. If M1 and M2 are not parallel, curved bands are obtained which are converted to straight bands when d = O. 1.4 Interference of Two Waves with Different Frequencies We will now consider the case of two frequencies with wavenumbers k l and k 2 that together follow two different paths with a difference of fjx. The sum of the waves with different amplitudes at point x along the x -axis is given by: (10) E (k - k ) If we let a = _2 and define Ok = 1 2 , after a lot of algebra, we can write the EI 2 intensity (E r ' Er ) as: 2(1 +a+a 2 +acos2&~x+(1 +aXcoskl~x+acosk2~X)) (II) The expected signal which consists of a fast oscillation as well as slow oscillation characteristic of Ok is shown in Fig. 3. s Figure 3: Beat Signal from two input frequencies Interferometer 9 into a Michelson CHAPTER 2: EXPERIMENTAL SETUPS FOR REFRACTIVE INDEX MEASUREMENT 2.1. Measuring the refractive index by counting the number of fringes. S 633nm (35mVV) He-Ne source Screen L1 o Camera lens 25mm (F1.4) 8 8K7 sample L2 Camera lens 55mm (1 :1.7) /'" Manual rotation stage Figure 4: Michelson Interferometric setup for measuring refractive index of BK7 sample by manually counting the number of fringes collapsing or appearing. 10 The optical setup for the Michelson interferometer is shown schematically in Fig. 4. Light from a Helium Neon source S (wavelength of 633nm) is deflected through a mirror M3 and strikes the beam splitter B. The beam splitter is a half-silvered glass plate (silvered on the back side) which reflects half of the light toward mirror M2 and transmits half of the light (but the entire cross section) toward mirror M,. The distance, or path length as it is called, between each mirror and the beamsplitter should be the same. These distances can be determined with a tape measure, and should be as long as possible for the table size. The interferometer's sensitivity increases the farther the mirrors are from the beamsplitter. Both mirrors then renect their respective beams back to the beamsplitter and strike the beamsplitter at the original incident beam's position. Part of mirror MI's reflected beam will then be reflected by the beamsplitter to the screen S and part of mirror M2'S reflected beam will be transmitted by the beamsplitter to the screen S. Two beanl spots that are visible on the screen can then be superimposed to view the interference pattern. This is achieved by moving either mirror, M1 or M2, slightly up and down and/or sideways. The BK7 Windows sample is then mounted on a manual rotation stage and inserted in one of the two paths; this changes the optical path difference between the two paths. It is then slowly rotated carefully noting dO\m the angles so that fringes start collapsing or appearing on the screen. The angle through which the sample has to be rotated for a set of 5. 10. 15 .... fringes to collapse or appear is noted dO\\ll. This is II then subsequently used in the fonnula which relates the refractive index and thickness of the sample with the angle of rotation to obtain the refractive index to a high degree of accuracy. The spots on the screen can be more precisely superimposed by placing a diverging lens L1 between the laser and the beamsplitter BS. The diverging beam lens is placed close to the beamsplitter BS. This increases the diameter of the two spots on the screen so they are easier to superimpose. Another diverging beam lens L2 may also be used to view the interference pattern more clearly while counting the number of fringes. A compensating glass of identical composition and thickness to the beam splitter is necessary to produce white-light fringes so that each of the two beams (paths d/ and d2 in Fig. 2) passes through the san1e thickness of glass. This is due to the fact that because of dispersion, the optical path lengths will be different for different wavelengths. Note that otherwise the bean1 that travels along path BM/BO (dJJ would pass through a thickness of glass three times while the beam that travels along the other path BM2BO (d2) would pass through the same thickness of glass only once. It would not be needed if one only worked with highly monochromatic light where optical path lengths of the two aons of the interferometer can be set to be equal for a particular wavelength. 12 2.1.1 Procedure for alignment: The alignment of the Michelson interferometer can be accomplished in several simple, logical steps. With the Helium Neon source S centered at one end of the table, it is positioned to a height above the optical table that allows the beam to be approximately 6-8 inches above the table surface. The source is then turned on, waiting several minutes for the light to reach full intensity. The mirror M3 is used to deflect the light to the beamsplitter; it is carefully aligned so as to reflect the whole light to the major part of the optical table so that there is sufficient space for the whole setup. The beamsplitter is taken out of the way of the laser beam initially. Making use of two pinholes kept a long distance away from each other, it is made sure that the laser beam is straight and at the same height. The stationary mirror MI is now mounted and its position is adjusted until the beam reflected from it re-enters the laser. That is, the mirror reflects the light perfectly back into the barrel of the laser. It is always a good idea to have the laser dot in the center of the mirror rather than the edge for proper alignment. Putting a screen behind the mirror, it will be possible to see several red dots surrounding a bright red spot. These several dots are focused to one spot by adjusting the knobs on the back of the mirrors. Now the movable mirror is exactly perpendicular to the laser beam. The beam-splitter is now mounted in between mlITors MJ and MI on a manual rotation stage and is rotated 45° so that the laser beam is now split into two beams. 13 One beam is transmitted through the beamsplitter to mirror M1 and the other beam is reflected at 90 degrees to mirror M2. Again mirror M2 is mounted in such a way that the reflected beam hits the mirror at the center. The two paths BM1 and BM2 are made equal to within a millimeter, using a white index card or small ruler. The reflected beams from both the mirrors pass through the beamsplitter again and hit the screen. By means of the two adjusting mirror screws in mirrors M1 and M2, both points of light are made to coincide. The test sample (BK7 windows) is then mounted on a manual rotation stage and inserted into the optical path BM I. By making use of the micrometer screws in the sample mirror mount, the height of the reflected beam from the sample is adjusted in reference to the interfering dots on the screen. The two dots from the reflection of the sample is then superimposed with the already superimposed dots due to the reflected beam from the mirrors to align the sample perpendicular to the path BM 1 and to obtain the zero fringe position. A diverging camera lens (25 mm) is now placed in the light beam between mirror M3 and the beamsplitter to expand the beam so that the points of light are enlarged and the interference pattems are observed on the screen (bands, circles). By careful readjustment, an interference image of concentric circles will be obtained. The fringes may. when first observed, appear rather closely spaced as shO\\l1 in Fig. Sa. While looking at the fringes. carefully adjust the screws on mirror M2 so that the fringes become circular with their common center lying in the center of your field of view as ShO\\l1 in Fig. 5b. At this point. the two mirrors .\11 and .\h. should have their planes perpendicular to one another. that is. B.\/] is perpendicular to B.\h 14 (a) (b) (c) (d) Figure 5: Successive fields of view in interferometer alignment Normally before any adjustment of the interferometer takes place, the two path lengths BMl and BM2 are very likely unequal. Moving mirror M( back and forth along BM( will change the difference in path lengths and, in fact, permit us to reduce the difference to very nearly zero. As mirror M( is moved so as to approach equal path lengths, the fringes move inward toward the center. The center fringe collapses and eventually disappears altogether while outer fringes start appearing at the edge and move towards the center as shown in Fig. 5c. The camera lens L2 is placed between the beamsplitter B and screen 0 and this lens combination of L1 and L2 is adjusted to view only the center fringe. Now the micrometer screw of mirror M1 is carefully turned in a direction that causes the fringes to move toward the center and disappear. This is continued until the field of view is entirely dark as in Fig. 5d. (This condition may be difficult to achieve since air currents or small table vibrations constantly cause small random changes in the path length. Nevertheless. one should approach the "dark-field" condition as closely as possible.). The "dark-field" condition is extremely useful in detennining the zero fringe position of the sample.that is sample is exactly perpendicular to the optical path in which it is inserted. 15 The BK7 sample is now rotated manually towards one side so that the fringes start collapsing or appearing on the screen. The angle of rotation through which the sample has to be rotated for a set of 5 fringes to appear or collapse is noted down carefully. This procedure is continued for la, 15, 20 ... fringes till the maximum fringes for which the fringe pattern is clearly visible. Now the same procedure is repeated for the opposite direction and both the values compared. The index of refraction of the sample is thus calculated from the no. of interference fringes shifted during the rotation of the glass plate. The following equation is used to calculate the index of refraction of the sample 6•7 : (21 - NA)(1 - cos i) 21(1- cosi) - NA (12) 11=------- where N is the number of shifted fringes, e is the angle of rotation of the sample and t is the thickness of the sample. Once the interference fringe pattern is achieved, we can touch the table, the mirrors, or the beamsplitter and watch them oscillate. These interference fringe patterns can be used to analyze any vibrations occurring on the table surface and/or any movement of thc optical components. It follows that if the interfercnce pattern is stationary, there are no vibrations and/or component movcment occurring in or on thc system. If the pattcrn moves rapidly and then settles dO\\ll. the table is rcceiving ground vibrations. These types of vibrations may be caused by moving vehicles. peoplc walking in an adjaccnt room. c1c\·ators. 16 dishwashers, etc. It is best to run this analysis in the evenings or on the weekend when the surrounding environment is quieter. If the fringe pattern moves slowly back and forth, it is due to air currents in the room. If central air conditioning is present, the thermostat or input vent to the room should be turned off. This type of pattern movement may also be caused by optical components that are not locked tightly into position. So is made sure that all thumb screws on the optical mounts are tightened. 2.1.2 Theory behind rotating the sample: If a plane parallel plate of index of refraction n is inserted normal to the path of one of the beams of light traversing the arms of a Michelson interferometer, the increase of optical path introduced will be 2(11 - 1), where t is the thickness of the plate. The factor 2 occurs because light traverses twice through the glass piece. For light of wavelength A, the difference of path introduced is NA, where N is the number of fringes displaced when the plate is inserted. If the plate is rotated through a small measured angle, the path of the light will be changed, and the no. of fringes N corresponding to this change is counted. This is because of the fact that as the plate is rotated, the length of the glass in the path will increase and therefore no. of wavelengths in that path will increase. This will change the interference pattern. 17 '-------,-----' t d' d d'=~ d'>d cose Figure 6: Demonstrating the change in path length As e increases, cos edecreases, so d' increases. Thus we can see that the path length through the glass plate increases as we rotate it through some angle. When projected on a screen, it can be seen that the fringes collapse as the glass plate is rotated. The change of the path through the glass plate depends upon the thickness of the plate, the angle through which it is rotated and the index of refraction. So the index of refraction may be calculated if the other two values are known. IS 2.1.3. Derivation of the relationship between the refractive index, number of fringes collapsing and the angle of rotation of the sample o A / ( "- / / , "- "- , '..[J/r.://.' / d-/-"/ / / t b ::r.------"'····c I i p "'" "- ~,:T, - " " Figure 7: Analysis of the sample being rotated 6 The expression for index of refraction of the sample in tenns of the number of fringes collapsing and the angle by which the sample has been rotated is given by G.S.Monk 6 Let OP be the original direction of light normal to plate of thickness' l' as shown in Fig. 7. The total optical path between a and c initially for the propagation of light is 111 + be where '11' is the index of refraction of the plate. When plate is rotated through an angle' i ., optical path is increased to ad * 11 + de. So the total increase in optical path is e5) -e5 2 = 2(ad* 11+de-111-be) = ;V), (13) where N is the number of fringes collapsing or appearing on the screen and i, is the wayclength of the light source used. 19 But we have I ab=-cosr (14) where' r ' is the angle of refraction of the light beam. We also have de = dc sin i = (fc - bc = - I- - I fd) sin i = I tan isin i-I tan r sin i (15) (16) cosi Substituting Egs. (14-16) into (13), we have 5: 5: VI - V 2 ' . =-?(-III- + I tan I.SIn I- cosr Now using Snell's Law 11 . . I) I tan r SIn I - 111 - - - . + I (17) COSI 11 sin,. = sin i , we get the expression for index of refraction as (21 - N).. )(1- cos i) =-'---------''--'----------'- (18) 2/(1- cosi) - N).. as already given in Eg. (12) This can also be rewritten as N = 2/(1- cosi)(n-I) n}, - (19) },(I - cos i) 20 2.2 Measuring the refractive index from the intensity of the fringe pattern: S 633nm (35mVV) He-Ne source L1 Camera lens 25mrn (F1.4) '- AGILENT DSC6054A OsciDoscope M1 Figure 8: Michelson Interferometric setup for measuring refractive index of BK7 sample by measuring the intensit)' of the interference fringe pattern Thc sctup is vcry similar to Fig. 4. Instcad of thc manual rotation stagc uscd thcrc. an automatcd rotation stagc controllcd by a Nc\vport ESP300 controllcr is uscd. It has a vcry good rcsolution of 0.00 1°. this hclps in providing a vcry accuratc valuc of thc anglc of rotation. Oncc thc intcrfercncc pattem is obtaincd aftcr recombining both the retlected beams. the center of the fringe pattem is then passed through a pinhole P and neutral density filters and then detected using a InGaAs detector D. If the intensity of the signal is too 21 high, the detector mechanism would not have the capability to detect it, hence the neutral density filters are used to reduce the intensity of the signal. The detector 0 is then connected to an oscilloscope to view and store the real-time wavefonn. 2.2.1 Procedure for Alignment: Instead of the manual rotation stage in the previous case, the sample is mounted on an automated rotation stage Newport SR50CC which is compatible with the Newport ESP300 Controller. Once the fringes are seen on screen, the screen is then removed and a pinhole P is fixed so that only the center of the fringe pattern goes through the hole. Since this may have a very high intensity, neutral density filters are used to reduce the intensity of the signal. The InGaAs detector is mounted carefully so that it catches the center fringe and transmits the signal to the oscilloscope. We use an Agilent DSC6054A Oscilloscope which has an attached USB which facilitates easy storing of the wavefoml in any jump drive. The BK7 Windows sample, after it has been aligned perpendicular the aml is then rotated in either direction through any desired angle. The real-time wavefoml trace as the sample rotates from the left to right is taken. observed in the oscilloscope and then stored in the jump drive for analysis. This process is repeated a number of times to get more number of measurements for accurate analysis. The velocity and acceleration of the SR50CC rotation stage can be controlled through the controller to see the effects of changing these parameters in the final stored wavefonn. .,., CHAPTER 3: DERIVATIONS AND RESULTS 3.1 Measuring the refractive index of BK7 Windows: 3.1.1 Plotting angle of rotation with respect to the number of fringes: The angle of rotation through which the sample has been rotated is calculated and noted down for every 5 or 10 fringes collapsing. It is then plotted with respect to the number of fringes and a non-linear curve fitting is performed to get the value of refractive index. The expression for refractive index derived in Eq.( 18) is then used as the formula for performing the non-linear curve fit. In the Figs. 9(a-d), 'a' denote the thickness of the sample and 'I' denotes the wavelength of the light source used which corresponds to the HeNe wavelength 633nm. Both these parameters are kept fixed and the refractive index 'n' is kept as a floating parameter to get an accurate fit to the experimental values. This fit value of refractive index obtained is highlighted in blue in all the plots. This value is then compared with the expected value of refractive index for BK7 windows at 633nm and room temperature 25°C which is 1.51509. The average of all the values obtained from the fits for refractive index is then calculated and standard deviation from the expected value is then calculated and reported in Table 1. 1.000 Data:Data1_B Model: ref-index Ch i"2 R"2 o.gga -;- = 6 .0507E-10 = 0 .gOOg2 1.51313 6.33 E·7 0.00507 0.g06 :0.00100 :0 :0 Ol C n '"0 0.g04 (J 0.g02 O.gOO -t---r---.---,.--,.----,,----,---..,.----.---.--.....---, o 10 20 40 30 50 No.offrlnges Figure 9(a): Plotting cosine of the angle of rotation with respect to the number of fringes collapsing or appearing. Refractive index of n = 1.51313 obtained. Dab Dab1_B o.goa Mod el: ref- ind ex • 0.006 -;Ol C n '"0 (J Chi"2 R"2 • 5.6653E-g = 0 .00051 n 1.50407 6.33E-7 0.00507 :000332 :0 :0 0.004 0.002 0.000 0.088 10 20 30 50 eo No. of fringes Figure 9(b): Plotting cosine of the angle of rotation with respect to the number of fringes collapsing or appearing. Refractive index of n 24 = 1.50497 obtained. 1.000 Data: D alal_B ~'" 0998 ~ 099t:l '6l c .!!. 1.3024E·9 C hi"2 ~ R"2 = 0.99983 15282t:l t:l .33E·7 :l;0.001t:l9 :1;0 0.00507 :1;0 "- ..., 0.994 0 Model: ret index (,) ~ 09Q2 0.990 ~". -t--..--,---.---,--.--,-----.--,---,...--..,---, 40 o 10 20 30 50 No.offringes Figure 9(c): Plotting cosine of the angle of rotation with respect to the number of fringes collapsing or appearing. Refractive index of n = 1.52826 obtained. 1.000 D ala: D atal_C Model: ref index • 0.998 0.99992 n _ ell 55878E·l0 0996 153106 10.00112 6.33E·7 10 0.00507 10 0) C " c; Og94 (,) 0.992 .. O. 99 0 -t--,....--.---..-----,r-----.--,----r----.----r---.--------, o 10 40 30 50 No. of fringes Figure 9(d): Plotting cosine of the angle of rotation with respect to the number of fringes collapsing or appearing. Refractive index of n = 1.53106 obtained. 25 Measurement no. Refractive Index(n) 1 1.51313 2 1.50497 3 1.52826 4 1.53106 Mean of all measurements = 1.519355 Standard deviation = 0.015706523 Table 1: The measured values of refractive indices presented in Figs. 9(a-d) along with the mean and standard deviations 3.1.2 Difference Angle derivation: The zero position of the fringe is problematic and if not known exactly may lead to missing a fringe thereby leading to inappropriate values. So inorder to exclude this from consideration, we derive a difference angle formula where the value of refractive index just depends upon the difference between the angles through which the sample has been rotated and the thickness of the sample. We have from (18). (20) Also 11 N~).)(I- cos0 2 ) 21(1- cosO~) - N~). = (21 - (21 ) 26 where NI' N 2 denote the number of fringes and (JI' (J2 denote the angle through which the sample has been rotated from the center. We can write this in terms of sine and cosine of angles as (22) I cos(fl) = (4(11 -1)1 + N A(2 - 11))2 2 2(2(11 -1)1 + NIA 1 (23) (24) I cos(!l) = (4(11 -1)( + N A(2 - 11))2 2 2(2(11-1)/+N 2 A 2 (25) (26) So substituting equations (22) to (25) , we have co { OI-(2) 2 4(n-l)(+NIA(2-11) =1 ----'---- o -0 T=cos- 4(n-l)1 + 2i\V- I 4(n-l)/+N2A(2-n) \ 4(n-l)( + 2N2 A [ 4(n-))t+N 1 ),(2-n) 4(n-!)t+N 1),(2-n) + 4(n-))t+2N I ), \ 4(ll-))t+2N 1), + I I NInA 4(n-l)( + 21\V- ~ 4(11-1)( + 2N/ NInA. f N1n), \ 4(ll-))t+2N 1 ),\4(n-))t+2N), (27) 27 N 2nA ] 3.1.3. Plotting difference III angle of rotation with respect to the number of fringes Figure 10 shows the simulation for ideal value of refractive index for BK7 sample. Here a fringe spacing of lOis fixed so that M1 and (M I+10) gives the fixed fringe separation. Diff (M I) represents half of the difference in angle through which the sample is rotated to achieve this fringe spacing (that is ()I - ()2 ). 2 This is then plotted with respect to the number of fringes to give the ideal difference angle plot. I "jIJlIO -3 D" Ul.lW at 633nm, 25deg celsIus room temp fOI BK7 \'I'here" denotes thickness of sample and?' denotes wa\'l!length of the laser light source used om, / I /' / I I om: I I D:T{Ml) I ,I O%~ I I I O%;~ :~'---_-----'_ _---I. .. I I -_ _.L-I_~ [ Figurc 10: Simulation for idcal yalucs of rcfractin index for BK7 sample 28 For a fringe spacing of 10 fringes (that is M 2 = M, + 10), 8) and 82 are measured and noted down. The cosine of half of the difference angle is calculated and ploned with respect to the number of fringes. This information is free of the zero position and yields an effective way to accurately measure the refractive index. In the Figs. I I(a-h), 'a' denotes the thickness of the sample and 'I' denotes the wavelength of the light source used which corresponds to the HeNe wavelength 633nrn. These parameters are kept fixed and the refractive index 'n' is kept as a floating parameter to get an accurate fit to the experimental values. This value is again compared with the expected value of refractive index for BK7 windows at 633nm and room temperature 25°C which is 1.51509. The average of all the values obtained from the fits for refractive index is then calculated and standard deviation from the expected value is then calculated and reported in Table 2. 29 ~ o.gggn .~.," ::::;:,., / O.ggggO C hi·2 R·2 • 2.7504E·13 • 0.ggg52 1.51517 O.33E·7 0.00507 Ogggg3 Ogggg2 :to.0042 :to :to . / 10 15 20 30 25 35 40 No.offringcs Figure II(a): Plotting cosine of the difference in angle of rotation with respect to the number of fringes collapsing or appearing. Refractive index of n = 1.51517 obtained. OggggB ~ o gggg7 _ 0 ggggO «II Cl c: " • ~--------, D.h D.I.l_B .: 0 gggg5 t.Aodtl diN-.oglt .. 0 gggg4 Chi"2 R"2 ;; 'lJ o u • 47405E·13 • Og.gl1 151571 000507 o 32BE·7 o gggg3 / to 00552 to to • 10 15 20 25 30 35 No.of fringes Figure 11(b): Plotting cosine of the difference in angle of rotation with respect to the number of fringes collapsing or appearing. Refractin index of n obtained. 30 = 1.51571 0.00008 0.09997 00gg90 ~ tll l: I'J - Data: Data1 B 0.09995 Mo del diN·a ng le2 .E ~ Chi"2 RA2 ~ 0.09004 III = 1 .7253 E·13 = 0.0907 0 (J 0.09993 1.51355 ±O .0033 0.33E· 7 ±O 0.00507 ±O 009992 10 15 20 30 No.offrlnges Figure 11 (C): Plotting cosine of the difference in angle of rotation with respect to the number of fringes collapsing or appearing. Refractive index of n = 1.51355 obtained. 09999g5 ~~-=--=. o .gggggO // -;- o .ggg985 tll l: I'J I I o gg99BO .E ~ o gg9975 '"0 C hi A2 R A2 t ~ (J Data Data1_B Model diN-angie o g99970 o 9g9905 • 18780E· 12 • 098350 1 51734 000507 033E·7 iOO~7 ±O 10 1 o gggg:lO 0 10 20 40 30 50 No. of fringes Figure 11(d): Plotting cosine of the difference in angle of rotation with respect to the number of fringes collapsing or appearing. Refractin index of n obtained. 31 = 1.51734 Oggggg5 OgggggO :§" 0 ggggS5 0010: 00101_8 Cl c: Modtl diH·onglt n ~ O.ggggSO C hi A2 RA2 ~ " - • 2.7502E·13 • a.ggOSg 0.gggg75 1.512g0 0.33E·7 0.00507 OIl o <.J Ogggg70 to.Ol10g to to Ogggg05 a 10 20 30 50 Ho. of hinges Figure l1(e): Plotting cosine of the difference in angle of rotation with respect to the number of fringes collapsing or appearing. Refractive index of n == 1.51296 obtained. OgggggO ~ a oh g> OggggSO c: - Ogggg75 ~ VI 0 gggg70 "o O.t.l_8 Model diH.• ngle n Chi'2 R A2 <.J • 5200BE·13 • 0.gg010 1 50037 to oogOl 0.33E· 7 000507 to to OggggOO • o gggg55 -t--......----,--.....--...--.....--,.--.,....--,.--.--------, 10 30 o 20 50 Ho. offringes Figure 11(f): Plotting cosine of the difference in angle of rotation with respect to the number of fringes collapsing or appearing. Refractive index of n obtained. == 1.50637 0.IlIlIlQ8 ~ O.IlIlIlQ7 -;- 0.IlQIlQ6 A - t a - :-Da-t-a2-8------1- 'El c: " .S 0.IlQIlQ5 ~ 'C iii o 0.IlQIlQ4 o 0.IlQIlQ3 0.IlQIlQ2 / 10 - / 15 Model: ditt·angle 20 Chi"2 : 4.B025E·13 RA 2 : 0.11111112 25 1.52317 to.005114 6.33E· 7 0.00507 to to 30 35 No.offriJlges Figure II(g): Plotting cosine of the difference in angle of rotation with respect to the number of fringes collapsing or appearing. Refractive index of n = 1.52317 obtained. O.IXlIXlQ O.IXlIXlB 0.1Xl1Xl7 -;~ O.IXlIXlO n C ~ 0.IXHXI5 !!. ., o Chr"2 • 18113E·12 R"2 • 0.QQ078 0 OQ0Q4 U I 5254Q fO OlOQ8 000507 fO 033E· 7 fO • 10 30 40 No. of fringes Figure 11(h): Plotting cosine of the difference in angle of rotation with respect to the number of fringes collapsing or appearing. Refractive index of n = 1.52549 obtained. The bar graph shown in Fig. 12 gives the approximate mean value of refractive index calculated from these set of measurements represented in Figs. 11 (a-h). The standard deviation from the mean value obtained from the graph is also calculated. 1.525 1.520 x CIJ 'tJ ..5 ~ 1.515 :p (.) ~a::: 1.510 1.505 1.500 +""~"-r 2 3 4 6 7 8 9 Measurement No. Figure 12: Refractive Index bar graph for Figs. 11(a-h) From the bar graph, the Mean of the refractive index was measured to be 1.51622 with a standard deviation of 0.00561. This method of measuring the refractive index manually counting the number of fringes shifting corresponding to the angle of rotations proves very cumbersome because of the following disadvantages: a) There is a possibility of missing some fringes because of the disadvantage of measurement-by-eye 34 b) Refractive Index precision of 2-3 digits might not be achieved if there is any small error in the measurement of the angle c) The zero position of the fringe is a very abstract concept and it is very difficult to reference it to a fixed position, so some fringes may be missed while doing the manual measurement. In order to overcome all these disadvantages, an automated method of measuring refractive index is proposed wherein the sample is mounted on top of an automated rotation stage whose movement is precisely controlled by a controller. The controller velocity and acceleration can be set to any desired value. The sample is rotated through any desired angle and a complete waveform of the intensity change is measured in the oscilloscope. We will propose a way to take this measured waveform and do a non-linear curve fitting with the proper derived intensity formula and get the refractive index to a high level of accuracy. 3.1.4 Derivation of the expression for intensity in terms of refractive index, thickness of the sample and the angle of rotation of the sample We have from Equation (6), (27) where I denotes the intensity of the resultant wave. A is the amplitude of the resultant waw and 9), and 9)2 are phase angles corresponding to each of the optical paths. 35 The expression for intensity can now be written as (28) Now substituting the value for optical path difference from (15) and using Snell's law for expanding, the expression for intensity comes out to be of the form 2 l(a,b,O):=a+b cos 2'11 2 n .\ (29) 3.1.5 Analysis of the stored waveform: When executing a move command in the controller to move the rotation stage by a specified angle, the stage will accelerate until the velocity reaches a pre-defined value. Both this acceleration and constant velocity can be set to a specified value from the controller. Then at the proper time, it will start decelerating so that when the motor stops, the stage is at the correct final position. The velocity plot of this type of motion will have a trapezoidal shape as shO\\11 in Fig. 13. 36 Desired Velocity Time Figure 13: Trapezoidal Motion Profile for the controller So inorder to account for this trapezoidal motion, the stored waveform in the oscilloscope has been modified in accordance with the principles of angular velocity and acceleration to account for the working operation of the controller. So since the difference between successful positions of the stage at continuous angles (!18 ) is not constant as a function of time, from the whole trace stored in the oscilloscope, the accelerating and decelerating part of the wavefonn is cut out and only the constant velocity part used for analysis. Analysis of BK7 Window sample for a particular case is shO\m below. If the sample is rotated through an angle of 9 degrces, the constant angular acceleration a is sct to 2 dcgls 2 and final angular velocity OJ of the rotation stage is set to 0.5 deg/s, the calculations involn::d in analyzing the wavcform can bc summarizcd bclow: Using the cquations for constant angular acccicration. wc havc 37 OJ where = OJo + al OJ o (30) is the initial angular velocity and t is is the time taken to reach the constant angular velocity OJ Substituting the values in Eq.(30), we have 0.5 = 2t So the time taken for the rotation stage to achieve constant angular velocity is 0.25 sec. Similarly time taken for the stage to decelerate to zero final velocity is also 0.25 sec. The distance covered by the stage in this time frame () can be calculated using the relation 1 2 () = OJol + -al (31 ) 2 Plugging in the values of a = 2deg/s 2 and t = 0.25 sec in Eq.(31), we get () = 0.0625 0 (32) So the linear angular distance traveled by the rotation stage once it has reached a constant angular velocity of (t) = 0.5 deg/ s can be given as 0/ = 9 - 2 * (0.0625) = 8.875 0 So 8.875 0 1/ = 0.5 deg/ s * 1/ ' (33) this gives the time taken for linear angular motion as = 17.75 sec So the total time taken 1=1/ + 0.25 + 0.25 = 18.25sec (34) 38 The analysis now for a particular plot is done as follows: The data for the plot contains 905 data points overall and this now corresponds to the total time of 18.25sec. 905 po int s ¢:> 18.25 sec Ipoint ¢:> 0.02016575sec . . . I 0.25 sec ¢:> So mltIa 0.25 0.02016575 ::::: 13 th po mt ' Linear angular motion continues till 18 sec So deceleration starts from 18 : : : 893 rJ po int 0.02016575 So only the portion of the waveform from the 13 th point to 89rd point is considered for analysis, rest of the waveform is cut and excluded from the analysis. Therefore linear angular motion corresponds to 880 po int s ¢:> 8.875° Ipoint ¢:> 0.010085227° Each data point is then multiplied by this value to obtain the exact angular position of the rotation stage relative to the intensity measured at each point. 39 3.1.5 Measuring the refractive index of BK7 Windows by plotting the intensity of the fringe pattern with respect to the angle of rotation: t= 0.9410- 3 J..= 63310- 9 n= 1.46 2 7,.-----.----.----,-----.,.-----.---,-------,-----,---,.------, 6 5 I(24,4J,e) 4 3 \ 2 -10 -8 -6 -4 -2 0 2 4 I \ l 8 10 e Figure 14: Intensity Fringe Pattern Simulation for measuring refractive index Figure 14 shows a simulation for the intensity of the measured signal with respect to the angle of rotation. An arbitrary yalue of thickness'( of 0.94 mm and a refractiyc index of n= 1.46 arc chosen. Here' a' and 'b' represents scaling factors that control 40 the height and amplitude of the waveform respectively. Appropriate values of a and b are chosen to represent the signal effectively. Figures 15(a-u) represent the observed wavefonn for the BK7 sample having thickness'd' of 5.17mm. A non-linear curve fit is made on the intensity pattern using Eq. (29) which gives the relation between the intensity of the fringe pattern and the angle of rotation. Appropriate values of a and b are chosen for scaling the fit properly with the signal. 8 9 10 11 12 8 Data: B K7 Sample Mode I: inte ns ity s imu lation C hi"2 R"2 \ = 1.74315 = 0.81119 a 7 b n ±O ·7 1.51395 ±0.00073 d 0.00517 ±O ±O II o ~ \ 4 Angle of Rotation Figure 15(a): Plotting intensity with respect to the angle of rotation. Non-linear curve fitting yields a refractive index of 1.51395 Thc wa\'cfonn pcaks towards both sidcs of thc ccntcr zcro fringc. corrcsponding to zcro dcgrcc anglc of rotation. arc numbercd in ordcr. as ShO\\11 in Fig. 15(a-u). Fig. 15(a-b) corrcspond to thc samc data. non-lincar fits arc madc on pcaks 8-12 in Fig. 41 15(a) and peaks 10-12 in Fig. 15(b) keeping a, band d parameters fixed and refractive index n as a floating parameter. Refractive index values of 1.51395 and 1.51489 are obtained which corresponds to an accuracy of two decimal places. 10 11 12 6 A Data: BK7 Sample Mod el: intens ity s im ulatio n o V. I Chi"2 R"2 = 1.62224 = 0 .66142 7 :1:0 b ·7 :1:0 n 1.5148Q :1:0.00083 d 0.00517 :1:0 v V V ~ 4 Angle of Rotiltion Figurc 15(b): Plotting intensity with respect to the angle of rotation. Non-Iincar cun'c fitting yields a refractive index of 1.51489 42 -16 -15 -14 6 Data BK7 Sample Mo de I: inte ns ity s im ulatio n Chr"2 R"2 = 0.46303 = 0 .89791 7 ±O ±O ·7 1.51501 0.00517 v o v I v ±0.00038 ±O v ·4 Angle of Rotation Figure 15(c): Plotting intensity with respect to the angle of rotation. Non-linear curve fitting yields a refractive index of 1.51501 In Fig. 15(c), a non-linear curve fit is made on peaks 14-16 towards the left of the center fringe and a refractive index value of 1.51501 is obtained. Comparing with the ideal refractive index value for BK7 glass at room temperature- 1.51509. it is observed that this is the best fit achieved and the most accurate value of index obtained from the measurement. 1" 'tJ -14 -13 -12 6 A A ~ D ala: BK7 Sample Mod el: inlens ity s im u lalion C hi"2 R"2 = 0.79726 = 0.84132 a 7 ±O ±O ·7 1.50921 0.00517 b n d ~0.00052 ~O I) v o ~ V V \J ·4 Angle of Rotation Figure 15(d): Plotting intensity with respect to the angle of rotation. Non-linear curve fitting yields a refractive index of 1.50921 A non-linear curve fit is made on peaks 12-14 towards the left of the center fringe in Fig. l5(d). The scaling factors a and b, as well as thickness d are kept fixed and refractive index n is defined as a floating parameter to obtain a value of 1.50921 for the index. 44 7 8 9 ... ... 17 I 1\ 8 \ If 1 Data· BK7 Sample Model: Inlensrty simulation R"2 = 1.05040 = 0.75Q22 a b n d 0.5 to ·0.5 to 1.51763 0.00517 Chi"2 , o ~ to .00030 to V 4 Angle of Rotation Figure 15(e): Plotting intensity with respect to the angle of rotation. Non-linear curve fitting yields a refractive index of 1.51763 13 14 15 16 8 D ala: BK7 Sample Model intensity simulation C hi'2 R '2 = 0.(/472 = 0.BB403 (/5 ·(/.5 to to 151757 000517 to to 00035 o 4 Angle of Rot.ltion Figure 15(f): Plotting intensit)· with respect to thc anglc of rotation. Non-lincar cun'c fitting yiclds a rcfractive indcx of 1.51757 45 Fig. 15(e-g) correspond to the same measurement. Non-linear curve fits are made on peaks 7-17 in Fig. 15(e), peaks 13-16 in Fig. 15(f) and peaks 11-13 in Fig. 15(g). Scaling factors and thickness are kept as fixed parameters and refractive index n is defined as a floating parameter. Refractive index values of 1.51763, 1.51757 and 1.519 are obtained respectively which again corresponds to an accuracy upto two decimal places. 11 12 13 ( Data: BK7 Sample Model: intensity simulation Chi"2 R"2 n = 0.92082 = 0 .90091 9.5 10 ·9.5 10 1.519 10.00046 0.00517 10 o 4 Angle of Rotiltion Figure 15(g): Plotting intensif)' with respect to the angle of rotation. Non-linear cun'e fitting yields a refractive index of 1.519 Fig. 15(h) corrcsponds to a diffcrcnt mcasurcmcnt. a non-linear curyc fit is made on peaks 13-15 towards the right of the ccntcr fringc and a rcfracti,·c index yalue of 1.50691 is obtained. 46 13 14 15 h D ala: B K7 Sample Model: intensity simulation C hi"2 R"2 = 1.11914 = 0.84037 7 ·7 n o ±O ±O 1.50691 0.00517 ±O .00061 ±O v v 4 Angle of Rotatio n Figure 15(h): Plotting intensity with respect to the angle of rotation. Non-linear curve fitting yields a refractive index of 1.50691 Fig. 15(i-k) correspond to the same measurement. Non-linear curve fits are made on peaks 4-12 in Fig. 15(i), peaks 6-8 in Fig. 150) and peaks 11-13 in Fig. 15(k), all towards the left side of the center fringe, keeping a, band d parameters fixed and refractive index n as a floating parameter. Refractive index values of 1.50985. 1.50854 and 1.50953 are obtained which again corresponds to an accuracy upto two decimal places. It is observed again that these values for index are very close to the ideal index value of 1.51509 for BK7 Windows at room temperature. 47 g ·12 .. .. ·5 -4 M 6 J 03t3: BK7 Sampl~ Model: intensrty simulation Chi'2 R'2 = 0.68268 • 0.00230 3 b n d 7 :to -7 :to 1.50085 0.00517 :1:0.00038 :1:0 o -4 -2 Angle of Rotation Figure 15(i): Plotting intensity with respect to the angle of rotation. Non-linear curve fitting yields a refractive index of 1.50985 g ~ 6 ·8 ·7 f ~ -6 ~ Data: BK7 Sample Model: intensity simulation i:' Cht'2 RA 2 ·iii ~ 1: 3 7 :1:0 ·7 10 150854 000517 IV 0 • 0.501171 e 0.Q2721 Vi V y 10000611 10 v ·2 Angle of Rol.11ion Figurc 15m: Plotting intcnsit)- with rcspcct to thc anglc of rotation. Non-Iincar cun-c fitting yiclds a rcfractive index of 1.50854 48 ·13 ·12 ·11 f\ 1\ ~ A " Data: BK7 Sample Model: intensity simulation C hi"2 R"2 = 0.4361 = 0.94043 a 7 -7 b 1.60963 ±0.00042 0.00617 ±O n d V ~ ~ o ±O ±O ~} V ~J ~ -4 Angle of Rotlltion Figure 15(k): Plotting intensity with respect to the angle of rotation. Non-linear curve fitting yields a refractive index of 1.50953 -10 -9 -8 ~ A o.t" 8 K7 Sample Modol: inhnsly simul.tion o Chi'2 • 0.24371 R'2 • 0.Q0701 • 8 i to n -8 to 150071 d 000517 b to .00034 to } o V ~ V ~ V AIl!}le of ROt.ltioll Figure 15(1): Plotting intensit)- with respcct to the anglc of rotation. Non-linear cun-c fitting yields a refractive index of 1.50071 49 A non-linear curve fit is made on peaks 8-10 towards the left of the center fringe in Fig. 15(1). The scaling factors, a and b, which are chosen to be 8 and -8, as well as thickness d of 5.17mm, are kept as fixed parameters, and refractive index n is set to be a floating parameter. A value of 1.50071 for the index is obtained from the fit. -18.. .. -14 h ! 8 D ala: B K7 Sample Model: intensity simulation C hi"2 R"2 = 1.5581 = 0.80857 a 9.5 b n d ·9.5 10 1.5242 10.00044 0.00517 10 \ I I 10 I V o ·4 Angle of Rotation Figurc 15(m): Plotting intcnsity with rcspcct to thc anglc of rotation. Non-lincar cun'c fitting yiclds a rcfractivc indcx of 1.5242 In Figs. 15(m) and 15(n), which corrcsponds to two scparatc mcasurcmcnts. a constant vclocity of 0.25 dcg/s and an acccicration and dcccleration of I deg/s 2 is set to be the controller paramcters. Analysis is donc using thescs valucs to see the effect of velocity regulation in the valucs for refractivc index. Non-linear curve fits are madc on peaks 14-18 in Fig. 15(m) and peaks 10-13 in Fig. 15(n). both towards the 50 left of the center fringe. Keeping a, b and d parameters fixed, refractive index values of 1.5242 and 1.52252 are obtained respectively. It is observed that using a lower velocity and acceleration value for the controller, there are more jerks in the measured waveform and corresponding refractive index values slightly deviate away from the ideal value. -13 -12 -11 -10 Data: BK7 Sample Model: inte nsity simulation = Chi'2 R'2 = 0.81Q71 a Q5 b n d ·Q.5 to 152252 0.00517 J v 15677 to to .00065 to V ~' ·2 ·4 Anule of Rotation Figurc 15(n): Plotting intcnsity with rcspcct to the anglc of rotation. Non-linear cun'c fitting yields a rcfractivc indcx of 1.52252 A non-linear curve fit is made on peaks 3-17 towards the left of the center fringc in Fig. 15(0). Thc scaling factors, a and b, which arc choscn to bc 9.5 and -9.5, as wcll as thickncss d of 5.17mm, arc kcpt as fixcd paramctcrs and rcfractivc indcx n is set to bc a floating parameter. A value of 1.52755 for the indcx is obtained from thc fit. 51 -17... .. -4 -3 ; I ~ 8 I D31a: BK7 Sample Model: inlensity simul31ion Chi"2 R"2 = 2.36462 = 0.72007 a g .~ b n d .Q.~ 1~27~~ :t00004Q :to 0.00517 ~ ~ o :to :to ~ V W ·2 ·4 Angle of Rotation Figure 15(0): Plotting intensity with respect to the angle of rotation. Non-linear curve fitting yields a refractive index of 1.52755 Fig. 15(p-q) correspond to the same measurement. Non-linear curve fits are made on peaks 5-9 in Fig. 15(p), peaks 13-15 in Fig. 15(q), both towards the left side of the center fringe keeping a, band d parameters fixed and refractive index n as a floating parameter. Refractive index values of 1.5210 I and 1.52646 are obtained respectively which still corresponds to an accuracy upto two decimal places. Thus we can observe that two digits of accuracy is continuously obtained thereby proving that the measurements are reproducible. 52 - 9. 8 7 - .0-5 ~ 1\ Data: BK7sampie Model: intensity simulation Ii Chi'2 R'2 = 5.74082 = 0.27787 a g .5 ±O .g .5 ±O 1.52101 0.00517 b n d \ V \ J ±0.00138 ±O I V V ·2 Angle of Rot.:ltlon Figure 15(p): Plotting intensity with respect to the angle of rotation. Non-linear curve fitting yields a refractive index of 1.52101 ·15 -14 ·13 Q {\ r .t" 8 K7 Sample Model r,hnsity S:~'TIul.ltion D (\ Chi'2 = 1.33523 R '2 • 0.B23Q3 Q5 .Q 5 ±O ±O 15~(\40 3 000517 to to 00040 \ \ o ·4 Angle of Rot.ltion Figurc 15(q): Plotting intcnsit)· with rcspect to the angle of rotation. Non-lincar cun·c fitting yields a refractive index of 1.52M6 53 8 8 9 10 D ala: BK7 Sample t Model: intensity simulation C hi"2 R"2 ; 0.04753 0.88101 a 0.5 ±O -0.5 ±O 1.53007 0.00517 ; b n d \ V ~ o V V ±O .00061 ±O \ -2 Angle of Rotatio n Figure 15(r): Plotting intensit)' with respect to the angle of rotation. Non-linear curve fitting yields a refractive index of 1.53097 -15 ·14 -13 -12 ) 8 Data BK7 Sample i\ Model intensity simulation ~ 'Vi t 1: C hi'2 R'2 = 0.86778 = 0.80107 a 0.5 10 -0.5 10 1.53005 0.00517 b n 4 d 0 I~ 10.0 0046 10 \ : , , ~ V V V -4 Angle of Rot"tion Figurc 15(s): Plotting intcnsity with rcspcct to thc anglc of rotation. Non-linear cun-c fitting yiclds a rcfractive indcx of 1.53095 54 Fig. 15(r-s) correspond to the same measurement. Non-linear curve fits are made on peaks 8-10in Fig. 15(r), peaks 12-15 in Fig. 15(s). Keeping a, b and d parameters fixed and refractive index n as a floating parameter, index values of 1.53097 and 1.53095 are obtained respectively. A non-linear curve fit is made on peaks 14-16 towards the right of the center fringe in Fig. 15(t). The scaling factors, a and b, which are chosen to be 9.5 and -9.5, as well as thickness d of 5.17mm, are kept as fixed parameters and refractive index n is set to be a floating parameter. A value of 1.50408 for the index is obtained from the fit. 14 15 16 8 Dolo: B K 7 Sample Mod 01: Inlon.ly • 1m ulolio n C hi'2 R'2 = 0.73573 • o.g 1784 Q.5 ·Q.5 :to :to 150408 0.00517 :to 00034 :to o 4 An!lle of Rotation Figurc 15(t): Plotting intcnsit)· with rcspect to the anglc of rotation. Non-lincar cun'c fitting yields a rcfractivc indcx of 1.50408 55 -15 ... .. -5 -4 ~ 8 1\ Data: B K7 Sample Model: intensity simulation Chi·2 R·2 ; 1 .~094B = 0.81607 a b n d g.~ I .g.~ 1.~4189 O.OO~17 l) I V o ±O ±O ±O .00046 ±O V ·2 ·4 Angle of Rotiltion Figure 15(u): Plotting intensity with respect to the angle of rotation. Non-linear cun'e fitting yields a refractive index of 1.54189 Fig. 15(u) represents a non-linear curve fit on peaks 4-15 towards the left side of the center fringe for a certain specific measurement. Since the peak amplitudes are very uneven because of the jerks in the motion of the rotation stage, the fit gives a value of refractive index which is quite deviated from the ideal value. The bar graph shO\\11 in Fig. 16 gives the approximate mean value of refractive index calculated from these set of measurements represented in Figs. 15 (a-u). The standard deviation from the mean value obtained from the graph is also calculated. 56 1.545 1.540 1.535 1.530 X C1I ".5 ~ 'Q (,) ~ ~ 1.525 1.520 1.515 1.510 1.505 1.500 10 15 20 Measurement No. Figure 16: Refractive Index bar graph for Figs. 15(a-u) From the bar graph, the Mean of the refractive index was measured to be 1.51773 with a standard deviation of 0.0 I023. 3.2 Measuring refractive indices of core and cladding sodium ZIDC tellurite glass samples Tellurium oxide based glasses are potential materials for optical device applications due to their nonlinear optical properties. They show high refractive index. and consequently high third order nonlinear susceptibility x'. and high dielectric constant 9. 1O • Also. they are transparent over the visible and near infrared regions. Kim cl alII have measured for a pure Te02 glass a value of X' = 1.41 x 10 -1: CSIi • however 5i it is not easy to obtain this pure glass. Tellurite-based glasses are also promising candidate materials for high Raman amplification applications, due to their high gain coefficients I2 -18 . They have a broad emission band around 1.5 Ilm and are promising for development of broadband integrated optical amplifiers. They exhibit a wide transmission range (0.35 to 5Ilm), the lowest vibrational energy (about 780 em-I) among oxide glass formers leading to fluorescence from additional energy levels compared to silicate or phosphate glasses. They have a low process temperature, good chemical durability, and exhibit significant nonlinear properties I9 '2o .. The zinc tellurite glasses have a higher density and refractive index and are considerably softer than silicate glasses 12 • Tellurite glasses have also been prepared by Stanworth 21 and Yakhkind 22 , who investigated glasses in various binary systems and in several tricomponent systems. Optical constants, densities, and spectral properties were given, together with experimental and calculated regions of glass formation. Tellurite glasses having semiconducting properties have also been reported, particularly in the V205-Te02 system 23 . The glasses obtained have a yellowish color, even when using spectrochemically pure Te02. This color was thus assumed to be a property of the pure glass as with the A1 20 3-Te02, glass prepared by Stanworth 21 . CI&J;5 eompolit.ioll (~ TtOs) 65 70 75 Dault, IfC1ll 2 .~·d 5.33 :i: 0.03 5.35:i: 0.03 5.37:i: 0.03 RdrattlTc indo: ~DOOp (~A.2J·C) bardDea 2.036 265.9:i: 6 263.4:i: 4 274.9:i:15 58 Table 2: Density, Hardness and Refractive Index of Zinc Tellurite Glasses 12 • Gla!>!> compo~illon 30Na0l1' 70TeO~ lONaOl,'2·20ZnO· 70Te02 30ZnO· 70TeO~ tim II) C6.J d 33 (pm/Vi 2.00 2.02 2.05 1.95 1.97 2.00 0.082 0.23 0.45 Table 3: Refractive Indices at 532 and 1064 nm and d 33 values of poled Na 20ZnO-Te0 2 glasses 24 • The second-order nonlinear optical coefficient increases with a replacement ofNaOI/2 by ZnO. This increase can be attributed to the rise in the content of nonbridging Oxygens 25 which contribute to the dipole moments oriented under the dc electric field, and lead to the anisotropic structure. From this viewpoint, a glass containing a large number of nonbridging oxygens is likely to present a large anisotropy. In addition, the high mobility of Na+ enhances the electrochemical reactions at high temperatures, leading to a decrease in the second harmonic intensity as well as the second-order nonlinear optical coefficient observed for the glasses with a large amount of Na20. The glass composition in our case is 75Te02-20ZnO-5Na20-6KNb03 for the core glass sample and 75TeOr20ZnO-5Na20 for the cladding glass sample. Figs. 16 (a-I) gives the obseryed value of refractive index for the core glass sample having thickness'd' of 1. 73 mOl. This is achieved by perfonning a non-linear cUI'\'e fit on the intensity pattern using Eg. (29) which gives the relation between the intensity of the fringe pattern and the angle of rotation. The scaling factors a and b arc kept as 59 floating parameters and appropriate values are chosen for scaling the fit properly with the measured signal. 3.2.1. Intensity plots for measuring refractive index of core glass sample: Fig. 17(a-b) corresponds to the measurement of the intensity pattern as a function of the angle through which the core glass sample has been rotated. Non-linear curve fits are made on peaks 9-12 in Fig. 17(a) and peaks 7-12 in Fig. 17(b). The difference between the two graphs here is that the scaling factors a and b along with the refractive index n are kept as floating parameters and thickness d is kept as a fixed parameter while performing the fit. So appropriate values of a and b are chosen for good fits and refractive index values of 2.01832 and 2.01379 are obtained respectively which again corresponds to values of index upto two decimal places of accuracy. 9 10 11 12 12 DOlO: Core G135S sample Modtl: intt nsity simulJtion Chi'2 R'2 • 0.38030 = 0.g0731 12.oa708 ·5301gg 201832 000173 10 11148 10.1754 10.000g0 10 Angle of Rot.ltion Figure 17(a): Plotting intensit)· with respect to the angle of rotation for the core glass sample. Non-linear cun'e fitting yields a rcfractin index of 2.01832 60 7 12 8 9 10 11 12 Data: Cote Glass Sample Model: into nsity simulation Chi0 2 R0 2 = 1.37417 • 0.08gg0 b -4.84417 2.0 137g 0.00173 11.4777~ n ±0.1 071~ ±0.2 O~g4 ±O.oOI77 ±O I~ \ 4 Angle of Rotiltion Figure 17(b): Plotting intensity with respect to the angle of rotation for the core glass sampleo Non-linear curve fitting yields a refractive index of 2001379 7 6 8 12 R A 9 If\ A r\ \ h M Dan: Cor e GI ass Sam pi e Model: intensrty simulation Chi 0 2 R"2 ~ b 0 '- \.: IV V 4.0 ~ ~ V = 0.g4g31 = 0.7g551 0.07240 5.32201 2.0256g o 00173 a Vi 5i 1; .~ n d ±0.15240 ±O .25272 ±O .00346 ±O l 5 6 4.8 Angle of rotation Figure 17(c): Plotting intensit)- with respect to the angle of rotation for thc corc glass samplc. Non-Iincar cun-c fitting yiclds a refractive indcx of 2002569 61 A non-linear curve fit is made on peaks 6-9 in Fig. 17(c) and on peaks 4-7 in Fig. 17(d), both towards the right of the center fringe. The scaling factors, a and b, are again set as floating parameters along with the refractive index n and arbitrary values are obtained from the fit conforming to a good fit. An index value of 2.02569 and 2.02137 is obtained from the fit which again conforms to an accuracy upto two decimal places. 12 4 5 7 6 D ala: Core Glass Sample M 0 dol inlo ns Ity s Imu lalion C h i"2 R "2 n d = 072508 = 082733 5.73423 5.17207 :l:0.11Q15 10.20041 202137 0.00173 :1:002085 :1:0 4 4 An !.lIe of rotiltio n Figure 17(d): Plotting intensif)' with respect to the angle of rotation for the core glass sample. Non-linear cune fitting yields a refractive index of 2.02137 In fig. 17(e). only a is kept as a floating parameter and b is chosen to be -6 to see the effect of keeping b as a fixed parameter in the final yalues of rcfractiye index. An 62 index value of 2.04862 is obtained which depends on the accuracy of the waveform measured rather than changing or fixing the values of scaling factors. 9 12 A 10 11 12 r~ Data: Data2_B Model: intensity simulation Chi'2 R'2 = 1.70807 = 0.57854 a 12.73460 ·6 ±O 2.04862 0.00173 b n d ±0.13412 ±O .00185 ±O 6 6 Angle of Rottltion Figure 17(e): Plotting intensity with respect to the angle of rotation for the core glass sample. Non-linear curve fitting yields a refractive index of 2.04862 63 7 6 12 8 9 ~ '[ Data: C ore Glass sample Mod e I: Intens tty s 1m ulatio n ~ III c: C hi"2 R "2 8 1.36467 0.70867 12 QI 1: V v V V 4- :to ·8 :to 210047 0.00173 n :to.00 18 :to IV \I V I 4 6 Angle of rot\ltion Figure 17(f): Plotting intensity with respect to the angle of rotation fo~ the core glass sample. Non-linear curve fitting yields a refractive index of 2.10047 4 12 5 6 7 D.I•. COlt GI ... S.rnpl. Modol: inhnsiysimul.lion ~ III c ... .5 I I 8 III J J I Chi A 2 RA 2 n e e 2.02608 0.5287'1 12 to to ·6 2 12777 000173 to.OO371 to 4 4 Angle of Rot.ltioll Figure 17(g): Plotting intcnsity with rcspcct to the angle of rotation for the core glass samplc. Non-lincar cun'c fitting yiclds a rcfractivc indcx of 2.12777 64 5 12 7 6 I~ f\ 1'1 o ala: C ore Glass Sample Mod el: inlens ity s im ula lio n C hi"2 i::' Il R"2 = = 0.70e1l4 0.831l05 'iii lD 1: \ 6 ~ a 12 b ·6 ±O n ±O 2.101l51l ±O .00 1114 d 0.00173 ±O IJ ' - - - - - - -... V 4 An gle of rotatio II Figure 17(h): Plotting intensity with respect to the angle of rotation for the core glass sample. Non-linear cun'e fitting yields a refractive index of 2.10959 Fig. 17(f-h) correspond to the same measurement, non-linear curve fits are made on peaks 6-9 in Fig. 17(f), peaks 4-7 in Fig. 17(g) and peaks 5-8in Fig. 17(h). In all the three graphs, the scaling factors a and b along with the thickness d of the sample, are kept as a fixed parameters, and refractive index n is kept as a floating parameter, while perfom1ing the fit. Refractive index values of2.10047, 212777 and 2.10959 are obtained respectively. It is observed that because of the random variations in the amplitude and spacing between alI the peaks, a comparatively higher valuc of refractive index is obtaincd from this set of data. 65 7 12 8 10 9 A f Data: Core Glass sample Mod e I: int.ns ity 9 ~ s im u lation C hi"2 = O.934Q3 R"2 "0.79092 'Vi lD 1: \ \ 6 ~ OJ ~ V n ~ ~ d 11.42255 ·5.17484 :1:0.41948 :1:0.8117 2 .04Q 18 0.00172 :1:0.96485 :1:0.00077 4 6 Angle of rolalio n Figure 17(i): Plotting intensity with respect to the angle of rotation for the core glass sample. Non-linear curve fitting yields a refractive index of 2.04918 8 12 9 10 11 Dato: Cor. Ola.. Sam pl. Modll: inhns i!y simuLltion g Chi"2 R"2 = 2 .86826 • 030056 12 to ·8 to 2 02384 to.00218 000173 to 6 4 Angle of Rotation Figure 17(j): Plotting intensit)' with respect to the angle of rotation for the eore glass sample. Non-linear eun'e fitting yields a refractin index of 2.0238~ 66 Fig. 17(i) and 170) again correspond to another set of scanned waveforms from the same measurement, the fit values of refractive index are observed to be 2.04918 and 2.02384 for all other parameters kept fixed and n being kept as a floating parameter. Non-linear curve fitting is performed on another set of data in Figs. 17(k) and 17(1) where refractive index values of2.03984 and 2.00683 are obtained. v 4 Angle of Rotation Figure 17(k): Plotting intensit)' with respect to the angle of rotation for the core glass sample. Non-linear cun'e fitting yields a refractivc index of 2.03984 67 -7 ·6 -5 fI V. 9 Data: Co,e Glass Sample Mo de I: inte ns Ity s im u lalion Chi"2 R"2 10 to ·7 to 2.00683 0.00173 b n d v 3 = 1.08558 = 0 .79941 to .00177 to IV ·2 ·4 Angle of rotation Figure 17(1): Plotting intensity with respect to the angle of rotation for the core glass sample. Non-linear curve fitting yields a refractive index of 2.00683 From the measurements of refractive index done on the core glass sample in Figs. 17(a-I), a bar graph is constructed to calculate the mean concentration of the refractive index .The bar graph shown in Fig. 18 gives the approximate mean value of refractive index calculated from these set of measurements and their standard deviation. 68 2.05 2.04 x C1I 2.03 ".5 ~ 'Q (,) i 2.02 0::: 2.01 2 .0 0 -f-LL~"'" 2 3 4 6 7 8 g Measurement no. Figure 18: Refractive Index bar graph for Figs. 17(a-l) From the bar graph, the Mean of the refractive index was measured to be 2.0275 with a standard deviation of 0.0 1509. 3.2.2 Intensity plots for measuring refractive index of cladding glass sample: Figures 19(a-g) gives the observed value of refractive index for the cladding glass sample having the composition 75TeOr20ZnO-5Na::!O and a thickness'd' of approximately 0.84mm. This is achieved by perfornling a non-linear curve fit on the intensity pattern using Eq.(29) which gives the relation between the intensity of the fringe pattern and the angle of rotation 69 In Figs. 19(a) and 19(b), corresponding to the same measurement, appropriate scaling factors of 2 and -1 are chosen for a and b respectively, thickness of the sample is kept as a fixed parameter and refractive index values of 1.99372 and 2.00347 are obtained from the fits. 7 6 2.0 tl r. ~ 1.6 ~ J!J 9 II ~ Data: Clad G lass Sample Mo de I: inte ns ity s imu lation 'Ii A " VI C 8 I Chi"2 R"2 a C lV IV IV 1.2 V V 'J = 0 .04188 = ·0.55561 2 to ·1 to 1.gg372 0.00084 V to.00202 to ~ 6 8 Allgle of Roti'ltioll Figure 19(a): Plotting intensit)' with respect to the angle of rotation for the cladding glass sample. Non-linear curve fitting yields a refractive index of 1.99372 70 4 5 2.0 ~ (, A ~ 113 8 7 6 Data: Clad Glass Sample Model: intensity simulation ~ '" Chi'2 R"2 ~ Vi ±!l ·1 ±!l 2.00347 0.00084 ii 1: 1.2 4 j IV V V \J 1\ = 0.04415 = ·0.27272 I~ 6 :1:0.00227 :1:0 8 Angle of rotation Figure 19(b): Plotting intensity with respect to the angle of rotation for the cladding glass sample. Non-linear curve fitting yields a refractive index of 2.00347 In Figs. 19(c) and 19(d), it is observed that the waveform obtained is not so good compared to other set of data measured. Infact, in general, the traces scanned for the cladding glass sample is not as sharp and clear as the ones obtained for the core glass sample. This is due to the fact that here the sample is very thin (O.84mm). so the optical path length difference is very sensible to even small changes in the angle of rotation of the sample. Hence the waveform trace obtained is more random compared to the earlier traces for both the core and BK7 samples. Refractive indices of 1. 99834 and 1.98967 are obtained from these two graphs after perfomling the non-linear curve fitting. 71 4 ~ 4.5 ~ '0:; ~ 1: 3.6 5 \ 6 7 A \ ! r~ \ ~ 0.10: CI.d gllSs SImple ~ 0 de L in te os ity s imu Ijtion C hl"2 RA2 • -1.31007 5 10 10 ,2 ~ m = 0.25638 1 QQ834 10.0043 000084 10 v v 4 6 Angle of 10t.1llon Figure 19(c): Plotting intensity with respect to the angle of rotation for the cladding glass sample. Non-linear curve fitting yields a refractive index of 1.99834 6 8 7 1\ 48 ~ 40 Vi ~ ~r ~ lD 1: 3.2 ~ I~ V I~ J n I V o.to: CI.d Olus S.mple Mod.1 intensity simuillion C hi A 2 RA 2 • 02081Q • ·1 10363 5 ,2 10 10 , Q8Q67 1000385 000084 10 V e Angle of lolalio n Figure 19(d): Plotting intensif)" with respcct to the angle of rotation for the cladding glass sample. Non-lincar cun"c fitting yiclds a refractiyc index of 1.98967 Similarly refractive index values of 1.99537 and 1.99315 are obtained from Figs. 19(e-f) which correspond to the same measurement. Curve fitting is performed on peaks 5-8 in Fig. 19(e) and peaks 6-10 in Fig. 19(f). An accuracy of upto two decimals is observed. 5 6 {\ f\ 4.8 7 8 A \ r ~ 'in 4.0 ID 1: I~ !~ \ , V ~ A ~ Data:Clad Glass Sample Mod el: intens ity s im ulatio n C hi"2 0.3068 R"2 -2.86367 a b n 5 ±O -2 ±O 1.99537 d 0.00084 to .0031 9 to 3.2 v IJ 6 8 Angle of rotation Figure 19(e): Plotting intensity with respect to the angle of rotation for the cladding glass sample. Non-linear cun'e fitting yields a refractive index of 1.99537 73 7 f\ 6 1\ 8 A 4.8 ~ m 1: 3.2 ~ ,J \l V 10 1\ 1\ II 1\ ~ ~ 'Vi 4.0 9 ~ Data: C lad Glass Sample Model: intensity simulation l~ V V Chi"2 RI\2 = 0.2Q257 = ·2.72736 a b n 5 ±O ±O ·2 1.QQ315 ±0.00256 d 0.00084 ±O ~ 6 8 Angle of Rotation Figure 19(1): Plotting intensity with respect to the angle of rotation for the cladding glass sample. Non-linear curve fitting yields a refractive index of 1.99315 Finally, in Fig. 19(9), non-linear curve fitting is performed on peaks 6-9 on the right side of thc centcr zero fringe. For scaling factors a and b choscn to be 5 and -2 rcspectively, an index value of2.01567 is observed. 74 6 7 8 9 5 Data: Clad Glass Sample Model: intensity simulation Chi"2 0.21593 RA 2 ·1.51313 a b n d 5 5 ·2 ±O ±O 2.01557 ±0.0025 0.00084 ±O 8 Angle of Rotation Figure 19(9): Plotting intensit)' with respect to the angle of rotation for the cladding glass sample. Non-linear curve fitting yields a refractive index of 2.01567 A bar graph is constructed from these measured vales in Figs. 19(a-g) to measure the mean and standard deviation of the refractive index for the cladding glass sample, as shO\\11 in Fig. 20 75 2.020 2.015 2.010 X 2.005 C1I ."s ~ 2.000 'Q (,) ~ 1.995 ~ 1.990 1.985 2 3 4 6 7 Measurement no. Figure 20: Refractive Index bar graph for Figs. 19 (a-g) From the bar graph, the Mean of the refractivc index was mcasured to be 1.99848 with a standard dcviation of 0.00874. Hence it is observed that cladding refractivc index is less than the core refractivc index by 1.43%. 76 Chapter 4: CONCLUSION: We have successfully demonstrated a Michelson Interferometric technique for measuring the refractive index of sodium zinc tellurite glasses based on the concept of changing the path length of one arm of the interferometer by rotating the test sample introduced into that optical path. The intensity of the waveform recorded is plotted against the angle of rotation of the test sample and non-linear curve fitting is done to obtain the refractive index to two digits of precision. The average value of Refractive Index for BK7 Windows sample was measured from the fits to be 1.51622 from the difference angle plots and 1.51773 from the intensity plots with a standard deviation of 0.01023. This value is then compared with the ideal value of refractive index of 1.51509 at room temperature for the BK7 sample and it is observed that a precision of two digits of accuracy has been successfully attained. The average value of refractive index for the core glass tellurite sample having the composition 75Te02-20ZnO-5Na20-6KNb03 is observed to be 2.0275 with a standard deviation of 0.0 1509. Similarly the average value of refractive index for the cladding glass tellurite sample having the composition 75Te02-20ZnO-5Na20 obser..ed to be 1.99848 with a standard deviation of 0.00874. 77 IS Chapter 5: FUTURE WORK This work can be extended to develop a white-light interferometric technique to measure the absolute phase shift and the refractive index for sodium zinc tellurite samples. White light interferometry with the use of a tunable monochromator is a very powerful technique which helps in determining the refractive index for different wavelengths and is a very effective tool in determining the variation in index with respect to wavelengths for these glasses. A double interferometer based system which couples this scannmg Michelson interferometric setup to a modified Mach-Zehnder interferometer (MZI) system has been developed to obtain refractive index directly to four digits of precision 26 . 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Stanworth, "Tellurite Glasses," Nalure, 169 [4301]581-82 (1952). (b) 1. E. Stanworth, "Tellurite Glasses," 1.SOc. Glass Teclmol., 38 [183] 425-35T (1954). 22) A. K. Yakhkind, "Tellurite Glasses," 1. Am. Ceram. Soc., 49 [12] 670-75 (1966). 23) P. L. Baynton, H. Rawson, and 1. E. Stanworth; pp. 52-61 in Proceedings oflhe Fourlh Infernalional Glass Congress, Paris, 1956. Imprimerie Chaix, Paris, 1957. 24) Aiko Narazaki, Katsuhisa Tanaka, Kazuyuki Hiraa, and Naahiro Saga, Journal 0.( Applied Physics, 83,8,3986-3990 (1998) 25) A. Osaka, J. Qiu, T. Fujii, T. Nanba, J. Takada, and Y. Miura, 1. Soc. Maler. Sci. Jpn 42.473 (1993) 26) M.Galli. FJv1arabelli. G.Guizzetti. Applied Oplics. 42,19.3910 (2003) 81 VITA Deepak. N. Iyer was born to Padma Narayanan and A.N.Iyer on May 14, 1983 in Cannanore, India. He completed his high school from Modern School, Nagpur in 1998 with a position in the merit list being ranked 11 th in the national level CBSE Exams. He completed his HSC from Shivaji Science College, Nagpur in 2000 with distinction, being in the merit list. He obtained his Bachelors Degree in Electronics and Telecommunication from Atharva College of Engineering, Mumbai University in June 2004. In pursuit of higher education in the field of Optics, he was accepted into the Masters Program in Electrical Engineering in Lehigh University in August 2004. Since then he has been at Lehigh, working on this research work among other things. 82 END OF TITLE