A Michelson Interferometric Technique for Measuring Refractive Index of sodium zinc tellurite glasses

8/13/2019 A Michelson Interferometric Technique for Measuring Refractive Index of sodium zinc tellurite glasses

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Lehigh University

Lehigh Preserve

Teses and Dissertations

1-1-2006

A Michelson interferometric technique formeasuring refractive index of sodium zinc tellurite

glassesDeepak N. IyerLehigh University

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Recommended CitationIyer, Deepak N., "A Michelson interferometric technique for measuring refractive index of sodium zinc tellurite glasses" (2006). Tesesand Dissertations. Paper 954.
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Iyer Deepak NMichelsonInterferometricforMeasuringRefractive Index

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A MICHELSON INTERFEROMETRIC TECHNIQUE FORMEASURING REFRACTIVE INDEX SODIUM ZINC

TELLURITE GLASSES

ByDeepak N Iyer

ThesisPresented to the Graduate and Research Committee

ofLehigh Universityin Candidacy for the Degree of

Master of Sciencen

Electrical Engineering

Lehigh UniversitvDecember 2 6


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edic ted to p rents


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KNOWLEDGEMENTS

I would like to take this opportunity to thank my advisor Dr Jean Toulouse forproviding me an opportunity to pursue research under his supervision His advancedand profound knowledge in optics has been a constant source invaluable guidancewithout which this work would never have been accomplished

I am indebted to Dr Radha Pattnaik for his immense support throughout this workHis technical input has been an invaluable asset and he has been a source ofinspiration for me at every step of my work

This will be incomplete without thanking my parents Padma Narayanan andA Narayanan Iyer for their constant support love and encouragement without whichI would not have come this far I also thank my sister for her continuous support andmotivation Thank you for being there for me

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T BLE OF CONTENTS

CERTIFIC TE OF PPROV L

CKNOWLEDGEMENTS

T BLE OF CONTENTS

LIST OF FIGURES

LIST OF T LES

STR CT

CH PTER MICHELSON INTERFEROMETER THEORY

Introduction 2 Michelson Interferometer 3 Interference o waves with single frequency

3 Condition for interference monochromatic light

4 Interference o two waves with different frequencies

IV

V

l ll

x

2

235

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CHAPTER 2 EXPERIMENTAL SETUP RREFRACTIVE

INDEX MEASUREMENTS

2 Measuring refractive index by counting the number of fringes

2 Procedure for alignment

2.1.2 Theory behind rotating the sample

3

7

2 3 Derivation of the relationship between the refractive indexnumber of fringes collapsing and the angle of rotation of the sample 9

2 2 Measuring the refractive index from the intensity of the fringe pattern 2 2 Procedure for alignment 22

CHAPTER 3 DERIVATIONS AND RESULTS 233 Measuring the refractive index of BK 7 Windows 23

3.1.1. Plotting angle of rotation with respect to the number of fringes 233.1.2. Difference Angle derivation 263.1.3. Plotting difference in angle of rotation with respect to thenumber of fringes 28

3.1.4. Derivation of the expression for intensity in tenus ofrefractive index. thickness of the sample and the angle ofrotation of the sample3.1.5. Analysis of the stored \\ avefonn

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3 6 Measuring the refractive index BK7 Windowsby plotting the intensity of the fringe pattern with respectto the angle rotation

3 2 Measuring refractive indices core and cladding sodiumzinc tellurite glass samples

3 2 1 Intensity plots for measuring refractive index core glass sample3 2 2 Intensity plots for measuring refractive index ofcladding glass sample

H PTER ON LUSIONH PTER FUTURE WORKREFEREN ESVIT

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40

57

60

69

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LIST O IGUR S

Figure 1: Schematic illustration a Michelson interferometerFigure 2: Formation circles on interferenceFigure 3: Beat Signal from two input frequencies into aMichelson Interferometer

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9

8

9

10

Figure 4: Michelson Interferometric setup for measuring refractiveindex BK7 sample by manually counting the number fringescollapsing or appearingFigure 5: Successive fields of view in interferometer alignmentFigure 6: Demonstrating the change in path lengthFigure 7: Analysis the sample being rotatedFigure 8: Michelson Interferometric setup for measuring refractiveindex BK7 sample by measuring the intensity the interferencefringe patternFigure 9 a-d : Plotting cosine the angle rotation with respect tothe number of fringes collapsing or appearing 24Figure 10: Simulation for ideal valucs ofrefractivc index for BK7 sample 28Figure II a-h : Plotting cosine of thc difference in angle rotation withrespcct to the number of fringes collapsing or appearing 30Figure 12: Refractive Index bar graph for Figs. II a-h 34Figure 13: Trapezoidal ~ o t o n Profile for the controller 37Figure 14: Intensity Fringe Pattern Simulation for mcasuring rcfractivc index 40

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Figure 15 a-I : Plotting intensity with respect to the angle rotationfor the BK7 glass sampleFigure 16: Refractive Index bar graph for Figs. 15 a-uFigure 17 a-I : Plotting intensity with respect to the angle rotation for thecore glass sample 60Figure : Refractive Index bar graph for Figs. 17 a-l 69Figure 19 a : Plotting intensity with respect to the angle rotation for thecladding glass sample. 70Figure 20: Refractive Index bar graph for Figs. 9 a-g 76

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LIST OF T LES

Table : The measured values refractive indices presented in Figs. 9 a-dalong with the mean and standard deviations 6Table 2: Density, Hardness and Refractive Index Zinc Tellurite Glasses 58Table 3: Refractive Indices at 532 and 64 nm and d33 values poled

Na20-ZnO-Te02 glasses 59


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STR CT

The Michelson interferometer is an extremely versatile instrument that can be used tomake an accurate comparison of wavelengths measure the refractive index gasesand transparent solids and determine small changes in lengths quite precisely hemain purpose our research is to develop a Michelson Interferometric setup tomeasure the refractive index of Sodium Zinc Tellurite glasses of known thickness.This is accomplished by rotating the test sample by means an automated rotationstage connected to a controller. The intensity the fringe pattern is measured withthe help of an oscilloscope and non linear curve fitting is done with the appropriateformulae to accurately obtain the value of refractive index. is successfullydemonstrated that refractive index can be obtained to two digits of precision andsamples ranging from a few microns in thickness to a few millimeters can bemeasured.


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CH PTER : MICHELSON INTERFEROMETER THEORY

Introduction

There are in general a number types optical instruments that produce opticalinterference. These instruments are grouped under the generic name ofinterferometers. The Michelson interferometer is perhaps the best known and mostbasic in a family interferometers which includes the Fabry-Perot interferometerthe Twyman-Green interferometer and the Mach-Zehnder interferometer.

The Michelson Interferometer first developed by Albert Michelson in 1881 hasproved vital importance in the development modem physics and is an opticalinstrument high precision and versatility. This versatile instrument was used toestablish experimental evidence for the development for the validity the specialtheory relativity to detect and measure hyperfine structure in line spectra tomeasure the tidal effect of the moon on the earth and to provide a substitute standardfor the meter in temlS of wavelengths light1.2. Michelson himself pioneered muchof this work.

is generally uscd in invcstigations that involvc small changes in optical path lengthsand is ablc to dctcct very small movements on the order nanometers . WithMichelson intcrferomcter. circular and straight-line fringes of both monochromaticlight and white light can bc produced. These fringes can be uscd to make an accurate


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comparison of wavelengths measure the refractive index gases and transparentsolids and determine small changes in length quite precisely. The instrument can beused as a stable mode selecting resonator element in laser cavities as well.

Interferometry has been used for thickness and index measurements thin films 3 4and optical fibers 5. In principle these measurements can be made to great precisiontypically on the order nanometers since interferometric technique uses theinterference nanometer wavelengths. We propose a Michelson Interferometrictechnique to measure the refractive index Sodium Zinc Tellurite bulk glasses. Thisis achieved by rotating the test sample in one arm the interferometer therebycreating a path length difference between the two optical paths the interferometer.Non linear curve fitting is performed on the measured waveform to relate its intensityto the refractive index the sample thereby obtaining values refractive index totwo digits accuracy.

2 Michelson Interferometer

The Michelson interferometer causes interference by splitting a beam light into twoparts Each part is made to travel a different path and brought back together wherethey interfere according to their path length difference 1 2 A diagram the apparatusis shO\m in Fig 1


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reen

M1I.

IILight sour e IC jd

igure Schematic illustration of a Michelson interferometer

The Michelson interferometer operates on the principle of division of amplituderather than on division wavefront. Light from a light source strikes the beamsplitter designated by S and is split into two parts. The beam splitter allows 50 ofthe radiation to be transmitted to the translatable mirror MI. The other 50 of theradiation is reflected back to the fixed mirror M2. Both these mirrors MI and M2 arehighly silvered on their front surfaces to avoid multiple internal reflections. Thecompensator plate C is introduced along this path to have the same optical path lengthwhen MI and M2 are of same distance from the beam splitter. After returning fromMI 50 the light is reflected toward the frosted glass screen. Likewise. 50 ofthclight returning from tvb is transmitted to the glass screen. Thc two beams arcsuperposed and one can obser.e the interference fringe pattern on the screen. Thecharacter the fringes is directly related to the differcnt optical path lengths travcledby the two bcams and. thcrcforc. is relatcd to whatevcr causcs a diffcrcncc in thcoptical path lcngths.

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3 Interference Waves With a Single Frequency

If two waves simultaneously propagate through the same regIOn of space, theresultant electric field at any point in that region is the vector sum of the electric fieldof each wave. This is the principle of superposition. it is assumed that all waves havethe same polarization . If two beams emanate from a common source, but travel overtwo different paths to a detector, the field at the detector will be determined by theoptical path difference, which we will denote by /j x =x2 XI

So if two waves of the same frequency, OJ but of different amplitude and differentphase impinge on one point they are superimposed, or interfere, so that:

I

where 1 and are the amplitudes of both the waves and a l and 2are the phaseangles at any time IThe resulting wave can be described as

y = sin OJI- with A being the resultant amplitude and a the resultant phase.

where is the phase difference which is givcn by

whcre is thc wavelcngth of the light source uscd.

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2

3

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In a Michelson interferometer, light is split into two beams by the beamsplitter(amplitude splitting), reflected by two mirrors, and passed again through the glassplate to produce interference phenomena behind it. A lens is inserted between thelight beam and the beamsplitter so that the light source lies at the focal point, sinceonly enlarged light spots can exhibit interference rings. If the actual mirror Mz isreplaced by its virtual image Mz which is formed by reflection at the glass plate, apoint P of the real light source is formed as the points and P of the virtual lightsources L and Lz as show n in Figure 2.

p 2d p--... . -::::J= -2 d C O S 9 ~ 8L, ...1 L2 ... M Mz . . . . ~ I 1 I p I ( ~ 1l/t\,11 rT1\ 11 I \1 8 r l I ~ f f : : : : E _- H - H t l f - 1 ~ f - - - - - ; L . . . - + - - H - - + - = = = - - - - - - f f - - - - + - - H -1\ \ 1; 11\1 \.t1I I dl ;d--r--

Figure 2: Formation of circles on intcrfercnce

So based on the different light paths, the phase difference, using the symbols ofFigure 2, is:

2ii 2 7i p = tl = dcosOI. ;. (5)

The intensity distribution for a = a = a according to (3) is:, , , DqJI ::::: AL =4aL sL-

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1 3 1 Condition for interference Monochromatic Light

Constructive interference Maxima occurs when

I rp = nm = 0, l 2, 3

So we have from equation 5 ,

dcosO = l lA =0, I, 2, 3

7

8that is circles are produced for a fixed value of and since remains constant see

Fig. 2 . So if both the optical path lengths are the same or if these two paths differ byan integral number of wavelengths the condition for constructive interference ismet. 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 , where = I, 3, 5, and so on, the condition for destructive2

interference is met and dark fringes will be formed. So destructive interference occurswhen

I rp = 2 + I =0, 1 2.3 9This might appear at first sight to violate conservation of energy. However energy isconserved. because there is a re-distribution of energy at the detector in which theenergy at the destructive sites arc re-distributed to the constructive sites. The effect ofthe interference is to alter the share of the reflected light which heads for the detector

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and the remainder which heads back in the direction o the source.

If the position o the movable mirror M1 is changed so that d for example decreasesthen, according to Eq 8 , the diameter of the ring will also decrease since is fixedfor this ring. A ring thus disappears each time d is reduced by half. The ring patterndisappears d = O IfM and M are not parallel, curved bands are obtained which areconverted to straight bands when d = O

4 Interference of Two Waves with ifferent requencies

We will now consider the case o two frequencies with wavenumbers k l and k thattogether follow two different paths with a difference of jx The sum of the waveswith different amplitudes at point x along the x -axis is given by:

10E - k If we let a and define 1 after a lot of algebra, we can write theE 2

intensity rE as:2 + a c o s 2 ~ x + 1 a X c o s k l ~ x a c o s k 2 ~ X I I

The expected signal which consists o a fast oscillation as well as slow oscillationcharacteristic of is shown in Fig 3

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Figure : Beat Signal from two input frequencies into a Michelsonnt rf rom t r


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CHAPTER : EXPERIMENTAL SETUPS FOR REFRACTIVE INDEXMEASUREMENT

2 Measuring the refractive index by counting the number fringes

633nm 35mVVHe-Ne sourceL

Camera lens25mm F1.4reeno

K sample

/Manual rotation stage

2Camera lens55mm :1.7

Figure : Michelson Interferometric setup for measuring refractive index ofBK7 sample by manually counting the number of fringes collapsing orappearing

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The optical setup for the Michelson interferometer is shown schematically in Fig. 4.Light from a elium eon source S wavelength of 633nm) is deflected through amirror M3 and strikes the beam splitter B. The beam splitter is a half-silvered glassplate silvered on the back side) which reflects half of the light toward mirror M2 andtransmits half the light but the entire cross section) toward mirror M. Thedistance, or path length as it is called, between each mirror and the beamsplittershould be the same. These distances can be determined with a tape measure, andshould be as long as possible for the table size. The interferometer s sensitivityincreases the farther the mirrors are from the beamsplitter. Both mirrors then renecttheir respective beams back to the beamsplitter and strike the beamsplitter at theoriginal incident beam s position. Part of mirror MI s reflected beam will then bereflected by the beamsplitter to the screen S and part mirror M2 S reflected beamwill be transmitted by the beamsplitter to the screen S. Tw o beanl spots that arevisible on the screen can th en be superimposed to view the interference pattern. Thisis achieved by moving either mirror, M or M2, slightly up and down and/orsideways.

The BK7 Windows sample is then mounted on a manual rotation stage and inserted inone of the tw o 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 startcollapsing or appearing on the screen. The angle through which the sample has to berotated for a set 5. 10. 15 .... fringes to collapse or appear is noted dO\\ll. This is


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then subsequently used in the fonnula which relates the refractive index and thickness the sample with the angle of rotation to obtain the refractive index to a high degreeof accuracy.

The spots on the screen can be more precisely superimposed by placing a diverginglens L between the laser and the beamsplitter BS The diverging beam lens is placedclose to the beamsplitter BS This increases the diameter the two spots on thescreen so they are easier to superimpose. Another diverging beam lens L2 may also beused to view the interference pattern more clearly while counting the number fringes

A compensating glass identical composition and thickness to the beam splitter isnecessary to produce white-light fringes so that each of the two beams paths d andd2 in Fig 2 passes through the san1e thickness glass. This is due to the fact thatbecause dispersion the optical path lengths will be different for differentwavelengths. Note that otherwise the bean1 that travels along path BM BO dJJ wouldpass through a thickness of glass three times while the beam that travels along theother path BM2BO d2 would pass through the same thickness glass only once. twould not be needed if one only worked with highly monochromatic light whereoptical path lengths the two aons the interferometer can be set to be equal for aparticular wavelength.


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2 Procedure for alignment:

The alignment of the Michelson interferometer can be accomplished severalsimple logical steps.

With the Helium Neon source S centered at one end the table it is positioned to aheight above the optical table that allows the beam to be approximately 6 8 inchesabove the table surface. The source is then turned on waiting several minutes for thelight to reach full intensity. The mirror M3 is used to deflect the light to thebeamsplitter; it is carefully aligned so as to reflect the whole light to the major part ofthe optical table so that there is sufficient space for the whole setup. The beamsplitteris taken out the way the laser beam initially. Making use two pinholes kept along distance away from each other it is made sure that the laser beam is straight andat the same height. The stationary mirror M is now mounted and its position isadjusted until the beam reflected from it re enters the laser. That is the mirror reflectsthe light perfectly back into the barrel of the laser. is always a good idea to have thelaser dot in the center the mirror rather than the edge for proper alignment. Puttinga screen behind the mirror it will be possible to see several red dots surrounding abright red spot. These several dots are focused to one spot by adjusting the knobs onthe back the mirrors. Now the movable mirror is exactly perpendicular to the laserbeam.

The beam splitter is now mounted in between mlITors MJ and M on a manualrotation stage and is rotated 45 so that the laser beam is now split into two beams.


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One beam is transmitted through the beamsplitter to mirror M and the other beam isreflected at 90 degrees to mirror M2. Again mirror M2 is mounted in such a way thatthe reflected beam hits the mirror at the center. The two paths M1 and M2 are madeequal to within a millimeter, using a white index card or small ruler. The reflectedbeams from both the mirrors pass through the beamsplitter again and hit the screen.By means of the two adjusting mirror screws in mirrors M and M2, both points oflight are made to coincide. The test sample BK7 windows is then mounted on amanual rotation stage and inserted into the optical path BM By making use themicrometer screws in the sample mirror mount, the height of the reflected beam fromthe sample is adjusted in reference to the interfering dots on the screen. The two dotsfrom the reflection the sample is then superimposed with the already superimposeddots due to the reflected beam from the mirrors to align the sample perpendicular tothe path BM and to obtain the zero fringe position.

A diverging camera lens 25 mm is now placed in the light beam between mirror M3and the beamsplitter to expand the beam so that the points light are enlarged andthe interference pattems are observed on the screen bands, circles . By carefulreadjustment, an interference image of concentric circles will be obtained. The fringesmay. when first observed, appear rather closely spaced as shO\\l1 in Fig. Sa. While

looking at the fringes. carefully adjust the screws on mirror M so that the fringesbecome circular with their common center lying in the center of your field view asShO\\l1 in Fig. 5b. At this point. the two mirrors .\11 and h should have their planesperpendicular to one another. that is. B \ ] is perpendicular to \h


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a) b) c) d)Figure 5: Successive fields view in interferometer alignment

Normally before any adjustment of the interferometer takes place, the two pathlengths and BM are very likely unequal. Moving mirror M back and forth alongBM will change the difference in path lengths and, in fact permit us to reduce thedifference to very nearly zero. As mirror is moved so as to approach equal pathlengths, the fringes move inward toward the center. The center fringe collapses andeventually disappears altogether while outer fringes start appearing at the edge andmove towards the center as shown in Fig 5c The camera lens is placed betweenthe beamsplitter B and screen 0 and this lens combination L and L is adjusted toview only the center fringe Now the micrometer screw mirror M1 is carefullyturned 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 conditionmay be difficult to achieve since air currents or small table vibrations constantlycause small random changes in the path length. Nevertheless. one should approachthe dark-field condition as closely as possible.). The dark-field condition isextremely useful in detennining the zero fringe position of the sample.that is sampleis exactly perpendicular to the optical path in which it is inserted.


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The BK7 sample is now rotated manually towards one side so that the fringes startcollapsing or appearing on the screen The angle of rotation through which the samplehas to be rotated for a set of 5 fringes to appear or collapse is noted down carefully.This procedure is continued for l 15 20 ... fringes till the maximum fringes forwhich the fringe pattern is clearly visible. Now the same procedure is repeated for theopposite direction and both the values compared. The index of refraction of thesample is thus calculated from the no of interference fringes shifted during therotation of the glass plate. The following equation is used to calculate the index ofrefraction of the sample67:

21 NA 1 cos i=21 1 osi NA 2

where N is the number of shifted fringes e is the angle of rotation of the sample andt is the thickness of the sample.

Once the interference fringe pattern is achieved we can touch the table the mirrorsor the beamsplitter and watch them oscillate. These interference fringe patterns can beused to analyze any vibrations occurring on the table surface and/or any movement ofthc optical components.

It follows that if the interfercnce pattern is stationary there are no vibrations and/orcomponent movcment occurring in or on thc system. If the pattcrn moves rapidly andthen settles dO\\ll the table is rcceiving ground vibrations. These types of vibrationsmay be caused by moving vehicles. peoplc walking in an adjaccnt room c1c\ators

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dishwashers etc. It is best to run this analysis in the evenings or on the weekendwhen the surrounding environment is quieter.

If the fringe pattern moves slowly back and forth it is due to air currents in the roomIf central air conditioning is present the thermostat or input vent to the room shouldbe turned off This type of pattern movement may also be caused by opticalcomponents that are not locked tightly into position. So is made sure that all thumbscrews on the optical mounts are tightened.

2 2 Theory behind rotating the sample:If a plane parallel plate of index of refraction n is inserted normal to the path of one ofthe beams of light traversing the arms of a Michelson interferometer the increase ofoptical path introduced will be 2 1 where t is the thickness of the plate Thefactor 2 occurs because light traverses twice through the glass piece For light of

wavelength the difference of path introduced is where is the number offringes displaced when the plate is inserted.If the plate is rotated through a small measured angle the path of the light will bechanged and the no of fringes corresponding to this change is counted. This isbecause of the fact that as the plate is rotated the length of the glass in the path willincrease and therefore no of wavelengths in that path will increase. This will changethe interference pattern.

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------- -----

d ~os

Figure 6: Demonstrating the change in p th length

d d

s e increases cos edecreases so increases Thus we can see that the path lengththrough the glass plate increases as we rotate it through some angle When projectedon a screen it can be seen that the fringes collapse as the glass plate is rotated Thechange o the path through the glass plate depends upon the thickness of the plate theangle through which it is rotated and the index o refraction So the index orefraction may be calculated if the other two values are known

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2.1.3. Derivation of the relationship between the refractive index,num er of fringes collapsing nd the angle of rotation of the sample

o

t::r.c

-d / -

/ I/ i p

A/ / ,

/ -( -/, / .[J/r :/ b

Figure 7: Analysis of the sample being rotated6

The expression for index of refraction of the sample in tenns of the number of fringescollapsing and the angle by which the sample has been rotated is given by G.S.Monk6

Let OP be the original direction of light normal to plate of thickness as shown inFig 7 The total optical path between a and c initially for the propagation of lightis + be where is the index of refraction of the plate. When plate is rotatedthrough an angle i ., optical path is increased to ad + de.So the total increase in optical path is

e5) -e52 =2 ad* 11+de-111-be = ;V), (13)where N is the number of fringes collapsing or appearing on the screen and i, is thewayclength of the light source used.


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But we havea b -cosr

where r is the angle of refraction of the light beam.We also have

de =dc sin i =fc - fd sin i = tan isin i tan r sin ibc =cosi

Substituting Egs. 14-16) into 13), we have5: 5: ? III . . . . VI V =- tan I SIn I - tan SIn I - - .cosr COSI

14)

15)

16)

17)

Now using Snell s Law 11 sin,. =sin i , we get the expression for index of refraction as2 N 1- cos i

11 = 2/ 1- cosi) - Nas already given in Eg. 12)This can also be rewritten as

N = 2/ 1- cosi n-I} }, I - cos i

20

18)

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2.2 Measuring the refractive index from the intensity of the fringepattern

S633nm 35mVVHe-Ne sourceLCamera lens25mrn F1.4

AGILENT DSC6054AOsciDoscope

M igure 8: Michelson Interferometric setup for measuring refractive index ofBK7 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. anautomatcd rotation stagc controllcd by a Nc\vport ESP300 controllcr is uscd. t has avcry good rcsolution of 0.00 this hclps in providing a vcry accuratc valuc of thcanglc 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 densityfilters and then detected using a InGaAs detector D If the intensity of the signal is too

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high the detector mechanism would not have the capability to detect it hence theneutral 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 timewavefonn.2 2 Procedure for Alignment:Instead the manual rotation stage in the previous case the sample is mounted on anautomated rotation stage Newport SR50CC which is compatible with the NewportESP300 Controller.Once the fringes are seen on screen the screen is then removed and a pinhole P isfixed so that only the center of the fringe pattern goes through the hole. Since thismay have a very high intensity neutral density filters are used to reduce the intensityof the signal. The InGaAs detector is mounted carefully so that it catches the centerfringe and transmits the signal to the oscilloscope. We use an Agilent DSC6054AOscilloscope which has an attached USB which facilitates easy storing thewavefoml in any jump drive. The BK Windows sample after it has been alignedperpendicular 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. Thisprocess is repeated a number of times to get more number of measurements foraccurate analysis. The velocity and acceleration the SR50CC rotation stage can becontrolled through the controller to see the effects of changing these parameters in thefinal stored wavefonn.


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CHAPTER 3: DERIVATIONS N RESULTS3 Measuring the refractive index of BK7 Windows:

3 Plotting angle of rotation with respect to the number offringes:

The angle of rotation through which the sample has been rotated is calculated andnoted down for every 5 or 10 fringes collapsing. It is then plotted with respect to thenumber of fringes and a non-linear curve fitting is performed to get the value ofrefractive index.The expression for refractive index derived in Eq. 18 is then used as the formula forperforming the non-linear curve fit. In the Figs. 9 a-d), denote the thickness of thesample and I denotes the wavelengthof the light source used which corresponds tothe HeNe wavelength 633nm. Both these parameters are kept fixed and the refractiveindex n is kept as a floating parameter to get an accurate fit to the experimentalvalues. This fit value of refractive index obtained is highlighted in blue in all theplots. This value is then compared with the expected value of refractive index forBK7 windows at 633nm and room temperature 25C which is 1.51509. The averageof all the values obtained from the fits for refractive index is then calculated andstandard deviation from the expected value is then calculated and reported in Table 1.


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1 000

o gga

0 g06;OlCn 0 g040 J

0 g02

Data:Data1_BModel: ref index

Ch i 2 = 6 0507E 10R 2 = 0 .gOOg2

1 513136 33 E70 00507

:0 00100:0: 0

O.gOO - t---r--- . -- - . -- . ---- ---- --- . . . ---- . --- . --.....- - -o 10 20 30 40 50No offrlnges

Figure 9 a): Plotting cosine of the angle of rotat ion with respect to the number offringes collapsing or appearing Refractive index of n =1.51313 obtained

Dab Dab1_BMod el: ref ind exChi 2 5 6653E gR 2 = 0 .00051

o goa 0 006

; 0 004OlCnJ 0 002

0 000

0 08810 20

n

30No of fringes

1 504076 33E 70 00507

50

:000332:0:0

eo

Figure 9 b): Plotting cosine of the angle of rotation with respect to the number offringes collapsing or appearing Refractive index of n =1.50497 obtained

24


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1.000 Data: D alal_B

Model: ret indexChi 2 1.3024E90998 R 2 = 0.99983

15282t:l :l; 1t:l9t:l .33E7 :1;0

099t:l 0.00507 :1;06l 0 0.994,)

~9Q20.990 - t - - . . - - - - - . - - - - - . - - - - - - - . - - - - - . . . - - .. - - -

o 1 2 30 40 50No offringes

Figure 9 c : Plotting cosine the angle rotation with respect to the number offringes collapsing or appearing Refractive index of n 1 52826 obtained

D l Datal_CModel: re f index

1.000

0.998

_ 0996ell0Cc; Og94,

n

55878E l00.99992

1531066.33E70.00507

10.001121010

0.992

O. 99 0 - t - - . . . . - - . - - - . . - - - - - r - - - - - . - - - - - - r - - - - . - - - - r - - - . - - - - - - - -o 1 30

No of fringes40 50

Figure 9 d : Plotting cosine the angle of rotation with respect to the number offringes collapsing or appearing Refractive index of n 1 531 6 obtained

25


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easurement no. Refractive Index n1 1 513132 1 50497

3 1 528264 1 53106

Mean of all measurements =1.519355tandard deviation =0.015706523

Table 1: The measured values of refractive indices presented in Figs. 9 a-dalong 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 tomissing a fringe thereby leading to inappropriate values. So inorder to exclude thisfrom consideration, we derive a difference angle formula where the value ofrefractive index just depends upon the difference between the angles through whichthe sample has been rotated and the thickness of the sample.We have from 18 .

Also = 2 N ~ . I - cos02 21 1- c o s ~

26

20

21


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where NI N 2 denote the number of fringes and JI J denote the angle throughwhich the sample has been rotated from the center.We can write this in terms of sine and cosine of angles as

Icos fl) =4 11 1 1 + N1A 2 - 11 2

2 2 2 11 1 1 + NIA

Icos l) =4 11 -1 + N 2A 2 - 11 2

2 2 2 11-1 /+N2A

So substituting equations 22 to 25) , we have

{ OI-(2) 4 n-l +NIA 2-11 4 n- l /+N2A 2-nco =1 +2 4 n-l 1 +2i\V \ 4 n-l +2N2A

22)

23)

24)

25)

26

NInA I N2nA4 n-l + 2 \V 4 11-1 + N

o -0 [T cos I 4(n-)) t+N 1) ,(2-n) 4(n- ) t+N 1),(2-n) + I NInA. f N1n), ]4(n-))t+2N I \ 4(ll-))t+2N 1 \ 4(ll-))t+2N 1 , \ 4 n - t+2N ,27)

27


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3 3 Plotting difference angle rotation with respect to thenumber 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 I10 gives the fixed fringeseparation. 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 . This is then plotted2with respect to the number of fringes to give the ideal difference angle plot.

-3I j Jl O Ul.lW

\'I'here denotes thickness of sample and?' denotes wa\'l length of the laser l ight source usedat 633nm 5deg celsIus room temp f BK7

o

D:T{Ml)

///IIIom: III,III

: ~ _ I ..I L I _ ~[

Figurc : Simulation for idcal yalucs of rcfractin index for BK7 sample28


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For a fringe spacing of 1 fringes that is 2 =M , 10), and 2 are measuredand noted down. The cosine of half of the difference angle is calculated and plonedwith respect to the number of fringes.

This information is free of the zero position and yields an effective way to accuratelymeasure the refractive index. In the Figs I I a-h), denotes the thickness of thesample and I denotes the wavelength of the light source used which corresponds tothe HeNe wavelength 633nrn. These parameters are kept fixed and the refractiveindex n is kept as a floating parameter to get an accurate fit to the experimentalvalues. This value is again compared with the expected value of refractive index forBK7 windows at 633nm and room temperature 25C which is 1.51509. The averageof all the values obtained from the fits for refractive index is then calculated andstandard deviation from the expected value is then calculated and reported in Table 2

29


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o gggn

O ggggO

Ogggg3

Ogggg2

/ . ~ . ::::;:Chi2 2.7504E13 R 2 0.ggg521.51517 :to.0042O.33E7 :to0.00507 :to

10 15 20 25No.offringcs

30 35 40

Figure II a : Plotting cosine the difference in angle of rotation with respect tothe number of fringes collapsing or appearing. Refractive index of n 1.51517obtained.

OggggB

o gggg7

_ 0 ggggOClc

0 gggg5;;l J 0 gggg4

~ h D.I.l_B

t.Aodtl diN .oglt

Chi 2 47405E 13R 2 O g . g l 1

o o gggg3/

10 15 20 25No of fringes

30

1515710 0 0 5 0 7o 32BE7

35

to 00552toto

Figure 11 b : Plotting cosine of the difference in angle of rotation with respect tothe number of fringes collapsing or appearing. Refractin index of n 1.51571obtained.

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0.00008

0.09997

00gg90~t ll Data: Data1 BI J 0.09995 -.E Mo de l diNa ng le2

Chi 2 = 1.7253 E13~ 0.09004III RA2 = 0.09070 1.51355 O .00330.09993 0.33E7 O0.00507 O

009992

10 15 20No.offrlnges

30

igure11 C : Plotting cosine of the difference in angle of rotation with respect tothe number of fringes collapsing or appearing Refractive index of n = 1.51355obtained.

09999g5

o .gggggO ~ ~ = =-;- o.ggg985tl l Data Data1_B Model diN angieI J o gg99BO C hiA2 18780E 12 R 098350~ o gg9975 1 51734 i O O ~ 70J o g99970 000507 O033E7 10

o 9g9905 1o gggg:lO0 10 20 30 40 50No. of fringes

igure11 d : Plotting cosine of the difference in angle of rotation with respect tothe number of fringes collapsing or appearing Refractin index of n = 1.51734obtained.

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to.Ol10gtoto

1.512g00.33E70.00507

0010: 00101_8 odtldiHonglt

C hiA2 2.7502E13RA a ggOSg

OgggggO

Ogggg70

Ogggg05

Oggggg5

0 ggggS5c:

O ggggSO0.gggg75

lo< J

a 10 20 30 50Ho of hinges

Figure l1 e : Plotting cosine o the difference in angle of rotation with respect tothe number o fringes collapsing or appearing. Refractive index of n 1.51296obtained.

to oogOltoto

1 500370.33E 7000507

Chi 2 5200BE13RA 0.gg010

ao O.t .l _8M o d e l diHngle

OgggggO

OggggOO

OggggSO

c: Ogggg75

I 0 gggg70o

< J

o gggg55 -t--. . . . . .---- --. . . . .--. . .--. . . . .-- .--. . . . .-- .--.--------o 10 20 30 50

Ho offringes

Figure 11 f : Plotting cosine of the difference in angle of rotation with respect tot number o fringes collapsing or appearing. Refractive index of n 1.50637obtained.


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to.005114to

1.523176.33E 70.00507

A ta Da t a2 8/ Model: dittangleChi 2 : 4.B025E13RA2 : 0.11111112/

O IlIlIlQ7

0 IlIlIlQ8

0 IlQIlQ3

-; - 0 IlQIlQ6Elc:S 0 IlQIlQ5 Cii i 0 IlQIlQ4oo

0 IlQIlQ2

10 15 20 25 30 35No offriJlges

Figure II g : Plotting cosine of the difference in angle of rotation with respect tot number of fringes collapsing or appearing. Refractive index of n = 1.52317obtained.

O.IXlIXlQ

O.IXlIXlB

0.1Xl1Xl7 ;

O.IXlIXlOnC 0.IXHXI5 . 0 OQ0Q4oU

Chr 2 18113E12R 2 0.QQ078

I 5 5 Q000507033E 7

fO OlOQ8fOfO

10 30No of fringes

40

Figure 11 h : Plotting cosine of t difference in angle of rotation with respect tot number of fringes collapsing or appearing. Refractive index of n = 1.52549obtained.


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The bar graph shown in Fig 12 gives the approximate mean value of refractive indexcalculated from these set of measurements represented in Figs. a-h . The standarddeviation from the mean value obtained from the graph is also calculated.

1.525

1.520

x IJt J 1.515p.

1.510a

1.505

1 5 ~ - r2 3 4 6 7 8 9

e surement o

igure : Refractive ndex bar graph for Figs. 11 a-hFrom the bar graph, the Mean of the refractive index was measured to be 1.51622with a standard deviation of 0.00561.This method of measuring the refractive index manually counting the number offringes shifting corresponding to the angle of rotations proves very cumbersomebecause of the following disadvantages:

a There is a possibility of missing some fringes because of the disadvantage ofmeasurement-by-eye

34


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b Refractive Index precision of 2 3 digits might not be achieved if there is anysmall error in the measurement of the angle

c The zero position the fringe is a very abstract concept and it is very difficultto reference it to a fixed position, so some fringes may be missed while doingthe manual measurement.

In order to overcome all these disadvantages, an automated method of measuringrefractive index is proposed wherein the sample is mounted on top an automatedrotation stage whose movement is precisely controlled by a controller. The controllervelocity and acceleration can be set to any desired value. The sample is rotatedthrough any desired angle and a complete waveform the intensity change ismeasured in the oscilloscope. We will propose a way to take this measured waveformand do a non-linear curve fitting with the proper derived intensity formula and get therefractive index to a high level accuracy.

3 1 4 Derivation of the expression for intensity terms of refractiveindex thickness of the s mpl and the angle of rotation of the sampleWe have from Equation 6 ,

27

where I denotes the intensity the resultant wave. A is the amplitude of the resultantwaw and 9 and 9 2 are phase angles corresponding to each the optical paths.

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The expression for intensity can now be written as

28)

Now substituting the value for optical path difference from 15 and using Snell s lawfor expanding, the expression for intensity comes out to be of the form

2l a,b,O :=a+b cos2n .\

293 5 Analysis the stored waveform:When executing a move command in the controller to move the rotation stage by aspecified angle, the stage will accelerate until the velocity reaches a pre-definedvalue. Both this acceleration and constant velocity can be set to a specified valuefrom the controller. Then at the proper time, it will start decelerating so that when themotor stops, the stage is at the correct final position. The velocity plot of this type ofmotion will have a trapezoidal shape as shO\\11 in Fig. 13.

36


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esired Velocity

ime

igure : Trapezoidal otion Profile for the controllerSo inorder to account for this trapezoidal motion, the stored waveform in theoscilloscope has been modified in accordance with the principles of angular velocityand acceleration to account for the working operation the controller. So since thedifference between successful positions of the stage at continuous angles 18 ) is notconstant as a function time, from the whole trace stored in the oscilloscope, theaccelerating and decelerating part the wavefonn is cut out and only the constantvelocity part used for analysis.Analysis BK 7 Window sample for a particular case is shO\m belowIf the sample is rotated through an angle 9 degrces, the constant angularacceleration a is sct to 2 dcgls2 and final angular velocity of the rotation stage isset to 5 deg s the calculations involn::d in analyzing the wavcform can bcsummarizcd bclow:Using the cquations for constant angular acccicration. wc havc

37


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= o al 30where o is the initial angular velocity and t is is the time taken to reach the constantangular velocity

Substituting the values in Eq. 30), we have0.5 = 2t

So the time taken for the rotation stage to achieve constant angular velocity is 0.25sec. Similarly time taken for the stage to decelerate to zero final velocity is also 0.25sec.The distance covered by the stage in this time frame can be calculated using therelation

1 2 = ol l2Plugging in the values a = 2deg/s2 and t = 0.25 sec in Eq. 31), we get

=0.0625 0

31 )

32So the linear angular distance traveled by the rotation stage once it has reached aconstant angular velocity of t =0.5 deg/ s can be given as

=9 - 2 0.0625) =8.875 0 33So 8.875 0 =0.5 deg/ s 1 this gives the time taken for linear angular motion as1 = 17.75 secSo the total time taken

1=1/ 0.25 0.25 =18.25sec38

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The analysis now for a particular plot is done as follows:The data for the plot contains 9 5 data points overall and this now corresponds to thetotal time of 18 25sec

9 po int s 18 25 sec point 0 02016575sec

S I 0 25 25 3 th o mltIa sec po mt0 02016575Linear angular motion continues till 18 secSo deceleration starts from 18 893 rJ po int0 02016575So only the portion of the waveform from the 13 th point to rd point is consideredfor analysis rest of the waveform is cut and excluded from the analysisTherefore linear angular motion corresponds to88 po int s 8 875Ipoint 0 010085227Each data point is then multiplied by this value to obtain the exact angular position ofthe rotation stage relative to the intensity measured at each point

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3 5 Measuring the refractive index BK7 Windows by plotting theintensity the fringe pattern with respect to the angle rotation:

t= 0.9410- 3 J = 63310- 9 n= 1 46

7 . . . . . . .

2

6

5

I 24,4J,e)

4

3\

2-10 8 -6 -4 -2 0 2 4 8 e

Figure : Intensity Fringe Pattern Simulation for measuring refractive index

Figure 14 shows a simulation for the intensity of the measured signal with respect tothe angle of rotation. An arbitrary yalue of thickness of 0.94 mm and a refractiycindex of n= 1 46 arc chosen. Here a and b represents scaling factors that control

40


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the height and amplitude of the waveform respectively. Appropriate values of a and bare chosen to represent the signal effectively. Figures 15(a-u) represent the observedwavefonn for the BK7 sample having thickness d of 5.17mm. A non-linear curve fitis made on the intensity pattern using Eq. 29 which gives the relation between theintensity of the fringe pattern and the angle of rotation. Appropriate values of a and bare chosen for scaling the fit properly with the signal.

8

o

8 2Data: BK SampleMode inte ns i ty simu lat ion

\C hi 2 = 1 74315R = 81119a 7 Ob 7 On 1 51395 73d 517 O

\

4Angle o Rotation

Figure 15(a): Plotting intensity with respect to th angle of rotation Non linearcurve fitting yields a refractive index of 1 51395

Thc wa\ cfonn pcaks towards both sidcs of thc ccntcr zcro fringc. corrcsponding tozcro 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.


54/95

15 a and peaks 10-12 in Fig. 15 b keeping a band d parameters fixed and refractiveindex n as a floating parameter. Refractive index values of 1.51395 and 1.51489 areobtained which corresponds to an accuracy of two decimal places.

2

6 AData: BK7 SampleMod el intens ity s im ulatio n

o

Chi 2 = 1 62224R 2 = 0 66142

: :b 7 : :n 1 5148Qd 51 7

v V4

Angle of Rotiltion

:1: 83: :

Figurc 15 b : Plotting intensity with respect to the angle of rotation. Non-Iincar un fitting yields a refractive index of 1.51489

4


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-16 -15 -146

Data BK7 SampleMode inte ns i ty simulationChr 2 = 463 3R 2 = 0 89791

7 1 515 1 517

38O

o

v I vAngle of Rotation

Figure 15 c : Plott ing intensity with respect to the angle of rotation. Non-linearcurve fitting yields a refractive index of 1.51501In Fig. 15 c , a non-linear curve fit is made on peaks 14-16 towards the left of thecenter fringe and a refractive index value of 1.51501 is obtained. Comparing with theideal refractive index value for BK7 glass at room temperature- 1.51509. it isobserved that this is the best fit achieved and the most accurate value of indexobtained from the measurement.

t


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14 13 126

o

D l B SampleMod el inlens ity s im u lalionChi 2 = 79726R 2 = 84132 7 Ob 7 On 1 5 921d 51 7

v V V4Angle of Rotation

Figure 15 d): Plotting intensity with respect to the angle of rotation. Non-linearcurve 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 inFig. l5 d). The scaling factors a and b as well as thickness d are kept fixed andrefractive index n is defined as a floating parameter to obtain a value of 1.50921 forthe index.

44


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17I

\ 1Data BK7 SampleModel:Inlensrty simulationChi 2 = 1.05040R 2 = 0.75Q22a 0.5 tob 0.5 ton 1.51763 to .00030d 0.00517 to V

7 8

o

8

4Angle of Rotation

Figure 15 e : Plotting intensity with respect to the angle of rotation. Non-linearcurve fitting yields a refractive index of 1.51763

3 14 5 68

D ala: BK 7 SampleM o d e l intensity simulation

C hi 2 = 0. /472R2 = 0.BB403

/5 to /.5 to151757 to 00035000517 to

o4Angle of Rot.ltion

Figure 15 f : Plotting intensit with respect to thc anglc of rotation. Non-lincarcun c fitting yiclds a rcfractive indcx of 1.51757

45


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Fig. 15 e-g) correspond to the same measurement. Non-linear curve fits are made onpeaks 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 isdefined as a floating parameter. Refractive index values of 1.51763, 1.51757 and1.519 are obtained respectively which again corresponds to an accuracy upto twodecimal places.

11 12 13

o

Data BK SampleModel intensity simulation

Chi 2 = 0 92082R 2 = 0 90091

9 5 109 5 10

n 1 519 10 000460 00517 10

Angle of Rotiltion

Figure 15(g): Plott ing intensif) with respect to the angle of rotation. Non-linearcun e fitting yields a refractive index of 1.519

Fig. 15 h corrcsponds to a diffcrcnt mcasurcmcnt. a non-linear curyc fit is made onpeaks 13-15 towards the right of the ccntcr fringc and a rcfracti,c index yalue of1.50691 is obtained.

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13 14 15h

D l BK7 SampleModel: intensity simulation

Chi 2 = 1 11914R 2 = 0 84037

7

n 1 50691 610 00517

o v4

vAngle of Rotation

igure 15 h): Plotting intensity with respect to the angle of rotation. Non-linearcurve fitting yields a refractive index of 1.50691

Fig. 15 i-k) correspond to the same measurement. Non-linear curve fits are made onpeaks 4-12 in Fig. 15 i , peaks 6-8 in Fig. 150) and peaks 11-13 in Fig. 15 k , alltowards the left side of the center fringe, keeping a, band d parameters fixed andrefractive 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 twodecimal places. It is observed again that these values for index are very close to theideal index value of 1.51509 for BK7 Windows at room temperature.

47


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g

6

12 5 4M

o4

Angle of Rotation

03t3: BK7 m p l ~Model: intensrty simulationChi 2 = 0 68268R 2 0 002303 7 :tob 7 :ton 1 50085 :1:0.00038d 0 00517 : :

Figure 15 i : Plott ing intensity with respect to th e angle of rotation. Non-linearcurve fitting yields a refractive index of 1.50985

Data: BK7 SampleModel: intensity simulation

g8 7

f 6i:iii1: 3

IV V y0

Cht 2 0 .501171RA 2 e 0 Q2721

7 : : 7 101 5 0 8 5 4000517

2

1000061110

vAngle of Rol 11ion

Figurc 15m: Plotting intcnsit - with rcspcct to th c anglc of rotation. Non-Iincarcun-c fitting yiclds a rcfractive index of 1.50854

48


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13 12 11

o

A Data: BK7 SampleModel: intensity simulation

Chi 2 = 0.4361R 2 = 0.94043a 7 b 7 n 1.60963 0.00042d 0.00617

V V ~-4

Angle of Rotlltion

igure 15 k : Plott ing intensity with respect to the angle of rotation. Non-linearcurve fitting yields a refractive index of 1.50953

-10 -9 -8

o

o

o t 8K7 SampleModol: inhnsly simul.tionChi 2 0.24371R 2 0.Q0701i 8 tob 8 ton 150071 to .00034d 000517 to

}

V V VAIl }le of ROt.ltioll

igure 15 1): Plott ing intensit - with respcct to the anglc of rotation. Non-linearcun c fitting yields a refractive index of 1.50071

49


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A non-linear curve fit is made on peaks 8-10 towards the left of the center fringe inFig. 15 1 . The scaling factors, a and b, which are chosen to be 8 and -8, as well asthickness d of 5.17mm, are kept as fixed parameters, and refractive index n is set tobe a floating parameter. A value of 1.50071 for the index is obtained from the fit.

-18.. .. -14

8

o

Da la: BK7 SampleModel: intensity simulation

Chi 2 = 1 5581R 2 = 0 80857

a 9 5 1b 9 5 1n 1 5242 10 00044d 0 00517 10

I \ I IV

Angle of Rotation

Figurc 15 m): Plott ing intcnsity with rcspcct to thc anglc of rotation. Non-lincarcun c fitting yiclds a rcfractivc indcx of 1.5242

In Figs. 15 m) and 15 n), which corrcsponds to two scparatc mcasurcmcnts. aconstant vclocity of 0.25 d g s and an acccicration and dcccleration of I deg/s2 is setto be the controller paramcters. Analysis is donc using thescs valucs to see the effectof velocity regulation in the valucs for refractivc index. Non-linear curve fits aremadc on peaks 14-18 in Fig. 15 m) and peaks 10-13 in Fig. 15 n). both towards the

50


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left of the center fringe. Keeping a b and d parameters fixed, refractive index valuesof 1.5242 and 1.52252 are obtained respectively. It is observed that using a lowervelocity and acceleration value for the controller, there are more jerks in the measuredwaveform and corresponding refractive index values slightly deviate away from theideal value.

-13 -12 11 -10

Data BK SampleModel inte nsity simulationChi 2 = 1 5 6 7 7R 2 0.81Q71a Q5 tob Q.5 ton 152252 to 00065d 0 00517 to

4

J v 2

Anule of Rotation

Figurc 15 n : Plott ing intcnsity with rcspcct to the anglc of rotation. Non-linearcun 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 inFig. 15 0 . Thc scaling factors, a and b which arc choscn to bc 9.5 and -9.5, as wcllas thickncss d of 5.17mm, arc kcpt as fixcd paramctcrs and rcfractivc indcx n is set tobc a floating parameter. A value of 1.52755 for the indcx is obtained from thc fit.

51


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8

o

-17 ...

.. 4 -3I ;

D31a: BK7 SampleModel: nl ns ty simul31ionChi 2 = 2 36462R 2 = 0 72007a g :tob :ton ~ 7 ~ ~ :t00004Qd 0 00517 :to

V W

Angle of Rotation

Figure 15 0 : Plotting intensity with respect to the angle of rotation. Non-linearcurve fitting yields a refractive index of 1.52755

Fig. 15 p-q correspond to the same measurement. Non-linear curve fits are made onpeaks 5-9 in Fig. 15 p , peaks 13-15 in Fig. 15 q , both towards the left side of thecenter fringe keeping a, band d parameters fixed and refractive index n as a floatingparameter. Refractive index values of 1.5210 I and 1.52646 are obtained respectivelywhich still corresponds to an accuracy upto two decimal places. Thus we can observethat two digits of accuracy is continuously obtained thereby proving that themeasurements are reproducible.

52


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987 0 51\

Data: BK7 s a mp i eModel: intensity simulation

Chi 2 = 5.74082R 2 = 0.27787a g 5 b g 5 On 1.52101 0.00138d 0.00517 O

J \ I\ V2

Angle of Rot :ltlon

Figure 15 p): Plotting intensity with respect to the angle of rotation. Non-linearcurve fitting yields a refractive index of 1.52101

15 -14 13Q

{D .t 8 K7 Sample od el r h n si tyS : ~ T I u l l t i o n

\

3

Chi 2 = 1 33523R 0 B23Q3

Q5 .Q 5

1 5 ~ \ 4000517

to 00040to

o\\

Angle of Rot.ltion

Figurc 15 q): Plott ing intcnsit) with rcspect to the angle of rotation. Non-lincarcunc fitting yields a refractive index of 1.52M6

5


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8 9 10

8

o

D ala: BK7 Sample Model: intensity simulationChi 2 ; 0.04753R 2 ; 0.88101a 0. 5 Ob 0.5 On 1.53007 O .00061d 0.00517 O

\ \ V V2

Angle of Rotatio n

Figure 15(r): Plotting intensit) with respect to the angle of rotation. Non-linearcurve fitting yields a refractive index of 1.53097

8

5 14 3 2

) Data BK7 SampleModel intensity simulationC hi 2 = 0.86778R 2 = 0.80107

i a 0. 5 10t 4 b 0. 5 10 : n 1.53005 10.00046d 0.00517 10

\VV V0

Angle of Rot tion

Figurc 15(s): Plott ing intcnsi ty with rcspcct to thc anglc of rotation. Non-linearcun c fitting yiclds a rcfractive indcx of 1.53095

54


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Fig. 15 r-s) correspond to the same measurement. Non-linear curve fits are made onpeaks 8-10in Fig. 15 r), peaks 12 15 in Fig. 15 s). Keeping a b and d parametersfixed 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 inFig. 15 t). The scaling factors, a and b which are chosen to be 9.5 and -9.5, as well asthickness d of 5.17mm, are kept as fixed parameters and refractive index n is set tobe a floating parameter. A value of 1.50408 for the index is obtained from the fit.

14 15 16

8Dolo: B K 7 SampleMod : I nl on l y ulolio nC hi 2 = 0 73573R o g 1784

Q 5 :toQ 5 :to150408 t 000340 00517 t

o

An lle of Rotation

Figurc 15 t): Plott ing intcnsit) with rcspect to the anglc of rotation. Non-lincarcun c fitting yields a rcfractivc indcx of 1.50408

55


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-15 ... 5 -4

8

o

Data BK7 SampleModel intensity simulationChi2 ; 1 ~ 9 4 BR = 0.81607

Ia b n 1 ~ 4 1 8 9 .00046d O O O ~ 1 7

I l4 2Angle of Rotiltion

Figure 15 u : Plotting intensity with respect to the angle of rotation. Non-linearcun e fitting yields a refractive index of 1.54189Fig. 15 u represents a non-linear curve fit on peaks 4-15 towards the left side of thecenter fringe for a certain specific measurement. Since the peak amplitudes are veryuneven because of the jerks in the motion of the rotation stage, the fit gives a value ofrefractive index which is quite deviated from the ideal value.The bar graph shO\\11 in Fig. 16 gives the approximate mean value of refractive indexcalculated from these set of measurements represented in Figs. 15 a-u . The standarddeviation from the mean value obtained from the graph is also calculated.

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1.545

1.540

1.535

1.530X I 1.525Q 1.520,

1.5151.510

1.505

1.5001 15 20

easurement 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 51773with a standard deviation of 0.0I 23

3.2 Measuring refractive indices of core and cladding sodium ZIDCtellurite glass samplesTellurium oxide based glasses are potential materials for optical device applicationsdue to their nonlinear optical properties. They show high refractive index. andconsequently high third order nonlinear susceptibility x and high dielectricconstant9. Also. they are transparent over the visible and near infrared regions. Kimcl al have measured for a pure Te02 glass a value ofX =1 41 x 1 : CSIi however

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it is not easy to obtain this pure glass. Tellurite-based glasses are also promisingcandidate materials for high Raman amplification applications, due to their high gaincoefficients I -8 . They have a broad emission band around 1.5 Ilm and are promising

for development of broadband integrated optical amplifiers. They exhibit a widetransmission range 0.35 to 5Ilm the lowest vibrational energy about 780 em-Iamong oxide glass formers leading to fluorescence from additional energy levelscompared to silicate or phosphate glasses. They have a low process temperature, goodchemical durability, and exhibit significant nonlinear properties I9 o The zinctellurite glasses have a higher density and refractive index and are considerably softerthan silicate glasses

Tellurite glasses have also been prepared by Stanworth and Yakhkind22 , whoinvestigated glasses in various binary systems and in several tricomponent systems.Optical constants, densities, and spectral properties were given, together withexperimental and calculated regions of glass formation. Tellurite glasses havingsemiconducting properties have also been reported, particularly in the V205-Te02system3 . The glasses obtained have a yellowish color, even when usingspectrochemically pure Te02. This color was thus assumed to be a property of thepure glass as with the A 20 3-Te02, glass prepared by Stanworth .

CI J;5 Dault RdrattlTceompolit.ioll IfC ll in o ~ D O O pTtOs 2 ~ A . 2 J C b rd e65 5.33 :i: 0.03 2.036 265.9:i: 670 5.35:i: 0.03 263.4:i: 475 5.37:i: 0.03 274.9:i:15

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Table 2: Density, Hardness and Refractive Index of Zinc Tellurite Glasses

Gla > > o m p o ~ l l o n tim C J pm Vi30Na0l1 T e O ~ 2.00 1.95 0.082lONaOl, 220ZnO 70Te02 2.02 1.97 0.2330ZnO T e O ~ 2.05 2.00 0.45

Table 3: Refractive Indices at 532 and 1064 nm and d33 values of poled Na 2ZnO-Te 2 glasses 24

The second-order nonlinear optical coefficient increases with a replacement ofNaOI/2by ZnO. This increase can be attributed to the rise in the content of nonbridgingOxygens5 which contribute to the dipole moments oriented under the dc electricfield, and lead to the anisotropic structure. From this viewpoint, a glass containing alargenumber of nonbridging oxygens is likely to present a large anisotropy. In addition, thehigh mobility of Na enhances the electrochemical reactions at high temperatures,leading to a decrease in the second harmonic intensity as well as the second-ordernonlinear 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 coreglass 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 havingthickness d of 1. 73 mOl. This is achieved by perfonning a non-linear cUI \ e fit on theintensity pattern using Eg. 29) which gives the relation between the intensity of thefringe pattern and the angle of rotation. The scaling factors a and b arc kept as

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floating parameters and appropriate values are chosen for scaling the fit properly withthe measured signal.3.2.1. Intensity plots fo r measuring refractive index of core glass sample:Fig. 17 a-b corresponds to the measurement of the intensity pattern as a function ofthe angle through which the core glass sample has been rotated. Non-linear curve fitsare made on peaks 9-12 in Fig. 17 a) and peaks 7-12 in Fig. 17 b . The differencebetween the two graphs here is that the scaling factors a and b along with therefractive index n are kept as floating parameters and thickness d is kept as a fixedparameter while performing the fit. So appropriate values of a and b are chosen forgood fits and refractive index values of 2.01832 and 2.01379 are obtainedrespectively which again corresponds to values of index upto two decimal places ofaccuracy.

9 10 11 1212

DOlO: oreG 5S sampleModtl int tnsity simulJtionChi 2 0 38030R = 0 g0731

12 oa708 5 3 0 1 g g201832000173

Angle of Rot.ltion

10 1 114 810 175410 000g010

F ig ur e 17 a ): P lo tt in g intensit) w it h r espe ct to t he a ng le of r ot at io n f or t he coreglass sample. Non-linear cun e fitting yields a rcfractin index of 2.0183260


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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, areagain set as floating parameters along with the refractive index n and arbitrary valuesare obtained from the fit conforming to a good fit. An index value of 2.02569 and2.02137 is obtained from the fit which again conforms to an accuracy upto twodecimal places.

2 4 5 6 7D ala : Core Glass SampleM 0 do l inlo ns Ity sImulalion

4

Chi 2 = 072508R 2 = 082733

5 734235 17207

n 202137d 0 00173

4An .lIe of rotiltio n

:l:0 11Q1510 20041:1:002085: :

Figure 17 d): Plotting intensif) with respect to the angle of r ot at io n for t he coreglass sample. Non-linear un 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 theeffect of keeping b as a fixed parameter in the final yalues of rcfractiye index. An

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index value of 2 4862 is obtained which depends on the accuracy of the waveformmeasured rather than changing or fixing the values of scaling factors

12

6

2

ngle of Rottltion

Data Data _BModel nt ns ty simulationChi 2 = 1 70807R 2 = 0 57854a 12 73460 0 13412b 6 On 2 04862 O 00185d 0 00173 O

6

Figure 17 e : Plotting intensity with respect to the angle rotation for the core

glass sample. Non-linear curve fitting yields a refractive index 2.04862


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12 7 8 9

[ Data C ore Glass sampleMod e I Intens tty s m ulatio nC hi 2 1.36467R2 0.70867 8IIIcQ

v nV V

\

I

Angle of rot\ltion

12 :to8 :to2100470.00173

6

:to.00 18:to

Figure 17 f): Plotting intensity with respect to the angle of rotation the coreglass sample. Non-linear curve fitting yields a refractive index of 2.10047

712

D.I. COlt GI S.rnpl.Modol: inhnsiysimul l ion

I ChiA2 e 2.02608 I RA 2 e 0.5287 1II I

c 8 12 toIII 6 to5 I n 2 12777 to.OO371000173 toAngle of Rot.ltioll

Figure 17 g): Plotting intcnsity with rcspcct to the angle of rotat ion for the coreglass samplc. Non-lincar cun c fitt ing yiclds a rcfractivc indcx of 2.12777

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125

6 7

oala Core G la ss S am pl eMod el: inlens ity s im ula lio n

IJ

C hi 2 = 0 70e1l4R 2 = 0 831l05 IliiilD1:

6 \

4

abnd

12 O6 O2 101l51l0 00173

O .00 O

An gle of rotatio II

Figure 17 h : Plott ing intensity with respect to th e angle of rotation fo r the coreglass 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 onpeaks 6-9 in Fig. 17 f , peaks 4-7 in Fig. 17 g and peaks 5-8in Fig. 17 h . In all thethree graphs, the scaling factors a and b along with the thickness d of the sample, arekept 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 areobtained respectively. It is observed that because of the random variations in theamplitude and spacing between alI the peaks, a comparatively higher valuc ofrefractive index is obtaincd from this set of data.

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12 7 8 9 10A Data: Core Glass sampleMod e I int ns ity s im ulationChi 2 = O 934Q3R 2 0 79092

9Vi \

\6

4

nd

11 422555 174842 04Q 180 00172

6

:1:0 41948:1:0 8117:1:0 96485:1:0 00077

Angle of rolalio n

Figure 17(i): Plotting intensity with respect to the angle of rotation for the coreglass sample. Non linear curve fitting yields a refractive index of 2.04918

8 9 10 12

g

6

4

Dato: Cor Ola Sam plodll inhns i y simuLltion

Chi 2 = 2 .86826R 2 030056

8 2 0 23 84 t o. 00 21 80 0 0 1 7 3

Angle of Rotation

Figure 17(j): Plott ing intensit) with respect to the angle of rotation for the eoreglass sample. Non linear eun e fitting yields a refractin index of 2 2 3 8 ~

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Fig. 17 i) and 170) again correspond to another set of scanned waveforms from thesame measurement, the fit values of refractive index are observed to be 2.04918 and2.02384 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 1where refractive index values of2.03984 and 2.00683 are obtained.

4Angle of Rotation

Figur e 17 k): Plott ing intensit) w it h respect to t he angle of ro ta ti on fo r th e coreglass sample. Non-linear un efitting yields a refractivc index of 2.03984


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n Figs. 19 a and 19 b , corresponding to the same measurement, appropriate scalingfactors 2 and are chosen for a and b respectively, thickness the sample keptas a fixed parameter and refractive index values 1.99372 and 2.00347 are obtainedfrom the fits.

6 7 8 92 0 tl

Data: Clad G lass Sample A Ii ode inte ns ity simu lation.6 Chi 2 = 0 04188 R 2 = 0 55561VICJ J 2 toC 1 tolV V 1 gg372 to 00202

1.2 V V 0 00084 V V

6Allgle of Roti ltioll

8

Figure 19 a): Plotting intensit) with respect to the angle of rotation for thecladding glass sample. Non-linear curve fitting yields a refractive index of1.99372

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4 5 7 82.0 A

Data: Clad Glass Sample

Model: intensity simulation113 Chi 2 = 0.04415R 2 = 0.27272Vi lii 1 l 2.00347 :1:0.00227IV 0.00084 : :1.2

V V

6

Angle of rotation8

Figure 19 b): P lot ti ng intensity with r espe ct to the angle of rotation for thecladding glass sample. Non-linear curve fitting yields a refractive index of2 00347In Figs. 19 c) and 19 d), it is observed that the waveform obtained is not so goodcompared to other set of data measured. Infact, in general, the traces scanned for thecladding glass sample is not as sharp and clear as the ones obtained for the core glasssample. This is due to the fact that here the sample is very thin O.84mm). so theoptical path length difference is very sensible to even small changes in the angle ofrotation of the sample. Hence the waveform trace obtained is more random compared

to the earlier traces for both the core and BK 7 samples. Refractive indices of 1.99834and 1.98967 are obtained from these two graphs after perfomling the non-linear curvefitting.

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4 5 6 7\ A

4.5

1 : C I d gllSs SImple

t os ity s imu Ijtion\ Chl 2 = 0.25638 \ RA2 -1.31007 :; 5 10 ,2 103.6 m 1 QQ834 10.0043000084 104

v v6

Angle of 1 t 1llon

Figure 19 c): Plotting intensity with respect to the angle of rotation for thecladding glass sample. Non-l inear curve fitting yields a refractive index of1.99834

6 7 8

8

n o to: CI d Olus S mpleMod 1 intensity simuillionChiA2 02081Q R A2 1 10363 V 5 101: 2 10Q8Q67 1000385

000084 10

3.2 V

eAngle of lolalio n

Figure 19 d): Plotting intensif) with respcct to the angle of rotation for thecladding glass sample. Non-lincar cun c fitting yiclds a refractiyc index of1.98967


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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 onpeaks 5-8 in Fig. 19 e and peaks 6-10 in Fig. 19 f . An accuracy of upto twodecimals is observed.

5 6 7 8{\ \ A4.8

r AData:Clad Glass Sample

\Mod el: intens ity s im ulation

Chi 2 0.3068in 4.0 R 2 2.86367 : V 5 O b -2 On 1.99537 to .0031 9d 0.00084 to

3.2 v 6

Angle of rotation8

igure 19 e : Plotting intensity with respect to the angle of rotation for thecladding glass sample. Non-l inear cun e fitting yields a refractive index of1.99537

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6 7 8 9 10 f A 1\ 1\4 8

II Data: Clad Glass Sample Model: intensity simulation Chi 2 = 0 2Q257Vi 4 0 RI\2 = 2 72736m : 5 O b 2 On 1 QQ315 0 00256d 0 00084 O

3 2l V V V

6 8Angle of Rotation

igure 19 1 : Plotting intensity with respect to the angle of rotation for thecladding glass sample. Non-linear urve fitting yields a refractive index of1.99315

Finally, in Fig 19 9 , non-linear curve fitting is performed on peaks 6-9 on the rightside of thc centcr zero fringe. For scaling factors a and b choscn to be 5 and 2rcspectively, an index value of2.01567 is observed.

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56 7 8 9

Data Clad Glass SampleModel: intensity simulation

Chi 2 0 21593RA 1 51313

a 5 b n 2 01557 0 0025d 0 00084

5Angle of Rotation

8

igure 19 9): P lot ti ng i nt ensit ) with r es pec t to t he angle of rotation for thecladding glass sample. Non-linear curve fitting yields a refractive index of2.01567

A bar graph is constructed from these measured vales in Figs. 19 a-g) to measure themean and standard deviation of the refractive index for the cladding glass sample, asshO\\11 in Fig 2

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2.020

2.015

2.010

X 2.005 Is 2.000,

1.995

1.990

1.985

2 3 4 6 7Measurement 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 99848with a standard dcviation of 0.00874. Hence it is observed that cladding refractivcindex is less than the core refractivc index by 1 43

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hapter ON LUSION

We have successfully demonstrated a Michelson Interferometric technique formeasuring the refractive index sodium zinc tellurite glasses based on the conceptof changing the path length of one arm of the interferometer by rotating the testsample introduced into that optical path The intensity the waveform recorded isplotted against the angle rotation the test sample and non linear curve fitting isdone to obtain the refractive index to two digits precisionThe average value Refractive Index for BK7 Windows sample was measured fromthe fits to be 1 51622 from the difference angle plots and 1 51773 from the intensityplots with a standard deviation 0 01023 This value is then compared with the idealvalue refractive index 1 51509 at room temperature for the BK 7 sample and it isobserved that a precision two digits accuracy has been successfully attained Theaverage value refractive index for the core glass tellurite sample having thecomposition 75Te02 20ZnO 5Na20 6KNb03 is observed to be 2 275 with astandard deviation 0 0 15 9 Similarly the average value of refractive index for thecladding glass tellurite sample having the composition 75Te02 20ZnO 5Na20 ISobser ed to be 1 99848 with a standard deviation 0 00874

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hapter FUTUR WORK

This work can be extended to develop a white-light interferometric technique tomeasure the absolute phase shift and the refractive index for sodium zinc telluritesamples. White light interferometry with the use a tunable monochromator is avery powerful technique which helps in determining the refractive index for differentwavelengths and is a very effective tool in determining the variation in index withrespect to wavelengths for these glasses.A double interferometer based system which couples this scannmg Michelsoninterferometric setup to a modified Mach-Zehnder

Documents

A Michelson Interferometric Technique for Measuring Refractive Index of sodium zinc tellurite glasses