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Design and development of an oblique incidence interferometer Bajaj, Vijay Kumar 1971

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DESIGN AND DEVELOPMENT OF AN OBLIQUE INCIDENCE INTERFEROMETER by VIJAY KUMAR BAJAJ B.Tech. (Hons.), Indian Inst, of Technology Kharagpur, India, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of Mechanical Engineering We accept t h i s thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA March, 1971  In presenting t h i s thesis i n p a r t i a l f u l f i l l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study.  I f u r t h e r agree that permission f o r extensive copying  of t h i s thesis f o r s c h o l a r l y purposes may be granted by the Head of my Department or by his representatives.  I t i s understood that  p u b l i c a t i o n , in part or in whole, or the copying of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission.  VIJAY KUMAR BAJAJ Department of Mechanical Engineering The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date  March, 1971  ABSTRACT  Interferometry o f f e r s great scope in the study of surface topography and small surface displacements.  The fundamentals of surface as  encountered i n engineering p r a c t i c e and a few of the o p t i c a l methods a v a i l a b l e f o r the metrology of surface have been reviewed.  In p a r t i c u -  l a r , Oblique Incidence Interferometry has been developed i n d e t a i l f o r the study of r e l a t i v e l y rough surfaces.  The r e l a t i o n s h i p between  t h i s method and the recent and more general method of holography has been explained. Some preliminary experiments were conducted to gain a f e e l i n g f o r the l a t t e r two methods and an oblique incidence interferometer was designed, constructed, and tested f o r i t s s u i t a b i l i t y f o r measuring indust r i a l surfaces of approximately 22 inches i n span from the f o l l o w i n g aspects: 1.  The determination of surface topography of rough surfaces;  2.  The measurement of small surface displacements;  3.  The execution of forward s c a t t e r and back s c a t t e r holography f o r normal q u a l i t a t i v e recording and subsequent comparison—holographic interferometry.  iii A rough turned surface of a 12 inch diameter c i r c u l a r aluminium plate was examined.  Small surface displacements of the same plate were  measured and compared with t h e o r e t i c a l p r e d i c t i o n s .  Further holographic  interferometry was performed on a turbine blade. Some of the d i f f i c u l t i e s encountered, such as the s i g n i f i c a n c e of d i f f r a c t i o n e f f e c t s at edges and marks on the surface and the determinat i o n of scale were studied and are discussed. and c y l i n d e r s are discussed.  Future studies on tubes  TABLE OF CONTENTS Page I.  II. III.  PHYSICAL ENGINEERING SURFACES  1  Introduction  1  Surface Topography  1  Optical Metrology  6  STATEMENT OF OBJECTIVES  13  BRIEF REVIEW OF INTERFEROMETRY  15  H i s t o r i c a l Review Electromagnetic Nature and Coherence of Light  IV.  V.  15 . . . .  17  P r i n c i p l e of Superposition  19  Interference i n A i r Films  21  Interference at Oblique Incidence  23  P r i n c i p l e of Holography  25  PRELIMINARY EXPERIMENTS  31  Oblique Incidence Interferometry  31  Instant: Holography  37  DESIGN AND DEVELOPMENT OF AN OBLIQUE INCIDENCE INTERFEROMETER Review of Various Optical Systems Component Design and B u i l d i n g of the Interferometer  45 45  .  .  50  V  Page VI.  PROVING EXPERIMENTS  60  Surface Topography of Rough Surfaces  .  .  .  .  The Measurement of Small Surface Displacements  .  .  .  60  . . . .  65  Forward S c a t t e r and Back S c a t t e r Holography VII.  CONCLUSIONS AND PROPOSED FUTURE STUDY  .  .  .  82 .  .  .  .  .  Conclusions Recommendations f o r Future Study REFERENCES  91 91  . . . .  92 97  BIBLIOGRAPHY  101  APPENDICES  104  I. II.  DESIGN DRAWINGS  105  THEORY OF BENDING OF CIRCULAR PLATES  113  LIST OF FIGURES Page 1.1  Talysurf Record of a Portion of a Real Plane  2  1.2  Typical Surface F i n i s h Traces  1.3  Surface as a Compounding of Frequencies  5  1.4  R e f l e c t i o n of Light on Rough and Smooth Surfaces  7  1.5  R e f l e c t i o n from a Glass Surface  8  1.6  Optics of the Fizeau Interferometer  g  1.7  Optics of the Shadow Moire Method  n  1.8  Optics of the Oblique Incidence Method  3.1  Graphical Representation of Electromagnetic Radiation  3.2  Interference in A i r Films  22  3.3  Interference at Oblique Incidence  24  3.4  Recording of a Hologram  26  3.5  Reconstruction from a Hologram  29  4.1  Preliminary Set-up f o r Oblique Incidence Interferometry  4.2  Schematic Diagram of the Interferometer  34  4.3  Interferogram of a Lapped Surface  35  4.4  D i f f r a c t i o n E f f e c t s from the Markings on a Surface  4.5  Internal R e f l e c t i o n at a Lens Surface  38  4.6  Schematic Diagram of t h e ' G a t e ' s Method  38  .  3  12 . . .  .  . . . .  17  .  32  35  vi i Page 4.7  Schematic Diagram--using Lens as a Mirror  42  4.8  Holographic Set up--using Lens as a M i r r o r  42  4.9  Reconstructed Images from the Hologram  5.1  Schematics of Wavefront D i v i s i o n Interferometers . . . . .  46  5.2  Schematic of Amplitude D i v i s i o n Interferometers  48  5.3  Schematic of the Modified Mach-Zehnder Interferometer  5.4  The Oblique Incidence Interferometer  51  5.5  S l i d i n g Mount  53  5.6  S p a t i a l F i l t e r Assembly  53  5.7  Optical M i r r o r Mount  55  5.8  Photographic System  55  5.9  Rough Surface Holder  57  5.10  Turbine Blade Holder  57  5.11  Laser Assembly--For Interferometry  58  5.12  Laser Assembly--For Holography  58  6.1  Experimental Set up f o r Surface Topography  61  6.2  Rough Surface Interferogram--Minimum Fringes  63  6.3  Rough Surface Interferogram--Strand-like Fringes  63  6.4  Compressed C i r c u l a r Observed F i e l d  64  6.5  The Test Plate  66  6.6  Symmetrical Loading  68  6.6.a  Signal Beam Record  68  6.6.b  'No Load' Interferogram  68  .  .  .  .  . . . . . . . .  . 4 3  49  vi i i Page 6.6.c  ' A f t e r Load' Interferogram--Straight Fringes  69  6.6. d  ' A f t e r Load' Interferogram--Strand-like Fringes  69  Non-Symmetrical Loading  71  6.7. a  Signal Beam Record  71  6.7.b  'No Load' Interferogram  71  6.7.c  ' A f t e r Load' Interferogram--Straight Fringes  72  6.7.d  ' A f t e r Load' Interferogram--Strand-like Fringes  72  6.8  Joyce-Dcbel Microdensitometer  73  6.9  Signal Beam Density Plots—Symmetrical Loading  75  6.10  'No Load' and ' A f t e r Load' Density Plots—Symmetrical Loading  76  6.11  Signal Beam Density Plots—Non-symmetrical Loading  77  6.12  'No Load' and ' A f t e r Load' Density Plots—Non-symmetrical Loading  78  6.13  D e f l e c t i o n Traverses showing comparisons with Theory of Thin Plates—Symmetrical Loading  80  6.14  D e f l e c t i o n Traverses showing comparisons with Theory of  6.7  . . . .  Thin Plates—Non-symmetrical Loading  81  6.15  Forward Scatter Holography Set up  84  6.16  Holographic Fringes due to Linear Displacements of the Blade  86  6.17  Holographic Fringes due to D e f l e c t i o n of the Blade  6.18  Back Scatter Holographic Set up  6.19  Reconstructed Image from the Hologram  89  7.1  Modified Layout of the Interferometer  94  7.2  Schematic Diagram f o r Comparison of Cylinders  95  7.3  Topography of Cylinders  95  7.4  Topography of Cylinders  96  .  .  .  .  .  .  . . . . .  .  .  .  87 88  ACKNOWLEDGEMENT  The author wishes to express his sincere g r a t i t u d e to Dr. J . P. Duncan f o r his advice and guidance throughout the study and preparat i o n of the t h e s i s .  His help and encouragement have been i n v a l u a b l e .  The assistance rendered by Dr. C. R. Hazell i s much appreciated. The author would also l i k e to thank Mr. John Hoar, Chief Techn i c i a n , and his s t a f f f o r t h e i r p r a c t i c a l help with the development of the Instrument. Support f o r t h i s study was provided by the Defence Research Board of Canada, through Grant Number 9601-07, f o r which thanks are given.  NOMENCLATURE wave transmitted through the hologram Flexural R i g i d i t y modulus of E l a s t i c i t y e l e c t r i c vector p a r a l l e l to x-axis energy of the r a d i a t i o n of frequency v e l e c t r i c vector i n t e n s i t y of l i g h t reaction at point support, f r i n g e spacing fringe deflection uniform loading radius of curvature of the lens Relative S e n s i t i v i t y time p e r i o d ; depth of deformation of an otherwise smooth wavefront amplitude of the l i g h t wave; radius of the plate amplitude of the reference wave amplitude of the object wave t o t a l amplitude of the l i g h t waves at the photo plate distance of the point support from the centre of plate v e l o c i t y of l i g h t  s p a t i a l frequency primary focal length of the lens secondary f o c a l length of the lens Planck's constant angle of incidence integer number angle of r e f r a c t i o n wave v i b r a t i o n s i n planes p a r a l l e l and perpend i c u l a r to the plane of incidence time, surface displacement d e f l e c t i o n of plate i n z d i r e c t i o n coordinate system phase of reference wave at point x at the photo plate angle between the reference and signal beam r e f r a c t i v e index oblique incidence Poisson's r a t i o ; frequency of v i b r a t i o n of l i g h t waves wave length of l i g h t Brewster's angle geometry of the surface expressed i n x plane geometry of the surface expressed i n y plane phase of object wave path d i f f e r e n c e between two rays of l i g h t reference number  DESIGN AND DEVELOPMENT OF AN OBLIQUE INCIDENCE INTERFEROMETER  I PHYSICAL ENGINEERING SURFACES  1.1  INTRODUCTION Physical surfaces, i n p a r t i c u l a r the bounding surfaces of engin-  eering components, are r a r e l y p e r f e c t .  Engineering design c a l c u l a t i o n s ,  to s a t i s f y various operating c r i t e r i a , frequently lead to a geometrical s p e c i f i c a t i o n of some three dimensional component and experimental procedures often lead to surfaces having unknown equations.  In g e n e r a l ,  the nature of physical f a b r i c a t e d or machined engineering surfaces  is  known only a f t e r they have been formed and measured with some appropr i a t e technique. The advent of modern machines has made i t f a i r l y economical and easy to r e a l i s e any macroscopic geometry.  However, the required surface  f i n i s h i s a very important element of the surface s p e c i f i c a t i o n .  The  importance of surface condition may e a s i l y be appreciated i n a p p l i c a t i o n s such as l u b r i c a t i o n , f a t i g u e , heat t r a n s f e r , aerodynamic design and many other places where the surface conditions a f f e c t the operating conditions.  1.2  SURFACE TOPOGRAPHY Most engineering components have a 'macroscopic  1  basic geometry  which may be described w i t h i n c a r t e s i a n space by functions of the form  2 F(x,y,z) = 0, where F may have e i t h e r known or unknown form, and an unwanted but i n e v i t a b l e roughness or texture (rough to f e e l ) from c u t t i n g t o o l s , which produce chips with p e r i o d i c or random d i s c o n t i n u i t i e s , the best example of which i s the surface cut by a b u i l t up edge  (B.U.E.).  This breaks away p e r i o d i c a l l y and leaves ' t a g s ' of d i s t o r t e d metal on the surface. The surface roughness i s commonly measured as a centre l i n e average (C.L.A.) departure from a mean, centre l i n e , shape as shown i n Figure 1.1 [ 1 ] .  R.M.S  F i g . 1.1  FROM  C.L.A-  T=i—"  T a l y s u r f Record of a Portion of Real Plane  A t y p i c a l , section of a physical surface r e s u l t i n g from a f i n e machining operation, and combining the e f f e c t s of several causes i s shown i n Figure 1.2.  The  ROUGHNESS  ( P R I M A R Y TEXTURE)  A  WAVINESS ( S E C O N D A R Y TEXTURE)  B  E R R O R O F FOR.M A surface texture representing the combined effects of sereral causes  IRAVtlll  DIAMOND t l f P I B  smut  40  Typical surface finish chart Ratio of magnification, vertical: horizontal — 50 J 1  F i g . 1.2  Typical Surface F i n i s h Traces  C  4 surface may have roughness, waviness or e r r o r of form representing geometrical waves of orders of frequencies ranging from high to low measured from some ideal l i n e .  Mechanical devices f o r recording such  departures often depend upon the t r a c i n g by a s t y l u s of minute radius guided by a SKID which follows a path roughly p a r a l l e l to the 'macros c o p i c ' form  of the surface.  The s t y l u s moves r e l a t i v e to the s k i d  at r i g h t angles to the l i n e of traverse and records the ' m i c r o s c o p i c ' shape.  'Talysurf  and 'Talyrond' measuring instruments are examples.  Hand-applied instruments of s i m i l a r type enable measurements to be taken over s h o r t , f i x e d base lengths.  The p r i n c i p l e s of making such measure-  ments and i n t e r p r e t i n g the r e s u l t a n t p r o f i l e are well o u t l i n e d i n B r i t i s h Standard S p e c i f i c a t i o n 1134: 1961 (UDC 620.179.6) e n t i t l e d 'Centre Line Average Height Method f o r the Assessment of Surface Texture' from which Figure 1.2 i s reproduced. Single point mechanical devices f o r measuring the z coordinate of a surface from a reference plane or f o r measuring curvature, such as spherometers, often employ elements which contact the surface over a macroscopic area and, therefore are i n s e n s i t i v e to microscopic perturbation.  These devices have t h e i r place and may often be more appropriate  than the o p t i c a l methods reviewed l a t e r . The geometry of the surface F(x,y,z) = 0 i s commonly defined i n terms of h e i g h t , . z , from a coordinate plane (x,y) expressed as two plane sections such that z = ^(x)  z = i|) (y) 2  5  ROUGHNESS (HIGH  FREQUE-NCY)  ERROR OF FORM (MEDIUM FORM (LOW  A  PORTION  OF  c,  flr A, MEP  FREQ.)  FREQ.)  f P r r r r  »+0Q  --^FOURIER TRANSFORM DISCRETE  cL(u) =  S  CONTINUOUS  FREQUENCY FOURIER  Fig.  1.3  FfcO i""Sx SPECTRUM  U.  TRANSFORM  OF  A GENERAL SURFACE  Surface as a Compounding of Frequencies  Either  of  these functions  would  have the character of Figure 1.3 and  would be a non p e r i o d i c , bounded function of e i t h e r x or y.  In general  such sections have a p o t e n t i a l l y i n f i n i t e range of s p a t i a l frequencies which can be represented by Fourier Transforms as shown in Figure The lowest frequency w i l l characterize the designed  Vorni of  1.3.  the surface.  A higher frequency w i l l usually come from p e r i o d i c machine tool errors and highest frequencies c a l l e d ' s p a t i a l ' noise are associated with random and p e r i o d i c features of cutting  1.3  processes.  OPTICAL METROLOGY In o p t i c a l metrology, l i g h t r e f l e c t e d or refracted from the  surface, containing information about i t s shape and conditional's examined. The scattered l i g h t from a rough surface, as shown in Figure 1.4,  results  in a d u l l or matt appearance of the surface where the law of r e f l e c t i o n applies l o c a l l y .  However by various processes of mechanical g r i n d i n g ,  lapping and p o l i s h i n g (or e l e c t r o p o l i s h i n g ) m i r r o r - l i k e surfaces can be obtained. Rays r e f l e c t e d specularly from these surfaces may then i n d i the cate/'macroscopic' nature of the surface. The c h a r a c t e r i s t i c s of the r e f l e c t e d and transmitted l i g h t depend on wavelength, p o l a r i z a t i o n  and angle of incidence.  F i g . 1.5 shows the rel  tion between the p o l a r i z a t i o n and incidence at a glass surface.  At one angl  of incidence, the Brewester angle (|), the r e f l e c t e d l i g h t i s completely plane-polarized with i t s e l e c t r i c vector perpendicular to the plane of  SCATTERED  REFLECT! ON  8  incidence.  At other angles the r e f l e c t e d l i g h t i s only p a r t i a l l y p o l a r -  ized with v i b r a t i o n s p a r a l l e l and perpendicular to the plane of i n c i dence [ 2 ] .  F i g . 1.5  R e f l e c t i o n from a Glass Surface  At normal incidence, where <|> = 0, the p a r a l l e l and perpendicular components have the same r e f l e c t a n c e of 4 per cent. drops and r  $  With increasing  r  r i s e s u n t i l at the p o l a r i z i n g angle t h e i r values are zero  and 15 per cent r e s p e c t i v e l y .  Thereafter the reflectances increase more  r a p i d l y u n t i l at 9 0 ° , i . e . , grazing incidence, both components are totally reflected. The f o l l o w i n g o p t i c a l methods have been used f o r revealing surface topography through the r e f l e c t i o n of l i g h t from the surface.  9  1.3.1  Fizeau Interferometry (Figure  1.6)  This i s used mainly f o r the examination of the f l a t n e s s of surfaces but d e f l e c t i o n s of a specimen due to loading may also be observed by comparing ' b e f o r e ' and ' a f t e r ' f r i n g e patterns.  Very small departures i n the top-  ography of surfaces of plates from a reference form, usually plane or spheri c a l , are revealed i n terms of contour i n t e r v a l s of up to 0.00001 i n c h . However, f o r large f i e l d of low curvature t h i s method requires p o l i s h i n g of the measured surface i f i t i s to be examined at normal incidence.  Q6SCRVE.R  F i g . 1.6  REFERENCE. GLASS  SURFACE  Optics of the Fizeau Interferometer  The r e s o l u t i o n and magnification of the observing o p t i c a l systems l i m i t s the curvature which can be revealed.  Thus on specular surfaces  small curvatures over large apertures and,on even rough surfaces,high curvatures w i t h i n the small apertures of an interference microscope may  10 be revealed.  Gate's Interferometer [3] and the Perkin-Elmer Spherometer  [4] are examples. 1.3.2  Shadow Moire Method (Figure  1.7)  This method, o r i g i n a l l y proposed by Weller et al. [5] and developed by Theocaris [ 6 ] , may be used to obtain contour maps of matt d i f f u s i n g surfaces of s i g n i f i c a n t curvature.  As shown in Figure 1.7, a grating i s  placed very near and nearly p a r a l l e l to the surface and i t i s i l l u m i n a t e d by a collimated beam i n c i d e n t at angle i .  When the grating and i t s  shadow are viewed together at an angle, o, fringes i n d i c a t i v e of v a r i a b l e i n t e n s i t y in the plane of i l l u m i n a t i o n and observation (containing the normal to the surface) also reveal points of known r e l a t i v e l e v e l , in terms of contour i n t e r v a l s of 0.01-0.001 inches, as is apparent from the figure. The permissible gap between the grating and the surface i s d i r e c t l y proportional to the p i t c h due to the d i f f r a c t i o n e f f e c t s which degrade r e c t i l i n e a r p r o j e c t i o n of shadows at a s u f f i c i e n t distance from the g r a t i n g .  Thus t h i s method depends on i n t e n s i t y v a r i a t i o n s r e s u l t i n g  from o p t i c a l transformation through complex apertures in the form of gratings, and not on phase comparison as in conventional interferometry. 1.3.3  Oblique Incidence Interferometry (Figure  1.8)  This is a two beam i n t e r f e r o m e t r i c method where a c i r c u l a r c o l l i mated beam scanning an e l l i p t i c a l f i e l d is l a t e r superimposed on the c i r cular reference beam as shown in Figure 1.8. cribed in d e t a i l l a t e r in t h i s t h e s i s .  The method w i l l be des-  F i g . 1.7  Optics of the Shadow Moire Method  12  P L A T E  E.LLIPT/CAL F7E.L.D  F i g . 1.3 Optics of the Oblique Incidence Method  Topographic measurements of non-specular surfaces may be made due to specular r e f l e c t i o n at oblique incidence.  However, t h i s advantage of  specular r e f l e c t i o n i s gained at the expense of s e n s i t i v i t y which decreases at higher incidences.  Contour i n t e r v a l s of up to 0.0001 inch  may be obtained which i s quite acceptable f o r engineering  purposes.  II STATEMENT OF OBJECTIVES  The Oblique Incidence method shares features of the two well known methods, the Fizeau method and the Shadow Moire method.  The com-  bination of Fizeau and Shadow Moire method was achieved i n an instrument designed by Duncan, J . P. [7] f o r commercial use.  However with the  rapid developments i n engineering optics some of the objectives of the Shadow Moire method are being achieved by phase interferometry by way of Oblique Incidence interferometry and i t s extension through Holography. The o b j e c t i v e of t h i s work was the design, development and proving of an Oblique Incidence Interferometer f o r executing any of the following i n t e r f e r o m e t r i c techniques: 1.  Surface topography of unpolished rough surfaces.  2.  The measurement  3.  The execution of forward s c a t t e r and back s c a t t e r holography f o r  of small surface displacements.  normal q u a l i t a t i v e recording and subsequent  comparison—holo-  graphic interferometry. Oblique incidence has s i x advantages:  (i)  i t permits measurements  on d i f f u s i n g surfaces of f a i r q u a l i t y and enhances r e f l e c t i v i t y due to obliquity; ( i i )  i t i s most s u i t a b l e f o r extended surface since a small  s i z e beam scans a long e l l i p t i c a l f i e l d ; ( i i i )  the use of a collimated  14 beam f a c i l i t a t e s q u a n t i t a t i v e a n a l y s i s ; ( i v ) required; (v)  no reference surface i s  i t provides an intermediate s e n s i t i v i t y , i n between nor-  mal incidence and Shadow Moire s e n s i t i v i t i e s as desired by i n d u s t r y ; and ( v i ) industrial  i t provides a system which can be used by s l i g h t adaption f o r holography.  Ill BRIEF REVIEW OF INTERFEROMETRY  3.1  HISTORICAL REVIEW The e a r l i e s t studies of interference i n optics are i n t i m a t e l y  bound up with the evolution of the theories of l i g h t propagation and u l timately s e t t l e d a longstanding l a r theories of l i g h t .  c o n f l i c t between the wave and corpuscu-  In 1665, Grimaldi discovered the d i f f r a c t i o n  fringes produced both by a narrow obstacle and by a s l i t and t r i e d to explain his observations by supposing that l i g h t consisted of a f i n e f l u i d i n a state of v i b r a t i o n .  In the same year Hooke attempted to ex-  p l a i n the colors of t h i n films (discovered two years e a r l i e r by Boyle) also by means of a crude wave theory. Huygens, i n 1678, then announced his p r i n c i p l e of wave propagat i o n through subsidiary wavelets and attempted to account f o r the double r e f r a c t i o n of c a l c i t e which had been discovered eight years e a r l i e r by B a r t o l i n u s .  Bartolinus f a i l e d to do t h i s because the  l o n g i t u d i n a l v i b r a t i o n s he postulated could not explain p o l a r i z a t i o n . During t h i s period (since 1666 i n f a c t ) Newton had been conducting his researches on l i g h t which he described i n his p u b l i c a t i o n "Opticks" in 1704.  Newton discussed both a wave a corpuscular theory of l i g h t but  was unable to r e c o n c i l e r e c t i l i n e a r propagation with a wave theory.  16 Optics i n the eighteenth century were dominated by the impressive authori t y of Newton--and no f u r t h e r progress was made u n t i l Thomas Young, i n 1802, proved that l i g h t i s subject to interference and i s propagated as a wave form. Young was the f i r s t to r e a l i z e both the P r i n c i p l e of Coherence and the P r i n c i p l e of Superposition, both of which are fundamentals in interference.  His work was followed by the outstanding and s t i l l  valid  developments of Fresnel who, in 1805, l a i d a foundation f o r a theory of l i g h t propagation and d i f f r a c t i o n . Other discoveries were r a p i d l y made in the f i e l d of i n t e r f e r e n c e . Herschel, in 1809, discovered highly sharpened fringes in a t h i n f i l m when l i g h t emerges at the grazing incidence (Herschel's f r i n g e s ) .  In  1817, Brewster reported the interference e f f e c t (Brewster fringes) which takes place when l i g h t passes through or i s r e f l e c t e d by two s i m i l a r plane p a r a l l e l plates s l i g h t l y i n c l i n e d to each other.  In 1837, Lloyd  performed an important interference experiment using a s i n g l e mirror and in 1849 Haidinger, using a piece of mica, observed and accounted f o r the rings now known by his name. From the middle of the nineteenth century onwards many new d i s coveries were made, e s p e c i a l l y by Fizeau.  New forms of interference  were studied and many of these were applied to widely d i f f e r e n t f i e l d s . A considerable body of p u b l i c a t i o n s grew up u n t i l now the subject of "Interferometry". i s one of much complexity and importance.  17 3.2  ELECTROMAGNETIC NATURE AND COHERENCE OF LIGHT According to the Quantum theory, energy i s radiated i n d i s c r e t e  pulses, c a l l e d photons.  These photons are emitted as the electrons in a  molecular structure change o r b i t s thus releasing or absorbing d i s c r e t e 'packets' or quanta of energy associated with frequencies of r a d i a t i o n given by Plank's Law AE = hv.  The electromagnetic r a d i a t i o n of an i n d i -  vidual photon, propagating as a succession of spherical wavefronts, may be represented symbolically and g r a p h i c a l l y as a sinusoidal wave motion of an e l e c t r i c vector E as shown in Figure 3.1  [8].  The emission from a t y p i c a l source i s composed of an immense number of photons of a mixture of wavelengths or frequencies, phases and planes of v i b r a t i o n .  If the r a d i a t i o n i s f i l t e r e d in some way so that  18 i t s constituent photons have the same wavelength, the l i g h t is said to be MONOCHROMATIC.  I f the r a d i a t i o n contains photons v i b r a t i n g in one  plane only, the l i g h t is said to be PLANE POLARISED.  I f , however, the  r a d i a t i o n contains vibrations in two planes, the l i g h t i s said to be ELLIPTICALLY POLARIZED. A general expression f o r the propagation of an electromagnetic r a d i a t i o n (and s p e c i f i c a l l y l i g h t ) in a three dimension f i e l d can be obtained from Maxwell's theory r e s u l t i n g in s i x vector equations  relat-  ing e l e c t r i c and magnetic disturbances i n the d i e l e c t r i c s in which radi a t i o n s are propagated by wave motion [ 9 ] .  These s i x equations may be  coordinated by the wave equation  ,2E-4+4 4- /4 +  3X  3y  3z  :l  c  (3-D  3t  where x, y , z are any convenient perpendicular coordinates.  For the one  dimensional case of plane wavefronts propogating in d i r e c t i o n z t h i s r e duces to  ^ = ^ ^ az  c  0.2) 3t^  For a point S on a plane wavefront, distance z from the source 0, the s o l u t i o n of equation 3.2 i s given by  E  X  = a cos | ^ ( c t - z ) A  (3.3)  where a and \ are the amptitude and wavelength of the wave and c is the v e l o c i t y of l i g h t .  In complex notations the above s o l u t i o n i s w r i t t e n as  19 2ni E = a e  (ct-z)  A  (3.4)  C l e a r l y , the phase at any point depends on the phase of the source.  Therefore i f the l i g h t beams from two independent sources  reach  the point S, i t w i l l be impossible to combine them to form s t a t i o n a r y waves due to absence of a f i x e d r e l a t i o n between the phases of the two beams.  However, t h e i r i n t e n s i t i e s combine l o c a l l y and such l i g h t beams  are termed INCOHERENT.  On the other hand two continuous beams from the  same source are c a l l e d COHERENT and these may be caused to superpose to produce v i s i b l e interference e f f e c t because t h e i r amplitudes, not t h e i r i n t e n s i t i e s , can combine.  When such coherent waves pass through a p o i n t ,  the v i b r a t i n g medium at t h i s point i s subjected to the combined, superposed e f f e c t of the two v i b r a t i o n s and under s u i t a b l e conditions t h i s leads to s t a t i o n a r y waves; such waves are c a l l e d INTERFERENCE FRINGES.  3.3  PRINCIPLE OF SUPERPOSITION Let two coherent waves of wavelength A and of d i f f e r e n t i n i t i a l  amplitudes a^, a  2  leave a point on the source and t r a v e l l i n g by d i f f e r -  ent path lengths, z-j, z , a r r i v e at a point S where superposition takes 2  place. y-j and y  The amplitude A of the r e s u l t a n t i s the sum of the amplitudes 2  at the point S given by:  = a, Cos  = a  2  A  (ct-zj (3.5)  Cos | i i ( c t - z ) 2  20 writina: ^  (ct- ) = * Z l  ( c t - z ) = <j> + 2  where:  then, A = a-j Cos <j> + a  2  Cos (<j> + e)  simplifying,  A = (a  2  +  + 2a^a  2  Cos e )  1 / 2  Sin (<f> + e) (3.6)  where. - a tan e =  ?  Sin e  a-j + &2 Cos  Thus, the e f f e c t at S i s such that the l o c a l v i b r a t i o n has a d i s placement with a period of the two component waves, but the phase i s determined by e and the amplitude by the term under the root. i n t e n s i t y I,  The  at S, is proportional to the square of the amplitude, i . e . ,  to 2 2 a-j + &2 + 2a-ja Cos e. 2  For the simple case in which both combining waves have the same amplitude, i . e . , a-| = a , the i n t e n s i t y 2  21 I = 4a^ Cos | I i s maximum f o r e = 0 ,  (3.7)  2n, 4n, e t c . , i . e . , 2 nn. Thus condition  for maximum i n t e n s i t y i s 2n (z-, - z ) 9  A  i.e.,  = 2nn  f o r (z-| - z^ ) = nA where n has a l l i n t e g r a l values. S i m i l a r l y I i s a minimum f o r 0 = n, 3n, 5n, e t c . , i . e . , f o r  (z-| - z ) = (n + l/2)x. 2  The simple physical meaning of t h i s i s that wave ' c r e s t s '  super-  impose and combine to a maximum when the path d i f f e r e n c e i s an i n t e g r a l m u l t i p l e of wavelength, w h i l s t ' c r e s t s ' and 'troughs'  f a l l together to  give zero e f f e c t when the path d i f f e r e n c e i s an odd m u l t i p l e of h a l f the wavelength.  3.4  INTERFERENCE IN AIR FILMS When a bundle of l i g h t rays i s i n c i d e n t at an angle (^ - a) to  the normal on  nearly p a r a l l e l r e f l e c t i n g surfaces, as shown in Figure  3.2, the elemental rays I and I tially  are each p a r t i a l l y r e f l e c t e d and par-  transmitted at surfaces AB and CD.  The ray I a f t e r r e f l e c t i o n at  i  0 meets the ray R, r e f l e c t e d part of I , at h^-  i  If the rays I and I  o r i g i n a t e d at the same source, the combination occurring at A  ?  will  22  Fig. 3.2  Interference i n A i r Film  r e s u l t i n interference according to the phase r e l a t i o n s h i p between and i n t e n s i t i e s of the wave motions reaching k^. The path d i f f e r e n c e between the i n t e r f e r i n g beam, A, i s given by  A  medium  < 1°  =  A  - [ 0 A l  Vdium  =  C  11 D  ^medium "  +  +  =  0A 2 t  2  C o s  - A  l C l  ]  ^ i V a i r m e d i u m  6  Equivalent path d i f f e r e n c e in a i r = 2yt Cos e Also the r e f l e c t i o n at A^ w i l l  (3.8)  introduce a natural phase change of n (or  a path d i f f e r e n c e of A/2) and hence the dark interference w i l l  appear  23 where the path difference i s an i n t e g r a l number of wave length, i . e . , 2yt Cos e = nx  (3.9)  and bright fringes form where 2yt Cos © = (n + J-) X  3.5  (3.10)  INTERFERENCE AT OBLIQUE INCIDENCE When a collimated c i r c u l a r beam of diameter, a, i s d i r e c t e d at a sur-  face at an oblique incidence 0, an e l l i p t i c a l f i e l d of minor a x i s , a, & major a x i s , a/Cos 0, i s i l l u m i n a t e d as shown i n Figure 3.3. 't'  For a displacement  the path length of the affected ray i s increased, from equation  by 2t Cos 0.  (3.8),  This w i l l lead to a phase change of 2n whenever 2t Cos 0 =  nx (where n = 1 , 2 , 3 , . . . , e t c . ) .  The s e n s i t i v i t y i s thus reduced by Cos 0  from the s e n s i t i v i t y at normal incidence.  The contour i n t e r v a l of the  interference f r i n g e s , appearing on the c i r c u l a r f i e l d , w i l l be  t  =  2Tc7s—  ( - > 3  1]  The r e l a t i v e s e n s i t i v i t y of an interferometer i s a constant and is defined as  [10]  Sb  where  Rel  = Afi£ T  =  1 X  (3 12)  24  F i g . 3.3  Interference at Oblique Incidence AP = f r i n g e excursion P = f r i n g e spacing  and T = depth of deformation of an otherwise smooth wavefront.  For a displacement ' t ' , the depth of deformation of  an incidence plane wavefront i s equal to 2t Cos 0.  Therefore, from  Equation 3.12, the displacement ' t ' in terms of f r i n g e displacement i s given by  b  Rel  =  AP/P 2t Cos 0  =  t - ( f ) ( T^e> 2  1 A  (3 = 1  25 However, due to o b l i q u i t y , the contours represented on the c i r c u l a r f i e l d of interferograms are i n planes p a r a l l e l to the mean plane of the examined surface and the plan view coordinate scale is  compressed  r e l a t i v e l y in the l o n g i t u d i n a l d i r e c t i o n in the r a t i o T/Cos e,  3.5  PRINCIPLE OF HOLOGRAPHY The p r i n c i p l e of holography was discovered i n 1948 from c e r t a i n  proposals made by Gabor [11] f o r improving the performance of the electron microscope.  But i t was not u n t i l the discovery of LASER (Light A m p l i f i -  cation by Stimulated Emission of Radiation) in 1960 by Javin and Maiman and pioneering work of L e i t h and Upatnieks [12,13,14] that holography became a new f i e l d i n i n t e r f e r o m e t r i c work. A hologram i s merely a complex interferogram in which a coherent reference beam i s superimposed on the coherent l i g h t scattered from each and every point of the i l l u m i n a t e d object f a l l i n g upon each and every point on the photographic p l a t e .  This interference e f f e c t i s recorded  on a photographic plate in the form of f i n e , complex s t a t i o n a r y i n t e r ference f r i n g e s .  When t h i s plate i s r e - i l l u m i n a t e d by the coherent' r e f -  erence beam a f t e r development, the developed plate acts as a complex transmission d i f f r a c t i o n grating and constructs the image of the o r i g i n a l object with a l l i t s three dimensional c h a r a c t e r i s t i c s .  The hologram  not only records'the i n t e n s i t y of the l i g h t scattered from an o b j e c t , as in normal photography, but also stores the phase structure of the s c a t tered l i g h t .  26 A simple mathematical concept, developed by A. E. Ennos [15], w i l l be used to explain the formation of a hologram and reconstruction of the image. For s i m p l i c i t y , a one dimensional treatment of the holographic p r i n c i p l e w i l l be considered.  However, the analysis holds good f o r any  three dimensional photographic emulsion and any three dimensional o b j e c t , both i n r e f l e c t i o n as well as  transmission.  PRISM  PHOTO PLATE.  F i g . 3.4  Recording of a Hologram  Coherent, monochromatic l i g h t from a l a s e r i s collimated a f t e r focussing, f i l t e r i n g and expanding through a s p a t i a l f i l t e r assembly as shown i n Figure 3.4.  The lower part undergoes a phase s h i f t in accor-  dance with the transmitting object information.  The upper part i s  27  deflected through angle 6 without any other change. %  - a  e^  s  and  { x )  \  Let  - a  r  e  i a X  represent object and reference beams, where  X- |5P  ax =  A  (3.14)  2en ——  a =  At the photographic plate t o t a l amplitude i s ,  a = a  r  +  t  s  = a e  i a x  r  +  a  e*<> 1  s  x  .  (3.15)  However, the photographic plate records the i n t e n s i t y I, proportional to square of the amplitude, i . e . ,  I T  i-h2 i i* |a| = i|a|i |a|  cc  = [a e-  = a :  2 r  + a  a e- * ][a e  +  i a x  r  2 s  1  s  +  (x)  r  +1ax  aa [ e ^ ^ - ^ r s  + a e s  +  e  1  + i  *  ( x )  ]  (3.16)  ^ " * ^ ]  Dark f r i n g e s , normally i n v i s i b l e to the naked eye, are formed p a r a l l e l to the b i s e c t o r of angle B.  The i n t e n s i t y of the l i g h t from  the object i s recorded as a modulation of the contrast of the fringes and the phase of l i g h t i s recorded as a modulation in f r i n g e spacing in the form of a d i f f r a c t i o n g r a t i n g .  28 To reconstruct the image the hologram i s i l l u m i n a t e d with the reference beam  and the wave transmitted through the hologram can be represented by:  A = ay I  = a  r  (a  2  + a) i  a  2  a  2  i  a  X  [ e ^ M  +  (3.17)  "  2 a X  ^ + i ^*)] 1  The f i r s t term represents, zeroth order d i f f r a c t i o n of the wave form, a pure i n t e n s i t y attenuation and i s transmitted without any devi a t i o n (Figure  3.5).  The second term containing e ' ^ * ^ i s an exact r e p l i c a of the 1  object with reverse curvature and term -i2ax e w i l l d i f f r a c t t h i s image by angle 6 so that the angle between the real object and photographic plate i s now 2p.  This reconstructed real image  of the object can be photographed without the use of any lens. The t h i r d term i s an exact r e p l i c a of the object with reduced i n tensity.  This i s a v i r t u a l image and can be seen when looking through  the hologram.  29  F i g . 3.5 Reconstruction from a Hologram  Response c h a r a c t e r i s t i c s of the photographic emulsion are taken into account f o r rigorous a n a l y s i s . the reconstructed image.  However, i t has l i t t l e e f f e c t on  I t i s important that the reference beam i s  approximately 5 times stronger than the object beam.  F i n a l l y , the r e -  cording f i l m must be capable of recording the f i n e d e t a i l s of the i n t e r ference pattern.  A hologram i s characterized by a s p a t i a l frequency  which is a function of the maximum angle 3 between the reference and signal beams and the highest s p a t i a l frequency i s given by [16]  f  m™  = f Sin  B/2  (3.18)  30 However, at these frequencies the contrast i s usually reduced by at l e a s t 20 per cent and the f i l m performance i s poor.  As a rule of thumb, the  r e s o l u t i o n l i m i t of an emulsion should be divided by a f a c t o r of three to a r r i v e at a p r a c t i c a l f i g u r e .  IV PRELIMINARY EXPERIMENTS  4.1  OBLIQUE INCIDENCE INTERFEROMETRY Some preliminary experiments i n executing oblique incidence i n -  terferometry were based on the o p t i c a l layouts of the Twymann-Green interferometer [17].  A modified version of t h i s interferometer (Skip  interferometer) has already been used by Perkin-Elmer Company of Norwalk [18] f o r small s c a l e , d e l i c a t e i n t e r f e r o m e t r i c measurements. The various o p t i c a l components used to assemble t h i s experimental o p t i c a l system were as f o l l o w s : 1.  A Helium-Neon gas Laser (1 mW Power);  2.  A s p a t i a l f i l t e r assembly ( i n c l u d i n g a microscope objective and a 10 micron p i n h o l e ) ;  3.  A plano-convex lens of 18.5 inch focal length;  4.  A beam s p l i t t e r (1 inch x 1 inch x 2 inch prism cube);  5.  Three o p t i c a l l y plane m i r r o r s ;  6.  A P o l a r o i d Camera f o r 4 inch x 5 inch f i l m s ;  7.  A double convex lens to magnify the f i e l d of view;  8.  A lapped s t e e l plate as t e s t specimen; and  9.  A v i b r a t i o n - i s o l a t e d granite table to reduce v i b r a t i o n e f f e c t s .  The elements were arranged on t h i s table as shown i n Figure 4.1.  33 4.1.1  Experimental Procedure and Results  The optics of the interferometer were as shown i n Figure 4 . 2 ( L a y o u t - l ) . The beam from the Laser was expanded and s p a t i a l l y f i l t e r e d through the s p a t i a l f i l t e r assembly.  The f i e l d was stopped with a c i r c u l a r card-  board aperture and the central part of the expanded beam was collimated by the plano-convex lens placed at i t s focal length from the s p a t i a l filter.  This beam was s p l i t , by the beam s p l i t t e r , i n t o two parts  forming the reference and signal beams.  Using m i r r o r , N^, the signal  beam was d i r e c t e d at a grazing incidence of 8.5 degrees onto the lapped steel p l a t e .  M i r r o r , M^, was arranged such that the l i g h t r e f l e c t e d  from the s t e e l plate retraced i t s path back to the beam s p l i t t e r where i t recombined with the reference beam, r e t r a c i n g i t s own path a f t e r r e f l e c t i o n at m i r r o r , M-j, to produce interference f r i n g e s . were recorded on p o l a r o i d f i l m .  These fringes  Some of the r e s u l t s are shown in Figure  4.3. Following points of i n t e r e s t were observed and considered during the f i n a l design of the instrument. 1.  During the i n i t i a l stages large movements of interference fringes occurred and these made t h e i r photography d i f f i c u l t .  However,  when the bags of leadshots were l a i d on the bases of the o p t i c a l components t h i s s i t u a t i o n improved.  It was c l e a r from t h i s that  adequate r i g i d i t y and s t a b i l i t y of a l l the o p t i c a l components was e s s e n t i a l f o r observing  fringes.  F i g . 4.2  Schematic Diagram of the Interferometer  .3  Interferogram of a Lapped Surface  D i f f r a c t i o n Effects from the Markings on the Surface  36 2.  Due to the f i x e d r e f l e c t i o n - t r a n s m i s s i o n r a t i o of the beam s p l i t t e r , contrast of fringes could not be much improved.  It was  considered necessary to use a v a r i a b l e beam s p l i t t e r in the f i n a l design of the instrument  to obtain any desired r a t i o of  the reference and signal beam. 3.  The lengths of the reference and signal beam-paths had to be made nearly equal to accommodate the f i x e d but l i m i t e d coherence length of the Laser.  I t was considered necessary to ensure that  the f i n a l design embodied means of adjusting path lengths to nearly equal values i n each mode of use. 4.  Interference fringes could also be observed at some convenient point before the c o l l i m a t i n g lens by placing a second beam s p l i t t e r as shown i n Fig.4.2(Layout 2).  However, t h i s i s not very  p r a c t i c a l due to s i g n i f i c a n t loss of i n t e n s i t y at various optical 5.  components.  In addition to the lapped s t e e l plate a f l a t , black granite plate was also examined.  Due to the low r e f l e c t i v i t y of the sur-  face the l a t t e r gave interference fringes only at a f a i r l y low grazing incidence. 6.  One of the main objectives of these preliminary tests was to search f o r some method of marking on the surface under t e s t to locate the observed r e s u l t s on the real surface and determine the scale of the photographs.  Marking methods t r i e d at t h i s  37 stage included masking tape, black i n k , scratches.  A l l these  gave r i s e to d i f f r a c t i o n e f f e c t s , as shown in Figure 4.4, which made i t d i f f i c u l t to determine features on the photographs exactly.  Further methods were t r i e d during the f i n a l develop-  ment of the interferometer as discussed l a t e r .  4.2  INSTANT HOLOGRAPHY In basic holography, wavefronts carrying information about an ob-  j e c t and a reference wavefront which i s to some degree coherent with the f i r s t f a l l simultaneously over the same area on the photographic recording medium.  To gain an understanding of the holographic p r i n c i p l e a  double focus coated lens system s i m i l a r to that used by Gates, and Bennett [19] was used. By making use of the second focus formed by l i g h t r e f l e c t e d i n t e r n a l l y at the lens surfaces, Figure 4.5, a lens can be used as a double focus lens.  A simple d e r i v a t i o n of the second focus of a t h i n double  convex lens f o r paraxial rays i s given below. An i n c i d e n t ray, I, a f t e r r e f r a c t i o n at a point A meets the second surface of the lens at point B where i t i s p a r t i a l l y r e f l e c t e d .  After  i n t e r n a l r e f l e c t i o n at point C, and r e f r a c t i o n at point D, i t focuses at the second focus of the l e n s , at E.  However, f o r a t h i n double convex  lens and considering paraxial rays only, hi = h  0  - h  0  * h  38  F i g . 4.6  Schematic Diagram of the Gate's Method  39 also Sin 0 =  5- T  Sin <f> =  l  Jj-  0 (4.1)  = <f>  Tan (J>2 ~ ^— -  where f-j and f  are the primary and secondary focal lengths and R i s the  2  radius of curvature of the lens.  The focal length, f-|, of a double con-  vex lens of radius of curvature, R, i s given by i i \ - ' R (P " D  1  2  j-  1_  or  R ~ " Zf^y  1  (4.2)  - 1)  From S n e l l ' s law at A _ Sin (e + ^) ^ (e + Sin U + r) ((j> + r)  y  or y r = 0 + * (1-y) = £ - + £ (1-y) = J ^ -  (4.3)  S i m i l a r l y , at point D, _ Sin (<fr -+ <j>) „ (<j> + j>) Sin (5> - r) " (5$ - r) 2  y  . _ , / r i\ v , _ h <t> - • (5y - 1) - yr - - ^ 2  7  (5y - 1) . ij ( y  h  40 (4.4)  From Equation  (4.1) h_ (j>2  f  - f  2  Thus the l i g h t focusing  (P -  1 (3  M  1) - 1)  (4.5)  at the second focus i s used as the r e f l e c t i o n  beam whereas the l i g h t passing through without any i n t e r n a l r e f l e c t i o n uniformly i l l u m i n a t e s the object.  However the i n t e n s i t y of doubly r e -  f l e c t e d l i g h t i s about 1/600 of the refracted wave but with s u i t a b l e coatings t h i s can be increased. 4.2.1  Experimental Procedure and Results  The optics of the experimental set-up i s shown i n Figure 4.6. The coated lens of 38.25 inch  focal length was placed at i t s focal  length from the s p a t i a l f i l t e r .  The roughly collimated beam from the  lens uniformly i l l u m i n a t e d the object (razor blade) and the beam due to internal r e f l e c t i o n was used as a reference beam by focussing onto a small mirror at the second focus.  The hologram was recorded on  Polaroid Type 55 P/N f i l m placed in a holder at approximately 20 inches from the object. F i r s t experiments were done using an e x i s t i n g lens with a bismuth oxide coating.  It was found that the reference beam i n t e n s i t y was very  41 poor and no successful holograms were obtained.  However, when t h i s lens  was replaced with a properly coated lens having a 30 per cent r e f l e c t i v i t y reasonably good holograms r e s u l t e d .  Holographic interferometry was  also performed by double exposure when the razor blade was unbent and then s l i g h t l y d e f l e c t e d .  Interference fringes between reconstructed  images of unbent and bent blade were seen when the r e s u l t i n g hologram was viewed in the reference beam from second focus a f t e r developing the polaroid f i l m .  This high speed Polaroid f i l m could be used to record an  interference pattern due to small o b l i q u i t y between the signal and r e f erence beams. I t was discovered that the reference beam could also be derived from a s i n g l e i n t e r n a l r e f l e c t i o n as shown in Figure 4.7.  Light r e f l e c -  ted from the f i r s t surface of the lens uniformly i l l u m i n a t e s the o b j e c t . The layout of various o p t i c a l components i s shown in Figure 4.8.  A  neutral density f i l t e r (N.D.0.9) was used to cover the reference beam so that at the recording f i l m , the i n t e n s i t i e s of the signal and reference beams were in the approximate r a t i o of about 1 to 5.  Exposures of 5 sec.  were required on P o l a r o i d Type 55 P/N f i l m s placed nearly 24 inch  from  the object. Holographic interferometry was also performed by the double exposure method on the new system, the second exposure being taken a f t e r the blade had been s l i g h t l y d e f l e c t e d .  When r e - i l l u m i n a t e d with the r e f e r -  ence beam the processed hologram revealed i n t e r f e r e n c e fringes due to displacement of the object.  42  LASER  F i g . 4.7  Schematic Diagram--using Lens as a M i r r o r  F i g . 4.8  Holographic Set up—using Lens as a M i r r o r  F i g . 4.9  Reconstructed Images from the Hologram  Figure 4.9 shows the reconstructed image of the razor blade as well as the interference fringes due to d e f l e c t i o n of the blade.  The  q u a l i t y of the reconstructed image was not very good due to l i m i t e d r e s o l u t i o n of the P o l a r o i d f i l m and the reconstructed images were only s l i g h t l y off axis.  However, the experiments were adequate to gain under  standing of the holographic p r i n c i p l e .  V DESIGN AND DEVELOPMENT OF AN OBLIQUE INCIDENCE INTERFEROMETER  5.1  REVIEW OF VARIOUS OPTICAL SYSTEMS Interferometers may be conveniently divided i n t o two main c l a s s e s ,  those based on ' d i v i s i o n of wavefront' and those based on ' d i v i s i o n of amplitude.'  In the former class the wavefront i s divided l a t e r a l l y i n t o  segments by mirrors or lenses.  It i s also possible to d i v i d e a wavefront  by p a r t i a l r e f l e c t i o n , i n which the two r e s u l t i n g wavefronts maintain the same width but have reduced amplitudes. The f o l l o w i n g i n t e r f e r o m e t r i c systems were studied during a search f o r the most s u i t a b l e layout f o r the f i n a l design of the interferometer. 5.1.1  Wavefront D i v i s i o n  Interferometers  Figure 5.1 shows three layouts employing the p r i n c i p l e of d i v i s i o n of wavefront.  In Layout 1, plane reference and signal wavefronts are de-  rived from a s i n g l e spherical wavefront. a  This makes i t necessary to  employ/very powerful source since the required small s i z e of aperture f o r such a spherical wavefront cuts o f f most of the i n t e n s i t y .  Further only  a part of the wavefront i s used f o r obtaining the reference and signal beams.  The Layout 2 i s s u i t a b l e f o r t r a n s m i t t i n g objects where the r e f e r -  ence and signal wavefronts are derived from the same plane wavefront.  46 LAYOUT-1  i  h 1 '/  ^  LASER SPATIAL FILTER AMMUY.  PHOTOPLATE SURFACE.  LAYOUT-2  LASER SPATIALFILTERASSMLY.  PHOTOPLATE  LAYOUT - 5  LASER SPATIAL FILTER  Fig. 5.1 Schematic of the Wavefront Interferometers  SURFACE  Division  PHOTOPLATE  47 The Layout 3, employing a lens less system, i s s i m i l a r to L l o y d ' s mirror system.  If the o b l i q u i t y of the i n c i d e n t wave-cone on the surface i s  large making angles a and e nearly the same, the forward scattered l i g h t i s nearly c o l l i m a t e d .  Since the d i r e c t beam i s also nearly collimated  under these conditions the two wavefronts can be arranged to i n t e r f e r e at a small angle. However, the large angle between the reference and signal wavefronts make these systems s u i t a b l e f o r executing high r e s o l u t i o n holography.  The p r a c t i c a l requirements of the instrument  such as  execu-  t i o n of interferometry which allows very small angles between the r e f e r ence and signal beams, independent control of the reference and signal beam i n t e n s i t i e s , and execution of i n d u s t r i a l holography, did not encourage f u r t h e r study of these systems. 5.1.2  Amplitude-Division Interferometers  Two of the layouts considered are shown i n Figure 5.2.  These are  the Twymann-Green Interferometer [17] and the Perkin Elmer I n t e r f e r o meter [18], (modified version of the Twymann-Green Interferometer).  The  preliminary experiments on the former type and the study of the l a t t e r proved these systems adequate f o r i n t e r f e r o m e t r i c work.  The execution of holo-  graphy in these systems w i l l probably require many changes i n the optics of the interferometer which makes i t rather d i f f i c u l t f o r i n d u s t r i a l applications. The modified Mach-Zehnder interferometer, as shown i n Figure 5.3, was considered most s u i t a b l e f o r the f i n a l design of the instrument.  48 MIRROR  1. TWYMANN - GREEN INTERFEROMETER  F i g . 5.2  Schematic of the Amplitude D i v i s i o n Interferometers  FOR WARP SCATTER AND ftACK 5CATTER F i g . 5.3  Mot  QftRAPuv  Schematic of the Modified Mach-Zehnder Interferometer  50 Following features of t h i s type of interferometer make i t quite acceptable f o r i n d u s t r i a l use. 1.  The symmetry of the system automatically makes the path lengths of the reference and signal beams almost the same.  2.  Independent controls of the reference and signal beams make r e l a t i v e l y easy alignment of the instrument f o r i n t e r f e r o m e t r i c work.  3.  The instrument can e a s i l y be set f o r forward s c a t t e r holography by s l i g h t m o d i f i c a t i o n as shown i n Figure 5.3.  4.  By r o t a t i n g the instrument about the o p t i c a l axes  by 90 degrees  i t can be used to execute normal back s c a t t e r holography. The interferometer, as f i n a l l y designed, i s intended f o r metrol o g i c a l uses i n conjunction with manufacturing a p p l i c a t i o n s . signed to examine c i r c u l a r discs of up to 22 inch  I t i s de-  by means of a 2 inch  diameter collimated beam (Drawing Number 413, Appendix I ) .  However, due  to l i m i t e d power of the Laser source, a one inch diameter beam was used to prove the instrument.  Design features of the various components are  described in the f o l l o w i n g s e c t i o n .  5.2  COMPONENT DESIGN AND BUILDING OF THE INTERFEROMETER The designed Oblique incidence interferometer i s shown in Figure  5.4.  The design of the various components of the interferometer i s des-  cribed below and d e t a i l drawings appear i n Appendix  I.  Fig. 5 . 4  The Oblique Incidence  Interferometer  52 5.2.1  Main Base Plate (Drawing Number 414)  Aluminum i s used to make a reasonably r i g i d ' L ' section base plate of 60 inch x 19 inch x 11 inch." The great length of the base plate i s necessary in achieving oblique incidence on the surface.  To  place c i r c u l a r discs of diameters up to 22 inch a rectangular cut-out of 18 inch x 4-1/2 inch i s cut in the v e r t i c a l plate as shown in the drawing and i n Figure 5.4.  S l i d i n g r a i l s are r i g i d l y f i x e d on t h i s plate f o r  mounting a l l o p t i c a l components which can be moved to required positions f o r the d i f f e r e n t configurations of the instrument. 5.2.2  S l i d i n g Mounts (Drawings Number 415 and 417)  These are designed to provide movements i n a l l three l i n e a r d i r ections and r o t a t i o n about the v e r t i c a l axis f o r the s p a t i a l f i l t e r assembly, c o l l i m a t i n g lenses, beam s p l i t t e r s , and photo-plate holder f o r holographic use, as shown in Figure 5.5.  For the plane mirrors the  mounts are designed s l i g h t l y d i f f e r e n t l y to accommodate e x i s t i n g motions in the mirror assemblies. 5.2.3  Beam S p l i t t i n g System  The instrument requires two beam s p l i t t e r s , one f o r s p l i t t i n g and the other f o r recombining the beams as shown i n Figure 5.4.  The former  one i s a v a r i a b l e beam s p l i t t e r (Jodon VBA 200 Beam S p l i t t e r ) and the l a t t e r one i s a simple p a r t i a l l y r e f l e c t i n g beam s p l i t t e r .  The v a r i a b l e  beam s p l i t t e r i s very convenient f o r obtaining the required r a t i o of the  F i g . 5.6  Spatial F i l t e r Assembly  54 two beams f o r i n t e r f e r o m e t r i c and holographic use of the instrument. 5.2.4  S p a t i a l F i l t e r Assembly (Drawing Number 416)  The reference and signal beams each require a s p a t i a l f i l t e r assembly. beam.  A microscope o b j e c t i v e (40 mm) i s used to expand the l a s e r For s p a t i a l f i l t e r i n g of the beam, a 10 micron pinhole mounted  on a three d i r e c t i o n a l stage mounting, as shown i n Figure 5.6, i s used. 5.2.5  C o l l i m a t i n g Lens Unit  Two lens-mounting units are designed (Drawing Number 415) f o r Wray compound lenses of 13.5 inch focal lengths.  These units are placed in  s l i d i n g mounts and can thus be moved in three l i n e a r d i r e c t i o n s or a rotated about/vertical a x i s . 2.000 inch  Three aperture stops of 1.000, 1.500, and  diameters are designed to f i t the lens in order to obtain  c i r c u l a r collimated reference and signal beams of required s i z e s .  The  interferometer i s designed f o r a 2 inch maximum aperture c i r c u l a r beam. However, due to l i m i t e d power of the Laser, only one inch apertures were used f o r the t e s t i n g of the interferometer. 5.2.6  Optical M i r r o r Mounts (Drawing Number 415)  Two o p t i c a l l y f l a t mirrors of 4 inch diameter are used f o r d i r e c t ing the reference and signal beams as shown i n Figure 5.4.  Sliding  mounts f o r these m i r r o r s , as shown in Figure 5.7, are designed s l i g h t l y d i f f e r e n t l y to accommodate e x i s t i n g motions in two planes of the m i r r o r assemblies.  F i g . 5.7  Optical M i r r o r Mount  F i g . 5.8  Photographic System  56 5.2.7  Photographic System (Drawings Number 418 and 415)  A unit f o r r i g i d l y holding the polaroid f i l m holder i s f o r i n t e r f e r o m e t r i c use (Figure 5.8).  designed  A 4 inch x 5 inch glass screen  with cross l i n e s at h a l f inch spacing i s mounted at the f r o n t and provides reference l i n e s on the interferograms to evaluate the r e s u l t s . This u n i t can be screwed along the axis of e i t h e r the reference or the signal beams.  Another u n i t i s designed to mount the holographic plate  holder (jodon X-Y M i c r o p o s i t i o n a b l e - P l a t e Holder). 5.2.8  Rough Surface Holder (Drawing Number 419)  The complete u n i t i s kinematically designed to r e s t on three points which can be adjusted to obtain the required incidence of a f i x e d signal beam on the t e s t surface.  This i s mounted on an indepen-  dent rotary unit s l i d i n g i n a ring as shown i n Figure 5.9.  Three f l e x u r e  pivots are used to j o i n the main plate and the ring and provide precise control of the oblique incidence.  The plate can be f i x e d by tightening  locking nuts once the system has been set f o r a p a r t i c u l a r incidence.  A  separate mask i s designed to locate the center of the t e s t surface on interferograms.... 5.2.9  Turbine Blade Holder (Figure  5.10)  For holographic examination turbine blades can be mounted on a platform screwed on to a b a l l j o i n t which can be clamped in any p o s i t i o n i n the space.  T h i s , in t u r n , i s screwed on to a three stage platform  57  F i g . 5.10  Turbine Blade Holder  F i g . 5.11  Laser Assembly—For Interferometry  F i g . 5.12  Laser Assembly—For Holography  59 r e s t i n g on three point supports. This u n i t can also be used f o r back s c a t t e r holography as shown i n sections to f o l l o w . 5.2.10  Laser Assembly  A 15mW Helium-Neon gas Laser i s placed behind the interferometer as shown in Figure 5.11.  When the instrument i s t i l t e d through 90 de-  grees f o r normal holographic use the l a s e r can s t i l l be placed, as shown in Figure 5.12, to keep scattered l i g h t o f f the instrument.  Two plane  mirrors with l i n e a r and angular motions are provided f o r d i r e c t i n g the las beam to the instrument. A l l these- components were b u i l t in the machine shop of the Mechanical Engineering department at the U n i v e r s i t y of B r i t i s h Columbia. Aluminum was used f o r almost a l l parts except where two parts had r e l a t i v e s l i d i n g motion, such as s l i d i n g r a i l s and s l i d i n g mounts and surface rotary table and. the ring in rough surface holder, and i n these cases combination of aluminum and brass was considered s u i t a b l e .  Before assem-  bling, a l l the components were d u l l black anodized f o r the aesthetic appeal of the instrument and to aid absorption of scattered l i g h t from various o p t i c a l  components.  VI PROVING EXPERIMENTS  6.1  SURFACE TOPOGRAPHY OF ROUGH SURFACES F i r s t experiments were performed to examine the topography of a  rough surface.  An aluminum c i r c u l a r plate of 12 inch  diameter turned  on a lathe was r i g i d l y mounted on the rotary table of the rough surface holder by 12 socket head screws around i t s circumference, as shown in Figure 4.1. The experimental setup i s shown i n Figure 6.1.  The beam, from the  Laser placed behind the interferometer, was d i r e c t e d to the v a r i a b l e beam s p l i t t e r .  The signal beam, transmitted through the beam s p l i t t e r ,  and the reference beam, r e f l e c t e d from the beam s p l i t t e r , were expanded and s p a t i a l l y f i l t e r e d i n the s p a t i a l f i l t e r assemblies.  One inch u n i -  formly collimated beams were achieved from 1.000 inch apertures mounted on compound lenses placed at t h e i r focal lengths from the s p a t i a l f i l t e r s . The reference beam p a r t i a l l y transmitted through the second beam s p l i t t e r was observed on the photographic plate mounted along i t s a x i s .  The  oblique incidence of the signal beam was so chosen that i t covered the edges of the c i r c u l a r rough surface.  This was f i r s t roughly arranged by  adjusting the three base screws of the rough surface holder. f i n a l adjustment was made with the flexure pivot system.  Finer  F i g . 6.1  Experimental Set up f o r Surface Topography  62 The signal beam r e f l e c t e d from the surface was d i r e c t e d to the second beam s p l i t t e r .  The r e f l e c t e d and transmitted parts of the signal  and reference beam from t h i s beam s p l i t t e r were superimposed on the photographic plate to produce interference fringes by adjusting the reference beam in X-Y plane.  These fringes i n d i c a t i v e of both the gen-  eral topography and the roughness of the surface were recorded on P o l a r o i d type 55 P/N f i l m and are shown in Figures 6.2 and 6.3. 6.1.1  Discussion of Results  Each interferogram represents an area i n the form of an e l l i p s e , one inch wide ( v e r t i c a l l y on the interferogram) and twelve inch  long  ( l a t e r a l l y on the interferogram), compressed o p t i c a l l y i n t o a c i r c u l a r observed f i e l d .  This i s i l l u s t r a t e d i n Figure 6.4.  F i r s t interferograms were recorded with a minimum number of fringes appearing i n the f i e l d , Figure 6.2, by making the reference and signal wavefronts nearly p a r a l l e l .  The central part of the i n t e r f e r o -  gram shows the combination of the macroscopic as well as microscopic form of the surface whereas the outer part merely indicates the macroscopic form.  The combination of macroscopic and microscopic form at the centre  makes i t d i f f i c u l t to i d e n t i f y the f r i n g e orders.  However, by wedging  the signal and reference beams, strand-1ike fringes along the major axis of the e l l i p t i c a l l y i l l u m i n a t e d area, Figure 6.3, were recorded. reveal both macroscopic and microscopic topography.  These  The scratches t r e n d -  ing v e r t i c a l l y , which are r e a l l y parts of elongated e l l i p s e s as i n d i cated in Figure 6.4, are more c l e a r l y revealed in t h i s interferogram  F i g . 6.2  Rough Surface Interferogram--Minimum Fringes  Y  F i g . 6.3  Rough Surface Interferogram--Strandl i k e Fringes  F i g . 6.4  Compressed C i r c u l a r Observed F i e l d  65 as l o c a l excursions.  Overall examination of the interferogram also  indicates uniform curvature of the c i r c u l a r plate which has a d i s placement at the centre with reference to edges of the plate c o r r e s ponding to 25 f r i n g e orders.  6.2  THE MEASUREMENT OF SMALL SURFACE DISPLACEMENTS The use of the oblique incidence interferometer f o r small surface  displacement measurements was i l l u s t r a t e d by examining the c i r c u l a r aluminum plate of Section 6.1 (Figure 6.5) under f o l l o w i n g loadings: 1.  A symmetrical loading by evacuating the c a v i t y below the plate using a vacuum pump.  2.  Non-symmetrical loading by introducing an a d d i t i o n a l , f i x e d l e v e l , point support at 3 inch  (2/3 radius of the plate) from  the centre of the plate and evacuating the c a v i t y below the plate. The plate was ground and moderately polished (not lapped) to obtain better r e f l e c t i o n .  Nominally f i x e d end-conditions were achieved  by designing an e l a s t i c mounting as shown in Figure 6.5 and b o l t i n g the plate r i g i d l y : to the rotary table of the rough surface holder by 12 socket head screws at i n t e r v a l s of 30 degrees around the circumference of the t h i n plate (d/t = 36).  An aluminum s l o t t e d mask (3/4 inch wide,  12 inch long), Figure 5.9, was designed to rotate f r e e l y on the circumference of the t e s t p l a t e .  This was to locate the centre of the p l a t e ,  F i g . 6.5  The Test Plate  67 the i n / Y - d i r e c t i on, on the interferograms as a reference point when analysing the the observed f r i n g e s .  To locate the centre i n / X - d i r e c t i o n a black paper  c i r c l e was stuck at the centre of the p l a t e .  A mercury manometer was  connected between the plate and the vacuum pump to record the vacuum in inches of mercury.  A valve was connected i n the vacuum l i n e to obtain  uniform vacuum by use of a c o n t r o l l e d leak.  However, the r e s t of the  experimental set-up was the same as f o r observation of the natural t o p ography of the rough surface. F i r s t the plate was examined under uniform symmetrical loading. The oblique incidence of the plate was so arranged that one inch diameter signal beam covered the 12 inch diameter plate along the s l o t t e d mask. The reference and signal beams were adjusted to produce interference were fringes on the photographic p l a t e .  The i n t e n s i t i e s of the two beams/made  approximately the same at the photographic plate by adjusting the variable beam s p l i t t e r to obtain the best contrast i n f r i n g e s .  The r e f e r -  ence beam was stopped with a cardboard to record the signal beam (Figure 6.6.a).  This was to locate the centre of the plate by recording the  d i f f r a c t i o n e f f e c t s at the edges of the mask and the image of a small c i r c l e on the p l a t e .  A f i e l d with minimum fringe density was then r e -  corded, by making the reference and signal beams nearly p a r a l l e l , to provide data f o r p l o t t i n g i n i t i a l curvature of the plate (Figure 6.6.b). F i n a l l y the plate was uniformly loaded by evacuating the c a v i t y below the plate up to 11.6 inches of mercury.  Two interferograms with symmetri-  cal and s t r a n d - l i k e fringes were recorded to p l o t f i n a l curvature of the plate (Figures 6.6.C and 6.6.d).  F i q . 6.6.a  Sicmal Beam Record  X—  F i g . 6.6.b  F i g . 6.6  'No Load' Interferogram  Symmetrical Loading  F i g . 6.6.c  ' A f t e r Load' Fringes  Interferogram—Straight  Y  F i g . 6.6.d  ' A f t e r Load* Interferogram--Strandl i k e Fringes  F i g . 6.6  Symmetrical Loading  70 The above plate was also examined f o r non-symmetrical loading by introducing an a d d i t i o n a l , f i x e d l e v e l , point support at 3 inches from the centre of the p l a t e .  The reference and signal beams were f i r s t  matched to give a f i e l d with minimum f r i n g e density.  The f i x e d l e v e l  point support was then provided by r o t a t i n g the screw (Figure 6.5) the fringes were j u s t disturbed.  until  This ensured that contact between the  plate and the point support had been made.  The signal beam and no load  fringes were recorded as in the f i r s t case (Figures 6.7.a and 6.7.b). F i n a l l y the plate was loaded by evacuating the c a v i t y below the plate up to 10.6 inches of mercury and two interferograms with symmetrical and S t r a n d - l i k e fringes were recorded (Figures 6.7.c and 6.7.d). Polaroid type 55 P/N f i l m was used f o r recording the i n t e r f e r o grams and exposures of 1/60 second f o r signal beam and 1/125 second f o r interferograms were required. 6.2.1  Evaluation of the  Interferograms  The centre of the plate on the interferograms was located by scanning the signal beam records (Figures 6.6.a and 6.7.a) under the Joyce-Dobel mic.rodensitometer (Figure 6.8).  The centre in the Y - d i r e c -  t i o n was located in between the same d i f f r a c t i o n orders from the edges of the s l o t t e d mask.  Its distance from a reference l i n e , imaged from,  the reference glass in the plate holder, was measured.  The l o c a t i o n of  the centre in X - d i r e c t i o n was found by scanning the same negative along the compressed e l l i p s e at a point showing the image of the small c i r c l e  F i g . 6.7.a  F i g . 6.7.b  F i g . 6.7  Signal Beam Record  'No Load' Interferogram  Non-Symmetrical Loading  72  F i g . 6.7.d  ' A f t e r Load' Interferogram—Strandl i k e Fringes  Fig. 6 . 7 Non-Symmetrical Loading  F i g . 6.8  Joyce-Dobel  Microdensitometer  •Hi  co  74 on the surface.  Its distance from the reference l i n e perpendicular to  the f i r s t was also measured.  Figures 6.9 and 6.11 show the density plots  of the signal beam records i n X and Y d i r e c t i o n s .  The centre on other  interferograms was located from these two reference l i n e s . To obtain d e f l e c t i o n s , the interferograms f o r the 'no l o a d ' and 'Ipaded'  conditions were scanned by the microdensitometer with 7.5 magni-  f i c a t i o n and the obtained density plots (Figure 6.10 and 6.12) were used to p l o t the f r i n g e orders on both sides of the centre up to 4.5 i n c h , the radius of the p l a t e .  The d e f l e c t i o n due to loading was obtained from the  difference between the ' i n i t i a l '  and ' f i n a l ' curvature of the p l a t e .  The l a t e r a l scale on the interferograms was obtained from the d i f f r a c t i o n rings of the reference beam aperture of one i n c h .  However, the  l o n g i t u d i n a l scale was measured as 1/12 of the l a t e r a l s c a l e . The contour i n t e r v a l of the interferograms was c a l c u l a t e d as f o l lows The wavelength of the Laser l i g h t = A = 6328 A = 2490 x 1 0 " The diameter:of the signal beam  8  inch  = 1 inch  The major axis pf the i l l u m i n a t e d e l l i p t i c a l f i e l d = 12 inch The oblique incidence, e, i s given by Cos 0 = 1/12 = 0.08333 Therefore from Equation (3.11), the contour i n t e r v a l , t , of the fringes, X  2Cos0  _ 2490 x I O "  8  2 x 0.08333  1  n  c  h  75  F i g . 6.9  Signal Beam Density Plots--Symmetrical Loading  76  'AFTER LOAD' FRINGE DISTRIBUTION (FiauRE-6-fed )  Fig. ,6.10 :  'Mo Load' and ' A f t e r ; Load' Density Plots --Symmetrical Loading  ON  LOCATION  NEG-  or C E N T R E  IN Y- DIRECTION  (MAGNIFICATION  7-5 J  (FIBUHE  67a)  ON NSO  LOCATION  OF  CENTRE IN X- DIRECTION  ( FIOCWE  6-7-a)  ( MAGNIFICATION - T 5 )  F i g . 6.11 ;  Signal Beam Density Plots—Non-symmetrical Loading  'NO-LOAD' FRINGE DISTRIBUTION (Piau«g6-7b)  'AFTER LOAD' FRINGE DISTRIBUTION (piewe 6-7 d)  F i g . 6.12  'No Load' and ' A f t e r Load' Density Plots —Non-symmetrical Loading  79 or t = 1.494 x 10~ 6.2.2  4  inch.  Results and Discussions  The d e f l e c t i o n s f o r a section in the symmetrical uniform loading and the section through the f i x e d level point support i n the nonsymmetrical loading, measured as the difference between the and ' f i n a l '  'initial'  curvature of the p l a t e , were p l o t t e d as shown in Figures  6.13 and 6.14.  Theoretical values of the d e f l e c t i o n f o r the plate were  c a l c u l a t e d using t h i n plate theory (Appendix The discrepancies  II).  between the t h e o r e t i c a l and experimental  results  are due to the physical nature of r e s t r a i n t s imposed on the boundary of the real p l a t e .  Even in very r i g i d plate mountings, the edge moments  and edge reactions required to maintain e q u i l i b r i u m of the plate develop local stress f i e l d s which thus s u f f e r displacements  [20].  These stress  f i e l d s are l i n e a r l y related to the load on the plate and make the mountings more e l a s t i c to provide boundary t r a c t i o n s to e s t a b l i s h in the system.  equilibrium  However, f o r t h i n plates with r i g i d mountings, the def-  l e c t i o n due to shear becomes i n s i g n i f i c a n t but the i n t e r a c t i o n of the t h i n plate and i t s mounting s t i l l prevents the r e a l i s a t i o n of t h e o r e t i c a l boundary conditions. However, i t i s d i f f i c u l t i n p r a c t i c e to have uniform values of e l a s t i c constants  in the material of real plates and the assumption of  constant values of modulus of e l a s t i c i t y , E, and Poisson's r a t i o , v, may  UNIFORM L O A D I N G  M  Q =  80  2953  * * * * • • * « * t• > * * * * * * » * * * * * »  i  FIXED END  RADIUS (INCH) «  O  I  2  THEORETICAL EXPERIMENTAL  F i g . 6.13  D e f l e c t i o n Traverses showing comparisons with Theory of Thin Plates—Symmetrical Loading  82 lead to some discrepancies  in the experimental and t h e o r e t i c a l r e s u l t s .  In the f i r s t case of the uniformly loaded state the experimental value of the d e f l e c t i o n at the centre of the plate is approximately 15 per cent higher than the t h e o r e t i c a l value.  In the second case of a  uniformly loaded plate with a f i x e d level point support the support point has a small d e f l e c t i o n due to a movement of the support when loaded.  Otherwise the experimental r e s u l t s f o l l o w  predictions c l o s e l y .  the t h e o r e t i c a l  However, with the above points taken i n t o account  the experimental errors are w i t h i n acceptable l i m i t s and lend reasonable support to the v a l i d i t y of t h i s interferometer f o r small surface d i s p l a c e ments  6.3  measurement.  FORWARD SCATTER AMD BACK SCATTER HOLOGRAPHY One of the features of the interferometer i s that i t can be set  up to execute forward s c a t t e r and back s c a t t e r holography f o r normal q u a l i t a t i v e recording and subsequent comparison through holographic i n t e r ferometry.  This would have useful a p p l i c a t i o n in the f i e l d of engineer-  ing inspection where a precise comparison pf the shape of a three dimensional object  with r e l a t i v e l y low r e f l e c t i v i t y  If the 'master'  with a 'master' could be made.  i s manufactured and tested by conventional means, the  geometrical shape of i t s surface may be stored i n a hologram.  A l l nom-  i n a l l y s i m i l a r shapes manufactured subsequently can then be checked against the holographically reconstructed 'master'  shape.  83 To demonstrate such a p o s s i b i l i t y a turbine blade was mounted on the turbine blade holder (Figure 6.15).  The interferometer was set up by  removing a l l the components a f t e r the s p a t i a l f i l t e r assemblies and placing the Jodon X-Y micropositionable plate holder along the reference beam as shown in Figure 6.15.  The expanded signal beam uniformly i l l u m -  inated the blade at oblique incidence of approximately 60 degrees and the v a r i a b l e beam s p l i t t e r was adjusted to give an approximately 1 to 5 r a t i o of the signal to reference beam i n t e n s i t i e s at the photographic plate. The photographic plates used were Agfa Gavaert type 10E75 4"x5" glass plates of.2800 lines/mm r e s o l u t i o n . found by t r i a l and e r r o r . the hologram recording.  The correct exposure time was  Exposures of 1 to 2 seconds were required f o r The exposed plate was developed in complete  darkness i n Metinol u developer at a temperature of 68 degrees F f o r four minutes.  I t was f i x e d in Hypo s o l u t i o n f o r four minutes and then washed  for two minutes-in Hypo-eliminator s o l u t i o n .  A f t e r washing in running  water f o r nearly f i f t e e n minutes the plate was f i n a l l y l e f t to dry f o r a couple of hours. When the processed hologram was replaced in i t s holder and both the reference and signal beam from the turbine blade were allowed to f a l l on i t , ' l i v e '  interference fringes appeared over the blade surface  due to the interference between the ' d i r e c t ' and ' r e c o n s t r u c t e d ' waves. However, by adjusting the hologram plate i n X-Y d i r e c t i o n s the ' d i r e c t ' and ' r e c o n s t r u c t e d ' waves brought i n t o near coincidence as indicated by  F i g . 6.15  Forward Scatter Holography Set up  85 the s p a r s i t y of the fringes on the blade as shown i n Figure 6.16.a. Live interference fringes due to movement of the blade in x, y and z d i r e c t i o n s were observed on the surface of the blade and fringes at convenient displacements were recorded as shown in Figure 6.16.b, c and d.  The comparison of two s i m i l a r objects was i l l u s t r a t e d by s l i g h t l y  d e f l e c t i n g the same blade at one end and the interference fringes i n d i c a t i v e of the displacements were recorded as shown i n Figure 6.17.  However,  the' blade mounting platform did not move s i g n i f i c a n t l y during d e f l e c t i o n of the blade as i s apparent from the lack of movement of fringes on the mounting. The back s c a t t e r holography was executed by turning the i n t e r f e r ometer by 90 degrees about the o p t i c a l axes.  A part of the signal beam  r e f l e c t e d from the mirror was allowed to i l l u m i n a t e the object (a car model) placed close to the photographic plate (Figure 6.18)  in order to  maintain the path d i f f e r e n c e between the signal and reference beam w i t h in coherence l i m i t of the l a s e r which was approximately 12 i n c h .  When  the processed hologram was replaced and i l l u m i n a t e d with the reference beam,a reconstructed image was seen and photographed as shown in Figure 6.19.  Exposures of 1 second were required f o r hologram recording on  Agfa Gavaert type 10E75 p l a t e s . Thus i t was shown that the interferometer can be used f o r both back s c a t t e r q u a l i t a t i v e holography and forward s c a t t e r holographic interferometry as applied to the comparison of two nearly s i m i l a r objects i n engineering i n s p e c t i o n .  A forthcoming report from the McGill  'No displacement'  fringes  Small displacement in X - d i r e c t i o n  Small displacement i n Z - d i r e c t i o n F i g . 6.16  Small displacement in Y - d i r e c t i o n  Holographic Fringes due to Linear D i s placements of the Blade  co  ction'  fringes  1.  'After deflection'  fringes  F i g . 6.19  Reconstructed Image from the Hologram  90 U n i v e r s i t y by Dr. Axelrad i s said to contain a complete s o l u t i o n of the i n t e r p r e t a t i o n of holographic i n t e r f e r o m e t r i c fringes in terms of the displacements.  VII CONCLUSIONS AND PROPOSED FUTURE STUDY  7.1  CONCLUSIONS A two-inch f i e l d Laser-based oblique incidence interferometer of  modified Mach-Zehnder layout f o r executing any of the f o l l o w i n g three i n t e r f e r o m e t r i c techniques has been designed, b u i l t and proved. 1.  The surface topography of rough surfaces of up to 22 inch long with 2 inch f i e l d .  2.  The measurement of small surface displacements under symmetrical and non-symmetrical  3.  loadings.  Forward s c a t t e r holographic comparison of curved surfaces and the execution of normal back s c a t t e r holography f o r q u a l i t a t i v e recording. The interferometer is designed f o r metrological use in conjunction  with manufacturing a p p l i c a t i o n s .  Oblique incidence in collimated l i g h t  leads to fringes"which can e a s i l y be evaluated f o r q u a n t i t i v e analysis and the s e n s i t i v i t y is reduced to a convenient value to s u i t usual manuf a c t u r i n g accuracies.  Surfaces do not require lapping and a rough s u r -  face turned on a lathe has been examined under the interferometer. C i r c u l a r discs of up to 22 inch diameter can be viewed by r o t a t i n g them about an axis so that the i l l u m i n a t e d e l l i p t i c a l f i e l d is aligned  92 with any selected diameter.  Fringes from a 22 inch long axis by 2 inch  wide axis e l l i p t i c a l f i e l d are recorded i n a 2 inch c i r c u l a r f i e l d and analysed by means of a microdensitometer. Holography can be executed by making the interference fringes  very  c l o s e l y spaced due to high o b l i q u i t y of the signal and reference beams and viewing the hologram i n the reference beam.  Holographic i n t e r f e r o -  metry has been executed on a turbine blade to i l l u s t r a t e the use of the interferometer f o r comparing any two, three dimensional objects with r e l a t i v e l y low r e f l e c t i v i t i e s .  This would have useful a p p l i c a t i o n i n the  f i e l d of engineering i n s p e c t i o n , where a precise comparison of a component with a 'master'  could be made.  The instrument i s arranged f o r normal use i n a v e r t i c a l plane so that g r a v i t y serves to locate the objects n a t u r a l l y under t h e i r own weight on k i n e m a t i c a l l y designed l o c a t o r s .  General holography  is  performed with the o p t i c a l beams of the interferometer in the horizontal plane, an arrangement achieved by r o t a t i n g the instrument by 90 degrees about i t s o p t i c a l , axes.  7.2  RECOMMENDATIONS FOR FUTURE STUDY The 15mW Helium-Neon l a s e r a v a i l a b l e f o r t h i s study was not s u f f i c -  i e n t l y powerful to t e s t the interferometer at a two inch aperture.  A  one inch c i r c u l a r collimated beam was used f o r proving the interferometer. To obtain s u f f i c i e n t i n t e n s i t i e s of the reference and signal beams in a  93 two inch f i e l d i t i s recommended that a 50 mW l a s e r be used. I t i s also recommended that f o r commercial use the o p t i c a l  layout  of the interferometer be modified to reduce i t s length to about h a l f . Figure 7.1 shows the basis of an arrangement.  By confining the input  o p t i c a l paths to the lower section and using the upper section f o r arranging the t e s t objects and recording system, the stray l i g h t from various components can also be avoided.  Further, alignment of the instrument should  be r e l a t i v e l y e a s i e r since an observer can reach to various o p t i c a l components while viewing through the photographic system. surfaces may be examined by increasing the f i e l d .  Also, larger  The o v e r a l l  dimensions  of the instrument may thus be reduced to about 3' x 2' x 1-1/2'. I t i s also recommended that f u r t h e r study of the s i g n i f i c a n c e of d i f f r a c t i o n at edges and at marks on the surface be made i n order to d i s cover other means of scale determination than those used i n the present study. F i n a l l y i t i s recommended that the p o s s i b i l i t y of inspecting  sur-  faces with non-zero curvature in one plane as exhibited by cylinders and tubes be examined.  The holographic interferometry f o r comparison of two  such cylinders as shown by Archbold, Burch and Ennos [21] may be executed in an adaption of the instrument by i l l u m i n a t i n g the surface of the c y l i n der at oblique ^incidence as shown in Figure 7.2.  The hologram can be  recorded in the.-.shape of an annul us where the reference and signal overlap on the photographic p l a t e .  •  beams •  94  M  A J ?  DETAILS  @  1  DETAILS @  &-6  A-A  U///////ns//UJ y yy yyy y/y y yyy/y > yy / J///J7J.  6-S  M •  /  j  / s //// J j y y / y > it J J /1 rr  DETAILS  @-CC  DETAILS @  M : M I R R O R j fi».5 - &EWV1 S P U T T E R ; -  F i g . 7.1  !222ZrZZZZZZ2ZrZ2ZZ2ZZZZZZ2ZZZZZ2  D-D  L". COLUMATING L E N S j A ' - SPATIAL PlLTER A S S E M B L V J C T C A M E R A .  Modified Layout of the Interferometer  C Y L I N D E R FROM  A  POINT  95  SOURCE  SIGNAL feEAM PHOTO PLATE.  F i g . 7.2  Schematic Diagram f o r Comparison of Cylinders  R £ F . BEAM  SIGNAL  ftEAM  CYLINDER  •SIGNAL B E A M  F i g . 7.3  Topography of Cylinders  When the processed hologram i s replaced i n i t s holder and both the reference and signal beam from the c y l i n d e r wall are allowed to f a l l on i t , ' l i v e '  interference fringes should appear when e i t h e r the  c y l i n d e r or the hologram i s d i s p l a c e d .  When the 'master' c y l i n d e r i s  replaced by another c y l i n d e r interference fringes i n d i c a t i v e of the d i f ference i n shape should appear. The topography of c y l i n d e r s and tubes may be examined by allowing the collimated signal beam to f a l l on a part of i t s surface (Figure 7.3). By r p t a t i n g the c y l i n d e r about i t s axis contour maps of the surface may be obtained.  However, the s i g n i f i c a n t curvature of the tubes w i l l a f f e c t  96 the r e f l e c t e d wavefront which may require use of c y l i n d r i c a l lenses f o r the c o r r e c t i o n of t h i s d e v i a t i o n .  PnoroPUVYE  MIRROR  SPECIAL LENS  F i g . 7.4  Topography of Cylinders  A modified o p t i c a l arrangement to examine topography of curved surface by use of an annular signal beam i s also recommended (Figure beam 7.4).  The annular signal/may be derived by using a c i r c u l a r prism and  when superimposed on the collimated c i r c u l a r reference beam should give fringes i n d i c a t i v e of the macroscopic shape of the surface i n the shape of an annul us. These fringes may be recorded on Polaroid type 55 P/N f i l m . The o b l i q u i t y at every point on the circumference of the c y l i n the the der w i l l be/same and/complete surface may be examined at one time.  REFERENCES  98  REFERENCES  1.  Duncan, J . P., "Surface Topography," book manuscript i n advanced stages of preparation, Dept. Mech. Eng., U.B.C.  2.  Jenkins and White, "Fundamentals of O p t i c s , " McGraw-Hill Co., New York, 1957.  3.  Gates, J . W., "An Interferometer f o r Testing S p h e r i c i t y , " published i n 'Optics in Metrology' by M o l l e t , Pergamon Press, New York, 1960, pp. 201-204.  4.  Duncan, J . P., "Topographical Survey of Curved Aerodynamic Surfaces," Final Report to D.R.B., Mech. Eng. Dept. U . B . C , 1970.  5.  W e l l e r , R. and B. M. Shepard,  "Displacement Measurements by Mechan-  i c a l Interferometry," Proc. SESA, V o l . 6, No. 1, 1948. 6.  Theocaris, P. S.,  "Moire Fringes i n S t r a i n A n a l y s i s , " Pergamon Press,  New York, 1969. 7.  Duncan, J . P., "Combined Fizeau and Shadow Moire Interferometer," I n s t r u c t i o n Manual, P o l a r i z i n g Instrument Company, New York, 1968.  8.  Duncan, J . P. and E. W. Johnson, "Basic Optics f o r Engineers," V o l . I and II,  Typescript, Dept. Mech. Eng., U.B.C.  99 9.  Strong, J . , "Concepts of C l a s s i c a l O p t i c s , " W. H. Freeman and Co., San Francisco, 1957.  10.  Langenbeck, P. H., "Multipass Interferometry," Applied O p t i c s , V o l . 8, No. 3, 1969, pp. 543-552.  11.  Gabor, D., "Microscopy by Reconstructed Wave Fronts," Proc. Roy. Soc. London A 197, 1949, pp. 454-487.  12.  L e i t h , E. N. and J . Upatnieks, "Reconstructed Wavefronts and Communi c a t i o n Theory," J . Opt. Soc. America, V o l . 52, No. 10, 1962, pp. 11.23-1130.  13.  L e i t h , E. N. and J . Upatnieks, "Wavefront Reconstruction with Continuous-Tone Objects," J . Opt. Soc. America, V o l . 53, No. 12, 1963, pp.  14.  1377-1381.  L e i t h , E. N. and J . Upatnieks, "Wavefront Reconstruction with Diffused I l l u m i n a t i o n and Three Dimensional Objects," J . Opt. Soc. America, V o l . 54, No. 11, 1964, pp.  15.  1295-1301.  Ennos, A. E., "Holography and i t s A p p l i c a t i o n s , " Contemp. Phys., V o l . 8, No. 2, 1967, pp. 153-170.  16.  Upatnieks, J . , "Experimental Holography," Eng. Summer Conf., Univ. of Michigan, 1969, Chapter 5.  17.  Commander Lab., Inc., "Laser Interferometry - i t s prac^tic^l a p p l i c a tions f o r i n d u s t r y , " Pamphlet No. 66-29 3-66.  100 18.  Bloxsom, J . T., J . p. Duncan, and J . B. Schroeder, " O p t i c a l Materials Study Program," Perkin-Elmer Report No. 8903, August 1967, pp. 71-77.  19.  Gates, J . W. C. and S. J . Bennett, "Holography with a Double Focus Lens: a Simple Lecture Demonstration," Nature, V o l . 218, No. 5145, 1968, pp. 942-943.  20.  Duncan, J . P. and T. E. T a y l o r , " E l a s t i c Restraints in the Flexure of C i r c u l a r P l a t e s , " J . Mech. Eng. Sc., V o l . 4, No. 2, 1962, pp. 143-148.  21.  Archbold, E., J . M. Burch and A. E. Ennos, "The A p p l i c a t i o n of Holography to the Comparison of Cylinder Bores," N.P.L., Teddington, Middlesex, To be published i n J . S c i .  22.  Instrum.  Timoshenkp and Woinowsky-Krieger, "Theory of Plates and S h e l l s , " McGraw H i l l Co., New York, 1959.  23.  M i c h e l l , J . H., "The Flexure of a C i r c u l a r P l a t e , " Proc. Lond. Math. Soc,  1902.  BIBLIOGRAPHY  102  BIBLIOGRAPHY  Abramson, N., "The 'Holo-Diagram'  - A P r a c t i c a l Device f o r the Making  and the Evaluation of Holograms,"  Proc. Univ. of S t r a t h c l y d e ,  Glasgow (Scotland), 1968, pp. 45-55. Brown, G. M., R. M. Grant and G. W. Stroke, "Theory of Holographic  Inter-  ferometry," J . Acoustical Soc. Amer., V o l . 45, No. 5, 1969, pp. 1166-1179. Candler, C ,  "Modern Interferometers," H i l g e r and Watts L t d , Univ.  Press,  Glasgow, 1951. Duncan, J . P., "Interferometry Applied to the Study of E l a s t i c F l e x u r e , " Proc. Inst. Mech. Engrs., V o l . 176, No. 16, 1962. Duncan, J . P. and P. G. Sabin, "Determination of Curvature in Flexed E l a s t i c Plates by the M a r t i n e l l i - R o n c h i Technique," Experimental Mechanics, 1963, pp. 285-293. Francon, M., "Optical Interferometry," Academic Press, New York, 1966. Frecska, S. A., " C h a r a c t e r i s t i c s of the Agfa-Gevaert Type 10E70 Holographic F i l m , " Applied O p t i c s , V o l . 7, No. 11, 1968, pp. 2312-2314.  103 Hidebrand, B. P. and K. A. Haines, "Surface Deformation Measurement Using the Wavefront Reconstruction Technique," Applied O p t i c s , V o l . 5, No. 4, 1966, pp. 595-602. Hockley, B. S. and J . N. B u t t e r s , "Coherent Photography (Holography) as an a i d to Engineering Design," J . Photo. Sc., V o l . 18, 1970, pp. 16-22. Lehmann, M., "Holography Technique and P r a c t i c e , " Proc. Univ. of S t r a t h clyde, Glasgow (Scotland), 1968, pp. 1-22. Stone, J . M., "Radiation and O p t i c s , " McGraw-Hill Co., New York. Tolansky, S.,  "An Introduction to Interferometry," Longmans, Green and  Co., New York, 1954. W i l l i a m s , R., " D e f l e c t i o n Surface of a C i r c u l a r Plate with M u l t i p o i n t Support," Perkin-Elmer Report No. 9136, 1968.  APPENDICES  APPENDIX I DESIGN DRAWINGS  OPTICAL ITEM LIST rrEM NO.  NO- REQO  '  '  2  z  S  2  MICROSCOPE  A  2  SPATIAL,  5  Z  COLLIMATINa COMPOUND ll**  6  t  fiCAM SPIITTCR  DESCRIPTION VAJOA&LE pmsirr BUM A'DIA.  PLAHE  eniTTtR  MIRRORS  OOJKTlVt*  FlLTLRM  7  1  POUROIP  D  t  /9ml*  FILM  Hl-HL  * PiA.  HOLDCR LAiiLft P L A C E D SEiliNp THE. I N T L R r t ROMLTCA  MECHANICAL ENGINEERING DEPARTMENT THE  UNIVERSITY  OF  BRITISH  C O L U M B I A  <-- y »«  M»v, 1970  OBUOUE  NCOENCE  OPTICAL LINE  WTERFEROMETER  DIAGRAM  4IS  o cri  Drawing Number 413 Optical Line Diagram  1  |- 4  h  6  _::...+ +  --+:.  + ill  ^  (—-til  1  6  =¥  51 •*  -  +—*  I»2 |  —1 i  -  :  1  * — + — 4 —Jlfcl-  :±... i  37 "  f 2 -  9i  —4-  tit  + •f  .+  • • 4 •  +  -• • + • .  •*-  ;  +  H  T  T  i_T  .I  ' .±  T«a  \ ^wS''///AAAS/AA'//AA -ii*r  Vt"  60  MECHANICAL ENGINEERING DEPARTMENT (HI  U N I V t l l l T f  OBLlOJE MAIN  Of  tllllStt  C O H 1 W I U  INCIOtNCE fffTERTEROMETCR  BAS£ PLATE  414  O  Drawing Number 414 Main Base Plate  Drawing Number 415  Optical Components  —yW|-  -  6it ' DETAILS & A  am 1  167 j -  1 1 .  '  -2» 1 l«W ' 1 I W I  -  u  •P •6375 -  •  .S  te ~ -  ' Q;  €r42 TAPTHBJ 2 HOUS TAP TtJTU 4 UGLCS  -6-P (Ml -  S4  -  1  X  T  a  a  a  ^  P  PCTAH5 g e  r  - • • » 0 - -'512 fa'  1 ' li)l» THREAP TP WITH  1  j  PE TAILS g ft"  MECHANICAL ENGINEERING DEPARTMENT THE UNIVERSITY O FB R I T I S H C O I U M B I A  no  J.PD. OBLIQUE 9WTIAI.  INCIDENCE FILTER  INTERFEROMETER  A53CM5LY  416 o LO  Drawing Number 416  S p a t i a l F i l t e r Assembly  Drawing Number 417  S l i d i n g Mounts  I  1  T  -J -ft- «  -5U-  LL|» I  -  -I  04  I 60  — 03.f  T  -OS-  9£CTWfi SCALE  I'.tf  4—»•  o  1  MECHANICAL ENGINEERING DEPARTMENT (HI  UNIVEIIIIT  — u (•• r •— MAY, WO  OBLIQUE  OF  IRllltH  «e*M  INCIDENCE INTERFEROMETER  PH0T08RAPHC PLATE HDLPER  Drawing Number 418 Photographic Plate Holder  COLUM1IA  J.P.C.  ~4ie  Drawing Number 419  Rough Surface Holder  APPENDIX  II  THEORY OF BENDING OF CIRCULAR PLATES  114 APPENDIX  II  THEORY OF BENDING OF CIRCULAR PLATES  I f d e f l e c t i o n s w of a plate are small i n comparison with i t s thickness  ' h ' , f o l l o w i n g assumptions  can be made f o r developing a very  s a t i s f a c t o r y approximate theory of bending of p l a t e s . 1.  The middle plane remains neutral during bending.  2..  Points of the plate l y i n g i n i t i a l l y on a normal-to-the-middle plane of the plate remain on the normal to the plate a f t e r bending.  3.  The normal stresses  i n the d i r e c t i o n transverse to the plate can  be disregarded. The f i r s t assumption does not hold any more i f , in addition to the l a t e r a l loads, there are external forces acting in the middle plane of the plate and i t i s necessary to take into consideration the e f f e c t on bending of the plate of the stresses plate.  acting i n the middle plane of the  The second assumption i s equivalent to  disregarding of the e f f e c t  of the shear forces on the d e f l e c t i o n of p l a t e s . I f a c i r c u l a r plate of radius  ' a ' and thickness h c a r r i e s a load  of i n t e n s i t y Q uniformly d i s t r i b u t e d over the e n t i r e surface of the plate and the plate i s clamped at the edges, the d e f l e c t i o n at any point r from the center of the plate i s given by [22].  115 Q  / 2  2x2  (A.l)  Flexural r i g i d i t y  where  .3. 12  (1  -  /)  (v = Poisson's Ratio of the plate material)  However i f the above plate i s supported by an a d d i t i o n a l point support as shown i n Figure A . l the f i n a l d e f l e c t i o n equation i s obtained by superimposing  the previous s o l u t i o n on M i c h e l ! ' s r e s u l t [23] f o r a  clamped p l a t e under a s i n g l e point load.  M  M  *  t t M M\  CASE-l.  M A \ \ 1 H ,H I t i t H t CA.SE-2.  T  P= 8 2 6 5 LBS. 21-*-)  9"  The d e f l e c t i o n of a clamped c i r c u l a r plate of radius  ' a ' loaded at (b,o)  by a load P was obtained by Michel! using an inversion method.  Michell's  116 r e s u l t i s given in [23] as  w (e,x)  ( l - x ) ( - l ) + { ( x + Y - 2 x C o s e ) L o g X} 2  2  2  where  x  and  X  r/a ;  (1+X  2  2  Y  = b/a  y  y  (A.2)  2  Y  -2XYCOS9)  X +Y -2X COS8 2  2  Y  w is measured in opposite sense to P. Therefore the d e f l e c t i o n of a plate under uniform loading with an a d d i t i o n a l f i x e d level point support is obtained by superposition of solutions  (A.l) and (A.2), i . e . , W(e ,x) = w+w(e,x) ol [ 1 - x ] 64D 2  =  (A.3)  2 +  (l-x )(Y -l)+{(x +Y -2x Cose)Log 2  2  2  2  Y  X}  However, t h i s i s a s t a t i c a l l y indeterminate problem and was solved as f o l l o w s : F i r s t the d e f l e c t i o n at point P due to uniform loading w i t h out any support at point P was c a l c u l a t e d from Equation ( A . l ) .  The r e -  a c t i o n , P, at the f i x e d l e v e l point support required to eliminate d e f l e c t i o n there due to Q, was then c a l c u l a t e d from Equation assuming the support had not moved during loading.  (A.2)  F i n a l l y the d e f l e c -  tions along the section through the support point were c a l c u l a t e d . Table A . l shows the c a l c u l a t e d values of the d e f l e c t i o n f o r  both loadings.  The modulus of e l a s t i c i t y , E , and the Poisson's r a t i o ,  v , were taken from manufacturers s p e c i f i c a t i o n s f o r the aluminum plate (E = 10 l b s / i n c h ; v = 0.33). 7  2  TABLE A . l  RADIUS ( i nch)  DEFLECTION x 10~ inch case - 1 case - 2 4  4.5  0.00  0.00  4.0  1.10  0.94  3.5  3.90  3.32  3.0  7.71  6.50  2.5  11.94  9.50  2.0  16.08  13.21  1.5  19.73  15 93  1.0  22.57  17.81  0.5  24.36  18.66  0.0  24.98  18.38  -0.5  24.36  16.94  -1.0  22.57  14.45  -1.5  19.73  11.07  -2.0  16.08  7.13  -2.5  11 .94  3.13  -3.0  7.71  0.011  -3.5  3.90  -0.614  -4.0  1.10  -0.288  -4.5  0.00  . +0.000  r  

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