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Holographic measurement of surface distortion in three dimensions
HOLOGRAPHIC MEASUREMENT OF SURFACE DISTORTION IN THREE DIMENSIONS J . W . C . GATES National P h y s i c a l L a b o r a t o r y , Teddington, M i d d l e s e x
Interferometric techniques f o r measuring the local displacements of the surface of a solid body depend on observation of two-dimensional interference patterns. Interference effects observable from holographic reconstructions, however, depend on path variation as a function of four variables, and cannot be completely represented by a single two-dimensional pattern. For an appropriate choice of plane of observation, it is shown that complete information about the component motions of the visible surface are obtainable from observations of the changes in the two-dimensional pattern as a function of viewing direction. Conveniently, the plane of observation can be chosen to be the plane of the hologram itself.
A F E W YEARS AGO, p u b l i c a t i o n s by C o l l i e r , D o h e r t y & Pennington (1965), Haines & H i l d e b r a n d (1966), Burch, Ennos & Wilton (1966) f i r s t pointed out the p o s s i b i l i t y of t e c h n i q u e s of h o l o g r a p h i c i n t e r f e r o m e t r y for m e a s u r e m e n t of the change in p o s i t i o n of s u r f a c e (or l o c a l r e g i o n s of s u r f a c e s ) of solid b o d i e s . Many d e s c r i p t i o n s of a p p l i c a t i o n s of these i d e a s have been p u b l i s h e d since, and the l i m i t a t i o n s of s o m e p a r t i c u l a r t e c h n i q u e s have been e x p l o r e d . In c a s e s where an i n t e r f e r e n c e p a t t e r n is o b s e r v e d a p p a r e n t l y c o i n c i d e n t with the r e c o n s t r u c t e d i m a g e of a s u r f a c e , the l i m i t a t i o n s a r e of t h r e e kinds.
at the t i m e of h o l o g r a m e x p o s u r e . These w a v e f r o n t s can be made to i n t e r f e r e with the w a v e f r o n t s r e f l e c ted f r o m the object; or with w a v e f r o n t s a l s o r e c o n structed from a second hologram integrally r e c o r d e d on the s a m e plate a f t e r the f i r s t h o l o g r a m was r e corded; o r with w a v e f r o n t s r e c o n s t r u c t e d f r o m a s e p a r a t e h o l o g r a m r e c o r d e d subsequent to the f i r s t . Any changes in p o s i t i o n of e l e m e n t s in the s u r f a c e s of the object v i s i b l e in the r e c o n s t r u c t i o n give r i s e to path d i f f e r e n c e s which show up as i n t e r f e r e n c e e f f e c t s f r o m which the change of f o r m of the s u r f a c e can be c a l c u l a t e d .
F i r s t , the s e n s i t i v i t y to d i s p l a c e m e n t s in c e r t a i n d i r e c t i o n s can be v e r y low. Second, the i n t e r f e r e n c e p a t t e r n m a y not be o b s e r v a b l e with good v i s i b i l i t y u n l e s s the p a t t e r n r e m a i n s e s s e n t i a l l y lnvarzant with r e s p e c t to the a r e a of the h o l o g r a m used for r e c o n s t r u c t i o n (i.e. u n l e s s the p a t t e r n i s l o c a l i s e d n e a r the s u r f a c e ) . And t h i r d , the d i s p l a c e m e n t of the p a t t e r n for a given change in s u r f a c e p o s i t i o n can be inconveniently l a r g e , as in the c a s e of m o s t e n g i n e e r i n g a p p l i c a t i o n s w h e r e the d i s p l a c e m e n t s c o r r e s p o n d to a l a r g e n u m b e r of wavelengths of light.
The technique is an e x t r e m e l y powerful one, but t h e r e m a y be d i f f i c u l t i e s in a s s i g n i n g exact v a l u e s to the d i s t o r t i o n s shown up in any p a r t i c u l a r choice of viewing c o n d i t i o n s . When the i n t e r f e r e n c e p a t t e r n is viewed f r o m a fixed point as a p r o j e c t i o n on the s u r f a c e being studied, a single i n t e r f e r e n c e p a t t e r n g i v e s a m e a s u r e of d i s p l a c e m e n t in a d i r e c t i o n b i s e c t i n g the angle between the d i r e c t i o n of the r e f e r e n c e b e a m and the d i r e c t i o n of viewing (Ennos 1968).
More g e n e r a l l y , c o n s i d e r a t i o n of i n t e r f e r e n c e p a t t e r n s l e a d s to a n u m b e r of p o s s i b i l i t i e s f o r obviating these l i m i t a t i o n s , and a technique is d e s c r i b e d in which l o c a l m o v e m e n t s of a s u r f a c e in any d i r e c t i o n a r e e a s i l y obtainable f r o m i n t e r f e r e n c e e f f e c t s o b s e r v a b l e by a single h o l o g r a m . The r a t i o of the o b s e r v e d effect to the m o v e m e n t of the s u r f a c e is c o n t r o l l a b l e by a s u i t a b l e choice of p r o p o r t i o n of h o l o g r a m s i z e and p o s i t i o n .
HOLOGRAM
INTERFEROME TRY
A hologram provides m e a n s of reconstructing wavefronts, however c o m p l i c a t e d , r e f l e c t e d f r o m an object
To e v a l u a t e o r t h o g o n a l d i s p l a c e m e n t s , a second r e c o n s t r u c t i o n f r o m a n o t h e r h o l o g r a m , involving diff e r e n t d i r e c t i o n of viewing, was u s e d by Ennos. Even this c o m b i n a t i o n will not r e v e a l d i s p l a c e m e n t s in the d i r e c t i o n o r t h o g o n a l to the plane containing the i l l u m i n a t i o n and viewing d i r e c t i o n , and a t h i r d r e c o n s t r u c t i o n d i r e c t i o n is n e c e s s a r y to p r o v i d e a c o m p l e t e t h r e e - d i m e n s i o n a l knowledge of the d i s t o r t i o n . These r e s t r i c t i o n which apply to the use of the i n t e r f e r e n c e p a t t e r n s a s d e s c r i b e d above do not apply g e n e r a l l y to a l l i n t e r f e r e n c e techniques. A method of i n t e r f e r o m e t r i c m e a s u r e m e n t is d e m o n s t r a t e d by which exact t h r e e - d i m e n s i o n a l evaluation of the d i s t o r t i o n of the s u r f a c e of a body can be obtained f r o m i n t e r f e r e n c e e f f e c t s p r o d u c e d by a s i n g l e p a i r of h o l o g r a p h i c r e c o r d s , such as, for e x a m p l e , a single doubly-exposed frozen-fringe hologram.
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Interference patterns produced from h o l o g r a m s In F i g . l a SS 1 i s a s u r f a c e in which an e l e m e n t a r y a r e a 1~ r e f l e c t s a c o h e r e n t l i g h t b e a m i n c i d e n t f r o m the d i r e c t i o n of O, and s o f o r m s a h o l o g r a m H on a p h o t o g r a p h i c p l a t e b y i n t e r f e r e n c e with a n o t h e r p a r t of the s a m e c o h e r e n t b e a m R, which i s a l s o i n c i d e n t on the p l a t e . The s u r f a c e SS 1 i s now d i s t o r t e d and P m o v e s to P ' . A s e c o n d h o l o g r a m H' i s r e c o r d e d at the s a m e p o s i t i o n a s H, on a s e p a r a t e p h o t o g r a p h i c p l a t e , o r on the s a m e p h o t o g r a p h i c plate. After processing the hologram, a real image of b o t h SS 1 and S'S 1' can b e r e c o n s t r u c t e d , f o r e x a m p l e (in t h e c a s e of a s i n g l e , d o u b l e - e x p o s e d h o l o g r a m ) b y d i r e c t i n g a r e p l i c a of the r e f e r e n c e b e a m R b a c k w a r d s t h r o u g h the p l a t e , a s shown in Fig. lb. A p h a s e d i f f e r e n c e e x i s t s b e t w e e n the two p a t h s HP and H P ' w h i c h v a r i e s a s a function of the p o s i t i o n of P on the d i s t o r t e d s u r f a c e , and so p r o d u c e s an i n t e r f e r e n c e p a t t e r n which c a n b e thought of a s p r o j e c t e d on the s u r f a c e f r o m a s e l e c t e d s m a l l r e g i o n of the hologram. The interference pattern gives contours of p a t h v a r i a t i o n r e l a t e d to the m o v e m e n t s of e l e m e n t s of the s u r f a c e , a s d e s c r i b e d b y E n n o s . T h i s i s the p a t t e r n f o r w h i c h the r e s t r i c t i o n s m e n t i o n e d a b o v e a p p l y , and in g e n e r a l the p a t t e r n i s not i n v a r i a n t with r e s p e c t s to the a r e a of the h o l o g r a m u s e d . The c o n s e q u e n t r e s t r i c t i o n which m u s t be i m p o s e d on the a r e a of the h o l o g r a m to o b t a i n a s h a r p i n t e r f e r e n c e p a t t e r n a p p a r e n t l y l o c a t e d on the s u r f a c e p r e c l u d e s the p o s s i b i l i t y of o b t a i n i n g i n f o r m a t i o n about m o v e m e n t in m o r e than one d i r e c t i o n . If the full a r e a of the h o l o g r a m i s u s e d , P and P ' w i l l b e i l l u m i n a t e d f r o m a r a n g e of a n g u l a r d i r e c t i o n s , and and t h e r e s u l t a n t c o m b i n a t i o n of p a t t e r n s b l u r s out the i n t e r f e r e n c e e f f e c t s . Confining a t t e n t i o n to a small area around P P', reconstructed wavefronts c o n v e r g i n g t o w a r d t h i s a r e a a p p r o x i m a t e to the s p h e r i c a l w a v e f r o n t s W W' ( F i g . 2). A s a r e s u l t , if the i n c i d e n t b e a m s a r e v i e w e d t h r o u g h a s m a l l a p e r t u r e a in the ( v i r t u a l ) p o s i t i o n of P and P ' , the i n t e r f e r e n c e p a t t e r n a p p e a r i n g to c o v e r the s u r f a c e of the h o l o g r a m i s a s e r i e s of c u r v e d b a n d s w h o s e s p a c i n g and o r i e n t a t i o n give the t r a n s v e r s e d i s p l a c e m e n t s of P ' with r e s p e c t to P in two o r t h o g o n a l d i r e c t i o n s and w h o s e c u r v a t u r e s h o w s the d i s p l a c e m e n t in the d i r e c t i o n of the c e n t r e of the r i n g p a t t e r n . T h u s t h r e e c o o r d i n a t e d i s p l a c e m e n t s can b e e v a l u a t e d f o r e a c h p b i n t P and a c o n t o u r m a p of the t h r e e d i m e n s i o n a l d i s t o r t i o n can be b u i l t up. A l t e r n a t i v e l y , if two s e p a r a t e h o l o g r a m s a r e u s e d ( G a t e s 1968), one m a y b e m o v e d in r e l a t i o n to the o t h e r to p r o d u c e a null, ' f l u f f e d o u t ' , f r i n g e p a t t e r n , in w h i c h e a s e the s p a t i a l r e l a t i o n of the two h o l o grams provides three coordinate displacements. When the r e f e r e n c e w a v e i s i n c i d e n t n o r m a l l y on e a c h h o l o g r a m , and the s e p a r a t i o n of the h o l o g r a m s i s m a d e without r o t a t i o n s , t h e c o o r d i n a t e d i s p l a c e m e n t s of the h o l o g r a m m u s t b e e q u a l to t h e t h r e e c o - o r d i n a t e s of the d i s t o r t i o n ( P P ' ) x , ( P P ' ) y , ( P P ' ) z . As a third possibility, a 'live fringe' technique can b e u s e d . A h o l o g r a m m a d e with the o b j e c t u n d i s t o r t e d i s u s e d with t h e o b j e c t i t s e l f r e - i l l u m i n a t e d to f o r m an i n t e r f e r e n c e p a t t e r n . If the i n t e r f e r e n c e e f f e c t s w h i c h a p p e a r to c o v e r the o b j e c t s e e n t h r o u g h the h o l o g r a m r e m a i n f i x e d in any s m a l l a r e a of the o b j e c t , i r r e s p e c t i v e of the p a r t of the h o l o g r a m through which they are viewed, the element must
248
Optics Technology
N o v e m b e r 1969
0
S
HandH'
Fig. la Recording a doubly-exposed hologram of a surface which shifts between exposures
( H+ H~) processed
"",.
~./
\
Fig. lb Reconstructing f r o m the resultant hologram
P
Fig. 2 I n t e r f e r e n c e between reconstructions f r o m an extended area o f the hologram plate
c o i n c i d e with the r e c o n s t r u c t i o n of the e l e m e n t f o r m e d b y the h o l o g r a m . If not, an a p p r o p r i a t e m o v e m e n t of the o b j e c t can be m a d e w h i c h r e s t o r e s t h i s c o n d i t i o n , and the m o v e m e n t i s a m e a s u r e of the d i s t o r t i o n at the e l e m e n t . The d i s t o r t i o n m o v e m e n t f r o m e l e m e n t to e l e m e n t can b e p l o t t e d in the s a m e way. A l t e r n a t i v e l y , the s t e p - b y - s t e p p l o t t i n g of the m o v e m e n t of the e l e m e n t s of a d i s t o r t e d o b j e c t c o u l d be c a r r i e d out with the o b j e c t fixed, and the h o l o g r a m m o v i n g to p r o v i d e the m e a s u r e m e n t . A f o u r t h p o s s i b i l i t y in w h e r e a s i m i l a r r e C o n s t r u c t i o n p r o c e s s i s u s e d , k e e p i n g t h e o b j e c t f i x e d in i t s o r i g i n a l p o s i t i o n (or, a l t e r n a t i v e l y , u s i n g a ' f r o z e n f r i n g e ' d o u b l y - e x p o s e d h o l o g r a m ) , but o b s e r v i n g the m o v e m e n t of i n t e r f e r e n c e b a n d s p a s t any s e l e c t e d p o i n t of the o b j e c t f o r a p r e s c r i b e d s h i f t of the v i e w ing a p e r t u r e a c r o s s the h o l o g r a m . T h i s i s the t e c h nique f o l l o w e d in the f i r s t e x p e r i m e n t s to v e r i f y the p r o p o s i t i o n , and an e x a m p l e i s given b e l o w .
EXPERIMENTAL
PROCEDURE
The s u r f a c e P (Fig. 3a) was one face of a p i e c e of b r a s s 1" angle, c l e a n e d up with fine e m e r y p a p e r but o t h e r w i s e u n t r e a t e d . This was mounted on a m i c r o m e t e r - c o n t r o l l e d t w o - c o o r d i n a t e s l i d e so that m o v e m e n t s of the s u r f a c e in i t s own plane (y d i r e c tion) and p e r p e n d i c u l a r to the plane (x d i r e c t i o n ) could be m e a s u r e d . Another block of b r a s s was f a s t e n e d to the b a s e a l o n g s i d e the moving c a r r i a g e to give a fixed r e f e r e n c e s u r f a c e of the s a m e s u r f a c e finish. A h e l i u m - n e o n l a s e r i l l u m i n a t e d these s u r f a c e s obliquely f r o m below, v i a a wedged b e a m s p l i t t i n g p l a t e B, which took out a p o r t i o n of the i l l u m i n a t i o n to p r o v i d e a r e f e r e n c e b e a m R. This b e a m was focused by a m i c r o s c o p e o b j e c t i v e M and then expanded to fill the a p e r t u r e of the holog r a p h i c s e n s i t i v e plate (Agfa 10E70), s e t p a r a l l e l with the s u r f a c e s of the b r a s s t e s t p i e c e s and s e p a r a ted f r o m them by 140mm. The b e a m i l l u m i n a t i n g the b r a s s t e s t p i e c e s was expanded and c o l l i m a t e d by a d i v e r g i n g l e n s and a concave m i r r o r (Fig. 3a). Each e x p o s u r e for the d o u b l y - e x p o s e d p l a t e was
R M
~hologrQm
reconstructed
images of objects
Q IYo.
PP"YpPto_
,~= d path difference at Q.- PQ- P~Q=yy~]d
-':I
change from Q1 to Q2-YYQ1Qzffim.( :-y, mXd Y•l Qz
]laser Fig. 3a Experimental arrangement for recording the hologram
Fig. 4 Reconstruction, and measuremenl of the inplace shift of P about 1 second in duration, and between the e x p o s u r e s the p o s i t i o n of the m o v a b l e b r a s s s u r f a c e was changed. Two, s e p a r a t e , e x p e r i m e n t s w e r e p e r f o r m e d (a)
and (b)
with a m o v e m e n t of 0.019mm between e x p o s u r e s in the y d i r e c t i o n (in the plane of the moving s u r f a c e ) with a m o v e m e n t of 0.250mm in the x d i r e c t i o n ( p e r p e n d i c u l a r to the plane of the moving s u r f a c e ) .
RECONSTRUCTION AND MEASUREMENT
X
Fig. 3b Detail of moving and fixed paris of surface P
In-plane movement A f t e r p r o c e s s i n g in the n o r m a l way, the doublye x p o s e d p l a t e was r e p l a c e d on the plate h o l d e r and i l l u m i n a t e d by the r e f e r e n c e b e a m alone. Viewing the r e g i o n of the r e c o n s t r u c t i o n of the fixed b r a s s s u r f a c e , F i g . 4, no v a r i a t i o n in the i n t e r f e r e n c e p a t t e r n could be s e e n as the eye moved a c r o s s the h o l o g r a m p l a t e . But a s the eye moved in the y d i r e c tion a s y s t e m of p a r a l l e l bands running v e r t i c a l l y , i . e . at right a n g l e s to the y d i r e c t i o n , a p p e a r e d to move p a s t the s m a l l r e g i o n P P ' in the r e c o n s t r u c t e d i m a g e of the d i s p l a c e d b r a s s s u r f a c e . Ten bands p a s s e d P P ' for a m o v e m e n t of the i n t e r s e c tion Q of the line of sight with the h o l o g r a m plane o v e r a d i s t a n c e QIQ2 of 52mm, f r o m which the y shift can be c a l c u l a t e d a s shown in Fig. 4. The r e s u l t is in r e a s o n a b l e a g r e e m e n t with the m o v e m e n t of the b r a s s s u r f a c e , to the a c c u r a c y of the m i c r o meter measurement.
Optics Technology
November 1969
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M o v e m e n t perpendicular to o b j e c t - s u r f a c e plane When the second plate is treated in this way, the pattern on the shifted b r a s s surface in one of curved bands, and these move with a quadratic relation of path difference with the t r a v e r s e of Q a c r o s s the hologram plate. In this case a shift of four bands was seen for a t r a v e r s e of 20mm out from the centre of the plate in each direction, which is again in a g r e e ment with the known change of position. The o b s e r vation is difficult to make quantitative, because of the n o n - l i n e a r relation, and one of the 'null movement' techniques suggested would certainly be m o r e suitable.
INTERFERENCE IN A THREE-DIMEI~SIONAL FIELD The foregoing is a n o n - m a t h e m a t i c a l presentation of some thought and experiments which attempt to extend holographic i n t e r f e r o m e t r y beyond its present twodimensional or one-dimensional limitations. Consideration of the general problem is likely to be ext r e m e l y rewarding and is being pursued, but is not appropriate to this note. The p r e s e n t experiments were stimulated by the paper of Ennos (1968) already quoted, and by the work of Vi~not, Froehly, Monneret & P a s t e u r reported at the Symposium on the Engineering Uses of Holography (Strathclyde University, 1968). The general problem is considered there, but the t r e a t m e n t concentrates on localisation of interference effects in a plane containing the source. A f u r t h e r publication has appeared m o r e recently (Froehly, Monneret, P a s t e u r & Vi~tnot, 1969 ). The relationship between the different interference patterns observed at the surface of the object, or elsewhere such as in the plane of the s o u r c e or in the plane of the hologram, r e s e m b l e s closely the well-known i n t e r - r e l a t i o n of the fringe patterns observed in classical i n t e r f e r o m e t e r s and the patt e r n s of ' s o u r c e - p l a n e ' f r i n g e s . These patterns are related by a t r a n s f o r m relationship in which spatial extent (aperture) in one pattern has an i n t e r p r e t a tion in phase variation (visibility of interference ) in the other pattern (Born & Wolf, 1965). The experimental technique described is, in fact, v e r y s i m i l a r to one s o m e t i m e s used in improving the visibility of interference bands in the F a b r y P e r o t i n t e r f e r o m e t e r based on this relation.
The principle underlying these m e a s u r e m e n t s is capable of application over a v e r y wide field. The choice of position for the hologram recording to give the optimum angle subtended at the test surface by the hologram provides a wide range of sensitivity. After recording, m e a s u r e m e n t by observation of the passage of interference bands can be made using all, or only part, of the hologram to obtain a convenient number of bands passing for each s u c c e s s i v e element of area. This facility is likely to be useful in engineering m e a s u r e m e n t , where the distortions are c o m monly many light wavelengths in magnitude and inconveniently large for direct i n t e r f e r e n c e m e a s u r e ment. Alternatively, the use of null methods reduces the m e a s u r i n g p r o c e d u r e s to standard techniques.
optics Technology
Subsequent development might lead to applications in which the interference patterns could be analysed photoelectrically to give controlling signals for s e r v o m e c h a n i s m s stabilising or guiding motions of a solid object without loading it with m e a s u r i n g devices or auxiliary optical units, and without need for any special m a r k s . Acknowledgement Discussion with J.M.Burch and A.E.Ennos on these p r o b l e m s is acknowledged. The work has been c a r r i e d out at the National Physical L a b o r a t o r y .
REFERENCES Born, M. & Wolf E. (1965). P r i n c i p l e s of optics. Third edition, p.264 et seq. Oxford: P e r g a m o n Press. Burch, J. M., Ennos, A. E. & Wilton, J. (1966) Nature, 209. 1015-~6. Burch, J.M., Tokarski, J.M.J. (1968) Optica Acta, 15. 101-11. Collier, R.J., Doherty, E.T. & Pennington, K.S. (1965). Appl. Phys. Lett. 7. 223-5. Debrus, S., Franqon, M. & May, M. (1969) Optical Instruments and Techniques (ICO Conference Reading).
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The f i r s t possibility of the m e a s u r e m e n t of distortion in three dimensions has been described here. There are s i m p l e r possibilities in which the sensitivity in only one or two directions can be retained. For example, a finely-ground diffusely-reflecting surface mounted on an object moving so that the surface is translated in its own plane could provide, in conjunction with a fixed hologram r e c o r d e d with the surface at some fiducial zero position, a distribution of intererence bands giving the change of position of the surface. This use "is comparable with the provision and reading of the position of r e f e r e n c e m a r k s on line standards, and simple calculation shows the precision expected is not inferior. In this respect, the proposal has s i m i l a r i t i e s with m o i r e - f r i n g e m e a s u r e m e n t (Guild, 1960). The great advantage would be in the much g r e a t e r durability and r e s i s t ance to accidental disturbance that the holographic s y s t e m might be expected to show. Considerable work in s y s t e m s of this sort has already been c a r r i e d out for purposes of theoretical optics by Burch & Tokarski (1968). and by Debrus, Fran~on & May (1969).
November 1969
Ennos A.E. (1968). J.Sci. Instrum. 1. 731-4. F r o e h l y C., Monneret J., P a s t e u r J & Vi~not J. Ch. (1969) Optica Acta, 16.343-62. Gates, J.W.C. (1968) Nature, 220. 473-4. Guild, J. (1960) Diffraction gratings as m e a s u r i n g scales. Oxford University P r e s s . Haines, K.A. ~ Hildebrand B.P. (1966)Appl. Optics, 5. 595-602. Vidnot J. Ch., Froehly CI., Monneret J. & P a s t e u r J. (1968). Engineering Uses of Holography, (Symposium).
Interferometric techniques f o r measuring the local displacements of the surface of a solid body depend on observation of two-dimensional interference patterns. Interference effects observable from holographic reconstructions, however, depend on path variation as a function of four variables, and cannot be completely represented by a single two-dimensional pattern. For an appropriate choice of plane of observation, it is shown that complete information about the component motions of the visible surface are obtainable from observations of the changes in the two-dimensional pattern as a function of viewing direction. Conveniently, the plane of observation can be chosen to be the plane of the hologram itself.
A F E W YEARS AGO, p u b l i c a t i o n s by C o l l i e r , D o h e r t y & Pennington (1965), Haines & H i l d e b r a n d (1966), Burch, Ennos & Wilton (1966) f i r s t pointed out the p o s s i b i l i t y of t e c h n i q u e s of h o l o g r a p h i c i n t e r f e r o m e t r y for m e a s u r e m e n t of the change in p o s i t i o n of s u r f a c e (or l o c a l r e g i o n s of s u r f a c e s ) of solid b o d i e s . Many d e s c r i p t i o n s of a p p l i c a t i o n s of these i d e a s have been p u b l i s h e d since, and the l i m i t a t i o n s of s o m e p a r t i c u l a r t e c h n i q u e s have been e x p l o r e d . In c a s e s where an i n t e r f e r e n c e p a t t e r n is o b s e r v e d a p p a r e n t l y c o i n c i d e n t with the r e c o n s t r u c t e d i m a g e of a s u r f a c e , the l i m i t a t i o n s a r e of t h r e e kinds.
at the t i m e of h o l o g r a m e x p o s u r e . These w a v e f r o n t s can be made to i n t e r f e r e with the w a v e f r o n t s r e f l e c ted f r o m the object; or with w a v e f r o n t s a l s o r e c o n structed from a second hologram integrally r e c o r d e d on the s a m e plate a f t e r the f i r s t h o l o g r a m was r e corded; o r with w a v e f r o n t s r e c o n s t r u c t e d f r o m a s e p a r a t e h o l o g r a m r e c o r d e d subsequent to the f i r s t . Any changes in p o s i t i o n of e l e m e n t s in the s u r f a c e s of the object v i s i b l e in the r e c o n s t r u c t i o n give r i s e to path d i f f e r e n c e s which show up as i n t e r f e r e n c e e f f e c t s f r o m which the change of f o r m of the s u r f a c e can be c a l c u l a t e d .
F i r s t , the s e n s i t i v i t y to d i s p l a c e m e n t s in c e r t a i n d i r e c t i o n s can be v e r y low. Second, the i n t e r f e r e n c e p a t t e r n m a y not be o b s e r v a b l e with good v i s i b i l i t y u n l e s s the p a t t e r n r e m a i n s e s s e n t i a l l y lnvarzant with r e s p e c t to the a r e a of the h o l o g r a m used for r e c o n s t r u c t i o n (i.e. u n l e s s the p a t t e r n i s l o c a l i s e d n e a r the s u r f a c e ) . And t h i r d , the d i s p l a c e m e n t of the p a t t e r n for a given change in s u r f a c e p o s i t i o n can be inconveniently l a r g e , as in the c a s e of m o s t e n g i n e e r i n g a p p l i c a t i o n s w h e r e the d i s p l a c e m e n t s c o r r e s p o n d to a l a r g e n u m b e r of wavelengths of light.
The technique is an e x t r e m e l y powerful one, but t h e r e m a y be d i f f i c u l t i e s in a s s i g n i n g exact v a l u e s to the d i s t o r t i o n s shown up in any p a r t i c u l a r choice of viewing c o n d i t i o n s . When the i n t e r f e r e n c e p a t t e r n is viewed f r o m a fixed point as a p r o j e c t i o n on the s u r f a c e being studied, a single i n t e r f e r e n c e p a t t e r n g i v e s a m e a s u r e of d i s p l a c e m e n t in a d i r e c t i o n b i s e c t i n g the angle between the d i r e c t i o n of the r e f e r e n c e b e a m and the d i r e c t i o n of viewing (Ennos 1968).
More g e n e r a l l y , c o n s i d e r a t i o n of i n t e r f e r e n c e p a t t e r n s l e a d s to a n u m b e r of p o s s i b i l i t i e s f o r obviating these l i m i t a t i o n s , and a technique is d e s c r i b e d in which l o c a l m o v e m e n t s of a s u r f a c e in any d i r e c t i o n a r e e a s i l y obtainable f r o m i n t e r f e r e n c e e f f e c t s o b s e r v a b l e by a single h o l o g r a m . The r a t i o of the o b s e r v e d effect to the m o v e m e n t of the s u r f a c e is c o n t r o l l a b l e by a s u i t a b l e choice of p r o p o r t i o n of h o l o g r a m s i z e and p o s i t i o n .
HOLOGRAM
INTERFEROME TRY
A hologram provides m e a n s of reconstructing wavefronts, however c o m p l i c a t e d , r e f l e c t e d f r o m an object
To e v a l u a t e o r t h o g o n a l d i s p l a c e m e n t s , a second r e c o n s t r u c t i o n f r o m a n o t h e r h o l o g r a m , involving diff e r e n t d i r e c t i o n of viewing, was u s e d by Ennos. Even this c o m b i n a t i o n will not r e v e a l d i s p l a c e m e n t s in the d i r e c t i o n o r t h o g o n a l to the plane containing the i l l u m i n a t i o n and viewing d i r e c t i o n , and a t h i r d r e c o n s t r u c t i o n d i r e c t i o n is n e c e s s a r y to p r o v i d e a c o m p l e t e t h r e e - d i m e n s i o n a l knowledge of the d i s t o r t i o n . These r e s t r i c t i o n which apply to the use of the i n t e r f e r e n c e p a t t e r n s a s d e s c r i b e d above do not apply g e n e r a l l y to a l l i n t e r f e r e n c e techniques. A method of i n t e r f e r o m e t r i c m e a s u r e m e n t is d e m o n s t r a t e d by which exact t h r e e - d i m e n s i o n a l evaluation of the d i s t o r t i o n of the s u r f a c e of a body can be obtained f r o m i n t e r f e r e n c e e f f e c t s p r o d u c e d by a s i n g l e p a i r of h o l o g r a p h i c r e c o r d s , such as, for e x a m p l e , a single doubly-exposed frozen-fringe hologram.
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Interference patterns produced from h o l o g r a m s In F i g . l a SS 1 i s a s u r f a c e in which an e l e m e n t a r y a r e a 1~ r e f l e c t s a c o h e r e n t l i g h t b e a m i n c i d e n t f r o m the d i r e c t i o n of O, and s o f o r m s a h o l o g r a m H on a p h o t o g r a p h i c p l a t e b y i n t e r f e r e n c e with a n o t h e r p a r t of the s a m e c o h e r e n t b e a m R, which i s a l s o i n c i d e n t on the p l a t e . The s u r f a c e SS 1 i s now d i s t o r t e d and P m o v e s to P ' . A s e c o n d h o l o g r a m H' i s r e c o r d e d at the s a m e p o s i t i o n a s H, on a s e p a r a t e p h o t o g r a p h i c p l a t e , o r on the s a m e p h o t o g r a p h i c plate. After processing the hologram, a real image of b o t h SS 1 and S'S 1' can b e r e c o n s t r u c t e d , f o r e x a m p l e (in t h e c a s e of a s i n g l e , d o u b l e - e x p o s e d h o l o g r a m ) b y d i r e c t i n g a r e p l i c a of the r e f e r e n c e b e a m R b a c k w a r d s t h r o u g h the p l a t e , a s shown in Fig. lb. A p h a s e d i f f e r e n c e e x i s t s b e t w e e n the two p a t h s HP and H P ' w h i c h v a r i e s a s a function of the p o s i t i o n of P on the d i s t o r t e d s u r f a c e , and so p r o d u c e s an i n t e r f e r e n c e p a t t e r n which c a n b e thought of a s p r o j e c t e d on the s u r f a c e f r o m a s e l e c t e d s m a l l r e g i o n of the hologram. The interference pattern gives contours of p a t h v a r i a t i o n r e l a t e d to the m o v e m e n t s of e l e m e n t s of the s u r f a c e , a s d e s c r i b e d b y E n n o s . T h i s i s the p a t t e r n f o r w h i c h the r e s t r i c t i o n s m e n t i o n e d a b o v e a p p l y , and in g e n e r a l the p a t t e r n i s not i n v a r i a n t with r e s p e c t s to the a r e a of the h o l o g r a m u s e d . The c o n s e q u e n t r e s t r i c t i o n which m u s t be i m p o s e d on the a r e a of the h o l o g r a m to o b t a i n a s h a r p i n t e r f e r e n c e p a t t e r n a p p a r e n t l y l o c a t e d on the s u r f a c e p r e c l u d e s the p o s s i b i l i t y of o b t a i n i n g i n f o r m a t i o n about m o v e m e n t in m o r e than one d i r e c t i o n . If the full a r e a of the h o l o g r a m i s u s e d , P and P ' w i l l b e i l l u m i n a t e d f r o m a r a n g e of a n g u l a r d i r e c t i o n s , and and t h e r e s u l t a n t c o m b i n a t i o n of p a t t e r n s b l u r s out the i n t e r f e r e n c e e f f e c t s . Confining a t t e n t i o n to a small area around P P', reconstructed wavefronts c o n v e r g i n g t o w a r d t h i s a r e a a p p r o x i m a t e to the s p h e r i c a l w a v e f r o n t s W W' ( F i g . 2). A s a r e s u l t , if the i n c i d e n t b e a m s a r e v i e w e d t h r o u g h a s m a l l a p e r t u r e a in the ( v i r t u a l ) p o s i t i o n of P and P ' , the i n t e r f e r e n c e p a t t e r n a p p e a r i n g to c o v e r the s u r f a c e of the h o l o g r a m i s a s e r i e s of c u r v e d b a n d s w h o s e s p a c i n g and o r i e n t a t i o n give the t r a n s v e r s e d i s p l a c e m e n t s of P ' with r e s p e c t to P in two o r t h o g o n a l d i r e c t i o n s and w h o s e c u r v a t u r e s h o w s the d i s p l a c e m e n t in the d i r e c t i o n of the c e n t r e of the r i n g p a t t e r n . T h u s t h r e e c o o r d i n a t e d i s p l a c e m e n t s can b e e v a l u a t e d f o r e a c h p b i n t P and a c o n t o u r m a p of the t h r e e d i m e n s i o n a l d i s t o r t i o n can be b u i l t up. A l t e r n a t i v e l y , if two s e p a r a t e h o l o g r a m s a r e u s e d ( G a t e s 1968), one m a y b e m o v e d in r e l a t i o n to the o t h e r to p r o d u c e a null, ' f l u f f e d o u t ' , f r i n g e p a t t e r n , in w h i c h e a s e the s p a t i a l r e l a t i o n of the two h o l o grams provides three coordinate displacements. When the r e f e r e n c e w a v e i s i n c i d e n t n o r m a l l y on e a c h h o l o g r a m , and the s e p a r a t i o n of the h o l o g r a m s i s m a d e without r o t a t i o n s , t h e c o o r d i n a t e d i s p l a c e m e n t s of the h o l o g r a m m u s t b e e q u a l to t h e t h r e e c o - o r d i n a t e s of the d i s t o r t i o n ( P P ' ) x , ( P P ' ) y , ( P P ' ) z . As a third possibility, a 'live fringe' technique can b e u s e d . A h o l o g r a m m a d e with the o b j e c t u n d i s t o r t e d i s u s e d with t h e o b j e c t i t s e l f r e - i l l u m i n a t e d to f o r m an i n t e r f e r e n c e p a t t e r n . If the i n t e r f e r e n c e e f f e c t s w h i c h a p p e a r to c o v e r the o b j e c t s e e n t h r o u g h the h o l o g r a m r e m a i n f i x e d in any s m a l l a r e a of the o b j e c t , i r r e s p e c t i v e of the p a r t of the h o l o g r a m through which they are viewed, the element must
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0
S
HandH'
Fig. la Recording a doubly-exposed hologram of a surface which shifts between exposures
( H+ H~) processed
"",.
~./
\
Fig. lb Reconstructing f r o m the resultant hologram
P
Fig. 2 I n t e r f e r e n c e between reconstructions f r o m an extended area o f the hologram plate
c o i n c i d e with the r e c o n s t r u c t i o n of the e l e m e n t f o r m e d b y the h o l o g r a m . If not, an a p p r o p r i a t e m o v e m e n t of the o b j e c t can be m a d e w h i c h r e s t o r e s t h i s c o n d i t i o n , and the m o v e m e n t i s a m e a s u r e of the d i s t o r t i o n at the e l e m e n t . The d i s t o r t i o n m o v e m e n t f r o m e l e m e n t to e l e m e n t can b e p l o t t e d in the s a m e way. A l t e r n a t i v e l y , the s t e p - b y - s t e p p l o t t i n g of the m o v e m e n t of the e l e m e n t s of a d i s t o r t e d o b j e c t c o u l d be c a r r i e d out with the o b j e c t fixed, and the h o l o g r a m m o v i n g to p r o v i d e the m e a s u r e m e n t . A f o u r t h p o s s i b i l i t y in w h e r e a s i m i l a r r e C o n s t r u c t i o n p r o c e s s i s u s e d , k e e p i n g t h e o b j e c t f i x e d in i t s o r i g i n a l p o s i t i o n (or, a l t e r n a t i v e l y , u s i n g a ' f r o z e n f r i n g e ' d o u b l y - e x p o s e d h o l o g r a m ) , but o b s e r v i n g the m o v e m e n t of i n t e r f e r e n c e b a n d s p a s t any s e l e c t e d p o i n t of the o b j e c t f o r a p r e s c r i b e d s h i f t of the v i e w ing a p e r t u r e a c r o s s the h o l o g r a m . T h i s i s the t e c h nique f o l l o w e d in the f i r s t e x p e r i m e n t s to v e r i f y the p r o p o s i t i o n , and an e x a m p l e i s given b e l o w .
EXPERIMENTAL
PROCEDURE
The s u r f a c e P (Fig. 3a) was one face of a p i e c e of b r a s s 1" angle, c l e a n e d up with fine e m e r y p a p e r but o t h e r w i s e u n t r e a t e d . This was mounted on a m i c r o m e t e r - c o n t r o l l e d t w o - c o o r d i n a t e s l i d e so that m o v e m e n t s of the s u r f a c e in i t s own plane (y d i r e c tion) and p e r p e n d i c u l a r to the plane (x d i r e c t i o n ) could be m e a s u r e d . Another block of b r a s s was f a s t e n e d to the b a s e a l o n g s i d e the moving c a r r i a g e to give a fixed r e f e r e n c e s u r f a c e of the s a m e s u r f a c e finish. A h e l i u m - n e o n l a s e r i l l u m i n a t e d these s u r f a c e s obliquely f r o m below, v i a a wedged b e a m s p l i t t i n g p l a t e B, which took out a p o r t i o n of the i l l u m i n a t i o n to p r o v i d e a r e f e r e n c e b e a m R. This b e a m was focused by a m i c r o s c o p e o b j e c t i v e M and then expanded to fill the a p e r t u r e of the holog r a p h i c s e n s i t i v e plate (Agfa 10E70), s e t p a r a l l e l with the s u r f a c e s of the b r a s s t e s t p i e c e s and s e p a r a ted f r o m them by 140mm. The b e a m i l l u m i n a t i n g the b r a s s t e s t p i e c e s was expanded and c o l l i m a t e d by a d i v e r g i n g l e n s and a concave m i r r o r (Fig. 3a). Each e x p o s u r e for the d o u b l y - e x p o s e d p l a t e was
R M
~hologrQm
reconstructed
images of objects
Q IYo.
PP"YpPto_
,~= d path difference at Q.- PQ- P~Q=yy~]d
-':I
change from Q1 to Q2-YYQ1Qzffim.( :-y, mXd Y•l Qz
]laser Fig. 3a Experimental arrangement for recording the hologram
Fig. 4 Reconstruction, and measuremenl of the inplace shift of P about 1 second in duration, and between the e x p o s u r e s the p o s i t i o n of the m o v a b l e b r a s s s u r f a c e was changed. Two, s e p a r a t e , e x p e r i m e n t s w e r e p e r f o r m e d (a)
and (b)
with a m o v e m e n t of 0.019mm between e x p o s u r e s in the y d i r e c t i o n (in the plane of the moving s u r f a c e ) with a m o v e m e n t of 0.250mm in the x d i r e c t i o n ( p e r p e n d i c u l a r to the plane of the moving s u r f a c e ) .
RECONSTRUCTION AND MEASUREMENT
X
Fig. 3b Detail of moving and fixed paris of surface P
In-plane movement A f t e r p r o c e s s i n g in the n o r m a l way, the doublye x p o s e d p l a t e was r e p l a c e d on the plate h o l d e r and i l l u m i n a t e d by the r e f e r e n c e b e a m alone. Viewing the r e g i o n of the r e c o n s t r u c t i o n of the fixed b r a s s s u r f a c e , F i g . 4, no v a r i a t i o n in the i n t e r f e r e n c e p a t t e r n could be s e e n as the eye moved a c r o s s the h o l o g r a m p l a t e . But a s the eye moved in the y d i r e c tion a s y s t e m of p a r a l l e l bands running v e r t i c a l l y , i . e . at right a n g l e s to the y d i r e c t i o n , a p p e a r e d to move p a s t the s m a l l r e g i o n P P ' in the r e c o n s t r u c t e d i m a g e of the d i s p l a c e d b r a s s s u r f a c e . Ten bands p a s s e d P P ' for a m o v e m e n t of the i n t e r s e c tion Q of the line of sight with the h o l o g r a m plane o v e r a d i s t a n c e QIQ2 of 52mm, f r o m which the y shift can be c a l c u l a t e d a s shown in Fig. 4. The r e s u l t is in r e a s o n a b l e a g r e e m e n t with the m o v e m e n t of the b r a s s s u r f a c e , to the a c c u r a c y of the m i c r o meter measurement.
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M o v e m e n t perpendicular to o b j e c t - s u r f a c e plane When the second plate is treated in this way, the pattern on the shifted b r a s s surface in one of curved bands, and these move with a quadratic relation of path difference with the t r a v e r s e of Q a c r o s s the hologram plate. In this case a shift of four bands was seen for a t r a v e r s e of 20mm out from the centre of the plate in each direction, which is again in a g r e e ment with the known change of position. The o b s e r vation is difficult to make quantitative, because of the n o n - l i n e a r relation, and one of the 'null movement' techniques suggested would certainly be m o r e suitable.
INTERFERENCE IN A THREE-DIMEI~SIONAL FIELD The foregoing is a n o n - m a t h e m a t i c a l presentation of some thought and experiments which attempt to extend holographic i n t e r f e r o m e t r y beyond its present twodimensional or one-dimensional limitations. Consideration of the general problem is likely to be ext r e m e l y rewarding and is being pursued, but is not appropriate to this note. The p r e s e n t experiments were stimulated by the paper of Ennos (1968) already quoted, and by the work of Vi~not, Froehly, Monneret & P a s t e u r reported at the Symposium on the Engineering Uses of Holography (Strathclyde University, 1968). The general problem is considered there, but the t r e a t m e n t concentrates on localisation of interference effects in a plane containing the source. A f u r t h e r publication has appeared m o r e recently (Froehly, Monneret, P a s t e u r & Vi~tnot, 1969 ). The relationship between the different interference patterns observed at the surface of the object, or elsewhere such as in the plane of the s o u r c e or in the plane of the hologram, r e s e m b l e s closely the well-known i n t e r - r e l a t i o n of the fringe patterns observed in classical i n t e r f e r o m e t e r s and the patt e r n s of ' s o u r c e - p l a n e ' f r i n g e s . These patterns are related by a t r a n s f o r m relationship in which spatial extent (aperture) in one pattern has an i n t e r p r e t a tion in phase variation (visibility of interference ) in the other pattern (Born & Wolf, 1965). The experimental technique described is, in fact, v e r y s i m i l a r to one s o m e t i m e s used in improving the visibility of interference bands in the F a b r y P e r o t i n t e r f e r o m e t e r based on this relation.
The principle underlying these m e a s u r e m e n t s is capable of application over a v e r y wide field. The choice of position for the hologram recording to give the optimum angle subtended at the test surface by the hologram provides a wide range of sensitivity. After recording, m e a s u r e m e n t by observation of the passage of interference bands can be made using all, or only part, of the hologram to obtain a convenient number of bands passing for each s u c c e s s i v e element of area. This facility is likely to be useful in engineering m e a s u r e m e n t , where the distortions are c o m monly many light wavelengths in magnitude and inconveniently large for direct i n t e r f e r e n c e m e a s u r e ment. Alternatively, the use of null methods reduces the m e a s u r i n g p r o c e d u r e s to standard techniques.
optics Technology
Subsequent development might lead to applications in which the interference patterns could be analysed photoelectrically to give controlling signals for s e r v o m e c h a n i s m s stabilising or guiding motions of a solid object without loading it with m e a s u r i n g devices or auxiliary optical units, and without need for any special m a r k s . Acknowledgement Discussion with J.M.Burch and A.E.Ennos on these p r o b l e m s is acknowledged. The work has been c a r r i e d out at the National Physical L a b o r a t o r y .
REFERENCES Born, M. & Wolf E. (1965). P r i n c i p l e s of optics. Third edition, p.264 et seq. Oxford: P e r g a m o n Press. Burch, J. M., Ennos, A. E. & Wilton, J. (1966) Nature, 209. 1015-~6. Burch, J.M., Tokarski, J.M.J. (1968) Optica Acta, 15. 101-11. Collier, R.J., Doherty, E.T. & Pennington, K.S. (1965). Appl. Phys. Lett. 7. 223-5. Debrus, S., Franqon, M. & May, M. (1969) Optical Instruments and Techniques (ICO Conference Reading).
APPL~A~O~
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The f i r s t possibility of the m e a s u r e m e n t of distortion in three dimensions has been described here. There are s i m p l e r possibilities in which the sensitivity in only one or two directions can be retained. For example, a finely-ground diffusely-reflecting surface mounted on an object moving so that the surface is translated in its own plane could provide, in conjunction with a fixed hologram r e c o r d e d with the surface at some fiducial zero position, a distribution of intererence bands giving the change of position of the surface. This use "is comparable with the provision and reading of the position of r e f e r e n c e m a r k s on line standards, and simple calculation shows the precision expected is not inferior. In this respect, the proposal has s i m i l a r i t i e s with m o i r e - f r i n g e m e a s u r e m e n t (Guild, 1960). The great advantage would be in the much g r e a t e r durability and r e s i s t ance to accidental disturbance that the holographic s y s t e m might be expected to show. Considerable work in s y s t e m s of this sort has already been c a r r i e d out for purposes of theoretical optics by Burch & Tokarski (1968). and by Debrus, Fran~on & May (1969).
November 1969
Ennos A.E. (1968). J.Sci. Instrum. 1. 731-4. F r o e h l y C., Monneret J., P a s t e u r J & Vi~not J. Ch. (1969) Optica Acta, 16.343-62. Gates, J.W.C. (1968) Nature, 220. 473-4. Guild, J. (1960) Diffraction gratings as m e a s u r i n g scales. Oxford University P r e s s . Haines, K.A. ~ Hildebrand B.P. (1966)Appl. Optics, 5. 595-602. Vidnot J. Ch., Froehly CI., Monneret J. & P a s t e u r J. (1968). Engineering Uses of Holography, (Symposium).