A b s tr a c t
A tim e-resolved holographic interferom eter specially suited for high-speed visu alization of th e gas flow in shock tu b e experim ents has been developed. H olographic interferom etry, which is based on th e recording of two coincident hologram s a t dif ferent tim es so th a t one of th em acts as a reference field, can accurately reveal th e density d istrib u tio n in a gas. T h e device described here fills the need for a p ractical m eth o d to record sh o rt sequences of holographic interferogram s docum enting th e evolution of shock wave reflections th a t are not self-similar in tim e.
M ultiple hologram recording was im plem ented on an existing holographic in- terferom etric system th ro u g h th e technique of sp atial frequency m ultiplexing, in which th e hologram s are overlaid b u t th e reference beam is angled differently for each exposure. Because th e object beam is not involved in the m ultiplexing process, the im aging optics of th e original system could be left unm odified. T h e up g rad e only entailed th e in tro d u ctio n of an angular sweeping system in th e reference beam p ath .
T h e beam m ultiplexing assem bly was initially based on a spinning m irro r design, which produced fairly satisfactory recordings of non-interferom etric holographic sequences b u t was incapable ox accurately overlaying a second set of exposures establishing th e reference field for each image. T he m echanical sweeping system h ad oth er draw backs as well, am ong th em th e tendency to create extraneous fringes in the holographic images because of th e unavoidable an g u lar m otion of th e reference beam over th e d u ra tio n of a laser pulse.
A solid-state m ultiplexing system was th en devised in which th e reference beam was split into several b ram hes, each aim ed a t th e film from a different direction and
was achieved by opening th e sh u tters in sequence as the laser was pulsed, b u t it was also possible to record th e reference exposure on all images sim ultaneously w ith a single laser pulse by having all sh u tte rs open a t th e sam e tim e. A pro to ty p e three- im age system was constructed a n d successfully tested by recording interferom etric sequences of a shock wave reflecting off a m odel at fram ing intervals down to 100 /ts.
Exam iners:
D r.Sf^M ^D iw ey, 'S u p e rv iso rT D ^ p a H r^ n t of Physic sj
D r. R. M. Clem ents, D ep artm en tal M em ber (D ep artm en t of Physics)
D r. G. B. Friedm ann, D ep artm en tal M em ber (D ep artm en t of Physics)
Dr. R. N. O ’Brien, O utside M em ber (D ep artm en t of C hem istry)
D r. P. van den Driessche, O utside M em ber (D ep artm en t of M athem atics)
Dr. D. K. A dditional .Member (D ep artm en t of Physics)
T a b le o f C o n te n ts
A b s t r a c t ... ii Table of C o n te n ts ... iv List of Figures ... vi A c k n o w led g e m e n ts... viii D e d ic a tio n ... ix 1 In tro d u ctio n 1 2 Shock w aves 5 2.1 N atu re of shock w a v e s ... 5 2.2 T he R ankine-H ugoniot e q u a t i o n s ... 62.3 O blique shock wave r e f le c tio n ... . 8
2.4 D escription of th e shock t u b e ... 12 2.5 V isualization m e t h o d s ... 13 2.5.1 Shadow an d schlieren p h o to g r a p h y ... 13 2.5.2 I n t e r f e r o m e t r y ... 14 3 H olograp h y 16 3.1 Intro d u ctio n to th e te c h n i q u e ... 16
3.2 H ologram form ation equations ... 17
3.3 H olographic i n t e r f e r o m e t r y ... 19
3.3.1 Principles an d a p p lic a tio n ... 19
3.3.2 Fringe in te r p r e ta tio n ... 22
3.4 T h e existing holographic in t e r f e r o m e te r ... 23
4 T im e-reso lv ed h olograp h y 27 4.1 O b je c tiv e s ... 27
4.2 H ologram m ultiplexing techniques ... 28
4.2.1 Spatial m u ltip le x in g ... 28
4.2.2 S p atial frequency m u l t i p l e x i n g ... 30
4.3 S p atial frequency m ultiplexing— an in-d ep th l o o k ...32
5 N 'echanical scan n in g 39 5.1 Basic c o n c e p t ...39
5.2.1 C o n stru ctio n d e t a i l s ...40
5.2.2 E l e c t r o n i c s ...43
5.2.3 E xposure s y n c h r o n iz a tio n ... 45
5.3 R e s u lts ... 4S 5.4 S h o r tc o m in g s ... 50
5.4.1 N on-continuous co v erag e... 50
5.4.2 A ngular reg istration in a c c u ra c y ... 54
5.4.3 Tim e s m e a r ...55 6 T im e sm ea r 57 6.1 G eneral equations ... 57 6.2 N um erical m o d e l ...60 6.3 A nalytical m o d e l ...61 6.4 D iscussion of th e m o d e ls ... 64
7 S o lid -sta te sca n n in g 67 7.1 G eneral a d v a n ta g e s ...67
7.2 Available beam ro u tin g te c h n o lo g ie s ... 67
7.3 Choice of sp u ^ al frequency m ultiplexing s t r a t e g y ... 69
7.3.1 Deflection of beam in to one of several p a t h s ... 69
7.3.2 S h u tterin g of m ultiple coexisting b e a m s ... 70
7.4 Im p le m e n ta tio n ... 71 7.4.1 O ptical s y s t e m ... 71 7.4.2 E lectro-optical s h u t t e r s ... 74 7.4.3 C ontrol c i r c u i t r y ... 78 7.4.4 E xposure s y n c h r o n iz a tio n ... 80 7.5 R e s u lts ... 82 7.6 Discussion of th e recording m e t h o d ...88
7.7 M ultiple-beam reference exposure—a theoretical view ...92
7.8 An altern ativ e d e s ig n ... 96
8 C on clu sion s 98
L ist o f F ig u r e s
2.1 T he geom etry of th e flow thro u g h an oblique shock... 7
2.2 R egular reflection of a plane shock off a plane wedge... 9
2.3 M acn reflection of a plane shock off a plane wedge... 11
3.1 O riginal layout of the holographic interferom eter...24
5.1 Layout of th e m ultifram e holographic recording a p p a ra tu s w ith a spinning-m irror reference beam sweeping sy stem ... 40
5.2 D etail draw ing of th e spinning-m irror beam sweeping assembly. . . 41
5.3 Schem atic diagram of th e m irror position optical pick-up circuit. . . 43
5.4 Schem atic diagram r f th e m irror face discrim inator circu it... 44
5.5 C ircuit diagram of the reference exposure delay u n it...45
5.6 Block diagram of the exposure synchronization electronics for the m echanical scanning sy stem ... 46
5.7 Sequence of holographically recorded schlieren im ages ob tain ed using the m echanical scanning system ... 49
5.8 Holographic interferogram from a double-sweep sequence o b tain ed using th e m echanical scanning sy stem ... 50
5.9 Possible design of a n u tatin g -m irro r reference beam sweeping assem bly for m uitifram e holographic recording... 53
5.10 Exam ple of a reconstructed image showing tim e sm ear effects. . . . 5 6 6.1 Profiles of laser pulse irradiance vs. tim e used to d e m o n strate the tim e sm ear m odels... 61
6.2 G rap h of hologram irradiance m odulation vs. la te ral displacem ent given by the num erical m odel... 62
6.3 G rap h of hologram irradiance m odulation vs. la te ral displacem ent given by the analytical m odel... 63
6.4 G rap h of hologram irradiance m odulation vs. lateral displacem ent given by the analytical model for a narrow pulse w id th ... 65
7.1 Layout of the m ultifram e holographic interferom eter w ith a solid-state reference beam m ultiplexing system ... 72
7.2 D etail draw ing o f th e three-channel beam m ultiplexing assem bly us ing liquid cry stal s h u tte rs ... 73
7.3 P h o to g rap h of the reference beam sp littin g and sh u tterin g optics. . . 74 7.4 S tru ctu re of a FLC cell in its two voltage-selected s ta te s ... 75 7.5 D iagram of a FLC s h u tte r in its closed and open configurations. . . 76 7.6 C ircuit diagram of th e sh u tte r sequencing system for three FLC light,
valves... 79 7.7 Block diagram of th e exposure synchronization electronics for the
solid-state scanning sy stem ... SI 7.8 Sequence of infinite-fringe holographic interferogram s recorded w ith
th e solid-state scanning system ... S3 7.9 D igitized sequence of infinite-fringe holographic interferogram s re
I wish to express my heartfelt th an k s to Dr. Jo h n Dewey, w ith w hom I first discussed this project and who guided and supervised m e in its developm ent. I am also in d eb ted to m y colleagues a t th e Shock Studies L aboratory, p articularly Alex van N etten, for their help and advice and for sportingly p u ttin g up w ith my som etim es unorthodox research habits. David Sm ith and P e te r W ard of the U niversity of V ictoria Physics m achine shop, David Searle of the U niversity of Vic to ria C hem istry glass shop, Jes Jessen of th e Dom inion A strophysical O bservatory optical shop, an d Luke R oosm a of Displaytech, Inc. provided invaluable technical assistance.
T h e financial su p p o rt of th e U niversity of V ictoria in th e form of a G rad u ate Fellowship is gratefully acknowledged. All of the equipm ent used in this project, and th e travel to in tern atio n al m eetings which allowed fruitful discussion of the concepts w ith a critical audience, were financed by an o p eratin g g ra n t from the N atu ral Sciences and Engineering Research Council of C anada.
I could n o t have persisted in this long task h ad it no t been for the unfailing affection an d su p p o rt of my wonderful family, n ear and far. In a special way I am than k ful to Bonny, now m y wife, who was always close to m e th ro u g h o u t this endeavour. My love an d g ra titu d e to all of them .
I n tr o d u c tio n
A variety of optical m ethods are used in the study of high-speed com pressible flows an d shock waves to g ath er inform ation as detailed as possible ab o u t th e local properties of a flow field. T he capability to visualize a n d record w ith very short exposure tim es th e prop ag ation of a disturbance thro u g h a gaseous m ass is p a rtic u larly im p o rta n t un d er th e nonsteady conditions associated w ith explosive flows. In the areas of shock tu b e an d blast wave research, high-speed photography has been team ed w ith visualization m ethods such as shadow graphy, schlieren, conventional interferom etry and, latest in developm ent, holographic interferom etry. A n exten sive review of these applications appears in a recent m onograph (Dewey, 1989). The above techniques, each one of which will be discussed a t least briefly in later chapters, rely on the change in the refractive index of a gas caused by a shock wave to visualize th e shock fronts or the density d istrib u tio n behind them . T h e stu d y of large scale b last waves m is t generally use visualization techniques based on the refraction of am bient light. Interferom etric m ethods are p articu larly suited to the controlled environm ent of shock tubes, which allows precise illum ination of a well denned test section w ith m onochrom atic light.
Interferom etry reveals a t every location in th e test field th e refractive index change relative to some reference point. In conventional (M ach-Z ehnder) interfer om etry th e optical length of a p a th passing th ro u g h th e test volume is com pared at
every point w ith the optical length of a reference p a th , w ith differences appearing as fringes in th e field of view. T h e m ethod is therefore affected by any discrepancy betw een the two p a th s introduced by im perfections in th e optical com ponents, and consequently requires prem ium quality optics th ro u g h o u t th e system. As will be seen later, holographic inteifero m etry does away w ith th is problem by com paring two wavefronts following a single p a th at different tim es, and m ay therefore be im plem ented w ith relative ease on optical system s of less th a n optim al quality
T h e p oten tial usefulness of holographic interferom etry in aerodynam ics research was illu stra te d in th e early n ap e by Heflinger et al. (1966). T he pop u larity of this technique for shock tu b e research has been steadily increasing since the initial work of researchers such as W ortberg (1973), especially in th e wake of the excellent recordings presented by Takayam a (1983). The optical equipm ent associated w ith th e shock tu b e in the U niversity of V ictoria Shock Studies L aboratory was recently converted to a holographic system (van N etten, 1988) to expand ongoing studies on th e reflection of weak shock waves trad itio n ally conducted by shadow graph and schlieren m ethods (W alker et al., 1982) and p article flow tracin g (Dewey et al., 1975).
T h e stu d y by optical m ethods of shock tu b e phenom ena th a t are no t self-similar in tim e, such as the reflection of a shock from a double or curved wedge, requires the recording of a series of im ages sep arated by tim e intervals of th e order of a few tens of m icroseconds. H olographic interferom uliic studies of such events are custom arily conducted by recording a single interferogram du rin g an experim ent and building a. tem p o ral sequence by progressively delaying the laser pulse over a series of sim ilar experim ents (Takayam a, 1983). T his approach is inconvenient, requiring the shock tu b e to be fired repeatedly, an d po ten tially in accu rate if the experim ental conditions cannov. b e kept absolutely constant. M oreover, it would be inapplicable to cases w here a m odel is m echanically affected by th e shock in a non rep eatab le m anner. A need therefore exists for a tim e-resolved holographic
inter-fercm etric system capable of recording several images during a single experim ent, and th e presen t work, was u n d ertaken to a tte m p t to fill th a t need.
In an endeavour to alter as little as possible th e existing optical layout, which allowed th e use of shadowgruphy, schlieren, p article tra c er photo g rap h y a n d holo graphic interferom etry, sp atial frequency m ultiplexing was chosen as th e m eth od to record m ultiple hologram s. T his technique required only th e reference b e am — th a t is, th e non-im age-carrying beam used in hologram recording— to b e angled differ ently for each im age in a sequence. T he p a th of th e im age form ing beam was not modified in any way from th e layout use '5 n th e original holographic system or in th e previous noi:-holographic configurations, an d full backw ard com patibility was therefore retained. A first im plem entation of th e m ultiplexing scheme based on a m echanical beam deflection system (R acca and Dewey, 1989a) gave some prom is ing results w hen recording non-interferom etric series of hologram s, b u t failed to produce reliable interferom etric sequences because of problem s w ith th e accurate overlaying of th e required two exposures p er image. T his difficulty am o th er sh o rt comings of th e m ethod, p articu larly th e appearance of spurious fringes due to the m in u te m otion of th e beam during exposures (R acca an d Dewey, 1989a., 1989b) shifted the research em . :asis tow ard a non-m echanical system.
Previous im plem entations of sp atial frequency m ultiplexing by solid s ta te m eans (H insch and B ader, 1974, L au terb o rn an d Ebeling, 1977, Y am am oto, 1989) were based on th e principle of redirecting th e entire energy of th e reference beam along one of several p a th s converging a t the film. These system s used acousto-optical or high-voltage electro-optical devices (see C h ap ter 7) as th e active com ponent, and were n o t rep o rted to have been applied to interferom etric work. Y am am oto (1989) also presented a system using two tw in-pulse lasers to record sp atial frequency m ultiplexed sequences of two holographic interferogram s. Unwieldy as th ey m ay be, m ultiple lasers using either sep arate lasing elem ents or different portio n s of the sam e lasing m edium have been employed to record sequences of holographic
inter-ferogram s in specialized applications since fairly early tim es (T hom as et al., 1972). In th e w ork presented here, it was chosen to explore th e capabilities as beam switchers of newly em erging low-^oltage electro-optical light m od u lato rs based on ferroelectric liquid crystals, and to pursue a novel approach to beam selection. T he reference beam was split in to several coexisting branches, each aim ed a t the film from a different direction an d individually sh u ttered by a liquid crystal light valve. M ultiplexing of th e beam was achieved by opening only one sh u tte r a t a tim e in sequence. If desired 'inwever, m ore th a n one beam could be allowed to reach th e film, and th is capability was used to record the em pty field exposure for interferom etry on all images a t once. Using this scheme, which offers several advantages th a t will be analysed later, three-im age p ro to ty p e has been built which satisfactorily records interferom etric sequences of shock tu b e experim ents.
C h ap ter 2 of this d issertatio n presents a brief background on shock waves and th e m ethods used to generate an d visualize them in th e laboratory, not including holographic interferom etry. C h ap ter 3 gives a theoretical in tro d u ctio n to holog rap h y and th e n focuses on th e kind of holographic interferom etry com m only used in shock tu b e research, including an overview of th e original holographic system used as th e basis for th is work. T h e concept of tim e-resolved holography and the possible approaches to it are discussed in C h ap ter 4, w ith an em phasis on spatial frequency m ultiplexing. T he subsequent two chapters discuss its im plem entation by m echanical scanning, including in C h ap ter 6 a m ath em atical analysis of the form ation of spurious fringes due to beam m otion. T he solid s ta te approach to beam m ultiplexing is tre a te d in C h ap ter 7, which goes from a bro ad evaluation of th e available technologies and scanning strategies to a detailed discussion of the present m eth o d and its results, possibilities and lim itations. C h ap ter 8 gives a closing view of th e work perform ed and its foreseen applications an d extensions.
C h a p te r 2
S h o ck w a v es
2.1
N a t u r e o f sh o c k w a v es
A m oving shock wave is form ed when a com pressional d istu rb an ce of finite am pli tude p rop ag ates th rou g h a gas. Such a disturbance m ay be produced by phenom ena such as th e d eto n atio n of an explosive, th e burstin g of a balloon, or th e m otion of a supersonic aircraft. In a supersonic flow tunnel, a statio n ary shock wave is form ed w here a vector com ponent of the flow makes a tra n sitio n from a supersonic to a subsonic regim e. Shock waves are characterized by a non-linear pressure profile w ith a sh arp leading edge, know n as th e shock fro n t, across which th e re is a v ir tually in stan tan eo u s change in the therm odynam ic p ro p erties of the m edium . T he thickness of a shock front is ab o u t 10 to ?0 m ean free p ath s, which for a gas initially at s ta n d a rd atm ospheric conditions -imounts to ab o u t 10~5 cm. As a m oving shock wave expands it weakens and eventually degenerates into a sound wave, which no longer significantly affects the properties of the m edium th ro u g h which it travels.
Interferom etric m ethods an d other experim ental techniques such as tra c er m o tion analysis can q u antitativ ely reveal changes in th e density of a gas th ro u g h o u t re gions w here it varies sm oothly, b u t cannot always do so across a near-discontinuity such as a shock. T he ra tio of th e therm odynam ic p ro p erties of th e m edium on th e two sides of a shock, however, can be derived analytically from the basic
conser-vation laws. T he resulting relations are fundam ental to th e experim ental stud y of shock waves.
2 .2
T h e R a n k in e -H u g o n io t e q u a tio n s
S ta rtin g from th e continuity, m om entum and energy equations for a compressible, inviscid and calorically perfect fluid, R ankine (1870) an d H ugoniot (1887) derived th e relations betw een th e gas p ro p erties on th e two sides of a normal shock— -a shock th a t changes th e p rop erties of th e flow in one co-ordinate direction only. Depending on th e p a ra m ete r in term s of which the ratios are expressed, these equations may take different b u t equivalent form s. T he appellation of Rankine-Hugoniot equations is often given to th e equations expressing th e ratio s across th e shock as a function of th e u p stre am M ach num ber, m easured in a fram e of reference where th e shock is at rest. To m ain tain generality, th e subscripts a and /? will be used in th is section to denote th e conditions ahead of and behind a shock. N um eric subscripts will be reserved for th e conditions in specific regions of a flow field.
T h e u p stre am M ach num b er M a is defined as th e ra tio of th e flow speed u a to th e local speed of sound aa ahead of th e shock wave (for an ideal gas, aQ = y/'jRT a , w here R is th e universal gas constant). T h e R ankine-H ugoniot equations for pressure, density, te m p e ratu re an d M ach num ber are:
I = l +
(2-D
a, _ <7+
m i
Pa T„ M jMl
i M l - (7 - l ) / 2 ’
w here 7 = cp/ c v is th e ra tio of specific heats for th e gas. A direct consequence of th e continuity equation is th a t th e ra tio of flow velocity across a norm al shock is th e reciprocal of expression 2.2.
2 + (7 - 1)MJ
f l + 27
7 +
1 + (7 - l )/2
2 + ( 7 - l ) A £ '
(7 + l ) M 2 (2.3)SHOCK F R ON T
Figure 2.1: The geom etry of the flow th ro u g h an oblique shock in th e fram e of reference in which th e shock is a t rest.
A norm al shock is a special case of th e family of oblique shock waves th a t occur in supersonic flows. Seen in a reference fram e w here it is a t rest, an oblique shock form s an angle (f> w ith respect to th e flow velocity vector u a u p stre am of it (Fig. 2.1). For a norm al shock, is equal to 7r / 2. A n analysis of th e continuity, m om entum and energy equations for th e com ponents of flow tan g en tial and norm al to th e shock front (A nderson, 1982) reveals th a t the tan g en tial com ponent is preserved across an oblique shock, an d th e norm al com ponent com pletely determ ines th e dow nstream flow properties. T a e R ankine-H ugoniot e q u a tio rs can therefore be applied to oblique shocks sim ply bv replacing M a in expressions 2.1 to 2.3 with its com ponent n o rm al to th e shock, M Qn — M a sin (f>. In equation 2.4, th e dow nstream M ach num ber Mp is also replaced by its norm al com ponent Mpn = Mp sm(<f) — 8). The reciprocal c f expression 2.2 in its new form gives th e ra tio of th e norm al com ponent of flow velocity across the shock. From th is ra tio a n d th e fact th a t th e tan gen tial com ponent of velocity is preserved, th e following relation for th e angle of flow deflection 8 can be derived:
M 2 sin2 <f> — \
ta n # = 2 — —^ --- — -cot<£. (2 .5) + cos 2<f>) + 2 Y v J
For a given M ach num ber, th e re is a m axim um deflection angle #max for which equation 2.5 has a solution. If th e flow geom etry is such th a t 9 > 0max th en no solution exists for a stra ig h t oblique shock. For 9 < 9m&x there are two values for the shock angle th a t will satisfy th e equation, the larger being called the strong shock solution and th e sm aller th e weak shock solution. T he weak shock solution is th e one th a t usually occurs in n atu re.
2 .3
O b liq u e s h o c k w a v e r e fle c tio n
C onsider an incident shock wave i of M ach num ber M, striking a rigid wedge of angle 0W as shown in Figure 2.2(a). This shock wave induces a net flow in the gas w ith a com ponent tow ard th e surface of the wedge. In order to satisfy the b o u n d ary condition th a t requires th e flow adjacent to a rigid surface to be parallel to it, a reflected shock r m ust develop which has th e correct stren g th and geom etry to b rin g th e flow to th e pro p er o rientation. This shock configuration, in which the reflection po in t lies a t th e surface of th e wedge, is known as regular reflection. To analyse the deflection angles, th e shocks are considered straig h t in proxim ity of the reflection p o in t G an d th e flow is assum ed to be self-similar (von N eum ann, 1943), th a t is, to become pseu do -statio n ary in a co-ordinate system ( x / t , y / t ) fixed to poin t G (Jones et al., 1951). In this fram e of reference (Fig. 2.2(b)) th ere is a stead y flow of M ach num ber Mo = Mi secdw, directed down th e wedge surface, which encounters a statio n ary oblique shock i at an angle (f>\ = n / 2 — 9W and is deflected by an angle 9i determ ined by equation 2.5. T he properties of th e flow in region 1, including its Mach num ber M i, are given by th e R ankine-H ugoniot equations for oblique shocks. T h e flow th en encounters th e reflected shock r and is deflected by 92. C learly 92 m u st equal 9\ in order to bring the flow to be once again parallel to th e wedge surface. T his determ ines th e shock angle <j>2 through eq u atio n 2.5 a n d hence com pletely defines th e shock geom etry an d ultim ately the p ro p erties of th e flow in region 2. In regular reflection, therefore, th ere are three
Figure 2.2: R egular reflection of a plane shock off a plane wedge of angle 8W: (a) lab fram e and (b) pseu d o -statio nary fram e of reference.
d istin ct regions having different therm odynam ic properties.
T h e above situ atio n fails if th e deflection angle 92 exceeds th e m axim um deflec tion angle 9miiX for w hich equation 2.5 has a solution at M ach num ber M i . This happens w hen th e shock strikes a wedge of sufficiently sm all angle Bw. In th a t case regular reflection cannot occur a n d a new configuration called Mach reflection is form ed (Fig. 2.3(a)), in which a shock m known as th e M ach stem connects the wedge surface to a trip le point T where the incident shock i an d the reflected shock r m eet. C onsidering th e shocks straight near th e triple point and assum ing self sim ilarity as before, th e flow is m ade pseudo-stationary by going to a co-ordinate system ( x / t , y / t ) a tta ch e d to p o in t T. In this fram e of reference (Fig. 2.3(b)) there is an incident flow of M ach num ber Mo = Mi sec (9W + x ), a portio n of which is deflected by th e statio n ary oblique shocks i and r th ro u g h angles 9\ and d2 in suc cession. T h e p o rtio n of th e incident flow encountering the statio n ary M ach stem m is deflected th ro u g h an angle 03. N ear the triple point th e b o u n d ary condition is th a t 03 = 9i — 92, which im plies th a t th e flows in regions 2 and 3 are parallel b u t n o t necessarily of th e sam e speed. A contact surface o r slipstream s separates the two regions. Across this b o u n d ary th ere is no change in pressure b u t th ere are changes in density, te m p e ratu re and entropy. M ach reflection therefore form s four regions of d istin ct th erm o d y n am ic properties.
A shock configuration th a t is tru ly self-similar expands in tim e according to a purely linear law, as if it was being photographically enlarged. T he experim ental stu d y of such phenom ena only requires th a t th e p ro p erties of the flow field be m easured a t one in s ta n t in tim e. Shock reflection off a single wedge, w hether regular or M ach, generally satisfies this condition a t least in th e proxim ity of the reflection p o in t o r trip le point. T h ere are types of shock wave phenom ena, however, where th e flow field is no t self-similar. As an exam ple, if an incident shock travels along a wedge whose angle changes, eith er ab ru p tly or gradually, th e reflection geom etry is continuously affected by a ‘m em ory effect’ of its p a st history th a t does
( 2 ) / W
m
/ s
(b )
Figure 2.3: M ach reflection of a plane shock off a p lane wedge of angle 0W: (a) lab fram e and (b) pseudo-stationary fram e of reference.
not become p seu d o -statio n ary in a tim e-dependent reference fram e ( x / t , y / t ) .
2 .4
D e s c r ip t io n o f t h e sh o c k t u b e
T he flows to b e stu d ied were generated in a shock tu b e (W h itten , 1969) w ith internal cross-sectional dim ensions of 7.65 cm w idth by 25.4 cm height. T he shock tu b e consisted of a com pression cham ber 1.04 m long th a t could b e filled w ith air pressurized u p to 6 atm ospheres, and an expansion cham ber 7.01 in long open to atm osphere. T h e air m th e com pression cham ber was confined by an acetate d iaphragm which could be b u rst by a needle driven by a solenoid. T h e breaking of th e d iap h rag m caused a shock wave to propagate down th e expansion section, and by th e tim e it reached th e test area in th e far end of ihe tu b e the shock front was p la n a r for all p ractical purposes. Two pressure tran sd u cers along th e expansion cham ber allowed the shock velocity to be m easured s " d also supplied a synchronization signal to th e laser system used to photographically record th e phenom enon.
T h e observation section n ear th e end of the shock tu b e consisted of two thick optical-qualitv windows m ounted flush w ith th e inner surfaces of th e tu b e sides. These windows allowed visibility of th e full height of th e tu b e cross section for a leng th of a b o u t 30 cm. A m odel off which th e incident shock would reflect, generally a single- or m ultiple-sloped stainless steel wedge bolted to th e shock tu b e floor, could be m o u n ted betw een the windows. T he m odel was uniform in shape across th e w id th of th e shock tube. For all practical purposes, the flow field in th e test section could be considered purely two-dim ensional in a plane parallel to th e windows.
2 .5
V is u a liz a tio n m e t h o d s
2 .5 .1 Shadow and schlieren p h otograp h y
T h e study of shock waves and th eir interactions requires th a t th e geom etry of the shock fronts b e known as a reflection evolves. T h e optical techniques of shadow gra- phy and schlieren photography visualize th e shocks as sh arp lines on a contrasting background, revealing th eir sh ap e over th e en tire field of view. Strong vortices a n d contact surfaces betw een regions of th e gas having different densities are also identified by these m ethods.
T h e sim plest form of shock visualization is th e direct contact shadow graph, in which a photographic film is directly illum inated by a beam of parallel light th a t traverses th e phenom enon. For tw o-dim ensional flow fields, th e illum ination is p erp en d icu lar to th e plane of th e flow. T h e light source gives a pulse short enough to freeze shock m otion. Because th e light rays are deflected by th e strong refractive index gradient at th e shock fronts, th e am o u n t of light falling on the film is reduced in th e region directly corresponding to a shock and increased in an adjacent region. Shock fronts ap p ear therefore on th e photographic record as th in double im ages opposite in exposure. It m ay be shown (see G oldstein, 1970) th a t th e change in intensity a t a point on th e im age plane relative to the intensity at th e corresponding p o in t of th e test section is p ro p o rtio n al to th e second p a rtia l derivative of refractive index w ith respect to distance in th e plane perpendicular to th e axis of illum ination.
Schlieren is sim ilar in principle to shadow graphy in th a t it is based on the deflection ox originally parallel light rays due to refractive index gradients. In a schlieren system , however, the light having traversed th e phenom enon is bro u g h t to a focal point before being im aged onto th e film. A knife edge precisely positioned at the focal point cuts off any light ray th a t has been deviated from its norm al p a th in such a v/ay as to be intercepted. Because these deflected light rays are com pletely
rem oved from th e im age, schlieren photography visualizes refractive index changes w ith g reater c o n trast th a n shadowgraphy. T he change in intensity a t a point on the im age plane relative to th e intensity a t the corresponding point of th e test section is p ro p o rtio n al to th e first p a rtia l derivative of refractive index w ith respect to distance in th e direction perpendicular to the axis of illum ination and th e knife edge (see G oldstein, 1970). For certain applications it is advantageous to be able to control th e directional sensitivity of th e system by th e orien tatio n of th e knife edge. If a uniform sensitivity is required in the plane p erp end icu lar to th e axis of illum ination, a narrow circular a p ertu re centered a t th e focal p o in t is used in place of th e knife edge.
2.5 .2 In terfero m etry
W hereas th e previously m entioned m ethods are only su itab le for th e visualization of shock fronts an d o th e r areas of large gradients in th e refractive index of th e gas, interferom etry gives a m uch g reater am ount of inform ation ab o u t the refractive index, and u ltim ately th e density, of th e gas a t every location in th e test field. T his technique depends on th e change in optical p a th len g th th ro u g h th e m edium as a function of refractive index, ra th e r th a n on the deflection of light rays. In fact, refraction effects are expected to be negligible in order for an interferogram to be accurate.
U ntil th e advent of holography, th e M ach-Zehnder interferom eter was th e m ost com m on typ e o f interferom etric system used for shock studies (H all, 1954). This device uses a single plane wavefront of m onochrom atic light w hich is split into two beam s by a beam splitter. T he two beam s are reflected by m irrors along sep arate p a th s, or arm s, and eventually recom bined by a second beam sp litter into a single beam th a t is im aged onto the film. By accu rate a d ju stm en t of the optical com ponents, it is possible to m ake th e optical p a th length along the two arm s exactly identical over the entire area of the beam , in which case the o u tp u t
beam will be uniform ly bright since th e two beam s will interfere constructively everywhere. If one of th e arm s passes through th e observation section containing th e phenom enon, th e n any change in refractive index occurring in th is section since th e interferom eter was aligned will cause th e form ation of interference fringes on th e uniform background of th e o u tp u t beam . As it will b e seen later, these fringes describe contours of constant refractive index and can be used to o b ta in a density m ap of th e gas. A lternatively, by slightly tilting one of the beam sp litters a p a tte rn of uniform , straig h t background fringes m ay be overlaid on th e o u tp u t beam w hen th e field is u n d istu rb ed . In th e presence of a d istu rb an ce these fringes will shift locally, th e am o u n t of shift being proportional to th e change in refractive index.
B ecause of th e requirem ent th a t th e optical p a th length along the two arm s be equal everywhere, the M ach-Zehnder interferom eter is relatively difficult to align an d requires optical com ponents of th e highest quality th ro u g h o u t th e system , including th e windows of th e observation section. T h e advent of holographic in ter ferom etry has provided a m uch m ore practical an d equally effective alternafciv. to this m ethod.
C h a p te r 3
H o lo g r a p h y
3.1
I n tr o d u c t io n t o t h e te c h n iq u e
Holography is an optical m eth o d th a t allows a wavefront of light to be stored on a photosensitive m edium and subsequently reconstructed. T h e wavefront m ay be represented as a complex quantity, having a t any point in space an am plitude and a phase. P h o to g rap hic em ulsions, being sensitive to irradiance, can only record th e real am p litu d e of th e light, w hilst the phase inform ation is irrevocably lost. T h e holographic process expands th e capabilities of th e photographic m edium by com bining th e wavefront to be recorded w ith a plane wave from th e sam e coherent source, and sto rin g th e resulting interference p a tte rn in a high-resolution emulsion. T h e reference wave effectively encodes in the stored p a tte rn th e local phase in form ation of th e object wavefront as well as its am plitude. In th e reconstruction process, a p lan e wave w ith th e sam e orien tatio n and w avelength as th e original reference wave is used fo illu m ira te th e developed emulsion. T he p a tte rn stored in th e emulsion, spatially m odulates the reconstruction wave so th a t th e resulting light is a replica of th e two waves originally interfering when the hologram was recorded. T h e object wave is therefore re stitu te d in its en tirety w hen th e hologram is reconstructed.
m onochrom atic light of very short coherence length, and h ad the disadvantage of having to be viewed by staring directly into th e reconstructing beam . T he holo graphic technique tru ly cam e of age w ith th e discovery of th e laser an d th e invention of off-axis holography (L eith and U patnieks, 1962, 1963, 1964), in w hich th e ref erence beam impinges on th e film from a different direction th a n th e object wave. T h e recon stru cted wavefront is then angularly sep arated from th e reconstructing beam a n d can be observed w ithout difficulty. This is by far th e m ost commonly used kind of holography, a n d th e basis for num erous variations.
3 .2
H o lo g r a m fo r m a tio n e q u a tio n s
In later p a rts of th is dissertation some aspects of th e recording technique used in th e p resent work will be analysed theoretically It is therefore useful to lay the groundw ork by briefly describing th e hologram form ation process from a m ath e m atical stan d p o in t. T he n o tatio n th a t will be followed here is th e one used in the classical book on holographic interferom etry by Vest (1979).
In th e recording of an off-axis hologram , as previously m entioned, th e referer j wave p ro p ag ates in a direction different from th a t of th e object wave. P o stu late a plane reference wave whose propagation vector is parallel to th e y-z plane and form s an angle 0R w ith th e norm al to the film plane z = 0. T he com plex am plitude (representing real am p litu d e and phase) of this wave a t z = 0 is
Uj*(®,y) = aR exp(i2irfyy), (3 .1)
w here f y = sin 0R/A is th e spatial frequency of th e reference wave An object wave U 0(.t, y) coherent w ith th e reference wave also im pinges on th e film. W hen th e two waves interfere at th e film plane, th e reference wave m ay be regarded as producing a set of “carrie r” fringes of sp atial frequency f y th a t is m o d ulated by th e object wave to yield th e interference p a tte rn con stitu tin g th e hologram . T his in te rp re ta tio n (L eith a n d U patnieks, 1962) is an exact analogy to o rd in ary com m unication theory,
in w hich a signal is encoded by m odulating a temporal carrier wave. T he irradiance at th e film plane from th e two waves combined is
I ( x ,y ) = |U c + U r | 2
= jU D + aR exp(i27r/j,y)|2
= |U 0|2 + aR + aRU 0 ex.p(—i 2 n f yy) + aR U* exp(i2irfyy). (3.2) A fter having been exposed to th e irradiance p a tte rn an d developed th e film has an am plitude tra n sm itta n ce t( x ,y ) th a t is proportional to J(x ,y ):
t ( x , y ) = t b + 0 [|U 0j2 + aR U 0e x p (-t2 T r/yy) + aRU* exp(i27r/vy)j , (3.3)
w here the con stan t te rm h as been absorbed insida tj. It is instru ctiv e to express U 0(a:,y) as a function of real am plitude and phase:
U 0(x, y) = a0(x, y) exp [~i<j>0{x, y ) ] , (3.4) and su b stitu te th e above into equation 3.3 to obtain, after com bining th e exponen tials,
t(x , y) = t b + 0 a 2o(x, y) + 2/3aRa0( x , y ) cos [27r/yy + <j>0(x, y)\ . (3.5) E qu atio n 3.5 clearly shows th a t th e holographic recording iu the em ulsion consists of a set of fringes of sp atial frequency f y m o d u lated in am p litu d e by a0(x, y) and in p h ase by <j>0(x,y ).
For reconstruction, th e hologram is illum inated by a plane wave having the sam e sp atial frequency as th e reference wave used in th e recording:
U c(at, y) = ac exp(i27r/yy). (3.6) T he resulting com plex am p litud e of th e light em erging th ro u g h th e film, in the vicinity of the hologram plane, is given by t(» , y )U c(a;, y):
T h e second term in th e above expression is th e only one of relevance for m ost applications, since it represents a diffracted wave which is a replica of th e original o bject wave. T he irradiance of this reconstructed object wave is
I ol{ x ,y ) = t32a2ca2R |U C|2 . (3.8) T h e o th e r two term s in equation 3.7 describe ad ditio n al reco n stru cted waves. T h e first te rm represents a p o rtio n of th e reconstruction wave which is tra n sm itte d by th e hologram w ith some irradiance m odulation, w hilst th e th ird te rm represents a wave which is th e conjugate of th e original object wave. All th e reco n stru cted waves pre angularly sep arated , an d th e conjugate wave is generally suppressed by diffraction effects in th e em ulsion unless th e angle O r is small.
3 .3
H o lo g r a p h ic in te r fe r o m e tr y
3 .3 .1 P rin cip les and ap p lication
T h ro u g h th e use of off-axis holography, it is possible to produce images overlaid by a p a tte rn of interference fringes th a t reveal changes in th e s ta te of th e object being holographed. These changes m ay consist of deform ation, displacem ent or ro ta tio n in th e case of an opaque, diffusely reflecting object, or variations in thickness or refractive index in th e case of a tra n sp aren t object. This ty p e of interferom etry is achieved by recording holographically th e wavefront of light from th e object at a given tim e an d com paring it interferom etrically w ith eith er th e direct wavefront from th e ob ject a t a la te r tim e or a holographic recording of it. T h e interferom etric com parison sim ply consists of reconstructing the hologram or hologram s in such a way th a t th e two wavefronts are in perfect sp atial reg istratio n and observing or p h o tog raph in g th e com bined wave. A m ajo r advantage of holographic interferom e try over conventional (M ach-Zehnder) interferom etry is th a t th e reference arm an d the te st arm a re spatially coincident b u t tem porally sep arated , so th a t p a th length
differences a re only in tro du ced by the tim e-varying phenom enon un d er study and n o t by characteristics of th e optical equipm ent.
T here are m any variations in th e technique of holographic interferom etry which are extensively described in th e aforem entioned book by Vest. Here th e atten tio n is lim ited to th e specific m eth od used in the present work, th a t is, two-exposure holographic interferom etry. M oreover, th e only case considered is w here th e object b eam is form ed by a plane wave propagating th ro u g h a refractionless tran sp aren t m edium (also known as a phase object). This situ atio n closely corresponds to the a ctu al experim ental conditions in our optical system : th e collim ated object beam is norm al to th e window surfaces in the test section, an d except a t the shock fronts th e refraction of the beam traversing the test volum e of gas can be shown to be negligible (van N etten, 1988).
In tw o-exposure holographic interferom etry, two successive recordings of the sam e object are m ade on the sam e film or plate, so th a t up o n reconstruction the wavefronts corresponding to th e two exposures interfere w ith each o th er and reveal changes in optical p a th length as a p a tte rn of fringes. T h e two exposures usually record a reference condition an d an altered state due to some physical phenom enon. For some studies it m ay be useful to record two different phases of an evolving phenom enon, in which case th e interferogram shows th e change betw een them (differential holographic interferom etry). In the present application, one hologram was recorded when no shock was present in th e test section a n d the o th er during shock passage. For an object b eam pro p ag atin g in th e 2 direction and traversing a p h ase object having refractive index d istrib u tio n n(x , y, 2), th e optical p a th length $ th ro u g h th e m edium is
$ ( x , y ) =
j
n ( x , y , z ) d z . (3.9) If th e refractive index d istrib u tio n is n i(x ,y , 2) du rin g th e first holographic expo sure a n d n 2( x ,y , 2) d u rin g th e second, th e object waves recorded on the film andth e n jointly reconstructed are .27r [ 2t t i — $ i ( x , y ) (3.10) an d U o2 = a 2(x ,y )e x p i ' ^ $ 2(:r,y) (3.11) A
w here and $ 2 are given by equation 3.9. A ssum ing for sim plicity th a t a x and a 2 in the reco n stru cted object waves are uniform , u n it am plitudes, th e irradiance I ( x , y ) = |U 0i + U o2|2 of the com bined reconstructed waveforms in th e im age plane m ay be expressed as
I ( x , y ) = 2 j l + cos ^ [$ 2(a\ y) - $ T(a r,y )]|
= 2 1 1 -f cos ^ A $ ( x , y ) j | . (3.12) In m ost applications th e refractive index during one of th e exposures, say th e first, is uniform a n d can be denoted by no. T h en th e optical p a th len g th difference betw een exposures is
A $ (x , y) =
J
[n(x, y, z ) - n 0] dz. (3.13) In th e experim ental configuration used in this work, the reflection of a plane shock wave m ay be tre a te d as a strictly tw o-dim ensional phenom enon in th e plane n o r m al to beam direction. T he refractive index d istrib u tio n induced by the shock is therefore a function of x and y only, and th e expression for th e optical p a th length difference is reduced toA $ ( x ,y ) = [n (z ,y ) - n 0] £ , (3.14) w here L is th e distance th a t th e light m ust travel in th e te st section.
A n im p o rtan t p a ra m ete r in th e recording of hologram s is th e ra tio of reference beam to object beam brightness, which affects th e diffraction efficiency for recon stru ctio n . T h e nonlinear response characteristics of th e em ulsion m ake it virtually
im possible to establish a clear cut value th a t will give best results under all cir cum stances. R eference-to-object-beam ratios of 3:1 to 10:1 are often suggested in ord er to keep th e m odu latio n w ith in th e linear p a rt of th e response curve. For holographic interferom etry, in which th e u ltim ate m otive is to produce interference fringes of high visibility in th e reconstructed image, a reference-to-object-beam ra tio of 1:1 is recom m ended (Vest, 1979) on th e basis of experim ental evidence, th o u g h th e ra tio can be higher if o th er conditions d ictate it.
3 .3 .2 Fringe in terp reta tio n
A ccording to equation 3.12, b rig h t fringes will form a t those positions where A $ = N X , o r su b stitu tin g 3.14,
[n(x,y) — n 0] L = NX. (3.15)
These infinite fringes describe contours of constant refractive index. Applying equation 3.15 a t any two points on th e reconstructed im age which lie on bright fringes, we o b ta in by su b tractio n a form ula relating th e difference in refractive index to th e fringe count N\ - 2 betw een th e two points:
A n L = iVj-2 A. (3.16)
T h e refractive index th ro u g h o u t th e field is m ap p ed by giving order num bers to th e fringes, assigning th e num b er N = 0 to a brig h t fringe in an area w here the refractive index n'0 is known. T h e centers of all subsequent brig h t fringes arc assigned num bers N = 1 , 2 , 3 , . . . consecutively, an d th e centres of th e interspacing d ark fringes axe assigned num bers N = 0 .5 ,1 .5,2 .5 , etc. At any location the refractive index is th en given by
N X
n = n'0 + ~ . (3.17)
B ecause changes in optical p a th length of — A $ an d A $ yield identical fringe p a t tern s, th ere is a sign am biguity in fringe order num bers. G enerally the experim enter
has sufficient knowledge of th e flow field to infer th e ap p ro p riate sign. If this is no t th e case, it is possible to resolve th e am biguity by intro d u cin g finite fringes in th e interferogram th ro u g h a sm all tilt of th e reference beam angle betw een th e two exposures (see Vest, 1979). F in ite fringes reveal changes in refractive index by th eir local displacem c.it from a regular straig h t line p a tte rn , a n d allow q u a n tita tiv e m ea surem ents to be m ade w ith no doubt ab o u t th e sign of th e change. O n th e other h an d , they yield a m uch less readily u n d erstan d ab le m ap of th e refractive index d istrib u tio n th a n infinite fringes. Only infinite fringe interferom etry was involved in th e work presented here.
From th e refractive index, the density of a gas m ay be o b tain ed using the Gladstone-Dale equation:
n — 1 = K p , (3.18)
w here K , th e G ladstone-D ale constant, is a p ro p e rty of th e gas. It is a weak function of rhe recording w avelength and is nearly ind ep en d en t of te m p e ratu re and pressu re un d er m oderate physical conditions. Com bining th e above w ith equation 3.17 gives th e following expression for th e density along a fringe of order N:
/ N X
P = P o + K L ’ (3 ' 19)
w here p'Q is th e density at the zeroth order fringe.
3 .4
T h e e x is t in g h o lo g r a p h ic in te r f e r o m e te r
T h e original optical system th a t was used as foun d ation for th e p resen t work is shown in Figure 3 .i. A thorough description of th e equipm ent is given by van Net- ten (1988), a n d only an overview will be presented here.
T h e source of light for th e holographic system was a ru b y laser. T h e laser cavity was bounded by two plane dichroic m irrors, having reflectances of 100% (back m irro r) an d 60% (front m irror) a t th e ru b y em ission w avelength of 694.3 nm . T h e lasing elem ent was a ruby ro d surrounded by a helical flashtube th a t
parobolic m irror
re fe re n c e
b ea m / o b je c tbeam
shock tu b e (en d sec tio n ) film
ruby laser
Figure 3.1: O riginal layout of th e holographic interferom eter used for producing infinite fringe interferogram s of shock waves.
provided th e optical pum ping for approxim ately 1 ms. T h e cavity also contained a Q -spoiling system com posed of an air-spaced polarizer and a transverse Pockels cell configured as a sw itchable quarter-w ave plate. T h e lasing action w ithin the p um p in g period could be enabled or inhibited by m odulation of th e Q-switch, generating tra in s of light pulses. To o b tain a p u re G aussian beam suitable for holographic applications, a 2 m m ap ertu re coaxial w ith th e rub y ro d was placed in th e cavity. By introducing large diffraction losses, this a p e rtu re purified the fu n d am en tal T E M 0o m ode of oscillation from higher transverse m odes th a t could be su stain ed in th e 10 m m diam eter ruby rod (B arnes, 1970).
T h e light from th e laser was divided in to two equal portio n s by a 50% prim ary beam sp litter. T h e reflected p o rtio n was expanded by a diverging lens so th a t it filled a p arabolic m irro r 30.5 cm in diam eter, in a slightly off-axis configuration.
T h e focal lengths an d spacing of lens an d m irro r were such th a t th e wide beam leaving the la tte r was parallel. T he light th en traversed th e test section of th e shock tu b e (see Section 2.4) a t norm al incidence to th e windows, an d was folded back onto itself by a plane stainless steel m irror located b eh in d th e back window. T h e fact th a t th e light travelled twice thro u g h th e test volum e classifies th is as a double-pass design. T he reflected light retraced its p a th to th e parabolic m irror and thence converged tow ard th e expanding lens. A secondary beam sp litte r deflected 50% of this re tu rn beam into a lens system th a t im aged it onto a segm ent of 35-m m holographic film held in a stretch m ount. N ote th a t this beam sp litte r unavoidably w asted h alf of th e outgoing beam on its way to th e parabolic m irro r by deflecting it ou t of th e a p p aratu s. Assum ing negligible losses in th e o th er p a rts of th e object beam p a th , therefore, 25% of th e light reflected by th e p rim ary beam s p litte r reached th e film.
T h e p o rtio n of laser light tra n sm itte d th ro ug h th e p rim ary beam sp litte r form ed the reference beam of th e holographic system . It was ro u ted v ia th ree m irrors along a p a th whose length m atch ed th a t of the object beam „o w ithin th e coherence length of the laser (several centim etres), a necessary condition for th e form ation of hologram s. Between the second an d th e th ird m irror, which directed th e light onto th e film from an angle of ab o u t 30° to th e norm al, was placed a diverging lens th a t expanded th e beam so th a t it covered th e whole im age area. Because of the 4:1 power im balance betw een th e two beam s due to the presence of th e secondary beam sp litter on the object beam p a th , th e local brightness ra tio a t th e film was reduced to th e desired value of ab o u t 1:1 by m aking th e size of th e reference beam projection p ro p o rtio n ately larger (van N etten, 1989).
H olographic interferogram s were recorded w ith two light pulses creat d by Q- sw itching th e laser cavity twice during a single firing of th e flashtube. Laser firing was triggered by th e passage of th e shock wave over a pressure tra n sd u ce r in th e wall of th e shock tu b e, th ro u g h a n ad ju stab le electronic delay. T h e triggering delay and
pulse spacing were set so th a t th e first hologram would record th e u n d istu rb ed field ju st before shock wave arrival in th e test section, and th e second would cap ture the shock in th e desired position. Typical intervals betw een th e two exposures were in th e ord er of 100 to 500 ps. T h e double-pass design of th is interferom etric system effectively doubled its sensitivity w ith respect to a single-pass system , since the density gradient per fringe is inversely proportional to th e distance th a t the light m u st travel in th e te st section (equation 3.19).
Because th e object beam pro jected a focused im age on the film, each point of th e im aged object corresponded uniquely to a po int on th e em ulsion, and the reco n stru cted holographic im age also appeared a t th e film plane. U nder such con ditions, a hologram can be reconstructed w ith a light source of wavelength different from th a t of th e recording laser w ithout introducing any disto rtio n s in the recon stru c ted im age or th e interferom etric fringes. T he expanded beam from a low-power Helium -N eon laser, having a w avelength of 632.8 nm , was generally used for the reco n stru ctio n an d photo g rap h in g of th e hologram s, b u t th e w hite light from a slide p ro jecto r was also a convenient an d equally effective reconstruction source. D etails of th e optical set-up for photographing th e reco n stru cted images are given by van N etten (1988).
C h a p te r 4
T im e -r e so lv e d h o lo g ra p h y
4 .1
O b je c t iv e s
H olographic interferom etry has assum ed a position of im p o rtan ce as a tool for the stu d y of com pressible flows, b u t it still widely lacks one feature th a t m ost other m ethods can offer: th e ability to record sequences of im ages in order to follow th e evolution of a phenom enon in tim e. One way to achieve tim e-resolved recordings is a hy brid m eth o d based on real-tim e holographic interferom etry. A single hologram of th e u n d istu rb e d field is exposed, developed an d replaced in th e exact position w here it was recorded (or even developed in situ). T h e original reference beam then works as a reco n stru ctio n beam and creates a replica of th e object wave u n d er no flow conditions. D uring th e phenom enon, the direct beam from th e optical system and the reco n stru cted object beam from the hologram in teract w ith each oth er to give a real-tim e interferogram . T he interference p a tte rn m ay be recorded by conventional high-speed cinem atography. T his technique, aside from its technical complexity, requires th a t the reference exposure be taken several m inutes before th e phenom enon. E xtraneous changes to the field m ay easily occur du rin g such a long gap.
T h e goal of th e p ro ject described in th is d issertation was to develop a tim e- resolved holographic interferom etric ap p aratu s th a t would be as simple to use as
th e existing system an d yield a series of individually reconstructible tw o-exposure hologram s th ro u g h a process of m ultiplexing. An additional requirem ent was th a t th e changes to th e original optical layout be kept to a m inim um , especially w ith regards to th e object beam p ath . T he optical equipm ent in th e Shock Studies L ab o rato ry h ad undergone an evolutionary process from a schlieren system to a holographic interferom etric system th a t had preserved all the original ch aracter istics, allowing th e form er m ode of operation to be easily restored if desired. It was hoped th a t th is tre n d could b e m ain tain ed in the upgrade to time-resolved operation.
4 .2
H o lo g r a m m u lt ip le x in g te c h n iq u e s
4 .2 .1 S p atial m u ltip lexin g
T h e m ajo rity of existing im plem entations of tim e-resolved holographic system s ad o p t in one form o r a n o th er th e technique of spatial multiplexing, w hereby each hologram is recorded on a sep arate area of film by successive laser pulses. Spatial m ultiplexing does n ot rely on o ptical properties unique to th e holographic recording process: ord in ary cinem atography is nothing b u t a non-holographic form of this technique. T h e recording of each hologram on a fresh area of photographic emulsion m ay be achieved by deflecting the light, by m oving th e film, or by a com bination of th e two m ethods.
T h e m a jo r challenge of the first approach is to m ake the object beam and th e reference beam coincide at th e film plane. However, if the object is diffusely illu m in ated th e object wave m ay b e allowed to fail on th e entire surface of the film w hilst th e reference beam illum inates different areas a t different tim es. G ates et al. (1968, 1970) d em o n strated a m ethod in which a scatter p la te was used to diffusely tran sillu m in ate th e scene whilst a p o rtio n of th e light was tra n sm itte d w ith o u t scatterin g th ro ug h th e p late and form ed the reference beam . Tim e-resolved
sequences were obtained by illum inating d ife re n t areas of th e sca tte r p late throu g h a ro ta tin g prism or n u ta tin g m irro r so th a t th e reference beam pro jectio n on the film moved w ith tim e. A m odified version of th e ir original scanning m eth o d (Hall et al., 1970) selected th e illu m in ated area on th e sca tte r p late by m eans of an a p e rtu re in a ro ta tin g disc placed in front of a wide beam . T h e second m ethod relaxed th e need for extrem ely short exposure tim es by effectively elim inating the m otion of th e beam during exposure. By synchronizing th e laser pulsing w ith th e angular position of th e scanning device, those au th o rs were able to ob tain tim e-resolved sequences of interferogram s by exposing each im age twice over two sep arate sweeps. M ore complex ro ta tin g m asks have been used b y workers such as D ubovik e t al. (1977) to generate differential interferogram s. Feldm an (1970) presented a sp atial m ultiplexing system using a solid-state acousto-optical deflector (see C h ap ter 7) to redirect th e light to different areas of film. For th e holography of tra n sp a re n t subjects, he used an arrangem ent th a t concurrently displaced b o th o b ject and reference beam s to th e sam e spot on th e emulsion to m inim ize dispersion of ob ject light. T hom as et al. (1972) used th ree sep arate lasers to im plem ent a high speed m ultiplexing system . T he light from each laser was divided into a reference b eam th a t reached a photographic p la te and an object beam th a t trav ersed the phenom enon along a p a th com m on to all the lasers. T h e com m on ob ject beam p a th was th e n split in to three branches th a t illum inated all th e plates, th e two additio n al ob ject beam s giving incoherent exposures th a t did not significantly affect the hologram .
System s achieving sp atial m ultiplexing by displacing th e film have th e po ten tial for recording very long sequences of fram es. A reel of film in a conventional movie cam era tra n sp o rt m ay be used in applications requiring an extended ru n len g th at m o d erate fram ing ra te s (Decker, 1982, Smigielski et al., 1985). For high-speed holo graphic cinem atography, a rapidly ro ta tin g holographic p late or a film m o u n ted on th e circum ference of a spinning d ru m have been used. H entschel a n d L a u terb o rn
(1985) com bined this technique w ith an acourto-optical beam deflection system to fu rth e r increase th e m ultiplexing ra te by recording q u ad ru p lets of fram es in a di rection p erpendicular to the film m otion. This com posite system achieved fram ing rates up to 300 kHz an d series lengths u p to ab o u t 4000 images. For holographic interferom etric applications, however, a m ethod relying on film displacem ent would not b e suitable because of th e g reat difficulty of exactly repositioning the moving m edium over successive sweeps in order to overlay th e two req u h ed exposures.
4 .2 .2 S p atial frequency m u ltip lex in g
T h ere is a second m eth o d of recording sequences of individually reconstructible hologram s, tak in g advantage of a unique pro p erty of th e hologram form ation pro cess. In C h ap ter 3 it was described how a hologram is recorded by m odulation of a set of “carrier” fringes having a specific spatial frequency. T h e analogy w ith com m unications theory m ay he carried fu rth er. In th e sam e way as com m unications signals having different carrier frequencies can be freely m ixed and still rem ain separable, so can hologram s w ith different “carrier” sp atial frequencies be su p er im posed w ith o u t losing individuality. This is the underlying principle of spatial frequency multiplexing. By L .v in g th e reference beam fall on the film from a dif ferent direction for each exposure, recordings of the object beam m ay be overlaid on th e same area of em ulsion a n d subsequently reco n stru cted individually. The selection is achieved by shining th e reconstruction beam a t th e sam e angle as the reference b eam used in the desired exposure. This hologram m ultiplexing tech nique offers several advantages. The object beam , w h eth er diffused or n o t, does not need to b e repositioned in any way for different exposures. T he size of the hologram s is n o t restricted by th e need to crowd spatially sep arated recordings in a given area of em ulsion as w ith m ost spatial m ultiplexing schemes. O bservation an d p h o to g rap h y of th e reco n stru cted hologram s is sim ple because th e film and th e eye or cam era m ay be placed in a fixed position an d only th e reconstruction
