Nonlinear effects in holography due to caustics

Volume 63, n u m b e r 3

OPTICS C O M M U N I C A T I O N S

1 August 1987

N O N L I N E A R EFFECTS IN H O L O G R A P H Y DUE TO CAUSTICS A SUBBARAO and A S PARASNIS l Department of Phystcs and Centrefor Laser Technology, Indtan Instttute of Technology Kanpur, Kanpur 208 016, India Received 22 September 1986, revised manuscript received 7 January 1987

Nonhnear effects, resulting in the appearance of higher order images on reconstructmn, which arise because of the enhanced lrradlance due to caustics at the holograpMc plane during recording are described and analyzed The possible connectton of the fringe patterns seen with catastrophe optics and wavefront dlslocaUons is pointed out

1. Introduction Holographic tmages are aberrated if the dynamic range of lrradlance falls outside the linear portion of the amplitude transmittance versus exposure curve of emulsions [ 1-4] In reconstruction this is manifested as extra images, autocorrelatlons, autoconvolutlons, ambiguity functions of the object and occasionally false tmages It is clear that in practice one should and does confine exposures to the linear portion In holography, lenses are often used for colhmatlon, beam expanslon, spatial filtering,etc Thelr mounts are usually blackened mslde Nevertheless some hght can reach the blackened surface at large enough angles so that thc reflected wave has sufficlcnt trradtance at the hologram plane, or light m a y be reflected from the rims of the lens components themselves Consequently, spurious wavefronts m a y reach the holographic emulslon This paper is a first report describing the effectsof such wavcfronts

oplng was in Kodak D- 19 (5 mln) and bleachmg was by acid dichromate solution The holographic set-up was originally for studying certain properttes of hquld crystals and other materials It was soon realized that the spurious effects occasionally seen were due to emulsmn nonlinearity and attention was focussed on them Thus no reference is being made herein to that study In order to have a suitable reference wave a well corrected projection lens was used after the corresponding spatim filter Eventually it was replaced by an FT lens but the spurious effects were still seen

3. Observations To start wlth, the lens(es) had been arbitrarily oriented, aligned/adjusted When spurious effects were noticed, adjustments were made so as to accentuate them 3 1 Causttcs

2. Experimental set-up Holograms were taken on Agfa-Gevaert 8E75 or 10E75 film or plates, using a 15 mW He-Ne laser, the entire setup being on Newport vtbratlon-lsolatlon table The nominal beam ratio was 2 DevelAll correspondence should he addressed to this author

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The &verging cone of light from the spatial filter, being wider than the input aperture of the colhmatmg lens, forms caustics (C) in the image space whose lrradIance distribution would depend upon the location and tilt of the lens, and reflectlvlty and finish of the inner surface of the lens barrel. They are superposed on the reference wave (R) passing through the lens By means of an ins diaphragm between the spa141

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i

Ftg 1 Ordinary enlargementof a hologram w~thcaustics (3 5 × ) tial filter a n d the lens the hght cone could be d e h m lted so as to avoid the caustics altogether By adjusting the lens the cusp of a caustic could be focussed at any desired plane If the cusp is faint it is not seen by the naked eye, yet its effects are seen in the hologram before and after bleanchlng We shall generally refer to the bleached hologram Fig 1 is an enlargement o f a caustic recorded on a hologram To bring out the radial e m a n a t m n s from the cusp more clearly laser shadowgraphs were taken (fig 2) In either there is no noticeable difference between holograms O + C + R and C + R though obviously there must be some at the micro-level A hologram is a negative of the wavefield so a print like fig 1 is a positive

Fig 2 Laser shadowgraph of a hologram with caustics

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Fig 3 (a) First order real ~mageformed when the readout beam passes through the far field of the cusp (b) Higher order images formed when the readout beam passes through the near field of the cusp (c) More higher order imagesformed when the readout beam passes through the very near field of the cusp (d) General blurring which occurs when the readout beam passes exactly through the cusp

3 2 Reconstruction of the obJect wave

Looking through the cusp a n d the area i m m e d i ately a r o u n d it, one sees multiple virtual images The n o r m a l first order ~mage alone ~s seen when looking through the regtons away from the cusp These effects are most strikingly seen in the real image W h e n the readout He-Ne laser beam passes through regmns far away from the cusp the normal first order real linage alone is seen (fig 3a), progressively more higher order images are seen (fig 3b, c) when the beam passes near the cusp and some twenty images are seen when it passes very close to the cusp (c f seven seen m ref [2]), when the beam passes through the cusp this n u m b e r increases indefimtely so there is general blurring (fig 3d) At the same t~me, on another screen placed normal to the readout b e a m are seen concentric circular bright rings which are very sharp and spaced widely apart, their n u m b e r being equal to that of the ~mages seen Indeed their appearance tells that the beam is passing through the cusp

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Fig 5 Photomicrograph of holographic fringes &splaying several scales of spatial frequency (91 × ) "dark" should be treated with care because of the well known Becke p h e n o m e n o n [ 5 ] in an optical microscope U n d e r a microscope the volume holographic fnnges give a lensllke effect so that " b n g h t " or "dark" appearance may depend upon focussing

Fig 4(a, b) Photomicrographs showing the fringe structure of fig 2 In (b) the microscope is focussed 14,um above the plane of(a) (53)<)

3 3 Structure of the hologram fringes The radial structure is seen more prominently after bleaching Microscopic examination enables one to study the fine structure o f the fringes having unusual patterns at and around the caustic The general structure at low magnification is closed loops o f fringes around the cusp, broken often on opposite sides by what m a y m analogy with ships be termed "wakes" (fig 4a, b, same cusp as fig 1, 2) The appearance varies through the depth in the emulsion but the layout remains the same The fringe loops around the caustic are broken also by the radial structure Where this perturbation IS not strong, there is still some modulation and at times the fringe shift is not uniform along a given flare (middle left, fig 4) A hologram IS the negative o f the wavefield and the photomicrograph is a positive, therefore the prints o f fig 4 are negatives Thus what appears bright in a print was dark in the actual wavefield at the hologram plane and vice versa The terms "bright" and

3 4 Structure ofthefrmges m thefarfield regtons The structure o f the fnnges far away (1-2 cm) from a cusp is basically the same, viz closed loops centred at the cusp The effect o f the cusp is seen less and less though the modulation is still noticeable Very far away, where there is presumably not much effect o f the cusp, the typical holographic fringes are seen, several scales of spatial frequency are dlscernable (fig 5) We have not found m the literature such an example though every hologram ever made must have such several scales o f frequencies

3 5 Structure of the cusp and wakes The appearance of the cusp at the hologram plane was like that of the cusp formed in a cup o f tea Depending upon the lens tilt the shape was curvihnear triangular (fig 4) or quadrilateral in outline, the dimension across being about 0 25 m m It is filled generally with hexagons closely packed together They are much larger than the grams o f the fine gram holographic emulsions, about 28000 n m versus about 40 n m Thus they must be part o f the interference pattern and contain information about the caustic 143

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surfaces and a b o u t the object when it is present This conclusion is the relevant point, not the exact shape o f the cusp region The structure o f the wakes is quite different from that o f the non-wake part a r o u n d the cusp The loop fringes seem to continue, albeit in a b r o k e n way, through the wakes They consist o f the hexagonal f o r m a t i o n as in the cusp So the wakes must also cont a m i n f o r m a t i o n about the caustic surfaces

4. Discussion

4 1 Multiple images Following Yu [3] we m a y express the t r a n s m i t tance o f the a m p l i t u d e hologram o f a l u m i n o u s p o i n t as a F o u r i e r sum o f an infinite n u m b e r o f slnusoidal Fresnel zone lenses

T ( l , k ) = T o + ~ Tn(l,k)

(1)

tl--I

( F o r the meanings o f the symbols see ref [3] ) The focal lengths o f the zone lenses are then given by f,=R/n, where R is the distance o f the p o i n t object from the emulsion Each lens gives its own real and virtual images on reconstruction Since each p o i n t o f the object would be similarly reconstructed the a p p e a r a n c e o f higher o r d e r images is not unusual in the present work W h a t is unusual is the n u m b e r o f images seen, viz over 20 against 7 o f ref [2] Even more unusual is the circumstance u n d e r which the nonlinear effects have arisen Clearly it IS the caustics which are responsible for multiple images (fig 3) as in their absence no n o n h n e a r l t y is encountered The irradmnce is so high at the cusp region that m a n y F o u r i e r coefficients o f eq (1) are large enough to reconstruct images, hence the fuzzy appearance (fig 3d) It then becomes necessary to caution all holographers and c o m p o n e n t designers to limit the cones o f light falling on the lenses so that caustics do not arise Suitable stops are all that is n e e d e d to e l i m i n a t e caustics Caustics can also arise from refractive index gradients (see the discussion later), however, today's high precision c o m p o n e n t s have no such p r o b l e m s Occasionally a hologram has given multiple images even m the far field regions o f the cusp, which indi144

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cates the strong irradxance o f caustics throughout the space o f which the hologram plane takes b u t a section The caution therefore, we believe, is meaningful

4 2 Far field regtons The cusp itself, if a p p r o x i m a t e d to a point, must arise as a result o f a focussed, a p p r o x i m a t e l y spherical wave Then the interference O + C + R (or C + R ) would result in concentric l o o p h k e fringes Even though the a s y m m e t r i c branches o f the caustic field spoil the fringe s y m m e t r y they still m a i n t a i n the closed loop a p p e a r a n c e (fig 4) At times a caustic field has more than one cusp, this is a situation which m u s t give rise to a p a t t e r n similar to the Young pattern ( m i d d l e left in fig 4)

4 3 Causttcs and wavefront dtslocattons Caustics have been analyzed by means o f variational methods, the first m a j o r a t t e m p t having been by Airy over a h u n d r e d years ago Nye, Berry a n d coworkers [ 6 - 1 2 ] have analyzed their intricate interference patterns in great detail by using T h o m ' s catastrophe theory, from this analysis has arisen C a t a s t r o p h e Optics Here we wish to p o i n t out a possible connection o f the present observations with catastrophe optics T h e i r e x p e r i m e n t is simple an He-Ne laser b e a m is passed through a d r o p l e t o f water on a dirty glass slide The wavefront is distorted by the irregular droplet which m a y have l n h o m o g e n e o u s internal pressure Intricate interference patterns are f o r m e d in the image space which basically are due to the caustic fields Caustics are images o f contours o f principal curvature The other concept to which we refer is that o f wavefront dislocations [ 10-12] A n u m b e r ( b u t not all) o f the morphological properties o f crystal dislocations are possessed by wavefront dislocations Berry, Nye, Wright et al have observed wavefront dislocations In the interference patterns o f water droplets a n d Berry [10] has p o i n t e d out that the concept is quite general, o f use in acoustics, water waves and q u a n t u m mechanics as well as optics They m a y be defined as follows a complex scalar function (here, an electromagnetic field) is expressed as

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q/(r, t)=p(r, t) e x p [ i z ( r , t)]

(2)

The contour surfaces o f the phase X are the wavefronts The most i m p o r t a n t features o f wavefronts are their singularities, which c o r r e s p o n d to singularitles o f the phase function z(r, t) T h e y occur where ~' = 0, this requires two conditions to be satisfied, vtz Re ~t = 0 and I m ~u= 0 So the singularittes are lines in space, or points in the plane [ 10] Nye a n d Berry [ 11 ] t e r m e d t h e m " w a v e f r o n t dislocations" The strength o f a wavefront dislocation is given by

1 Sc =~--~n~ d z c

(3)

Indeed, in fig 4b can already be seen some o f the features o f the p h o t o g r a p h s published by Berry, N y e et al These features are being analyzed

4 4 Caustics, catastrophes and holography F r o m our present p o i n t o f view what is i m p o r t a n t is that wavefields m u s t possess singularities, essentially phase singularities To study t h e m would require p h a s e - m a p p i n g T h e inability o f detectors to detect and hence m a p phase directly in itself makes holography indispensable in m a n y well k n o w n ways In this context we consider all holographic fringes on a hologram as p h a s e - m a p p i n g H o l o g r a p h y thus provldes a pre-reqmsIte to observing a n d studying wavefront dislocations The water d r o p l e t technique cannot legitimately be called p h a s e - m a p p i n g F u r ther, the very fine resolving p o w e r o f holographic emulsions o f t o d a y would enable far m o r e detailed studies o f fine features to be m a d e Catastrophe optical p h e n o m e n a are not confined to liquid droplet caustics Caustics are quite general and must occur in the very p h e n o m e n o n o f wave p r o p a g a t i o n for they are smgularttles o f gradient maps The relation o f caustics to diffraction a n d to wavefront dislocations is best u n d e r s t o o d In terms o f the scales o f wavelength If the wavelength is small c o m p a r e d to the variation scales o f objects/media the light field is d o m i n a t e d by caustics The existence o f fractals and dlffractals in an e n v i r o n m e n t precludes caustics f o r m a t i o n It ts when one handles the scale o f the wavelength itself that dtslocations a p p e a r as wavefront singularities [ 9 ] Since objects a n d m e d i a

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are not necessarily o f single "scale" only, we feel that, in general, all p h e n o m e n a might coexist In that case, p r o p e r p h a s e - m a p p i n g with a d e q u a t e resolution is even m o r e advantageous Work is proceeding in this direction as well as in the direction o f m a p p i n g caustic fields in a controlled way

5. Conclusions It is shown that caustics give rise to such effects as the a p p e a r a n c e o f higher o r d e r tmages in holography The a p p e a r a n c e o f higher o r d e r images is usual in n o n l i n e a r recording but the circumstance here is different, the high n o n h n e a n t y arising due to the high l r r a d i a n c e in a n d a r o u n d the caustics W h e n the caustics and reference wave are recorded without the object a c o m p h c a t e d fringe system is still seen though there is no object i n f o r m a t i o n in it A microscopic study o f it d e m o n s t r a t e s the existence o f a detailed structure o f the causttcs, viz the cusp, wakes, near a n d far fields o f the cusp and a radial pattern It is suggested that holography would thus be a powerful technique for studying singularitles in caustic patterns by virtue o f it being essentially high resolution p h a s e m a p p m g Equally well, wavefront dislocattons could be studied in greater detail This is the first time such an exotic use has been m a d e o f the holographic p n n c l p l e

References [ l ] J W Goodman and G R Kmght, J Opt Soc Am 58 (1968) 1276 [2] O Bryngdahl and A Lohmann, J Opt Soc Am 58 (1968) 1325 [3] F T S Yu, Optical reformation processing (Wdey-Intersoence, New York, 1983) Ch 11 [4] P Hanharan, Optical holography (Cambridge University Press, Cambridge, 1986) p 82 15] N H Hartshorne and A Stuart, Crystals and the polarizing microscope (Edward Arnold, London, 1970) p 258 [6] M V Berry, J F Nye and FJ Wright, Phd Trans R Soc Lond A 291 (1979) 453 [ 7 ] J F Nye, Optical caustics and &ffract~on catastrophes, m Structural stability m physics, eds W Gunmger and H Elkemoer (Sprlnger-Verlag, Berhn, 1979) p 54 [8] J F Nye, Proc R Soc Lond A 403 (1986) 1 [9] M V Berry and C Upstlll, Progress in optics, XIII (North145

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Holland Publshmg Company, Amsterdam, 1980) pp 257-346 [ 10] M V Berry, Singularities m waves and rays, m Physics of defects, Les Houches 1980, eds R Bahan et al (NorthHolland Pubhshmg Company, Amsterdam, 1981 ) p 453

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[ 11 ] J F Nye and M V Berry, Proc R Soc Lond A 336 (1974) 165 [ 12 ] F J Wright, Wavefront &slocatlons and their analysis using catastrophe theory, m Structural stabd~ty m physics, eds W Guttmger and H Elkemeler (Sprmger-Verlag, Berlin, 1979) p 141