Holographic particle image velocimetry using Bacteriorhodopsin

HOLOGRAPHIC

PARTICLE IMAGE VELOCIMETRY

USING

BACTERIORHODOPSIN

Wouter Koek February 2006

About this thesis 1 Holography 5

Bacteriorhodopsin 29 Volatility 51

Digital holography 79

Holographic particle image velocimetry 93

Holographic Particle Image Velocimetry

using Bacteriorhodopsin

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkema, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 13 februari 2006 om 10:30 uur

Dit proefschrift is goedgekeurd door de promotoren: Prof. dr. ir. J.J.M. Braat

Prof. dr. ir. J. Westerweel

Samenstelling promotiecommissie:

Rector Magnificus, voorzitter

Prof. dr. ir. J.J.M. Braat, Technische Universiteit Delft, promotor

Prof. dr. ir. J. Westerweel, Technische Universiteit Delft, promotor

Dr. N. Bhattacharya, Technische Universiteit Delft

Prof. dr. ir. P.G. Bakker, Technische Universiteit Delft

Prof. dr. ir. I.T. Young, Technische Universiteit Delft

Prof. dr. D.J. Broer, Technische Universiteit Eindhoven

Dr. J.M. Coupland, Loughborough University, United Kingdom

Prof. dr. ir. L.J. van Vliet, Technische Universiteit Delft, reservelid

Dr. N. Bhattacharya heeft als begeleidster in belangrijke mate aan de totstandkoming van het proefschrift bijgedragen.

This work is part of the research programme of the “Stichting voor Fundamenteel Onderzoek der Materie (FOM),” which is financially supported by the “Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO).”

ISBN-10: 90-78314-01-X ISBN-13: 978-90-78314-01-1

Contents

Summary

Samenvatting

1

About this thesis

1.1 Background 1

1.2 Objectives and outline of this thesis 2

1.3 References 3

2

Holography

2.1 Introduction 5

2.2 Concepts of holography 5

2.2.1 Grating formed by interference 5

2.2.2 Huygens’ principle 8

2.2.3 The complex amplitude approach 9

2.3 Holographic recording materials 10

2.3.1 General requirements 11 2.3.2 Silver-Halide 12 2.3.3 Bacteriorhodopsin 12 2.3.4 Photo-polymers 13 2.3.5 Digital recording 13 2.4 Diffraction Efficiency 13 2.4.1 Thin holograms 14

2.4.1.1 Sinusoidal thin amplitude grating 14

2.4.1.2 Sinusoidal thin phase grating 14

2.4.1.3 General thin amplitude grating 15

2.4.2 Thick holograms 15

2.4.2.1 Recording a volume holographic grating 16

2.4.2.2 Reconstructing a volume holographic grating 17

2.4.2.3 Transmission and reflection gratings 20

2.4.2.4 Diffraction efficiency of volume holograms 20

2.4.2.4.1 Volume transmission phase grating 21

2.4.2.4.2 Volume transmission amplitude grating 22

2.4.2.4.3 Mixed volume transmission grating 23

2.5 Polarization holography 23

3

Bacteriorhodopsin

3.1 Introduction 29

3.2 Halobacterium Salinarium 29

3.3 BR structure 30

3.4 BR photocycle 31

3.5 Absorption and phase properties of a BR film 33

3.6 Polarization properties of a BR film 36

3.7 Polarization holography in BR 40 3.8 Polarization multiplexing 44 3.9 Conclusions 47 3.10 References 47

4

Volatility

4.3.2 Experiments and results 53

4.3.2.1 Rate of photo-induced erasure 55

4.3.2.2 Spectral dependence of the amount of reconstructed energy 58

4.3.2.3 Geometry for wavelength-shifted reconstruction 58

4.4 Cooling the BR film 64

4.4.1 Moderate cooling 64

4.4.2 Extreme cooling 67

4.5 Useful features of volatile recording media 68

4.5.1 Signal retrieval with an increasing inline background 69

4.5.2 Signal retrieval with an increasing off-axis background 71

4.5.3 Positioning by spatial correlation of the background 73

4.5.4 Independent storage of two data-sets: η(t)-multiplexing 73

4.5.5 Intensity stepping 76 4.6 Conclusions 77 4.7 References 78

5

Digital holography

5.3 Noise in digital holography 80

5.3.1 Theoretical analysis of SNR 81

5.3.2 Numerical simulation 86

5.3.3 Experimental benchmark 87

5.4 Conclusions 91

6

Holographic Particle Image Velocimetry

6.1 Introduction 93

6.2 Important issues for HPIV 94

6.2.1 NA 94

6.2.2 Aberrations 95

6.2.3 Data extraction 97

6.3 State-of-the-art HPIV systems 98

6.3.1 Near-forward scattering, dual-view system using relay optics 98

6.3.2 Forward scattering, orthogonal-view system using spatial filtering 99

6.3.3 Side-scattering, high NA 101

6.3.4 Planar data, retrieved from virtual images 103

6.3.5 Object conjugate reconstrcution (OCR) 104

6.3.6 Light-in-flight particle holography 105

6.3.7 Limitations of current state-of-the-art 106

6.4 A novel HPIV system 107

6.4.1 Possible geometries 107 6.4.2 Recording setup 108 6.4.3 Reconstruction setup 111 6.4.4 Data analysis 113 6.5 Experiments 114 6.5.1 Experimental procedure 114 6.5.2 Experimental results 115

6.5.2.1 Particle image analysis 115

6.5.2.2 System accuracy / alignment test 118

6.5.2.3 Jet flow 119 6.6 Conclusions 120 6.7 References 121

7

Conclusion

A

System parameters

A.2 Experimental validation 132

A.3 Reference-to-object intensity ratio 133

A.4 Total laser power 135

A.5 Hologram aperture size 136

A.6 References 138

List of publications

Nawoord / Acknowledgments

Summary

The main goal of the research that is described in this thesis was the development of a holographic system that can be used to perform instantaneous three-dimensional flow measurements. Such a system may lead to new fundamental insights, particularly into the behaviour of turbulent flow.

Because flow media (such as water and air) are generally transparent, optical measurements in such media are enabled by adding small particles to the flow of interest. When these so-called tracer particles follow the flow, without influencing it, their movement is a reliable measure for the local flow vector.

Holography is a three-dimensional imaging technique, and as such is ideally suited for the study of inherent three-dimensional phenomena, such as turbulent flow. In the last decade several holographic systems have been presented that were used to successfully perform measurements on turbulent flow. However, this has not yet lead to the acceptance of HPIV as a commonly accepted research tool (HPIV: holographic particle image velocimetry). A possible explanation for this is that all previous HPIV systems relied on traditional silver-halide film. Such a film requires chemical development, thereby decreasing the practicality of the system. Furthermore, all previous systems in which multiple independent holograms can be recorded rely on angular multiplexing. This type of multiplexing increases the inaccuracy of the retrieved data, and has some practical disadvantages. Starting points in our research therefore were the application to HPIV of a recording material that does not have the disadvantages of traditional holographic film, and the development of new techniques to independently record multiple holograms.

The developed HPIV system uses the light-sensitive protein bacteriorhodopsin (BR) as the recording medium. After a BR molecule has absorbed a photon, it temporary becomes more transparent. The genetic variant BR D96N can maintain its transparency for several minutes at room temperature. Under normal circumstances a BR film may thus only be used for temporary storage of (holographic) information. The same film may be reused many times. We studied the properties of BR, with a focus on the practical applicability of this material and the development of new techniques. We have developed methods that allow us to increase the amount of information that may be retrieved from a BR film. Furthermore, we have developed several techniques that allow us to improve the signal-to-noise ratio of a holographic reconstruction with BR.

of multiplexing, a HPIV system can be made more robust and more accurate. Furthermore, polarization multiplexing allows to digitize the particle field images twice as fast.

The result of this research is a user-friendly and reliable HPIV system based on polarization multiplexing in BR. It was found that the multiplexed images do not suffer from an inherent mutual shift, thereby making polarization multiplexing especially favourable over angular multiplexing. To demonstrate the capabilities of the system, a number of measurements were performed on a multiphase jet. In such a measurement we determined the displacement of more than one thousand particles in a volume of less than

0.1 cm3. From this achieved particle density it is likely that in the near future our system

can be used to determine more than one million flow vectors in a volume of approximately

65 cm3.

Using our system, it was possible for one person to perform thirty multiphase jet measurements within eight hours. This would have been impossible with any other HPIV system. Because we have focused on the development of a practical HPIV system, it is now possible to rapidly perform three-dimensional flow experiments. As such, this research has made a significant step towards HPIV becoming a standard research tool.

Samenvatting

Het belangrijkste doel van het in dit proefschrift beschreven onderzoek was de ontwikkeling van een holografisch systeem voor het verrichten van instantane drie-dimensionale stromingsmetingen. Een dergelijke meettechniek kan leiden tot nieuwe fundamentele inzichten in de stromingsleer in zijn algemeenheid, en in het gedrag van turbulente stroming in het bijzonder.

Omdat stromende media als lucht en water over het algemeen transparant zijn, wordt de stroming meetbaar gemaakt door er kleine deeltjes, zogenaamde tracerdeeltjes, aan toe te voegen. Wanneer deze deeltjes de stroming goed volgen, maar niet beïnvloeden, is de beweging van een deeltje een goede maat voor de stromingsvector ter plaatse.

Holografie is een drie-dimensionale opname- en afbeeldingstechniek, en leent zich derhalve uitstekend voor het verrichten van metingen aan een inherent drie-dimensionaal verschijnsel zoals turbulente stroming. Alhoewel het afgelopen decennium meerdere holografische systemen zijn gepresenteerd waarmee succesvol metingen in turbulente stroming zijn verricht, heeft dit nog niet tot een algemene acceptatie van HPIV als meettechniek geleid (HPIV: holographic particle image velocimetry). Een mogelijke verklaring hiervoor is dat alle voorgaande HPIV-systemen gebruik maken van traditionele fotografische film. Dergelijke film dient chemisch ontwikkeld te worden, hetgeen de praktische bruikbaarheid van het systeem niet ten goede komt. Bovendien maken alle voorgaande systemen, althans die waarbij meerdere hologrammen gescheiden opgenomen kunnen worden, gebruik van hoek-multiplexing. Deze multiplexmethode doet de onnauwkeurigheid in de gemeten resultaten toenemen, en blijkt ook niet altijd even praktisch. Uitgangspunt van het hier beschreven onderzoek was dan ook het toepassen van een opnamemateriaal dat niet de nadelen van traditionele holografische film heeft, en het ontwikkelen van nieuwe technieken om meerdere hologrammen gescheiden op te kunnen slaan.

Het binnen dit onderzoek ontwikkelde HPIV-systeem gebruikt het licht-gevoelige eiwit

bacteriorhodopsin (BR) als opslagmedium. Nadat een BR-molecuul een foton heeft

geabsorbeerd wordt het gedurende enige tijd transparanter. De genetische mutant BR D96N kan bij kamertemperatuur zijn transparantie tot enige minuten na belichting behouden. Een BR-film kan onder normale omstandigheden dus gebruikt worden voor tijdelijke opslag van (holografische) informatie, waarbij dezelfde film vele malen gebruikt kan worden. Wij hebben de eigenschappen van BR bestudeerd, waarbij de praktische toepasbaarheid van dit materiaal en de ontwikkeling van nieuwe technieken hiervoor centraal stonden. Wij hebben methoden ontwikkeld waarmee de hoeveelheid data die uit een BR-film gereconstrueerd wordt kan worden vergroot. Ook hebben wij technieken ontwikkeld waarmee de signaal-ruisverhouding van een holografische reconstructie met BR verbetert.

dezelfde film gescheiden opgeslagen kunnen worden. Hiertoe hebben wij een nieuwe techniek ontwikkeld waarbij twee hologrammen met afwisselende circulaire polarisatie worden opgenomen. Het is zelfs mogelijk om twee datastromen gelijktijdig, doch met verschillende polarisatie, uit de BR film te reconstrueren. Het toepassen van

multiplexing voor HPIV kent een aantal grote voordelen. Doordat er bij

polarisatie-multiplexing aanzienlijk minder optische componenten nodig zijn dan bij traditionele multiplexing, maakt dit het HPIV-systeem robuuster en nauwkeuriger. Bovendien kan met behulp van polarisatie-multiplexing de reconstructie van de deeltjesvelden met een factor twee versneld worden.

Het resultaat van dit onderzoek is een gebruikersvriendelijk en betrouwbaar HPIV-systeem gebaseerd op polarisatie-multiplexing in BR. Het bleek dat de gemultiplexde opnames geen onderlinge verschuiving vertonen, hetgeen polarisatie-multiplexing bijzonder aantrekkelijk ten opzichte van hoek-multiplexing maakt. Om de werking van het systeem aan te tonen is een aantal metingen verricht aan een meerfase-jet stroming. Hierbij werd de verplaatsing

van meer dan duizend deeltjes binnen een volume van minder dan 0,1 cm3 bepaald. Uit

deze behaalde deeltjesdichtheid is het aannemelijk dat het binnenkort mogelijk is om meer

dan één miljoen stromingsvectoren te bepalen in een volume van circa 65 cm3.

About this thesis

1.1 Background

From large-scale weather systems the size of a continent, down to the delivery of oxygen through our smallest capillaries, flow phenomena continuously impact our lives. Man’s natural desire to understand the world he lives in, combined with its relevance for medicine and industry, has made fluid mechanics one of the longest and most intensively studied fields of physics. Although this massive research effort has led to a deep understanding of the underlying mechanisms, still many questions remain to be answered. As the knowledge of the subject increases, more sophisticated measurement techniques are required to better resolve certain phenomena, thereby answering old questions and raising new ones.

Over the past decades optical flow measurements have evolved from the age old qualitative observation using the human eye, through quantitative single point measurements (Laser

Doppler Anemometry; LDA)1 and quantitative two-dimensional measurements (Particle

Image Velocimetry; PIV),2 to quantitative instantaneous three-dimensional measurements

(Holographic Particle Image Velocimetry; HPIV).3 The first holographic three-dimensional

flow measurements date quite far back. In 1969(!) Trolinger successfully performed

holographic particle tracking.4 Holographic particle image correlation has been an active

area of research since the early 1990’s, with several successful laboratory systems.3,5-8

Although these systems produced convincing results, they typically were complex systems that could not be easily operated outside the specialized laboratory they were developed in. One of the common drawbacks of these systems was their reliance on silver-halide film as the holographic medium. The chemical processing associated with such films introduces distortion to the recorded hologram, requires a skilled person for the development, and is time consuming.

By the end of the 1990’s several people realized that HPIV could not become a common research tool until an easy to operate holographic system was available. This led some

research groups to focus on digital holography;9 the direct recording of a (particle)

hologram on a CCD.10-14 Although the technique is very user-friendly, the space-bandwidth

product of current state-of-the-art CCD’s is very small when compared to holographic film. This limits the number of flow vectors that can be obtained from a single recording. Other groups strived to replace the silver-halide film by a less laborious, preferably reusable, holographic material. Shortly after the development of a holographic flow measurement system had started at Delft University of Technology, Barnhart et al suggested the use of

Bacteriorhodopsin as a recording medium for HPIV.15 This reversible, photo-chromic

1.2 Objectives and outline of this thesis

The work that relates to this thesis focuses on the development of a holographic flow measurement system. Obviously the word ‘development’ is meant to be broader than determining type and position of the several components that form the system. It also implies knowing and understanding the properties of the recording material and other components. We ambitiously set out to develop a system that could record 1 million velocity vectors in a 40 mm x 40 mm x 40 mm measurement volume. Because we wanted to design a measurement tool that is easy to use, it was important to identify a suitable holographic medium. We thoroughly analyzed the properties of Bacteriorhodopsin. It is crucial to understand how these properties affect the performance of the system, and what new possibilities they might offer. We also investigated certain advantages and limitations of applying digital holography to the study of flow. Furthermore, we needed to collect and analyze the various factors that affect the system’s performance, so that others may successfully apply this system in actual flow measurements.

Besides providing the fluid mechanics community with a novel measurement tool, the work presented in this thesis may also benefit those working in other fields such as data storage, holographic interferometry, coherent optical imaging, and optical cryptology. Chapter 2 introduces some basic holographic concepts, leading to an introduction of topics in holography that are relevant for this thesis. We will introduce different types of holograms, and provide means to calculate how much energy diffracts from a given grating. Furthermore we will introduce polarization holography. This type of holography forms the basis of our novel holographic flow measurement system.

In Chapter 3 the photo-chromic protein Bacteriorhodopsin (BR) will be introduced, and its optical properties will be discussed in detail. From its properties can be understood that it can be used to record polarization holograms. Finally we show that two recordings can be multiplexed in the same film using a technique called polarization multiplexing. We present a new scheme for polarization multiplexing that allows to reconstruct two orthogonally polarized images simultaneously.

Subsequently, Chapter 4 discusses the volatile nature of BR. Several methods that can be used to prolong the storage life, or the amount of reconstructable information, of holograms in BR will be presented. We also present several noise-reducing techniques that take advantage of BR’s volatile nature.

Chapter 5 deals with digital holography: the recording of a hologram directly on a CCD. We discuss the ability to use BR as a temporary holographic buffer that is digitized later. Furthermore we will find an important relation that describes how the limited number of pixels influences the signal-to-noise ratio (SNR) of digital (particle) holograms.

Finally, in Chapter 7 we present the conclusions of the project, and the recommendations for further research are given.

1.3 References

1. Y. Yeh and H. Cummins “Localised fluid flows measurements with a He-Ne laser spectrometer,” Appl. Phys. Lett. 4, 176-178 (1964).

2. R.J. Adrian, “Particle-imaging techniques for experimental fluid mechanics,” Ann. Rev. Fluid Mech. 23, 261-304 (1991).

3. K. D. Hinsch, “Holographic particle image velocimetry,” Meas. Sci. Technol. 13, R61-R72 (2002).

4. J.D. Trolinger, R.A. Belz, and W.M. Farmer, "Holographic techniques for the study of dynamic particle fields," Appl. Opt. 8, 957-962 (1969).

5. D.H. Barnhart, R.J. Adrian, and G.C. Papen, “Phase conjugate holographic system for high resolution particle image velocimetry,” Appl. Opt. 33, 7159-7170 (1994).

6. J. Zhang , B. Tao, and J. Katz, “Turbulent flow measurement in a square duct with hybrid holographic PIV,” Exp. Fluids 23, 373 – 381 (1997).

7. Y. Pu and H. Meng, “An advanced off-axis holographic particle image velocimetry (HPIV) system,” Exp. Fluids 29, 184 – 197 (2000).

8. S. Herrmann, H. Hinrichs, K. D. Hinsch , and C. Surmann, “Coherence concepts in holographic particle image velocimetry,” Exp. Fluids 29, S108-S116 (2000).

9. U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179-181 (1994).

10. B. Skarman, J. Becker, and K. Wozniak, “Simultaneous 3D-PIV and temperature measurements using a new CCD-based holographic interferometer,” Flow Meas. Intrum. 7, 1-6 (1996).

11. G. Pan and H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt. 42, 827-833 (2003).

12. W. Xu, M. Jericho, and J.H. Kreuzer, “Tracking particles in four dimensions with in-line holographic microscopy,” Opt. Lett. 28, 164-166 (2003).

13. S. Coëtmellec, C. Buraga-Lefebvre, D. Lebrun, and C. Özkul, “Application of in-line digital holography to multiple plane velocimetry,”Meas. Sci. Technol. 12, 1392-1397 (2001).

14. M. Malek, D. Allano, S. Coëtmellec, C. Özkul, and D. Lebrun, “Digital in-line holography for three-dimensional-two-components particle tracking velocimetry,” Meas. Sci. Technol. 15, 699-705 (2004).

15. D.H. Barnhart, N. Hampp, N.A. Halliwell, and J.M. Coupland, “Digital holographic velocimetry with bacteriorhodopsin (BR) for real-time recording and numeric

reconstruction,” in 11th International Symposium on Applications of Laser Techniques

Holography

2.1 Introduction

The study of the three-dimensional motion of flow requires a three-dimensional imaging technique. Gabor conceived the technique of wavefront reconstruction: an imaging

technique that preserves the three-dimensional nature of the object that is recorded.1,2 Work

in the early 1960’s by Leith and Upatnieks demonstrated the practical feasibility and

importance of optical wavefront reconstruction,3-5 now known as holography. Since many

excellent textbooks exist on the topic,6-8 it is outside the scope of this chapter to supply the

reader with an in-depth background in the history and theory of holography. Instead the subject will be explained in three different conceptual approaches. After this conceptual framework has been constructed, some issues that are relevant for this thesis will be introduced.

2.2 Concepts of holography

The three different concepts that will be used to describe the fundamentals of holography each provide their own unique insight into the subject. One may be mathematically more precise, the other may give more physical insight for a particular application. At different times I have found the different conceptual views to be helpful in understanding a certain topic, and as such I find it useful to present them to the reader at this point.

2.2.1 Grating formed by interference

Essential to holography is the concept of interference. When two mutually coherent plane

waves of light intersect each other at an angle θi, a characteristic fringe pattern of bright

and dark regions will be observed in the region of overlap. As is well known, and is shown in Fig. 2.1, if the path length difference between the two beams is a multiple of the

wavelength of light, λ, then a bright region will occur. On the other hand, if the relative

path length difference between the two beams is λ/2, then a dark region is observed. For

the period of the resulting fringe pattern the following expression can easily be found

Fig. 2.1. Interference of plane waves A and B. For wave B both TE (transverse electric) polarization and TM (transverse magnetic) polarization have been indicated.

If the resulting intensity pattern of Fig. 2.1 is printed on a transparent substrate, an object with a sinusoidally varying transmission is obtained. Such an object is generally referred to as a grating. Upon illumination of a grating, higher order beams are generated due to diffraction (Fig. 2.2). It is well known that for the diffraction on a grating we can write

m d

mλ= sinθ , (2.2)

where m indicates the diffracted order. Apart from the undiffracted zeroth order beam, the two first order beams (m =+1 and m= –1) are likely to contain a substantial amount of light.

For the angle θ_±1 between the first order beams and the undiffracted beam we find

d λ θ±1 =

sin . (2.3)

Comparing Eq. (2.1) and Eq. (2.3) we see a striking resemblance. The angle under which a

grating diffracts light, is exactly the angle between the two beams that formed the grating.

This is the essence of holography.

Fig. 2.3. (a) Recording a hologram of a point source. (b) Reconstruction of the hologram. Both a virtual image (thick lines), and a real image (thin lines) are formed.

Obviously it is of little practical significance to record a hologram of two plane waves. Generally a hologram is recorded using a known reference wave and an unknown object wave. Figure 2.3(a) illustrates the recording of a hologram using a plane reference wave and a spherical wave coming from a point source object. Because the angle between the reference wave and the object wave varies over the film aperture, the resulting grating will have a spatially varying period. Upon reconstruction with the original plane reference wave, each local grating will diffract primarily into the first orders (Fig. 2.3(b)). The +1 diffracted order forms a continuation of the original object wave, resulting in a virtual image of the point source object (thick black lines in Fig. 2.3(b)). The –1 diffracted orders form a real image of the point source object, that has been deflected from the z-axis at an angle approximately twice that which the reference beam makes with the z-axis (thin black lines in Fig. 2.3(b)). As is shown in Fig. 2.3(c), alternatively the hologram may be reconstructed with the conjugate of the original reference beam. In this case the –1 diffracted orders form a real image at the location of the original object (thick black lines in Fig. 2.3(c)). The +1 diffracted orders now form a virtual image that has again been deflected from the z-axis (thin black lines in Fig. 2.3(c)). In Figs. 2.3(b) and 2.3(c) we see that the real image, the virtual image, and the reconstruction beam are spatially separated. This happens when the angle of the reference beam is large enough, and this type of holography is called off-axis holography. Alternatively, when the angle between the reference wave and the object wave is very small or even zero, spatial separation of the various wave components that exit the grating is not achieved. This type of holography is called in-line holography. Although in-line holography is easy to implement (the reference beam may also serve as object illumination beam) the presence of the twin image and the undiffracted reconstruction beam impose severe restrictions on its applicability.

We have seen that holography enables the reconstruction of both real and virtual images of point source objects. Because each physical object can be considered as an ensemble of point source objects it should be clear that a hologram can be recorded from any object, and that the three-dimensional structure of this object remains preserved.

interference is a vectorial effect, the state of polarization of the two interfering beams strongly influences the resulting intensity pattern. With reference to Fig. 2.1, if waves A and B have a polarization that lies in the plane of incidence, then the component of wave B along z cannot interfere with wave A and thus yields a constant offset in the intensity pattern. If wave A is polarized perpendicular to the plane of incidence and wave B has TM polarization, or vice versa, no modulation of intensity will be observed at all. Only if both waves are polarized perpendicular to the plane of incidence, all the energy contained in both waves will contribute to the intensity modulated interference pattern.

2.2.2 Huygens’ principle

An alternative way to look at holography is by means of Huygens’ principle. Huygens stated that every point on a primary wavefront serves as the source of spherical secondary wavelets, such that the primary wavefront at some later time is the sum of these wavelets. Practically this implies that if the wavefront is known in a plane the resulting wavefront in any other plane can be derived. Rather than holistically considering the hologram as a grating structure we may consider it as a point source based wavefront regeneration device; a view which proves very helpful when working with digital holography (Chapter 5). Consider the recording of a hologram as illustrated in Fig. 2.4(a). A reference wave, R, falls onto the holographic film at a certain angle. As a result it has a spatially varying phase on the film. At the same time an object wave, O, having a spatially varying phase and amplitude impinges on the film. At the plane of the hologram, H, the reference wave interferes with the object wave to produce the resulting intensity pattern I. The intensity has a minimal value at point 1 (destructive interference), and a maximal value at point 2 (constructive interference). After development of the hologram the intensity pattern I has resulted in an amplitude transmission pattern T, where, for now, we shall assume a linear relation between I and T. Thus, there where the reference phase matches the phase of the object wave the hologram has a high transmission. Where the two phases do not match, the hologram has a reduced transmission. As a result, when illuminating the transmission pattern T with the reconstruction wave R’ an interesting effect occurs (Fig. 2.4(b)). In the transmitted wave the regions where the relative phase relations of R’ did not match those of

O have been suppressed. The resulting wave O’ can thus be regarded as a collection of

secondary point sources whose relative phases match those of O. In effect, wave O’ will be similar to wave O.

(a) (b)

Fig. 2.5. Intensity as a function of phase difference. This cosine relationship introduces an ambiguity in the phase.

This is a very qualitative explanation and does not yield any insight into the exact nature of the various resulting wave components. However, it does yield insight into the question why a thin hologram always produces a real and a virtual image. After all, the better the phase of R corresponds to the phase of O the larger the amplitude of the secondary point source at that location will be during the reconstruction. As is illustrated in Fig. 2.5 the intensity at the time of recording has a cosine dependence on the phase difference between

O and R. This means that a value obtained for the transmission T would be obtained

equally if the phase difference between O and R, ϕ, had been its conjugate, ϕ*. As a result,

the hologram does not only amplify the relative strength of secondary point sources that resemble wavefront O but it does the same for its conjugate version. It is thus this ambiguity in the hologram transmission that causes both the original wave (virtual image) and its conjugate (real image) to be reconstructed simultaneously. With an additional

phase-stepped recording it is possible to discriminate between ϕ and ϕ*, and this yields the

possibility to selectively reconstruct either the virtual or the real image.

2.2.3 The complex amplitude approach

The third way in which the fundamentals of holography will be described is the one most frequently encountered in the textbooks. Consider the recording of a hologram as depicted in Fig. 2.4. For the complex object wave O(x,y) we may write

) , ( ) , ( ) , (_{x y} =_{O x ye}−iψo xy O , (2.4)

where O(x,y) is the absolute amplitude and ψo(x,y) the phase. For the reference wave R(x,y)

we find ) , ( ) , ( ) , (_{x y} =_{R x y e}−iψr xy R . (2.5)

For the light intensity of the resulting interference pattern we may write

where * denotes the complex conjugate, and the (x,y) indices have been dropped for brevity. This intensity pattern will influence the transmission of the holographic film. If the amplitude transmission t(x,y) is proportional to the incident intensity we find

) ( ) , ( O2 R2 OR O R t x y =β + + * + * , (2.7)

where β is a combined measure of the film sensitivity and the exposure time. If the

hologram is illuminated with a reconstruction wave, B(x,y), we may write for the complex amplitude behind the hologram

4 3 2 1 ) , ( ) , ( ) , ( u u u u y x y x y x + + + = =B t E , (2.8) where 2 1 =βBO u (2.9) 2 2 =βBR u (2.10) * BOR β = 3 u (2.11) R BO* β = 4 u (2.12)

Of particular interest are terms u3 and u4. When the reconstruction wave is chosen identical

to the original reference beam, it is found that

O R2 β = 3 u , (2.13)

which, apart from a constant β|R|2, represents the original object wave. When the reconstruction wave is chosen as the conjugate of the original reference beam, it is found that * 2 O R β = 4 u , (2.14)

which, apart from a constant β|R|2, represents the conjugate of the original object wave. It can thus again be found that both a virtual image and a real image may be reconstructed from the hologram.

2.3 Holographic recording materials

2.3.1 General requirements

Recording a hologram involves registering the intensity pattern that results from the interference between reference wave and object wave. Obviously this implies that the recording material has a physical response to the incoming light of wavelength λ. The complex amplitude transmittance of such a material can be written as

)] / 2 ( exp[ ) exp(−αd −i πnd λ = t (2.15) where α is the absorption constant of the material, d its thickness, and n its refractive index. If illumination only results in a change of α, we call the resulting hologram an amplitude hologram. In the case that n or d changes the hologram is called a phase hologram. The

response of the recording material, i.e. the change in transmission or optical thickness, is preferably high for low levels of light. The sensitivity, or required exposure, of a holographic medium expresses the amount of energy per unit area (normally called fluence) that is required to successfully record a hologram.

A further requirement is that the resolution of the material is sufficiently high such that the intensity pattern is sampled at least at the Nyquist frequency of the band-limited interference pattern. From Eq. (2.1) it can easily be found that the spatial frequency of an interference pattern that is formed by two overlapping beams that subtend an angle θi is

given by f=sin(θi)/λ. The minimal sampling frequency therefore is

λ θi s

f ,min =2sin . (2.16) When recording holograms with a large numerical aperture (NA) the material requires a resolution of several thousand line-pairs per millimeter (e.g. for λ = 532 nm and θi = 45° it

is found that fs,min ≈ 2600 mm-1). This goes well beyond the resolution of traditional

photographic film. For example, the resolving power of Fujicolor superia 100 ASA film is limited to 125 line-pairs per millimeter.

2.3.2 Silver-Halide

Silver-halide films are by far the most commonly used holographic recording medium. A silver-halide recording material is based on silver halide crystals (e.g. AgCl, AgBr, AgI) that are embedded in a gelatin layer, commonly known as the photographic emulsion.9 Silver-halide grain sizes vary from 10 nm for ultrafine emulsions to several micrometers for highly sensitive photographic emulsions. Because the silver salts are only sensitive in the spectral region between UV and deep blue, special sensitizers (dyes) must be used to prepare the films for use with commonly used visible lasers.

During exposure a latent image forms inside the silver-halide film. Chemical processing is required to convert the latent image into a silver image. The development process causes a gain in the order of 106 in terms of the number of silver-halide crystals that contribute to the recorded image, thereby making silver-halide films a sensitive recording medium.7 Despite this high gain, the chemical processing has some serious drawbacks. The development process is time consuming and produces varying results. Although this is usually no problem for the holography enthusiasts, it has proven to be a severe limitation in the general acceptance of holographic (measurement) techniques. A second drawback of the chemical processing is emulsion shrinkage. The thickness of the processed photographic emulsion layer is usually about 15 per cent less than its original thickness due to the removal of unexposed silver halide grains during fixation.7 Despite the supporting glass plate, serious shrinkage and distortion of the fringe pattern will also occur in the transverse directions. As a result, to obtain a diffraction limited reconstruction of a particle field it was found that the scattered light from a particle must subtend a region not greater than ~1 cm2 at the plane of the hologram.10

By bleaching the amplitude hologram it can be converted to a phase hologram. This is advantageous because phase holograms have a significantly higher diffraction efficiency (see section 2.4.1.2). A disadvantage of bleaching is that due to additional scattering, the signal-to-noise ratio (SNR) of the reconstructed image may be lower than in the case of an unbleached transmission hologram.7

One problem with silver-halide films has been its commercial availability. Large companies such as Kodak and Agfa have ceased to produce films. The Russia based company Slavich is the only large factory worldwide to currently produce viable quantities of both film and glass-based holography materials.11

2.3.3 Bacteriorhodopsin

2.3.4 Photo-polymers

The lack of suitable recording materials has long been an obstacle to the development of holographic data storage. Due to their large refractive-index contrast and good photosensitivity, combined with their ease of use, photopolymers were quickly recognized as attractive candidates for this application. The commercial potential of volumetric write-once-read-many-times (WORM) data-storage has sparked the development of several very promising materials. The biggest promise seems to be the Tapestry media that are developed by InPhase Technologies (formerly Lucent Technologies).13 Their film consists of a polymer system composed of two independently polymerizable and compatible chemical systems,14 and has a sensitivity comparable to that of BR. The Tapestry two-component system effectively eliminates shrinkage, does not require any development, and yields highly modulated phase holograms. The polymerization upon exposure is irreversible, and the material is thus not reusable. Combined with a relatively high price (~$100/film) this makes the material unattractive for use in holographic measurement systems where many measurements are made per day. InPhase has announced that it is developing a reusable film with similar specifications. When this material is readily available, or when mass production has seriously reduced the price of its WORM media, photopolymers could very well become the media of choice for holographic measurement systems.

2.3.5 Digital recording

Equation (2.16) suggests that if the angle between the reference beam and the object beam is very small, a hologram may be recorded with a material that has a relatively low spatial resolution. This inspired Schnars and Juptner to record a low NA hologram directly onto a CCD.15 The digitized fringe pattern can then be reconstructed numerically instead of optically. Digital holography has some obvious advantages and disadvantages. Its main advantage is the ease of use. Also, because the phase of the reconstructed wave is available, the recorded object may be analyzed in a more sophisticated manner.16 Disadvantages are the relatively low resolution and small space-bandwidth product of CCD sensors compared to traditional holographic film. Due to its promising possibilities for holographic measurement systems, digital holography will be treated in more detail in Chapter 5.

2.4 Diffraction Efficiency

2 2 Λ = n d Q πλ , (2.17) where λ is the illuminating wavelength, d the thickness of the recording material, n the refractive index, and Λ the spacing of the recorded fringes.17 Small values for Q (Q<<1) classify the transmission hologram as thin, whereas large values for Q (Q>>1) indicate a thick hologram. However, this criterion has been shown to be somewhat arbitrary.18,19

2.4.1 Thin holograms

The diffraction efficiencies for various types of thin gratings are well known and will be discussed in the following sections.

2.4.1.1 Sinusoidal thin amplitude grating

In a sinusoidal thin amplitude grating the amplitude transmittance is a sinusoidally varying function of position t(x)=t0+∆t sin(2πx/Λ). In this expression t0 is the mean transmission, ∆t

is the amplitude of the sinusoidal modulation of the transmission, x denotes the position along the axis in which the transmission varies, and Λ is the period of the grating. It is essentially a transmission grating that is formed by interference of two plane waves. This type of grating reconstructs two diffracted beams, denoted as the m=+1 and m=-1 orders, on either side of the undiffracted zeroth order beam (m=0). The diffraction efficiency, η, into the various orders is given by20

2 0 0 t m= = η (2.18) % 25 . 6 4 / 2 1=∆ ≤ ± = t m η (2.19) 0 1 | |m> = η (2.20)

In a transmission grating the diffraction efficiency is relatively low, and a substantial amount of the illuminating light is absorbed by the grating.

2.4.1.2 Sinusoidal thin phase grating

In a sinusoidal thin phase grating the optical thickness is a sinusoidally varying function of

position. As a result for the phase retardation that is experienced we may write that φ(x)=

φ0+∆φ sin(2πx/Λ). In this expression φ0 is the mean transmission, ∆φ is the amplitude of the

sinusoidal modulation of the transmission, x denotes the position along the axis in which

the transmission varies, and Λ is the period of the grating. For the diffraction efficiency

into the various orders it can be found20

) (

2

0 φ

% 8 . 33 ) ( 2 1 1= ∆ ≤ ± = φ ηm J (2.22) % 100 ) ( 1 2 0 ≤ ∆ − =

å

where Jn is the n-th order Bessel function of the first kind. A phase grating clearly has a

higher maximum diffraction efficiency than a transmission grating. However, even a perfectly sinusoidal grating will produce higher order diffracted beams (|m|>1). These higher order beams may contribute to additional noise in the reconstructed image.

2.4.1.3 General thin amplitude grating

The sinusoidal gratings described previously provide insight into the different behaviour of amplitude and phase gratings. However, in practical holography the amplitude transmittance consists of an apparent random fluctuation ∆t(x,y) around an offset tavg, thus

t(x,y)=tavg+∆t(x,y). The case of the general thin amplitude grating is of great importance to

digital holography (Chapter 5). Although it is impossible to give a general expression for the diffraction efficiency into the image forming orders (m=±1), the transmittance function can be used to compute the diffraction efficiency of the zeroth order and the sum of the higher orders20 2 0 avg m= = t η (2.24) 2 2 2 2 0 var ) , ( ) , ( t avg m m t y x t t y x t σ η = = ∆ = − =

å

where σt is the variance of the amplitude transmission.

The response of any photographic material can be expressed by means of a curve where the transmission t (or alternatively optical density) is plotted as a function of the exposure E (or its log). The diffraction efficiency of thin amplitude holograms can be shown to be proportional to the gradient of the t-log E curve.21 The maximum diffraction efficiency is thus obtained where the slope of the t-log E curve is steepest. This method can be used to optimize the performance of a hologram, and its validity was demonstrated.22 However, when the amount of available energy to record the hologram is limited, as is the case in pulsed laser holography, the maximum diffraction efficiency is obtained where the t-E curve is steepest.22

2.4.2 Thick holograms

2.4.2.1 Recording a volume holographic grating

Because during recording the fringes are formed by interference between the object wave and the reference wave, the propagation of these constituent waves within the media is of great interest. A common way to describe the propagation of a wave front is by means of its wave vector kH, where kHpoints in the direction of the wave front normal, and

λ π / 2 |

|kH = . The complex amplitude of the (plane) object and reference wave can thus be represented by r k i r r Ae r U H H H ₌ ⋅ ) ( (2.26) r k i o r ae o U H H H ₌ ⋅ ) ( (2.27)

where A and a denote the complex amplitude of the reference wave and the object wave respectively, kr

H

and ko

H

are the wave vectors of the reference wave and the object wave inside the medium, and rH is a position vector. For the resulting intensity pattern in the interference region of the two waves follows

(

)

[

]

where ϕ denotes the phase difference between phasors A and a. The grating can thus be described by means of a grating vector KH. As is illustrated in Fig. 2.6, it can be found that

o r k k KH=H −H , (2.29) with Λ = =K 2π KH , (2.30) where, when the angle between kr

H

and ko

H

subtends 2β, the fringe period Λ is given by

β λ sin 2 = Λ . (2.31)

All preceding equations are derived for the case of a plane wave hologram. However, as we have seen earlier, the case of the plane wave hologram is not as restrictive as it might appear; any arbitrary wave front may be locally approximated by a plane wave.

2.4.2.2 Reconstructing a volume holographic grating

The volume holographic grating is essentially a three-dimensional distribution of quasi-planes of high and low absorption or refractive index, also known as platelets (see Fig. 2.7). We may regard each of these planes as a partially reflecting mirror. Because it is required that the reflections from all these planes are to constructively interfere in order to construct a diffracted order, the reconstruction wave front must impinge the volume holographic grating at a specific angle. This requirement is called the Bragg condition. For the angle of incidence of the reconstruction wave is easily found that

Λ ± =

2

sinα λ (2.32) The Bragg condition defines a cone of reconstruction beams that may be used to reconstruct a hologram. This is, however, only the case for a plane wave hologram. In practice there are only two beams that fulfill the Bragg condition for every hologram location: the original reference beam and its conjugate. When the original reference beam is used, the Bragg-matched diffracted order forms a virtual image (also see Figs. 2.11(c) and 2.11(d)). When the conjugate of the original reference beam is used the Bragg-matched diffracted order forms a real image, such a reconstruction is called the conjugate reconstruction (also see Figs. 2.11(e) and 2.11(f)). A thick hologram thus reconstructs either a virtual or a real image, depending on the illumination.

The reconstruction beam and the resulting diffracted signal may be described by means of their wave vectors ρH and σHrespectively. The signal wave vector σH is forced by the grating, and is related to the wave vector of the reconstruction beam and the grating vector by

KH

H H_=ρ−

σ (2.33)

(a) (b)

Fig. 2.8. Vector diagrams for (a) exact and (b) near Bragg incidence.

Figure 2.8 shows the vector diagrams for exact and near Bragg incidence. Near Bragg incidence may occur for example when the reconstruction wavelength is slightly shifted, the reconstruction beam is slightly misaligned, or the hologram has experienced shrinkage or expansion during development. Because the Bragg condition is not exactly fulfilled the amplitude of the diffracted order will be dampened. A useful parameter for evaluating the effects of deviations from the Bragg condition is the dephasing parameter23

(

)

where n is the refractive index of the film, and φ and θ are as shown in Fig. 2.9. In the case of perfect Bragg-matching the dephasing parameter is zero. For small deviations ∆θ and ∆λ from the Bragg condition, Eq. (2.34) reduces to

n K

K φ θ λ π

θ

ζ =∆ ⋅ sin( − 0)−∆ ⋅ 2/4 . (2.35) The influence of the Bragg mismatch on the diffraction efficiency is usually described through the parameter

θ ζ χ cos 2 d ⋅ = , (2.36) where d is the thickness of the grating.

Fig. 2.9. Relation between the grating vector KH and the wave vectors of the reconstruction beam

(a)

(b)

Fig. 2.10. Normalized diffraction efficiency as a function of Bragg-mismatch parameter χ for (a) amplitude transmission holograms and (b) phase transmission holograms. The modulation parameters

Figure 2.10 shows how χ affects the diffraction efficiency for various levels of modulation in amplitude holograms (Fig. 2.10(a)) and phase holograms (Fig. 2.10(b)), where the modulation parameters Φ and Φa will be discussed in sections 2.4.2.4.1 and 2.4.2.4.2

respectively. In amplitude holograms the degree of modulation Φa hardly influences the

behaviour of the normalized diffraction efficiency. In phase holograms, however, the modulation Φ strongly determines the dependence of the normalized diffraction efficiency on the Bragg mismatch. The relation between Bragg mismatch and diffraction efficiency for weakly modulated phase and amplitude holograms is very similar. We will treat the diffraction efficiency of volume holograms in more detail in section 2.4.2.4. We will now first discuss two basic types of thick holograms: transmission versus reflection holograms.

2.4.2.3 Transmission and reflection gratings

The grating vector can have either one of two general directions. When the reference wave and the object wave during the recording impinge the hologram from the same side, the resulting grating vector(s) will be oriented more or less parallel to the film surface (see Fig. 2.11(a)). It is also possible that the reference wave and the object wave impinge the hologram from opposite sides; in this case the grating vector(s) will be oriented perpendicular to the film surface (Fig 2.11(b)). Because the diffracted order builds up by partial reflection on a multitude of quasi-planes, the general orientation of these planes is crucial in determining the location of the constructed image. Figures 2.11(c) and 2.11(e) show the Bragg-matched diffracted order when reconstructing the hologram of Fig. 2.11(a) with a normal and a conjugate reconstruction respectively. Figures 2.11(d) and 2.11(f) show the Bragg-matched diffracted order when reconstructing the hologram of Fig. 2.11(b) with a normal and a conjugate reconstruction respectively. In Figs. 2.11(c) and 2.11(e) the image wave front is constructed while traversing the film, and such a hologram is called a

transmission hologram. In Figs. 2.11(d) and 2.11(f) the image wave front is reflected from

the hologram; such a hologram is called a reflection hologram. Also note that for both the

reflection hologram and the transmission hologram a conjugate reconstruction yields a real image.

Because all thick holograms that are recorded during the experimental part of this thesis are transmission holograms, in the next section (when discussing the diffraction efficiency of volume gratings) we limit ourselves to the case of the transmission grating.

2.4.2.4 Diffraction efficiency of volume holograms

In a volume holographic grating the amplitudes R(z) of the reconstruction wave and S(z) of

the reconstructed signal are functions of the depth within in the hologram z. There is an

exchange of energy between the two waves due to diffraction on the grating, and a loss of energy due to absorption within the hologram. The interaction between the two waves and the holographic medium is described by Kogelnik’s coupled-wave equations.23

Fig. 2.11. Recording of a (a) transmission hologram and (b) reflection hologram. Normal reconstructions ((c) and (d)) resulting in virtual images ((c) and (d)). Conjugate reconstruction

resulting in real images ((e) and (f)).

2.4.2.4.1 Volume transmission phase grating

For the refractive index throughout a phase-only hologram we may write

r K n n

θ λ π cos 1d n = Φ , (2.39)

χ is as defined by Eq. (2.36), and α0 is the absorption constant. When the incident

amplitude of the reconstruction beam R(0) is assumed to be unity, we find for the

diffraction efficiency in the case of Bragg incidence (i.e. χ=0)

Φ = = −2 /cos 2 2 sin ) ( 0 θ α η d e d S (2.40)

For a lossless (α0=0) phase grating the diffraction efficiency reaches unity when Φ=π/2.

When the hologram thickness d or the modulation of the refractive index n1 is further

increased, energy is coupled back from the signal beam into the reconstruction beam and the diffraction efficiency decreases.

2.4.2.4.2 Volume transmission amplitude grating

In an amplitude grating the refractive index does not vary (i.e. n1=0). For the spatially

varying absorption constant may be written

r KH⋅H

+ =α₀ α₁cos

α . (2.41)

In this case, using Kogelnik’s coupled-wave equations, the diffracted amplitude is23

2 1 2 2 2 1 2 2 cos / ) / 1 ( ) sinh( ) ( 0 a a i d _e e d S Φ − − Φ − = − − χ χ χ θ α , (2.42) where θ α cos 2 1d a = Φ , (2.43)

and χ is as defined by Eq. (2.36). When the incident amplitude of the reconstruction beam

R(0) is assumed to be unity, then we find for the diffraction efficiency in the case of Bragg

incidence (i.e. χ=0) a d e d S Φ = = −2 /cos 2 2 sinh ) ( 0 θ α η (2.44)

Because it is required that α1≤α0, the highest diffraction efficiency is obtained when α1=α0,

and α1dcosθ=ln3. It is thus found that the maximum diffraction efficiency of a volume

In section 2.4.1.3 we mentioned that the maximum diffraction efficiency of a thin amplitude hologram is obtained where the slope of the t-log E curve is the steepest. It can be shown that this is also valid for thick amplitude gratings that are reconstructed at the

Bragg angle.24

2.4.2.4.3 Mixed volume transmission grating

If a holographic medium undergoes both a change in refractive index and a change in absorption upon illumination, a mixed grating will be recorded. This is the case when working for example with Bacteriorhodopsin (see Chapter 3). When the amplitude and phase grating are assumed to interact with the reconstruction wave independently, and the diffraction efficiencies are low, the amplitude of the diffracted signal may be found by summing Eqs. (2.38) and (2.42). When these complex amplitudes are in phase, as is the

case with unslanted gratings (φ=π/2), the diffraction efficiency is the sum of the amplitude

and phase related diffraction efficiencies.23

So far we have assumed that the modulation index of the grating is constant throughout the entire hologram. Because the light beams that are used to record the hologram are normally

attenuated inside the recording medium (α0>0), the modulation index (α1, n1) decreases

with increasing depth inside the hologram. In this case the results of the coupled-wave equations can still be applied. It can be shown that at Bragg incidence the effective

modulation index is given by its value averaged over the thickness of the grating.25

Alternatively, a thick hologram may be considered as a multitude of thin gratings, with each grating acting on the output of the preceding grating. This thin-grating decomposition

was first introduced by Alferness.26 It can be shown that this approach is analytically

equivalent to the coupled-wave theory.27

2.5 Polarization holography

Thus far we have implicitly assumed both interfering waves to have TE polarization. After all, two TM polarized beams that interfere at an angle would not be able to reach full extinction. When the two polarizations are fully orthogonal at the plane of the hologram no modulation of the intensity can be observed. However, the interference of the two beams does result in a varying state of polarization. Figure 2.12 shows this modulation of polarization as a function of the phase difference between (a) two orthogonal linearly polarized beams, and (b) two orthogonal circularly polarized beams.

Kakichashvili was the first to realize the additional possibilities that materials with

photoinduced anisotropy offer for holographic recording.28-30 He showed the feasibility of

recording a hologram in polarization-sensitive materials using an object wave and reference wave with orthogonal polarizations, both linear and circular, by recording a varying state of polarization rather than recording a varying state of intensity as is done in traditional holography. This process requires an optical material that is sensitive to the polarization of the incident light. Upon illumination the material will exhibit a photo-induced anisotropy. The photo-induced birefringence, in the case of a phase hologram, or the induced dichroism, in the case of an amplitude hologram, gives polarization holograms the unique ability to modify the polarization of the reconstruction beam. This allows for the reconstruction of the original object polarization.

The diffraction efficiency of a polarization hologram obviously depends on the amount of photo-induced anisotropy. In the case of an amplitude hologram we may describe the anisotropic behaviour of the film by means of the amplitude transmission for light that is

linearly polarized along the polarization direction that has induced the anisotropy T||, the

amplitude transmission for perpendicularly polarized light T⊥, and ∆T=( T||- T⊥)/2. Based

on the work of Nikolova and Todorov we are now able to calculate the diffraction

efficiency of a polarization hologram.31 For the case of a thin amplitude polarization

hologram that was recorded using two orthogonal linearly polarized beams we find

4 2 1 T m ∆ = ± = η . (2.45)

Referring to Eq. (2.19) it is found that the diffraction efficiency is equal to that of an

ordinary thin amplitude hologram, given that ∆T=∆t. For the case of a thin amplitude

polarization hologram that was recorded using a right-handed circularly polarized reference beam and a left-hand circularly polarized object beam, we find

2 2 2 2 2 1 4 y x y x y x m R R R iR iR R T + − ± + ± ∆ = ± = η , (2.46)

where Rx and Ry are the components that constitute the Jones vector∗ of the reconstruction

beam, the upper signs correspond to S+1, and the lower signs to S-1. Reconstruction with a

right-handed circularly polarized beam (Rx = 1, Ry = -i) thus yields η+1 = ∆T2 and η-1 = 0.

Compared to a polarization hologram that has been recorded using linearly polarized light, a polarization hologram recorded using circularly polarized light can have a four times higher diffraction efficiency. Polarization holograms formed by two orthogonal circularly polarized waves also have the ability of reconstructing an image in the +1 order only and not in the –1 order and vice versa. The relative diffraction efficiencies and reconstructed polarization states for various combinations of recording and reconstruction polarization are summarized in Table 2.1. It is worth noting that when a thin polarization hologram is reconstructed from the opposite side, due to the inversion of the phase relationships in the

∗ _{The Jones vector is a column matrix in which each element represents a specific linear mode of}

Jones matrix that describes the film, the expected polarization state and intensity between the +1 and the –1 order interchange.

For the case of a phase grating we define ∆ϕ=2π∆nd/λ, where ∆n =(n||-n⊥)/2, in which n||

and n⊥ are analogous to T|| and T⊥. For the case of a thin phase polarization hologram that

was recorded using two orthogonal linearly polarized beams we find an expression that is analogous to Eq. (2.22). In the circularly polarized case an expression identical to Eq.

(2.46) is found where ∆T2/4 must be replaced by sin2∆ϕ. Nikolova et al have shown how

the diffraction efficiency of a mixed phase-amplitude grating that has been recorded using

two orthogonal circularly polarized waves can be determined.33 Korchemskaya et al have

presented a numerical model to determine the complex amplitude of diffracted waves for

polarization gratings recorded in BR.34 Furthermore a number of methods concerning

beam-propagation and diffraction in anisotropic media is available.35-38

Table 2.1 Relative diffraction efficiencies of amplitude polarization holograms. R: reference wave polarization; O: object wave polarization; Recon: reconstruction wave polarization; P_±1: diffracted

polarization in the ±1 order;31 I_±1: normalized diffracted intensity in the ±1 order.

2.6 References

1. D. Gabor, “A new microscopic principle,” Nature 161, 777-778 (1948).

2. D. Gabor, “Microscopy by reconstructed wavefronts,” Proc. Roy. Soc. A 197, 454-487 (1949).

3. E.N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. 52, 1123-1130 (1962).

4. E.N. Leith and J. Upatnieks, “Wavefront reconstruction with continuous-tone objects,” J. Opt. Soc. Am. 53, 1377-1381 (1963).

5. E.N. Leith and J. Upatnieks, “Wavefront reconstruction with diffused illumination and three-dimensional objects,” J. Opt. Soc. Am. 54, 1295-1301 (1964).

6. J.W. Goodman, Introduction to Fourier optics (McGraw-Hill, New York, 1996).

7. P. Hariharan, Optical holography (Cambridge University Press, Cambridge, 1996).

8. R.J. Collier, C.B. Burckhardt, and L.H. Lin, Optical holography (Academic Press, San

Diego, 1971).

9. H.I. Bjelkhagen, “Review of current holographic materials: recording and processing,”

in Proceedings of the International Workshop on Holographic Metrology in Fluid

Mechanics, J. Coupland, ed. (Loughborough University, Loughborough, UK, 2003),

pp 29-40.

10. D. Barnhart, Whole-field holographic measurements of three-dimensional

displacement in solid and fluid mechanics (Ph.D. Thesis, Loughborough University,

Loughborough, UK, 2001). 11. www.slavich.com

12. T. Juchem, Systemintegration des biologischen Photochroms Bakteriorhodopsin:

Entwicklung und Charakterisierung eines holographischen Kamerasystems für die

interferometrische Analyse (Ph.D. Thesis, Philipps Universität Marburg,

Marburg/Lahn, Germany, 2001). 13. www.inphase-tech.com

14. L. Dhar , A. Hale , H.E. Katz , M.A. Schilling , M.G. Schnoes, and F.C. Schilling, “Recording media that exhibit high dynamic range for digital holographic data storage,” Opt. Lett. 24 , 487-489 (1999).

15. U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179-181 (1994).

16. G. Pan and H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt. 42, 827-833 (2003).

17. W.R. Klein and B.D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Transactions on Sonics & Ultrasonics, SU-14, 123-134 (1967).

18. R. Alferness, “Analysis of propagation at the second-order Bragg angle of a thick holographic grating,” J. Opt. Soc. Am. 66, 353-362 (1976).

19. R. Magnusson and T.K. Gaylord, “Analysis of multiwave diffraction of thick gratings,” J. Opt. Soc. Am. 67, 1165-1170 (1977).

20. S. Benton, MAS450 Class Notes (Massachusetts Institute of Technology, 2002).

21. F. G. Kaspar and R. L. Lamberts, Abstract WA14 OSA meeting Oct. 1966, J. Opt. Soc.

Am. 56, 1414A (1966).

22. K. Biedermann, “A function characterizing photographic film that directly relates to brightness of holographic images,” Optik 28, 160-176 (1969).

23. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J.

24. H.M. Smith, “Effect of emulsion thickness on the diffraction efficiency of amplitude holograms,” J. Opt. Soc. Am. 62, 802-806 (1972).

25. D. Kermisch, “Nonuniform Sinusoidally Modulated Dielectric Gratings”, J. Opt. Soc. Am. 59, 1409-1414 (1969).

26. R. Alferness, “Analysis of optical propagation in thick holographic gratings,” Appl. Phys. 7, 29-33 (1975).

27. R. Alferness, “Equivalence of the thin-grating decomposition and coupled-wave analysis of thick holographic gratings,” Opt. Comm. 15, 209-212 (1975).

28. Sh.D. Kakichashvili, “Polarization recording of holograms,” Opt. Spectrosc. (USSR)

33, 171-173 (1972).

29. Sh. D. Kakichashvili, “Method for phase polarization recording of holograms,” Sov. J. Quant. Electron. 4, 795-798 (1974).

30. Sh. D. Kakichshvilli and T.N. Kvinikhidze, “Polarized recording of holograms using a reference wave of arbitrary polarization,” Sov. J. Quant. Electron. 5, 778-780 (1975). 31. L.Nikolova and T. Todorov, “Diffraction efficiency and selectivity of polarization

holographic recording,” Opt. Acta 31, 579-588 (1984).

32. R.C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. 31, 488-493 (1941).

33. L. Nikolova, T. Petrova, N. Ivanov, T. Todorov, and E. Nacheva, “Polarization holographic gratings: Diffraction efficiency of amplitude-phase gratings and their realization in AgCl emulsions,” J. Mod. Opt. 39, 1953-1963 (1992).

34. E. Korchemskaya, D. Stepanchikov, and N. Burykin, “Potentials of dynamic holography on Bacteriorhodopsin films for real-time optical processing,” in

Bioelectronic Applications of Photochromic Pigments, A. Der and L. Keszthelyi, eds.

(IOS Press, Amsterdam, The Netherlands, 2001), pp. 74-89.

35. F. Castaldo, G. Abbate, and E. Santamato, “Theory for a new full-vectorial beam-propagation method in anisotropic structures,” Appl. Opt. 38, 3904-3910 (1999). 36. L. Thylen and D. Yevick, “Beam propagation method in anisotropic media,” Appl.

Opt. 21, 2751-2754 (1982).

37. G. Montemezzani and M. Zgonik, “Light diffraction at mixed phase and amplitude gratings in anisotropic media for arbitrary geometries,” Phys. Rev. E 55, 1035-1047 (1997).