Advanced shaping of light fields with cascaded spatial light modulators

Advanced shaping of light fields with

cascaded spatial light modulators

Dr.rer.nat. Alexander Jesacher

Thesis to obtain the degree “Doctor of Philosophy” at the Innsbruck Medical University,Innsbruck

Supervisor: o. Univ. Prof. Dr. Monika Ritsch-Marte

April 11, 2008

Acknowledgements

Many persons have supported me in carrying out the work presented in this thesis. I wouldlike to thank:

• My advisors Prof. Monika Ritsch-Marte and Prof. Stefan Bernet for numerous helpfuldiscussions, their superb scientific advice and support in many other concerns.

• The Austrian Science Foundation (FWF) for their financial support.

• My colleagues at the institute for their valuable contributions to this work, especiallyChristian Maurer.

• All members of the Division for Biomedical Physics for support in all kinds of ques-tions and a pleasant work environment.

Moreover, I would like to express my special thanks to my parents for supporting me in allrespects, my brother Marco for many interesting discussions, and of course Tamara, theperson who most of all gave me support and motivation, for (now even more) experiencedadventures and those to come.

Contents

1 Phase-only liquid crystal SLMs 41.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 The HEO 1080 P spatial light modulator . . . . . . . . . . . . . . . . . . . 51.3 Shaping light fields with SLMs . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Shaping light with two cascaded SLMs 112.1 Basic principles of optical trapping . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1 Practical implementation of single-beam gradient traps . . . . . . . 132.1.2 SLM-based optical tweezers . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 PAPER: Optical tweezers of programmable shape with transverse scatteringforces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 PAPER: Near-perfect hologram reconstruction with a spatial light modula-tor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 PAPER: Full phase and amplitude control of holographic optical tweezerswith high efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5 Correcting aberrations of the SLM “double-filter” setup . . . . . . . . . . . 39

3 Summary and Discussion 43

Bibliography 49

1

Introduction

Methods for shaping the spatial profile of light are used for many applications in optics,such as beam shaping (Dickey and Holswade, 2000), wavefront correction (Hardy, 1998;Wilson, 2004) and holography (Hariharan, 1996). Typically, the parameters of interestare amplitude and phase, although there are also applications which require light beamsshowing a specifically designed polarization profile (Maurer et al., 2007).

The first method to “store” and “read out” amplitude and phase of light simultaneouslywas developed by Dennis Gabor in 1948 (Gabor, 1948). His idea of holographic imagingwas originally developed to increase the resolution of electron microscopy and was honoredby the nobel prize in physics 1971. Since its invention, optical holography underwent aconstant technological development, leading for instance to the “off-axis reference beamtechnique” (Leith and Upatnieks, 1962, 1963) or computer-generated holography (Lohmannand Paris, 1967), which allows to generate arbitrary computer-designed amplitude andphase distributions. However, until digital high resolution displays became available, mostholographic techniques relied on photographic films or crystals as recording media, theproduction of which is relatively complex and time consuming.

Within the last decades, several types of electronic devices for wavefront shaping havebecome commercially available. Some of them, for instance specific types of micromir-ror arrays (MMA) (Mignardi, 1994) and liquid crystal based twisted nematic spatial lightmodulators (TN-SLM) (Lueder, 2001; Holoeye-2) have been developed for video projectiontasks and are thus amplitude modulators. On the other hand there are phase modulators,such as optically or electronically addressed parallel aligned nematic spatial light modulators(PAL-SLM) (Hu et al., 2004; Hermerschmidt et al.; Hamamatsu) or deformable mirrors(DM). All of these devices have the advantage of being real-time programmable, whichmeans that measured or computer-generated amplitude or phase profiles can be directlyapplied to a wavefront. This property is for example exploited for video projection, meth-ods of adaptive aberration correction (Hardy, 1998; Neil et al., 2000a; Wilson, 2004; Zhanget al., 2006) and optical manipulation of microscopic particles (Dufresne and Grier, 1998;Liesener et al., 2000) and atoms (McGloin et al., 2003).

This thesis examines the prospects of utilizing cascaded diffractive patterns for wave-front shaping and the applicability of such techniques in the field of holography and opticaltrapping. Our experimental approach is based on using two diffractive elements, displayedside by side at the panel of a single PAL-SLM. Optically the patterns are located in conju-

2

gate planes, where the second element acts as Fourier filter or optical processor (Reynoldset al., 1989).

We have demonstrated the creation of high power spiral-filtered light fields (Furhapteret al., 2005; Jesacher et al., 2005, 2006) by choosing the second filter pattern to be of theform exp (iφ), where φ denotes the azimuthal angle in a polar coordinate system. Such afilter element alters an incoming wavefront in a very specific way: the filtered wave showsamplified intensities at regions where the input wave has field discontinuities. Analogously,the intensity appears suppressed at regions where the wavefront has a “smooth behaviour”.Exploiting these effects, we could demonstrate the efficient generation of arbitrarily shapedline tweezers by displaying binary phase structures at the first diffractive pattern. Trans-verse phase gradients, which appear in spiral-filtered light fields, are supposed to give riseto transverse scattering forces (Guo et al., 2006). We could for the first time observe theinteraction of these scattering forces to microparticles. Moreover, our observations confirmthat the transverse momentum transfer depends on the geometric curvature of the linetrap.

We have shown that it is possible to create arbitrary complex light fields of high qualitywith a single SLM (Jesacher et al., 2008a). In this approach – which has already been sug-gested by H. Bartelt in 1984 (Bartelt, 1984, 1985) – the modulus of the Fourier-transformeddesired light field is reconstructed by the first diffractive pattern, while the second patternshapes its phase. We demonstrated the simultaneous creation of two light objects in dis-tinct axial planes and compared the performance of the method to that of single phasediffractive optical elements (DOE). We could show that the obtainable light efficiency ismainly limited by the absorption of the SLM.

We applied this technique for the first time to optical micromanipulation (Jesacheret al., 2008b). We have shown that it is possible to generate arbitrary amplitude andphase distributions with high purity and we could clearly demonstrate the effects of thecorresponding gradient and scattering forces to silica and polystyrene microparticles.

Finally, we showed how the optical path of our “double-filter” setup can be optimizedby detecting aberrations in a closed-loop procedure (Neil et al., 2000b; Booth et al., 2002)and compensating them by adding a corresponding corrective phase mask to the phasepatterns displayed at the SLM.

3

Chapter 1

Phase-only liquid crystal SLMs

1.1 Introduction

Liquid crystal (LC) spatial light modulators are miniaturized liquid crystal displays withpixel sizes in the range of about 10 microns. Originally, they have been developed for videoprojection tasks. Consequently, most SLMs are amplitude modulators, which modify lightintensity via a controlled polarization rotation caused by the birefringent liquid crystal incombination with light polarizers (Jesacher, 2007). The most common type of LC used insuch devices is called twisted nematic (Lueder, 2001), the molecules of which are rod-shaped(calamitic) and arranged in a helix. Increasing use of this LC type in electronic productshas made twisted nematic devices affordable and also interesting for various applicationsin optics, where they are increasingly used as real-time programmable diffractive devices.However, many applications like wavefront correction or temporal pulse shaping requiremodulations of the light’s phase and not of its amplitude. Moreover, phase diffractivepatterns show a higher light throughput, since they do not absorb light. Although twistednematic SLMs can be also utilized as phase modulators (Pezzanaiti and Chipman, 1993;Yamauchi and Eiju, 1995; Davis et al., 1998; Moreno et al., 2001), it turns out that theachievable phase modulation is always accompanied by an undesired residual polarizationmodulation (and thus amplitude modulation).

Polarization modulations do not appear within alternative LC configurations, where therod-shaped molecules are aligned differently, for instance in parallel (see Fig. 1.1). In suchconfigurations, the birefringent molecules rotate along an axis perpendicular to both thelight propagation direction and polarization when the cell voltage changes, which allowsan almost pure phase modulation.

In contrast to optically addressed phase-only SLMs, which have been introduced tothe market many years ago, some companies started relatively recently to offer also elec-tronically addressed high resolution devices. Examples for devices being currently offeredare the the X-Y phase series SLM of Boulder Nonlinear Systems, the X10468 SLM ofHamamatsu Photonics and the HEO 1080 P of Holoeye Photonics.

All experiments described in this thesis were conducted by using the HEO 1080 P

4

CHAPTER 1. PHASE-ONLY LIQUID CRYSTAL SLMS 5

director d

� , n

� , n

V1~

E

E

V2~

E

E

��

rod-shapedbirefringent liquidcrystal molecule

Figure 1.1: Schematic of a parallel aligned LC cell. Changing the voltage across the layer causesthe birefringent molecules rotate along an axis perpendicular to both the light propagation directionand polarization. Thus the polarization remains almost unaffected.

PAL-SLM of Holoeye Photonics.

1.2 The HEO 1080 P spatial light modulator

The HEO 1080 P phase-only SLM is a reflective liquid crystal on silicon (LCoS) (Lueder,2001) display, i.e., after passing the LC layer an incident beam is reflected at a siliconmicrochip, which also contains the addressing circuitry. It has a resolution of 1920×1080pixels, where a single quadratic pixel has a side length of 8 microns. Our display is opti-mized for wavelength in the near infrared, since our laser source (Ytterbium fiber laser fromIPG Photonics) emits linearly polarized light at 1064 nm. Additional technical informationis provided by Holoeye Photonics at their webpage.

Figure 1.2: Panel of the HEO 1080 P phase-only spatial light modulator.


The following section contains information about phase modulation properties and read-out light intensity threshold of the HEO 1080 P modulator. Further information aboutliquid crystal SLMs can be found in Jesacher (2007).

before linearization

after linearization

2�

gra

yle

vel

0

255

length

Figure 1.3: Phase shift characteristics at 1064 nm of the infrared-optimized HEO 1080 P SLM.

doubleaperture

microscopeobjective

lin. pol.,collimated

SLM

fixed gray value(e.g. black)

CCD

variable gray value

Figure 1.4: Setup to determine the phase shift characteristics.

Fig. 1.3 describes the phase-shift characteristics of the panel. The data was obtainedby interference experiments according to the setup sketched in Fig. 1.4. Two collimatedand horizontally polarized beams are reflected from two distinct display areas showingdifferent gray values. The display “converts” the gray shades of the computer image intocell voltages and hence into definite phase shifts. The relative phase difference between thetwo reflected beams can be determined by the spatial shift of their interference pattern at


the CCD. In this manner one can measure the complete phase-shift characteristics of thepanel by stepping through all 256 gray levels with one half of the display.

Every row of the two images in Fig. 1.3 represents a separate fringe pattern, which hasbeen averaged over all rows, and corresponds to a specific gray value difference betweenthe two display halves. Thus the total number of rows is 256. As can be seen from the leftimage, the relationship between gray shade and resulting phase shift is nonlinear. Moreover,the maximal obtainable phase shift is almost exactly 2π, which allows the generation ofoptimal phase diffractive elements. The images of Fig. 1.3 were obtained using the PhaseCam software of Holoeye Photonics. In order to linearize the mapping of gray shades tophase values, one can create a specific lookup tables, which compensates for the measurednonlinear behaviour. The right image of Fig. 1.3 shows the phase-shift characteristics ofthe panel after linearization. The “ripples” apparent in both images originate from thedisplay “flickering” (Jesacher, 2007): To avoid dissociation of the LC, the cell voltage isalternating, with a temporal average of zero. The resulting small rotational movementsof the LC molecules cause an oscillation in the resulting phase shift and the diffractionefficiency. The display “flickering” strongly depends on the actual gray value and can bereduced by a special display configuration provided by Holoeye Photonics.

After linearization of the phase modulation behaviour, the diffraction performance ofthe SLM at 1064 nm was evaluated. An important parameter is the light utilizationefficiency, i.e the portion of light being reflected when the display is switched off. Inour case, the light utilization efficiency is 55%. The losses of 45% are dominated byabsorption, however a small part is also due to the limited fill factor, i.e. due to diffractionat the “dead” space between the individual pixels. In order to obtain the diffractionefficiency, the power of the first diffraction order created by a displayed sawtooth grating(10 pixel grating period) was measured. The relative intensities of the four most intensediffraction orders are listed in table 1.1. The efficiencies are defined as the power ratios ofthe corresponding diffraction orders to the readout light, i.e. the light utilization efficiencyis already included. The theoretical power maximum of the first diffraction order of a10-pixel grating is approximately 97%.

diff. order -1 0 +1 +2

% 1 6 41 1

Table 1.1: Diffraction at a sawtooth grating with a grating period of 10 pixels. Denoted are therelative intensities of the four most intense diffraction orders with respect to the intensity of theincident laser beam (Yb fiber laser, 1064 nm wavelength).

Another important point to be discussed in context with SLMs is the maximal toleratedreadout light intensity, since too high intensities can cause irreversible thermal damageto the liquid crystal. The birefringence of the LC begins to decrease above a certaintemperature. At first, this effect is reversible, further heating may however cause the LCto boil, which results in irreversible damage. The accepted intensity threshold for CWlasers can be measured with the setup shown in Fig. 1.5. The principle was taken from an


application note to the LCOS-SLM X10468 of Hamamatsu Photonics. For pulsed lasers,it has to be taken into account that high field amplitudes may also cause abrasion damageto the reflective dielectric or aluminum surfaces.

Within the experimental configuration of Fig. 1.5, changes of the LC birefringence aredetected by monitoring the laser power after the analyzer. Slight changes indicate reducedbirefringence. The polarization of the incident beam must be 45 degrees rotated withrespect to the “design polarization” of the modulator, i.e., the polarization state where itacts as a pure phase modulator (0 degrees in the case of our light modulator). Duringthe measurement, in order to gain a high measurement sensitivity, the display should beaddressed with a gray value where the gradient ∆I/∆GV is high. Here, I denotes the powermeasured by the powermeter and GV the gray value. Therefore, in order to find out themost appropriate gray value, it is reasonable to map the function I(GV ) by continuouslystepping though all available gray values.

In this manner, we examined the CW intensity threshold for the HEO 1080 P at awavelength of 1064 nm. The measured power started to decrease slightly after about 5minutes of exposure to a 2 W beam with a waist of 2.5 mm. After about 20 minutes,the measured power remained stable at an about 5% reduced level. Blocking the incidentbeam for a minute led to a significant “recovery”, which is an indicator that the loss ofbirefringence was caused by the beam exposure.

SLM

Polarizer 45°

Analyzer -45°Powermeter

Figure 1.5: Setup to determine the maximal tolerated readout light intensity (principle taken froman application note of Hamamatsu Photonics).

1.3 Shaping light fields with SLMs

In the following section, the possibilities of shaping light with a SLM as phase modulator arediscussed. Unlike other user-programmable phase modulators, like for instance deformablemirrors, the dynamic range (“stroke”) of SLMs is typically restricted to phase values around2π. Phase masks which require higher modulation bandwidths are displayed modulo 2π,i.e. they are “cut” into slices of 2π rad. This is demonstrated in Fig. 1.6, which showsthis principle applied to an inclined phase plane and a lens, respectively. Inclined phaseplanes are used for redirecting beams into certain angles, while lenses are used to controlthe beam divergence. Such 2π-discontinuities would not affect a monochromatic wave if


the phase “reset” regions, i.e., the regions where the phase values step from 2π to zero,were infinitely small. In practice however, these regions have a certain spatial extension,which leads to a degradation of the obtained phase accuracy (Wang et al., 2005).

Figure 1.6: Inclined phase plane and lens (first and third image) and how they are displayed on aSLM (second and fourth image).

Generally, phase structures displayed on a SLM are spatially digitized (not indicated inFig. 1.6). This “pixelation” restricts the spatial frequencies of phase masks to the maximalvalue kmax < π/∆P , where ∆P is the width of a single pixel. Consequently, the largestbeam deviation which can be obtained with an 8-micron-pixelated SLM is about 3.8◦ at1064 nm wavelength. Higher spatial frequencies cause aliasing (Jesacher, 2007).

It is also possible to modify the amplitude of light with a pure phase modulator, providedthat the “plane of interest” has a certain distance to the modulator, such that the phase-modulated wavefront can evolve into the desired shape. There exist several algorithms forderiving the corresponding diffractive phase masks (Kress and Meyrueis, 2000; Gerchbergand Saxton, 1972; Seldowitz et al., 1987; Kirkpatrick et al., 1983). One example is theGerchberg-Saxton (GS) algorithm (Gerchberg and Saxton, 1972), which is – in its originalor modified form – often utilized because of its speed and simplicity. Fig. 1.7 outlinesthe working principle of the GS algorithm. The figure was adopted from (Jesacher, 2007).Basically, the algorithm mathematically goes back and forth between the SLM plane (alsocalled the hologram plane) and image plane, where the desired intensity structure shouldbe created. To simplify mathematical calculations, the image plane is usually chosen tobe conjugate to the SLM plane, i.e., the fields in both planes are connected by the Fouriertransform. First, an initial condition for the field in the image plane has to be chosen,which consists of the desired amplitude and usually a randomly chosen phase. In thisexample, the aim is to create a 3×3 matrix of optical traps. This initial field is inverseFourier transformed in order to obtain the corresponding field in the SLM plane. Then, theamplitude function is replaced by the actual SLM illumination condition (a constant valuein our example). Another Fourier transform brings us back into the image plane. Replacingthe amplitude function by the desired amplitude distribution completes one cycle of theiterative algorithm. With every cycle, the amplitude A(~k) becomes more similar to thedesired field, i.e. the trap matrix. The algorithm aborts, when the cost function ε fallsbelow a user-defined value. ε can be defined as

ε =∑

~k

[A(~k)− A(~k)

]2, (1.1)


where A(~k) describes the trap matrix in our example. The obtained phase function φ(x, y)represents the output of the algorithm, i.e., the “phase hologram” of the trap matrix.

SLM planeimage plane

�(x,y)a(x,y)�(k ,k )x y

IFFT

rand( )k ,kx y1

......

.........

IFFT...

phasehologram=�(x,y) � �< lim

no

FFT�(x,y)Â( )k ,kx y

�( )k ,kx y 1

initial condition

yes: abort

Figure 1.7: Working principle of the Gerchberg-Saxton algorithm.

Techniques like the GS algorithm can be used for deriving phase holograms of almostarbitrary two-dimensional and even three-dimensional (Whyte and Courtial, 2005) intensitydistributions. However, it is clear that it is not possible to determine both amplitude andphase within the entire image plane with a single phase hologram, since this requires fullcontrol over the complex light field in the SLM plane. This is only possible if another degreeof freedom is provided to the algorithm: For instance, one can define a certain part of theimage plane to be “free”, i.e. the field there is not replaced by A(kx, ky) exp (iΦ(kx, ky))after each cycle. This method is often used in order to obtain a high degree of uniformitywithin reconstructed light fields (Seldowitz et al., 1987; Kim et al., 2006).

Another approach uses a fast modulating “carrier” sawtooth grating superposed to theoriginal phase pattern (Kirk and Jones, 1971). By locally modulating the phase modulationdepth of this grating, it is possible to control the amount of light which appears in its firstdiffraction order. Hence, the method is similar to the way amplitude information is storedin recorded off-axis holograms. Nonetheless, the efficiency of this method strongly dependson the amplitude function and can be very small in the case of high-contrast amplitudes.A method for shaping complex wavefronts which is applicable to binary phase modulatorswas introduced by Neil et al. (Neil et al., 2000c). The method is also based on shaping thefield of a single diffractive order and has an accordingly low efficiency.

Chapter 2

Shaping light with two cascaded

SLMs

In this thesis, the prospects of shaping light fields by two cascaded SLMs have been exam-ined. It will be shown in the following that the subsequent arrangement of two SLMs (orphase masks in general) bears interesting possibilities.

For instance, by arranging both SLMs within a 2-f setup, we have demonstrated the gen-eration of user-programmable “helical light patterns”, i.e. light fields which show a helicaltransverse phase profile. In such light patterns, the photon momentum has a componenttransverse to the optical axis. When interacting with matter, this transverse photon mo-mentum gives rise to transverse scattering forces. In our experimental configuration, thetransverse intensity profile of the light fields is determined by the first SLM, while the heli-cal phase is obtained by the second modulator, which acts as spiral phase filter (Furhapteret al., 2005). Experimental results of our investigations are presented in chapter 2.2 in apublication in Optics Communications.

Moreover, we showed that such a 2-f setup allows to create arbitrary amplitude andphase distributions with theoretically unlimited efficiency: During our investigations werediscovered a technique the basic principle of which had already been introduced in the1980’s by H. O. Bartelt (Bartelt, 1984, 1985), who performed preliminary experiments withbleached films. For the first time, we applied the method to SLMs. This allows the flexiblegeneration of arbitrary computer-designed complex field amplitudes with high quality (seejournal papers in chapters 2.3 and 2.4).

Both techniques – the method to create “helical light patterns” and the method togenerate arbitrary complex field amplitudes – have been evaluated in optical trappingexperiments, by letting the generated light fields interact with microscopic particles. Be-cause SLMs are often utilized as “steering elements” in so-called holographic optical tweez-ers (Dufresne and Grier, 1998; Liesener et al., 2000), the methods might find applicationsespecially in the field of optical micromanipulation. Since the forces exerted on micropar-ticles are very sensitive to the amplitude and phase of light, optical trapping is also wellsuited to judge the quality of the generated light fields.

11

CHAPTER 2. SHAPING LIGHT WITH TWO CASCADED SLMS 12

In the following section, the basic principles of trapping microscopic dielectric particleswith light are outlined.

2.1 Basic principles of optical trapping

Light carries momentum. This momentum is transferred to matter when photons areabsorbed or emitted. Although the resulting forces remain mostly unnoticeable in themacroscopic world they can have significant effects on microscopic particles or atoms.“Optical tweezers” are instruments which exploit these forces to trap and manipulatemicroparticles with light.

Since optical tweezers were demonstrated for the first time in 1970 (Ashkin, 1970),they underwent a continuous development. Nowadays there exist various experimental ap-proaches. The most commonly used type of optical tweezers are so-called single-beam gradi-ent traps (Ashkin et al., 1986), where a transparent dielectric particle is three-dimensionallytrapped by a single strongly focussed laser beam. How the stabilizing forces arise withinsuch a trap, is explained in Fig. 2.1. The explanation is based on a simple ray-opticsmodel and thus only valid for particles much larger than the light wavelength. Within theray-optics model, absorption as well as wave properties such as interference and diffractionare neglected. Each beam is characterized by intensity, propagation direction, and polar-ization. The figure shows a beam of high NA (represented by two single rays) penetratingfrom below into a transparent sphere, partly being reflected and refracted due to the stepin refractive index. The forces arising from the reflection and refraction are also visible inthe sketch (blue and orange arrows, forces are not to scale). Multiple reflections occurringinside the sphere are neglected. The left image shows the sphere located within the energypotential minimum, where all emerging forces yield zero. If the particle is moved awayfrom this location – for instance by an external force or Brownian motion – a net forcearises, trying to push the particle back. Such situations are described by the other twoimages in Fig. 2.1, which show the sphere located above and to the right of the potentialminimum, respectively. It can be shown that – for the case of a trapped sphere – theenergy potential landscape in a single-beam gradient trap can be considered harmonic ingood approximation for particle displacements below one sphere radius (Ashkin, 1992).However, the potential shows an axial asymmetry, which is caused by radiation pressure.For particle shapes of low symmetry, it is generally not possible to derive the potentiallandscape analytically.

Optical trapping of particles much smaller than a wavelength cannot be described usingthe ray-optics model, which does not consider diffraction and interference of light. In suchcases, useful results can be delivered by using the Rayleigh approximation, which treatsdielectric particles as single dipoles. The force on the particle can be decomposed into twoparts: the scattering force, which acts in direction of the Poynting vector, and the gradientforce, which is proportional to the light intensity gradient. The scattering force can be


FF

Figure 2.1: Principle of a single-beam gradient trap. A beam of high NA (represented by twosingle rays) penetrates from below into a transparent sphere, partly being reflected and refracted.Left image: The sphere is located at the energy potential minimum. Middle and right image: Adisplaced sphere feels a restoring force.

expressed as (Harada and Asakura, 1996)

~Fscat = nm〈~S〉σc

, (2.1)

where 〈~S〉 denotes the time averaged Poynting vector, and nm the refractive index of thesurrounding medium. The scattering cross section σ depends on the particle radius r, thewave number k, and the relative refractive index nr = n/nm:

σ =83π (kr)4 r2

(n2

r − 1)2

(n2r + 2)2

. (2.2)

The gradient force can be written as:

~Fgrad =α

21

cnmε0~∇〈~S〉, (2.3)

where α denotes the polarizability, c the vacuum speed of light, and ε0 the vacuum per-mittivity. The polarizability is further given by (Harada and Asakura, 1996)

α = 4πε0n2mr

3

(n2

r − 1n2

r + 2

), (2.4)

A difficulty arises in the case of particles the size of which is in the range of a wavelength.In this regime neither the ray-optics model nor the Rayleigh approximation (Harada andAsakura, 1996) is applicable. Here, the Generalized Lorentz-Mie theory (Gouesbet, 1999)can be used to calculate the arising forces.

2.1.1 Practical implementation of single-beam gradient traps

Usually, the high beam aperture required for trapping microparticles with a single beamcan be only provided by microscope objectives. Within the Rayleigh approximation, it is


possible to express an analytical stability criterion for a gaussian trapping beam (Ashkinet al., 1986):

Fgrad

Fscat=

3√

364π5

n2m(

n2r−1

n2r+2

) λ5

r3w20

≥ 1, (2.5)

where w0 denotes the waist of the Gaussian beam. According to this equation, stabletrapping is only possible for nr > 1, i.e. when the particle’s refractive index is greater thanthat of the surrounding medium. Typically, particles in water (e.g cells) can be stablytrapped with a NA ≈ 1. Most commonly, objectives with NA = 1.2− 1.4 are utilized foroptical trapping experiments.

Fig. 2.2 gives an idea of how optical tweezers can be realized in practice. A collimatedlaser is directed towards a gimbal-mounted rotatable mirror, which is imaged onto theobjective back focal plane by a telescope. In this plane, mirror rotations only affect thebeam propagation direction but not the beam position. In this manner, trap movementsalong one dimension can be achieved.

laser

dichroicmirror

to CCDcamera

100x1.3NA

gimbal mountedmirror

Figure 2.2: Implementation of single-beam optical tweezers. A collimated laser is coupled into ahigh NA microscope objective via a two-lens telescope.

2.1.2 SLM-based optical tweezers

To meet the specific demands of different applications, several types of laser-based microma-nipulation systems have been developed in the past, such as interferometric traps (Chowd-hury et al., 1985; Casaburi et al., 2005; Rohner et al., 2006), fiber traps (Constable et al.,1993; Guck et al., 2001) or SLM-based systems (Curtis et al., 2002; Liesener et al., 2000;Lynge Eriksen et al., 2002; Jesacher et al., 2004). A popular member of the latter typeare so-called holographic optical tweezers (HOT), which utilize a user-programmable SLMinstead of the rotatable mirror shown in Fig. 2.2. Trap movements within the object planeof the microscope can be achieved by displaying sawtooth (also called “blazed”) phase grat-ings on the modulator panel, which diffract the incident laser into a user-defined direction.Even axial trap movements can be obtained by displaying Fresnel lenses. The general-ized phase contrast (GPC) method introduced by Jesper Gluckstad and co-workers (LyngeEriksen et al., 2002) represents another type of SLM-based laser tweezers. There a phase


contrast filtered image of the SLM surface is directly imaged into the microscope objectplane. Intensity structures are designed by displaying the corresponding phase structureson the SLM.


2.2 PAPER: Optical tweezers of programmable shape with

transverse scattering forces

Alexander Jesacher, Christian Maurer, Severin Furhapter, Andreas Schwaighofer, StefanBernet, and Monika Ritsch-Marte, Optics Communications, 281, 2207-2212 (2008).

2.2.1 Summary

Optical processing with a phase vortex filter has proven to be strongly and isotropicallyedge enhancing for amplitude and phase objects (Davis et al., 2000; Swartzlander, 2001;Crabtree et al., 2004; Furhapter et al., 2005). The filter has a complex transmission ofexp (iθ), where θ is the azimuthal coordinate of the filter plane. Moreover, the filterprocess has interesting effects on the light phase: Its local value depends on the geometricorientation of the edges within the object (Jesacher et al., 2006).

In this paper, we describe the creation of line tweezers via spiral phase filtering and theireffects on microparticles. We show that the curvature of a line determines the amount oftransverse photon momentum and document that it is possible to transport polystyrenemicrobeads along such lines by exploiting the corresponding scattering forces. The tweezerspatterns and the spiral filter were both displayed on a single SLM.

Author's personal copy

Optical tweezers of programmable shape with transversescattering forces

Alexander Jesacher *, Christian Maurer, Severin Furhapter, Andreas Schwaighofer,Stefan Bernet, Monika Ritsch-Marte

Division for Biomedical Physics, Innsbruck Medical University, Mullerstraße 44, A-6020 Innsbruck, Austria

Received 13 November 2007; received in revised form 4 December 2007; accepted 4 December 2007

Abstract

We propose a non-holographic method to create line traps of arbitrary shape in the sample plane. Setting the phase gradient alongtheses lines gives control over the transverse forces acting on the confined particles. Phase structures, displayed on a spatial light mod-ulator, are optically processed by a spiral phase filter and imaged onto the object plane of a microscope objective. The resulting brightline structures can be used to trap microparticles. Additionally, they exert transverse scattering forces, which can be exploited for induc-ing orbital motions or for creating ‘‘attracting” or ‘‘repelling” points, respectively. We give theoretical and experimental evidence thatthese scattering forces are proportional to the curvature of the line tweezers.� 2007 Elsevier B.V. All rights reserved.

PACS: 42.30.Kq; 42.79.Kr; 42.79.Ci; 87.80.Cc

Keywords: Optical tweezers; Spatial filtering; Spatial light modulator; Spiral phase

1. Introduction

In the last decade, holographic or diffractive opticaltweezers [1] have greatly enhanced the possibilities ofmanipulating microscopic particles with light. Diffractivelycreated patterns of light can be used to trap and manipu-late dielectric particles by intensity gradients. The general-ized phase contrast (GPC) method has been introduced asan alternative to diffractive tweezers by Gluckstad andco-workers [2]. Basically the method is based upon project-ing a phase-contrast-filtered image of a computer-steeredphase modulator display into the object plane of a micro-scope. Phase structures, displayed at the SLM panel, aredirectly transferred into intensity images by the phase-con-trast filter. The technique offers high intensity efficiencyand allows real-time steering, since moving of traps canbe directly achieved by moving the corresponding trap

structures on the light modulator panel. However, to datesuch a method cannot be used to shape the trapping beaminto certain interesting mode structures, such as Laguerre–Gaussian beams, which in holographic setups can be usedto transfer controlled transverse momentum to the trappedobjects. A related method for micromanipulation, whichwas recently proposed by Guo et al. [3], makes use of aspiral phase filter [4], instead of a phase-contrast filter.The filter operation performed by the spiral phase filterrepresents a two-dimensional generalization of the Hilberttransform [5] and creates field amplitudes which areroughly proportional to the phase and amplitude gradientsof the patterns displayed at the modulator panel [6]. Con-sequently, the edges of binary amplitude or phase struc-tures appear strongly enhanced. However, probably themost interesting property of a light field created by spiralphase filtering is that it shows a specific phase profile thatmay give rise to transverse scattering forces on microparti-cles. Unlike forces caused by intensity gradients, scatteringforces are generally not conservative, which allows, for

0030-4018/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.optcom.2007.12.042

* Corresponding author.E-mail address: [email protected] (A. Jesacher).

www.elsevier.com/locate/optcom

Available online at www.sciencedirect.com

Optics Communications 281 (2008) 2207–2212


instance, the realization of orbital motion. An example forlight fields creating such motion are Laguerre–Gaussianmodes, which show a helical phase structure of the formexp½ih�, where h is the azimuthal coordinate. The opticaltorque resulting from this helical phase acts as ‘‘opticalspanner” [7,8] and can induce a directed flow of micropar-ticles [9]. By modifying the helical phase profile ofLaguerre–Gaussian modes, it is also possible to modifytheir circular shape [10].

In this paper we present the experimental realization andevaluation of spiral phase tweezers, as proposed by Guoet al. [3], which – without using holographic methods –combine advantages of the GPC method like the easy gen-eration of arbitrary light structures and real-time steeringwith the possibility of transferring optical torque. We alsodemonstrate an interesting feature of this filter method,which has not been described in Ref. [3], namely that thetransverse scattering forces are proportional to the curva-ture of the projected light patterns.

2. Properties of spiral phase filtered light fields

To understand the effects of the spiral phase filter, let usconsider the point spread function (PSF) of the system. ThePSF describes how a point source dð~rÞ is imaged by a givensystem. In ‘‘filter-less” imaging systems like telescopes, thePSF usually corresponds to the Airy function. Analo-gously, the point spread function of a spiral phase imagingsystem is represented by a diffraction limited optical vortex,which shows the same helical phase profile as the spiral fil-ter. Its radial amplitude profile f ðrÞ can be described usingBessel and Struve functions [6,11].

When an extended object is imaged by a spiral phaseimaging system, every single point of this object appearsas a tiny optical vortex in the image plane (see Fig. 1).The interference of all vortices yields the complete spiral-filtered image. To achieve the resulting field at a certainpoint ~P , one has to integrate the contributions of all vorti-

ces to the field at this point. For simplicity, let us assumethat ~P coincides with the origin of our coordinate system,i.e., ~P ¼~0. Then, the vortex located at an adjacent point~Q ¼ ðrQ;/QÞ contributes to the field in point ~0 withEinðrQ;/QÞf ðrQÞ exp½iðpþ /QÞ�, where Ein is the originallight field (see Fig. 1). Note that here the geometric angle/Q appears in the phase of the light field.

Thus one can express the total output field at point~0 asfollows:

Eoutð~0Þ /Z 1

0

Z 2p

0

Ein r;/ð Þf ðrÞ exp½i/�r dr d/: ð1Þ

In practice, the extension of f ðrÞ, i.e. the PSF, is usuallymuch smaller than typical variations in Ein. In this case itis valid to expand the input light field in a Taylor seriesto first order:

Einð~rÞ � Einð~0Þ þ ~rEinð~rÞj~0 �~r

¼ Einð~0Þ þ ~gAð~0Þ exp½iWinð~0Þ� þ i~gPð~0ÞEinð~0Þ� �

�~r;

ð2Þ

and to restrict the r-integration to a limited interval ½0; q�,where q defines the approximate radius of the PSF. Inthe above equation, ~gA ¼ ~rjEinj and ~gP ¼ ~rWin representthe amplitude and phase gradients of the complexfield Ein ¼ jEinj exp½iWin�. Inserting Eq. (2) into Eq. (1)yields

Eoutð~0Þ / Einð~0ÞZ q

0

Z 2p

0

exp½i/�f ðrÞr dr d/

þ exp½iWinð~0Þ�Z q

0

Z 2p

0

~gAð~0Þ �~r exp½i/�f ðrÞr dr d/

þ iEinð~0ÞZ q

0

Z 2p

0

~gPð~0Þ �~r exp i/½ �f ðrÞr dr d/:

ð3Þ

Because of the integration over exp½i/�, the first term iszero. Evaluating the other two terms results in:

Fig. 1. Left: Every single point of the input light field (star) is imaged as diffraction limited optical vortex. Right: Sketch to visualize the contribution of anoptical vortex located at ~Q to the field at point ~P .

2208 A. Jesacher et al. / Optics Communications 281 (2008) 2207–2212


Eoutð~0Þ / exp½iWinð~0Þ�gAð~0Þ exp½idAð~0Þ�n Z q

0

Z 2p

0

f ðrÞr2 dr d/

þ exp½�idAð~0Þ�Z q

0

Z 2p

0

exp½i2/�f ðrÞr2drd/

�

þ iEinð~0ÞgPð~0Þ exp½idPð~0Þ�n Z q

0

Z 2p

0

f ðrÞr2 drd/

þ exp½�idPð~0Þ�Z q

0

Z 2p

0

exp½i2/�f ðrÞr2 drd/

�;

ð4Þ

where dA and dP are the geometric polar angles of theamplitude and phase gradients. In the above equation,the integrals containing exp½i2/� yield again zero, and theintegral

R q0

f ðrÞr2 dr yields a constant value. The filteredlight field can finally be written as [6]

Eoutð~P Þ / exp½iWinð~P Þ�gAð~P Þ exp½idAð~P Þ�þ iEinð~PÞgPð~PÞ exp½idPð~P Þ�: ð5Þ

This result is valid for an arbitrary point ~P . It shows thatthe filtered light field is intense at amplitude and phaseedges. In our experiments we restrict ourselves to phasestructures, which do not absorb light and thus allow higherefficiency. Assuming Ein to be a pure phase function, Eq. 6simplifies to

Eoutð~P Þ / gPð~P Þ exp½iðWinð~P Þ þ dPð~PÞÞ�: ð6ÞThe filtered light field shows an interesting phase profile: itsphase depends on dP, which is the geometric angle of the

phase gradient. Consequently, straight edges will show aconstant phase value, while a curve – where dP varies alongits edge – will show a transverse phase gradient the ampli-tude of which is proportional to the curve bending. Thisproperty is also the basis of interferogram deconvolutionin so-called ‘‘spiral interferometry” [12,13]. Using spiral-fil-tered light fields for micromanipulation, this means thatcurved parts within such structures exert transverse scatter-ing force to microparticles, while straight lines do not.

3. Experimental setup

The probably simplest way of realizing spiral phasetweezers is to utilize a computer steered spatial light mod-ulator (SLM) for displaying the phase structures and a sep-arate static vortex filter element in a Fourier plane (seeFig. 2a), which for instance can be manufactured byphoto-lithography [14]. Since such a filter element designedfor a wavelength of 1064 nm (the wavelength of our trap-ping laser) was not available in our case, we decided touse one single SLM to display both, the structures to be fil-tered and the spiral phase filter. The two correspondingphase functions are placed side by side at the SLM panel.A sketch of the setup is shown in Fig. 2.

The structure pattern (pattern P S) is fully illuminated byan expanded Ytterbium fiber laser at k = 1064 nm. The dif-fracted light is subsequently focussed onto the spiral filterfunction (pattern P F). Consequently, the Fourier Trans-form of P S emerges at the filter function P F. A sufficiently

filtered image of PS

PS . . . phase object (star)

collimated beam@1064nm

PF

. . . phase vortex filter

PAL-SLM

100xNA 1.3

dichroicmirror

to camera

f2=750 mm f3=400 mm

f1=2000 mm

spiral phase filter

Fig. 2. (a) Principle of optical processing with a spiral phase filter. (b) Sketch of the experimental setup. One SLM displays both, the phase structure to befiltered and the spiral phase filter. The patterns are arranged side by side at the SLM panel. The gray values of the patterns represent phase values from 0 to2p. Phase edges within P S appear as bright lines in the spiral-filtered image under the microscope.

A. Jesacher et al. / Optics Communications 281 (2008) 2207–2212 2209


large focal length of the Fourier Transforming lensðf1 ¼ 2000 mmÞ prevents the display from being damagedby too high laser intensity and leads to a strongly down-sized image of P S in the microscope object plane. This isdesired, if one wants to exploit the torque of the filteredlight field. Finally the filtered light is coupled into a micro-scope objective by a 2-lens telescope.

The utilized SLM is a HEO 1080 parallel aligned nema-tic device from Holoeye Photonics AG and shows a diffrac-tion efficiency of approximately 40% at the wavelength of1064 nm. The residual light is either absorbed (about50%) or appears in other, undesired diffractive orders. Toachieve a spatial separation of these orders from thedesired light, we superpose both phase patterns by inclinedphase planes (see Fig. 2). Regarding the limited phase mod-ulation depth of the light modulator, the functions P S andP F are displayed modulo 2p radian:

P Sðx; yÞ ¼ mod2p½Sðx; yÞ þ kSx�; ð7ÞP Fðx; yÞ ¼ mod2p½argðxþ iyÞ þ kFx�: ð8Þ

There, Sðx; yÞ is the structure pattern – in our case a binaryphase function. kS and kF are the gradients of the super-posed inclined phase planes.

4. Experimental results

Fig. 3 shows various phase structures P S, as they appearunder the microscope, when the filter pattern P F is a pureblazed grating without vortex. The dark lines dividingareas of different phase shift are explained by the low

numerical aperture of our system (NA = 0.002): most ofthe light scattered from these lines is not caught by the sub-sequent optics, which lets them appear dark in the image.Adding a phase spiral to P F results in strongly edgeenhanced images (second row), as expected from Eq. (6).The squares surrounding the individual structures markthe boundaries of the pattern windows at the SLM panel.

Although the vortex filter is a pure phase filter and henceis supposed to preserve total power, it may practically notact intensity preserving in every case. Light, which isfocussed into a small region around the singular filter cen-ter is strongly scattered [15] and may therefore be lost, if itis not captured by subsequent optics. Although this effect isof minor importance in specifically designed spiral phaseimaging setups [4], it is strongly pronounced in the setuppresented here, since the NA is very small. However, on

Fig. 3. Various binary phase structures, created by the SLM. The first row shows brightfield images of the structures as they appear under the microscope,if P F is a pure grating without vortex. The images of the second row represent the spiral phase filtered patterns. As expected, the phase edges are stronglyemphasized. As indicated by the images of the third row, they can act as optical traps for polystyrene microbeads (diameter 750 nm). The transversescattering forces cause the particles to move along the bright lines. The total laser power in the microscope object plane was about 70 mW.

T=0 T=0.3 s T=0.6 s

Fig. 4. Polystyrene beads (diameter = 750 nm), travelling in counter-clockwise direction along the edges of a spiral-filtered phase triangle. Themean bead velocity is about 1.5 lm per second at total power of about70 mW in the microscope object plane.

2210 A. Jesacher et al. / Optics Communications 281 (2008) 2207–2212


the other hand this ensures a high degree of isotropy in thespiral-filtered images [6].

The images in the third row of the figure demonstratethe ability of the bright edges to act as optical traps. Awater layer containing polystyrene microbeads (diameter750 nm) was sandwiched between two glass coverslips, suchthat the layer thickness was about 20 lm. The beads werelaterally trapped by the bright edges and simultaneouslypressed onto the upper coverslip by scattering forces. Theused objective is a ZEISS Neofluar with a magnificationof 100 and a NA of 1.3.

The lateral phase gradients of the light fields at the struc-ture edges cause the trapped beads to travel along thestructure boundaries, as indicated by the arrows inFig. 3. This effect is also demonstrated in Fig. 4 by meansof movie snapshots. The observed motion only occurswhen the structure is completely filled with microparticles,which is expected, since straight lines have a constant phaseand do not exert transverse scattering forces. Thus, a singlebead at any position on a straight line is not pushed bylight forces into any direction. This has also been experi-mentally checked by creating straight line traps with thedescribed method. On the other hand, if a closed line is‘‘filled” with beads, the resulting closed chain alwaysmoves, since beads in a region without local tangentialforces are pushed by neighbouring ones which are locatedat positions where a correspondingly higher momentumis transferred. Such a ‘‘closed chain” of beads also over-comes local stable trapping positions that can appear dueto inhomogeneities of the light intensity distribution alonga curve.

A further prediction of Eq. (6) is that the sign of theexerted optical torque depends on the sign of the line cur-vature. Hence, a turning point within a line trap shouldrepresent either a point of attraction or repulsion, depend-ing on whether the vortex filter is of the form exp½i/� orexp½�i/�. The experimental demonstration of this predic-tion is given in Fig. 5. The figure shows such a line trap,which was created by vortex filtering a specifically designedphase step (first and second image). As expected, a positivehelicity caused a ‘‘compression” of the beads at the turningpoint, while a negative filter helicity led to a repulsion. Cor-respondingly, a soft object (like a red blood cell) trapped atsuch a point will experience a stretching or a compressingforce, depending on the sign of the local curvature of itspath.

5. Discussion

The presented experiments describe the, to our knowl-edge, first realization of optical tweezers shaped by a spiralphase filter, similarly as proposed by Guo et al. [3]. Thesetup makes use of one single phase modulator to createboth, the trapping structures and the vortex filter. Thesetup shows some advantageous features of the GPCmethod [2], such as the straightforward non-holographicgeneration of light structures and real-time trap steering,with the additional feature of transverse momentum trans-fer. Our observations confirm that this transfer depends onthe geometric curvature of the line trap. However, due tothe structure of the vortex filter, the transverse phase gradi-ent necessarily has – independent of the trapping structure– a mean value of 2p=L, where L is the total length of theline trap. The transverse momentum transferred to polysty-rene microparticles in water has, according to our observa-tions, only a significant amplitude for relatively strongcurvatures with radii in the range of 1 lm. Thus the pre-sented method is probably only suitable for very specifictasks, but nevertheless represents a step towards the reali-zation of arbitrarily shaped tweezers, where gradient andscattering forces can be adjusted independently.

Acknowledgements

This work was supported by the Austrian Science Foun-dation (FWF), Project Nos. P18051-N02 and P19582.

References

[1] J. Liesener, M. Reicherter, T. Haist, H.J. Tiziani, Opt. Commun. 185(2000) 77.

[2] R.L. Eriksen, V.R. Daria, J. Gluckstad, Opt. Express 10 (2002)597.

[3] Cheng-Shan Guo, Yan Zhang, Yu-Jing Han, Jian-Ping Ding, Hui-Tian Wang, Opt. Commun. 259 (2006) 449.

[4] S. Furhapter, A. Jesacher, S. Bernet, M. Ritsch-Marte, Opt. Express13 (2005) 689.

[5] R.N. Bracewell, in: The Fourier Transform and its Applications,McGraw-Hill, New York, 1965.

[6] A. Jesacher, S. Furhapter, S. Bernet, M. Ritsch-Marte, J. Opt. Soc.Am. A 23 (2006) 1400.

[7] N.B. Simpson, K. Dholakia, L. Allen, M.J. Padgett, Opt. Lett. 23(1997) 52.

[8] A.T. O’Neil, I. MacVicar, L. Allen, M.J. Padgett, Phys. Rev. Lett. 88(2002) 053601.

Fig. 5. Turning points represent points of attraction or repulsion, depending on the sign of the filter helicity. The total laser power in the object plane wasabout 90 mW.

A. Jesacher et al. / Optics Communications 281 (2008) 2207–2212 2211


2.3 PAPER: Near-perfect hologram reconstruction with a

spatial light modulator

Alexander Jesacher, Christian Maurer, Andreas Schwaighofer, Stefan Bernet, and MonikaRitsch-Marte, Optics Express, 16, 2597–2603 (2008).

2.3.1 Summary

In 1984, H. O. Bartelt demonstrated the generation of complex amplitude fields with twophase masks arranged in a 2-f setup, i.e., the two masks were placed in the front andback focal plane of a lens, respectively. The phase masks were experimentally realized bybleached films. The most remarkable benefit of this approach compared to others is that itcan theoretically achieve a light efficiency of almost 100%. Only residual losses are inducedby the non-perfect transmissivity and the spatial digitization of the phase masks.

The paper describes how the approach of H. O. Bartelt can be applied to real-time pro-grammable SLMs and how a single SLM can be used to display both required phase masks.Results of holographic image creation are presented, which confirm the high efficiency ofthe method and reveal a high quality of the created images.

Near-perfect hologram reconstructionwith a spatial light modulator

Alexander Jesacher, Christian Maurer, Andreas Schwaighofer,Stefan Bernet and Monika Ritsch-Marte

Division for Biomedical Physics, Innsbruck Medical University, Mullerstrasse 44A-6020 Innsbruck, Austria

[email protected]

Abstract: We present an implementation method for noiseless holographicprojection of precalculated light fields with a spatial light modulator. Inthe reconstructed image, both the spatial amplitude and phase distributionscan be programmed independently. This is achieved by diffracting the lightfrom two successive phase holograms that are located in conjugate Fourierplanes. The light path is folded such that the two corresponding phasemasks can be displayed side by side at a single phase-only spatial lightmodulator. Such a device has relevant applications in holographic display-or projection systems, and for optical micromanipulation in laser tweezers.

© 2008 Optical Society of America

OCIS codes: (090.1995) Digital holography; (230.6120) Spatial light modulators

References and links1. P. Hariharan, Optical Holography: Principles, Techniques, and Applications (Cambridge University Press, 1996).2. B. Kress and P. Meyrueis, Digital diffractive optics (Wiley, 2000).3. H. Kim, K. Choi, and B. Lee, “Diffractive Optic Synthesis and Analysis of Light Fields and Recent Applications,”

Jpn. J. Appl. Phys. 45, 6555–6575 (2006).4. M.-L. Hsieh, M.-L. Chen, and C.-J. Cheng, “Improvement of the complex modulated characteristic of cascaded

liquid crystal spatial light modulators by using a novel amplitude compensated technique,” Opt. Eng. 46, 07501(2007).

5. J. P. Kirk and A. L. Jones, “Phase-Only Complex-Valued Spatial Filter,” J. Opt. Soc. Am. 61, 1023–1028 (1971).6. C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector

beams,” N. J. Phys. 9, 78 (2007).7. H. Bartelt, “Computer-generated holographic component with optimum light efficiency,” Appl. Opt. 23, 1499–

1502 (1984).8. H. O. Bartelt, “Applications of the tandem component: an element with optimum light efficiency,” Appl. Opt. 24,

3811–3816 (1985).9. Y. Roichman and D. G. Grier, “Projecting extended optical traps with shape-phase holography,” Opt. Lett. 31,

1675–1677 (2006).10. G. O. Reynolds, J. B. Develis, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier

Optics (SPIE, 1989).11. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and

diffraction plane pictures,” Optik 35, 237-246 (1972).12. S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys.

Rev. Lett. 57, 314–317 (1986).13. D. McGloin, G. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Applications of spatial light modulators in

atom optics,” Opt. Express 11, 158–166 (2003).14. J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, “Multi-functional optical tweezers using computer-

generated holograms,” Opt. Commun. 185, 77–82 (2000).15. A. Jesacher, C. Maurer, S. Furhapter, A. Schwaighofer, S. Bernet, and M. Ritsch-Marte, “Optical tweezers of

programmable shape with transverse scattering forces,” Opt. Commun. (to be published).

#90962 - $15.00 USD Received 17 Dec 2007; revised 25 Jan 2008; accepted 27 Jan 2008; published 11 Feb 2008

1. Introduction

In principle computer generated holography (CGH) should be easy: in order to obtain a perfectreconstruction of any precalculated light field distribution one just has to “somehow imprint”the two-dimensional Fourier transform of the desired holographic image (optionally superposedwith a parabolic phase term to control the imaging distance) onto an incoming wave.

However, in practice the “imprinting process” is difficult. One reason for this is that the two-dimensional Fourier transform of an image is generally complex, which means that both theamplitude and the phase of an incoming wave have to be controlled, while display elementsfor holographic reconstruction, like CGHs or real-time programmable spatial light modulators(SLMs) are typically phase-only modulators.

Although there are ways to encode light fields in a single phase-only diffractive optical ele-ment (DOE) [1, 2, 3], it generally turns out that this requires serious trade-offs, producing forexample increased speckle noise and loss of independent control over amplitude and phase ofthe reconstructed light field.

On the other hand exist methods where both, the amplitude and the phase of the incominglight wave are modulated, for example by using sophisticated photographic materials to producereferenceless on-axis complex holograms (ROACH) [1], or more recently, by using a phase andan amplitude SLM cascaded in a 4-f-system [4]. Further possibilities are to control the intensityof the diffracted light by modifying the local grating depth of an off-axis phase DOE [5, 6], orby properly dividing a DOE into assigned and unassigned regions [9]. However, typical Fourierholograms usually exhibit a huge intensity contrast of several orders of magnitude betweentheir central zero-order Fourier component, and the outer higher-order components. Techniqueswhich shape such a high contrast amplitude profile with an absorptive mask from a uniformbeam sacrifice most of the intensity.

Here we present a SLM-based practical realization of an approach, which was introduced in1984 by H. Bartelt [7, 8]. The method allows to shape arbitrary complex light fields by usingtwo subsequently arranged phase DOEs. Instead of producing the amplitude modulation with anabsorbing mask, this is achieved by redistributing the intensity of an incoming beam by on-axisdiffraction at a first phase-hologram. This projects the desired precalculated amplitude modula-tion with a theoretically unlimited high contrast, and without sacrificing any light intensity, to asecond phase-only display, which adjusts the phases of the respective Fourier components. Thusbehind the second phase mask the full complex amplitude of a precalculated field distributionis generated. Such a complete complex modulation produces a speckle-free hologram readoutwhere both, the amplitude and the phase of the reconstructed image can be adjusted indepen-dently. Although the hologram is reconstructed “on-axis”, the generated light field contains noundesired contributions from other diffraction orders, and no disturbing zero-order Fourier spotoften appearing in non-optimized DOEs. Since, in principle, no light is absorbed, the wholeincoming intensity is used for imaging, and the theoretically obtainable efficiency is 100%.

So far, implementations of this technique used static diffractive elements, such as bleachedfilms [8], the production of which is relatively complex and time consuming. Here we exper-imentally demonstrate the feasibility of creating arbitrary complex light fields of high qualitywith real-time programmable SLMs. Our proposed setup requires only one single SLM, whichdisplays both required diffractive patterns side by side. Moreover, we show that image recon-struction is even possible, if the two phase masks are only binary. This allows one to use themethod also with ferroelectric SLMs, which offer only binary phase modulation, but have theadvantage of a very high switching speed.


2. Principle of the method

Figure 1 outlines the principle of the technique schematically. The method requires two phasediffractive elements, placed in conjugate Fourier planes in the beam path (planes refereed to as“plane 1” and “Fourier plane” in the figure), which means that the light fields in these planesare mathematically “connected” by the Fourier transform [10]. In the figure, the two elementsare referred to as P1 and P2, respectively. The complex fields in the planes directly before andafter P2 are each represented by two images, showing the actual amplitude and phase.

lens

f

P1

f f fplane 1 Fourier plane object plane

P2

|A| � |A| �

lens

|a|

Fig. 1. Principle of the method. A specific complex light field in the object plane (rightplane) is created by two phase-diffractive patterns (P1 and P2 – gray shades in the figurecorrespond to phase values from 0 to 2π), which are placed in two conjugate planes (plane 1and the Fourier plane). P1 is iteratively optimized to create the amplitude |A| in the Fourierplane. There, the desired phase function Φ is shaped from the “noisy” phase Θ by patternP2.

Let us assume that one wants to create a specific light field a(x,y) in the object plane(plane on the right). In our example, the amplitude of this specific light field has the shapeof our department logo, and we chose its phase to be uniform. The creation of the fielda(x,y) = |a(x,y)| exp [iφ(x,y)] in the object plane demands a definite complex field A(u,v) =|A(u,v)| exp [iΦ(u,v)] in the Fourier plane (right pair of images in the Fourier plane), whichcan be expressed as:

A(u,v) =1

λ f

∫

x

∫

ya(x,y) exp

[−i

2πλ f

(ux+ yv)]

dx dy. (1)

Here, λ is the laser wavelength and f the focal length of the Fourier-transforming lens.The phase of the field A(u,v) could be directly shaped by the element P2 in the Fourier plane,

but not its amplitude. Thus the amplitude of A(u,v) must be created by P1, which is located


in plane 1. P1 can be iteratively optimized using specific phase DOE design algorithms [2],such that it creates the desired modulus |A(u,v)| in the Fourier plane when it is homogeneouslyilluminated. This even works if |A(u,v)| exhibits very high contrast, as typical for Fourier holo-grams. However, a consequence of this optimization procedure is that the field created by P1 inthe plane P2 will have a randomly distributed phase Θ(u,v,). Generally this “noisy” phase willnot correspond to the demanded phase function Φ(u,v). Although the phase Θ(u,v,) cannot becontrolled, since it is used as the optimization parameter to create |A(u,v)|, it is known. Now theidea is to modify Θ by P2 such that the result is Φ(u,v). Consequently, P2 has to be calculatedas follows:

P2 = mod2π{Φ−Θ}, (2)

where mod2π{...} symbolizes the “modulo-” operation that restricts the phase of P2 to an in-terval between 0 and 2π . Altogether, this procedure generates the desired complex field A(u,v)directly behind the second phase mask P2, and hence also a(x,y) in the object plane. Theoreti-cally, no light is lost since the method solely utilizes non-absorbing phase elements.

3. Experimental setup and results

Figure 2 illustrates how this concept can be realized with a single SLM. For our experimen-tal work, we utilized a HEO 1080 parallel aligned nematic SLM from Holoeye Photonics AG,which is a pure phase modulator with an active area of approximately 2 × 1 cm 2, consistingof 1920 × 1080 pixels. Each pixel measures 8×8 μm 2. The display has a light utilization effi-ciency (defined as the power ratio of zeroth diffraction order to incident beam when no patternis displayed) of 55%, which is mainly determined by absorption. Typical values of the zerothand first diffraction orders of displayed phase structures are 6% and 41%, respectively. Thevalues represent power ratios of the according diffraction order to incident light and correspondto a 10 pixel sawtooth grating (measured values). The zeroth order is mainly caused by thelimited fill factor of the SLM.

P2

sphericalmirrorf =300 mmM


PAL-SLM

P1

CCD

6°

4°

Fig. 2. Sketch of the experimental setup. The phase patterns P1 and P2 are displayed side byside at the phase modulator panel. The light diffracted from P1 is reflected back onto P2 bya slightly tilted concave mirror, which additionally performs the optical Fourier transform.P2 rearranges the light phase and creates the desired phase function.


The two patterns P1 and P2 (each measures 960×960 pixels) are displayed side by side atthe modulator panel. Almost all pixels can be used to display the square patterns, since thepanel dimensions show a relation of 16:9. P1 was obtained by 15 iterations of the Gerchberg-Saxton algorithm [11]. It is illuminated by an expanded and collimated Ytterbium fiber laser(horizontally polarized) emitting at λ=1064 nm. No polarizers are used in the present config-uration. The diffracted laser is subsequently reflected onto P2 by a concave mirror, such thatthe desired modulus |A(u,v)| emerges in the plane of P2. There P2 shapes the demanded phasefunction Φ(u,v) from the noisy, but known, wavefront of the incoming wave. Finally, the re-sulting complex field is again Fourier transformed by an imaging lens such that the desired lightfield appears at the CCD (DVC-1412 camera). In the present configuration, the two subsequentdiffraction processes restrict the maximal obtainable diffraction efficiency to only about 17%,which is mainly determined by the absorption of our SLM, but not by fundamental reasons.The red beam in the sketch of Fig. 2 indicates the laser path if the SLM is switched off, i.e.when it acts as a mirror. This path is identical to that of the zeroth diffraction orders of P1 andP2.

Strictly speaking, Eq. 1 is not applicable for the experimental situation described in Fig. 2,since the pattern planes and the concave mirror are slightly tilted with respect to their counter-parts in Fig. 1. However, the tilt angles are very small (maximal 3 degrees), and hence allowabandoning a corresponding modification of Eq. 1 without considerable quality reduction. Acrucial point for achieving good holographic reconstructions is the accurate alignment of P2,which should be exact within a few microns. In the experiment the alignment can be achievedwithout mechanical tools by electronically shifting the phase masks across the SLM display.The sensitivity of image quality to misalignments is determined by f M , the focal length of theconcave mirror, since this parameter determines the highest possible spatial frequency of the“noisy” phase Θ(u,v). A relative short focal length fM leads to high spatial noise frequencies

Inte

nsity

/arb

.units cross

sections

(c) binary(b) 256 gray levels (d) optimized kinoform(a) original

Fig. 3. Holographical reconstructions of a gray-scale image. (a) represents the original. (b)and (c) have been created with the presented method, using 256 or 2 distinct phase levelsfor displaying the phase patterns, respectively. (d) shows the experimental reconstruction ofthe image from a single optimized phase DOE (256 phase levels) for comparison. Clearlyvisible are the “noisy” texture and the zero diffraction order, which is represented by theintense spot in the image center. The plots below the images show the individual intensitiesalong a definite cross section (indicated by the yellow dashed line in (a))


and hence to a high sensitivity to misalignment, but, on the other hand, increases the resolutionof the reconstructed amplitude |A(u,v)|. Furthermore, the angles of incidence of the beam tothe display should be as small as possible. According to our experience, good results can beachieved with fM=300 mm.

A translation of the phase mask P1 across the SLM screen does not result in an image degra-dation, but just in a corresponding translation of the reconstructed image in the camera plane.This is due to the Fourier-shift theorem, which assures that a transverse shift of P1 does notlead to a shift of the reconstructed amplitude pattern |A(u,v)| in the plane P2, but just to alinear phase-offset of this pattern, which, due to the subsequent Fourier transform, finally trans-lates into a lateral image shift in the observation plane.

Figure 3 demonstrates the performance of the experimental setup. It shows three holographicreconstructions of our department logo, which have been created using different methods. Thegray-scale image on the left (a) represents the original logo. (b) is a holographically recon-structed image that was created by exploiting all available phase levels of our 8-bit display. P1and P2 were calculated “on-axis”, which in the case of “normal” Fourier-holograms typicallyleads to contributions of undesired diffraction orders, i.e. a bright spot of the zero order in thecenter. However, image (b) shows that this undesired effect does not occur in our setup. Themeasured light efficiency was 15%. The patterns used for this experiment are printed in Fig. 2as small gray scale images.

The third image (c) was created by “binarizing” the patterns used in (b) and hence demon-strates the applicability of the method to binary ferroelectric SLMs. Here, both patterns werecalculated “off-axis” in order to avoid image degradation by the now intense conjugate andzero diffraction orders. The measured light efficiency was 1.5%. Finally, the logo on the right(d) was shaped by a single kinoform, which has been optimized using the Gerchberg-Saxton al-gorithm [11]. The efficiency was close to the theoretical maximum. However, the image clearlyshows the typical “noisy” texture of a kinoform image, which is caused by the randomizedphase, and an intense zero order spot near the center.

A first step for using this setup for three-dimensional holographic display- or projectiontasks is to reconstruct two objects in two distinct axial planes. This is demonstrated in Fig. 4.The left image is a holographic reconstruction of an image containing a rabbit and an eagle,both in focus and (virtually) having an infinite distance to the camera. Then the hologram wasrecalculated, adapting the two phase masks at the SLM, such that the eagle had a virtual distance

Fig. 4. Left: Images of a rabbit and an eagle are reconstructed in a common axial plane.Thus both animals appear sharply. Middle and right image: The rabbit and the eagle emergein two distinct axial planes. Different focus settings allow to see only one animal sharplyat a time, while the other one appears blurred.


of now only about one meter from the camera. As a consequence, the eagle appears blurred inthe middle image of Fig. 4. Finally, the camera was focussed on the eagle (right image) whichleads simultaneously to a defocussing of the rabbit.

4. Summary and discussion

We have presented an SLM-based implementation of a method to create complex hologramsusing two cascaded phase DOEs. Compared to existing techniques, the method shapes the am-plitude profile by pure phase elements, which allows a theoretical light efficiency of almost100%. In addition, high contrast amplitudes, which usually appear in Fourier holograms [1],can be easily created. Although there are alternative methods to create speckle-free complexlight fields with a single phase diffractive pattern (for instance via a confined reconstructionarea), they are not lossless and imply side-effects such as the appearance of sometimes unde-sired light in higher diffraction orders.

Our implementation requires only a single SLM, which displays both diffractive patterns sideby side. We have compared the resulting image quality with that of an optimized kinoform.Furthermore, we have demonstrated the feasibility of the technique in view of binary DOEs,such as ferroelectric SLMs. In the present SLM configuration, the theoretically obtainable lightefficiency was limited by the absorption of the SLM display to about 17%. However, a valueclose to this theoretical maximum was also experimentally achieved, which indicates that theefficiency of an implementation with ideal DOEs would be close to 100%.

The possibility to project light with full efficiency into precalculated amplitude and phasefields has many practical applications. First, the method leads to speckle-free images, sincespeckles arise as a consequence of a speckled object phase and the discarded amplitude instandard phase holograms, kinoforms, or DOEs. Second, in holographic display- or projectionsystems, shaping not only the intensity but also the phase distribution of the reconstructed imageallows one to control the light field also in out-of-focus planes. For example, an image area witha flat phase profile has a large depth of sharpness – such areas will be sharply visible even inout-of-focus observation planes. On the contrary, image areas with a noisy phase distributionhave a low depth of focus and will be sharply reconstructed only in focus. Phase shaping alsoallows to determine the divergence and the direction of light behind (or in front) of the sharpimage plane. Altogether this is the basis to create holograms of artificial objects with differentsurface textures, like specular or diffuse reflecting areas.

Furthermore, independent intensity and phase control of reconstructed light fields is the ba-sis for artificial holography of three-dimensional objects. Since it is possible to generate anydesired light field distribution in the plane behind the second DOE, it is clear that also thelight field emitted from a real three-dimensional object can be simulated, and an observer willjust see the corresponding virtual object. The high efficiency of the projected light fields alsosuggests applications for optical trapping and micromanipulation in atom traps [12, 13] or dif-fractive laser tweezers [14]. Controlling phase and amplitude of the trapping light field meansto have full control over the exerted scattering and gradient forces. For example, the scatteringforce of light pushes particles in the direction of its phase gradient. Therefore the possibilityto project line patterns with a tailored transverse phase gradient allows to move particles alongprecalculated pathways, even within static light fields [15].

Acknowledgments

This work was supported by the Austrian Science Foundation (FWF), Project No. P18051-N02and Project No. P19582.



2.4 PAPER: Full phase and amplitude control of holographic

optical tweezers with high efficiency

Alexander Jesacher, Christian Maurer, Andreas Schwaighofer, Stefan Bernet, and MonikaRitsch-Marte, Optics Express, 16, 4479–4486 (2008).

2.4.1 Summary

Generally, the forces on microparticles arising from a light field depend on both its am-plitude and phase distribution. Hence, techniques that are able to shape both functionsindependently offer a higher flexibility in creating optical traps than purely “amplitude-shaping” methods.

The paper presented in chapter 2.3 addressed the application of a method to shapecomplex wavefronts to user-programmable SLMs. The efficiency and quality of holographicimage creation were evaluated. In this paper, we examine the method in view of opticaltrapping. In particular, we demonstrate a controlled transport of microspheres along pre-defined lines, which is purely driven by forces arising from also pre-defined transverse phasegradients.

Full phase and amplitude control ofholographic optical tweezers with high

efficiency

Alexander Jesacher, Christian Maurer, Andreas Schwaighofer, StefanBernet, and Monika Ritsch-Marte

Division for Biomedical Physics, Innsbruck Medical University, Mullerstraße 44A-6020 Innsbruck, Austria

[email protected]

Abstract: Recently we demonstrated the applicability of a holographicmethod for shaping complex wavefronts to spatial light modulator (SLM)systems. Here we examine the potential of this approach for opticalmicromanipulation. Since the method allows one to shape both amplitudeand phase of a trapping light field independently and thus provides fullcontrol over scattering and gradient forces, it extends the possibilities ofcommonly used holographic tweezers systems. We utilize two cascadedphase-diffractive elements which can actually be display side-by-side on asingle programmable phase modulator. Theoretically the obtainable lightefficiency is close to 100%, in our case the major practical limitationarises from absorption in the SLM. We present data which demonstrate theability to create user-defined “light pathways” for microparticles driven bytransverse radiation pressure.

© 2008 Optical Society of America

OCIS codes: (350.4855) Optical tweezers or optical manipulation; (230.6120) Spatial lightmodulators.

References and links1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156-159 (1970).2. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of s single-beam gradient force optical

trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).3. A. Constable, Jinha Kim, J. Mervis, F. Zarinetchi, and M. Prentiss, “Demonstration of a fiber-optical light-force

trap,” Opt. Lett. 18, 1867–1869 (1993).4. A. Chowdhury, B. J. Ackerson, N. A. Clark, “Laser-Induced Freezing,” Phys. Rev. Lett. 55, 833–836 (1985).5. A. Casaburi, G. Pesce, P. Zemanek, and A. Sasso, “Two- and three-beam interferometric optical tweezers,” Opt.

Commun. 251, 393-404 (2005).6. J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, “Multi-functional optical tweezers using computer-

generated holograms,” Opt. Commun. 185, 77–82 (2000).7. R. L. Eriksen, V. R. Daria, and J. Gluckstad, “Fully dynamic multiple-beam optical tweezers,” Opt. Express 10,

597–602 (2002).8. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175

(2002).9. A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Diffractive optical tweezers in the Fresnel regime,”

Opt. Express 12, 2243–2250 (2004).10. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt.

Commun. 124, 529–541 (1996).11. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys.

J. 61, 569–582 (1992).

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12. J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Kas, “The Optical Stretcher:A Novel Laser Tool to Micromanipulate Cells,” Biophys. J. 81, 767-784 (2001).

13. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angularmomentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997).

14. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, ”Optical alignment and spinningof laser-trapped microscopic particles,” Nature 394, 348–350 (1998).

15. P. Galajda and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett. 78, 249–251(2001).

16. J. P. Kirk and A. L. Jones, “Phase-Only Complex-Valued Spatial Filter,” J. Opt. Soc. Am. 61, 1023-1028 (1971).17. Y. Roichman and D. G. Grier, “Projecting Extended Optical Traps with Shape-Phase Holography,” Opt. Lett. 31,

1675–1677 (2006).18. M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and

point spread function engineering,” J. Microsc. 197, 219–223 (2000).19. R. Tudela, E. Martın-Badosa, I. Labastida, S. Vallmitjana, I. Juvells, and A. Carnicer, “Full complex Fresnel

holograms displayed on liquid crystal devices,” J. Opt. A 5, 189–194 (2003).20. M.-L. Hsieh, M.-L. Chen, and C.-J. Cheng, “Improvement of the complex modulated characteristic of cascaded

liquid crystal spatial light modulators by using a novel amplitude compensated technique,” Opt. Eng. 46, 07501(2007).

21. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and Extrinsic Nature of the Orbital AngularMomentum of a Light Beam,” Phys. Rev. Lett. 88, 053601 (2002).

22. J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. 28, 872–874 (2003).23. Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradi-

ents,” Phys. Rev. Lett. 100, 013602 (2008).24. H. Bartelt, “Computer-generated holographic component with optimum light efficiency,” Appl. Opt. 23, 1499–

1502 (1984).25. H. O. Bartelt, “Applications of the tandem component: an element with optimum light efficiency,” Appl. Opt. 24,

3811–3816 (1985).26. A. Jesacher, C. Maurer, A. Schwaighofer, S. Bernet, and M. Ritsch-Marte, “Near-perfect hologram reconstruction

with a spatial light modulator,” Opt. Express 16, 2597–2603 (2008).27. A. Jesacher, S. Furhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Reverse orbiting of microparticles in

optical vortices,” Opt. Lett. 31, 2824–2827 (2006).28. G. O. Reynolds, J. B. Develis, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier

Optics (SPIE, 1989).29. B. Kress and P. Meyrueis, Digital Diffractive Optics (Wiley, 2000).30. R. W. Gerchberg, and W. O. Saxton, “A practical algorithm for the determination of phase from image and

diffraction plane pictures,” Optik 35, 237-246 (1972).31. A. Jesacher, S. Furhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Holographic optical tweezers for object

manipulations at an air-liquid surface,” Opt. Express 14, 6342–6352 (2006).32. C. Bertocchi, A. Ravasio, S. Bernet, G. Putz, P. Dietl, and T. Haller, “Optical Measurement of Surface Tension in

a Miniaturized Air-Liquid Interface and its Application in Lung Physiology,” Biophys. J., 89, 1353–1361 (2005).33. M. A. Seldowitz, J. P. Allebach, and D. W. Sweeney, “Synthesis of digital holograms by direct binary search,”

Appl. Opt. 26, 2788–2798 (1987).

1. Introduction

Since the invention of optical tweezers in the 1970s [1, 2], trapping and manipulating dielectricmicroscopic particles with lasers have found numerous applications in biomedical research andmaterials science. Nowadays optical micromanipulation is carried out in many experimentalconfigurations, which create light traps using a variety of approaches, for instance using opticalfibers [3], direct beam interference [4, 5], or spatial light modulators (SLM) [6, 8, 7, 9].

In the context of optical micromanipulation the forces exerted by optical fields are oftenconceptually divided into two parts: the gradient force and the scattering force. Microscopicallythis distinction between the (conservative) dipole forces arising from intensity gradients andthe mechanical effects originating from the scattering of light by the microscopic particle is anatural “partition” of the forces. In the case of a single dipole, e.g., the scattering or “radiationpressure” force always points into the direction of the incident photon flux and the gradientforce into the direction of the intensity gradient [10]. For larger dielectric particles of arbitraryshape these definitions are less natural, and different communities may mean different things


by “scattering forces”, for example depending on whether the particle can be considered to be asmall perturbation for the light fields or not. For the trapping of large spherical particles whichcan approximately be described by the ray optics model, the scattering and gradient forces ofeach single ray basically act along or orthogonally to the ray propagation direction, i.e. can beinterpreted as the axial and transverse component, respectively [11].

Apart from specific configurations, where scattering forces can be exploited to trap and evenstretch particles [2, 12], most techniques rely on the gradient force to perform stable trapping.Especially within single-beam traps, scattering forces act destabilizing and thus are often un-desired. Nonetheless, for specific applications it has been shown that scattering forces can bequite useful. For instance, they can form “force vortices” which act as an “optical spanner” [13].They can also cause spinning of birefringent particles [14] or drive specifically manufacturedmicro-machines [15]. And it seems there will be an even broader spectrum of possibilities, ifone could only control the scattering force more precisely: Since the direction of the scatteringforce is – besides particle properties – determined by the local wavefront shape, controlling thelight field’s amplitude and phase is a precondition to maintain full control over the scatteringforce. Unfortunately, the use of phase-only spatial light modulators (SLM) in holographic aswell as direct projection methods does generally not allow one to control amplitude and phaseof the trapping field simultaneously. Although there are methods to shape the amplitude profileof the first diffraction order with a pure phase modulating element [16, 17, 18], and also meth-ods which employ two SLMs to generate complex-valued fields [19, 20], they usually imply asignificant loss of light.

Here we demonstrate the creation of optical “trapping patterns” which exert user-definedscattering forces with a phase-only SLM. In particular we demonstrate the creation of linetraps, which provide transportation of micro particles on freely designable pathways. Previousexperiments have also demonstrated particle transport driven by a phase gradient along the ring-shaped cross-section of Laguerre-Gaussian modes [21]. Another approach presented by Curtisand co-workers [22] allows the design of more general pathways, however still with the restric-tion to closed paths. Moreover, the transverse scattering force component is not independentof the spatial structure of the path. Recently, a method to create arbitrary light structures withfreely programmable phase profiles has been demonstrated [23]. Though powerful in manyrespects, the method might not be optimal for the design of high-contrast amplitudes, sinceamplitude-shaping is based on the omission of pixels.

We use a specific method for shaping complex light fields [24, 25] the applicability of whichto liquid crystal SLMs has been investigated recently by our group [26]. The technique is basedupon two cascaded phase diffractive elements, arranged in an optical 2-f setup, where the firstelement is designed to create the desired amplitude in the plane of the second element, whichfinally “imprints” the desired phase. The method is in principle lossless and can accommodatehigh amplitude-contrast requirements.

2. Methodic principle and practical implementation

Figure 1(a) outlines the principle of the technique. Amplitude and phase of the light field areshaped in two subsequent steps by two phase diffractive elements (P1 and P2). Both elementsare arranged in a 2-f setup, i.e., they are placed in the front and back focal plane of a convexlens. Consequently, the Fourier Transform of the field in the plane of P1 emerges in the planeof P2. The principle of the method is explained in the following practical example.

Let us consider that we want to create a light field a(x,y) = |a(x,y)| exp [iφ(x,y)] in the objectplane, which collects particles like a funnel, i.e. which “grasps” and guides them towards a spe-cific point in the object plane. Such a light field represents the “reticle” shown in Fig. 1(b). Thefour bright lines confine particles by gradient forces. To maintain the “collecting” property, we


phase (x,y)�

amplitude |a(x,y)|

(b)

��

lens

f

P1“amplitude shaper”

f f fplane of P1 plane of P2 object plane

P2“phase shaper”

|A(u,v)| �(u,v) �(u,v)

lens

|a|

(a)

|A(u,v)|

readoutlaser

Fig. 1. (a) Principle of the method. The reticle in the object plane are created by the twophase masks P1 and P2, which are subsequently arranged in a 2-f setup. P1 generatesthe modulus |A(u,v)| in the plane of P2, where P2 shapes the desired phase Φ(u,v). (b)Schematic of a trapping light field, the amplitude (upper image) and phase of which aredesigned to collect and guide microparticles to the center of the “crosshairs”.

design the phase of the light field to have a gradient pointing towards the reticle center, whichcorresponds to the transverse component of the linear photon momentum p t(x,y) = h ∇tφ(x,y),where the index t denotes the transverse component, i.e. the component within the microscopeobject plane. It should be mentioned that the resulting scattering force on trapped microscopicparticles strongly depends on specific particle properties, such as shape and refractive index. Insome specific cases, particles can also travel against the direction of pt [27]. However, such abehaviour is only expected from particles of highly irregular shape.

The creation of the complex field a(x,y) = |a(x,y)| exp [iφ(x,y)] in the object plane requiresa definite complex field A(u,v) = |A(u,v)| exp [iΦ(u,v)] in the plane of P2, which can be cal-culated by mathematically back-propagating a(x,y). In the most common case, this plane isconjugate to the object plane, thus A(u,v) corresponds to the optical Fourier transform [28]of a(x,y). Hence the task is to create A(u,v) in the plane of P2 in order to obtain the desiredcomplex field a(x,y) in the object plane. Since P2 is a phase diffractive element, it can be pro-grammed to shape the phase part Φ(u,v) directly, but unfortunately not the modulus |A(u,v)|.The key idea now is to design P1 such that a readout reproduces |A(u,v)| in the plane of P2 asexactly as possible. There exist a couple of so-called phase retrieval algorithms which can beutilized for this design process [29]. The “random” phase Θ(u,v), which usually results fromsuch optimization procedures, cannot be controlled, but it is known. Thus, in the plane of P2the desired phase Φ(u,v) can be restored by calculating P2 as follows:

P2(u,v) = mod2π{Φ(u,v)−Θ(u,v)}, (1)

where Θ describes the non-controllable phase that is produced by P1 in the plane of P2, andmod2π{...} symbolizes the “modulo-” operation that “cuts” the phase of P2 into slices of 2π .

Figure 2 illustrates how we realized a practical implementation of the method. By introducinga concave mirror into the beam path, it is possible to utilize one single phase modulator todisplay P1 and P2, which are adjacently arranged at the modulator panel. Our SLM is a HEO1080 P phase-only modulator from Holoeye Photonics AG, which can display phase patterns


P2“phase shaper”



PAL-SLM objective

dichroicmirror

to camera

f1=750 mm f2=400 mm

P1“amplitude shaper”

experiment:reticle focussed on mirror

5 µm

300 mm

Fig. 2. Experimental setup. The complex trapping field is created stepwise: Illuminatingpattern 1 with a collimated CW Ytterbium fiber laser creates the amplitude profile A(u,v)in the plane of pattern 2, which reshapes the “random” phase to Φ(kx,ky). The modulatedlaser beam is subsequently coupled into a microscope objective, finally taking the desiredform a(x,y) exp [iφ(x,y)] in the object plane.

showing up to 41% diffraction efficiency (defined as the intensity ratio of first diffraction orderto the incident light) at 1064 nm. The setup is almost identical to that introduced in Ref. [26].A minor but important difference is that we designed P1 as an “off-axis” structure this time,such that its zeroth diffraction order does not focus on P2. This is advantageous, because therelatively high laser power required for optical trapping could otherwise lead to irreversibledamage of the liquid crystal in the SLM. According to our measurements, slight (reversible)power-induced changes of the liquid crystal birefringence begin to appear after a few minutesof exposure to a Gaussian beam of 2 W CW power with a 2.5 mm waist. In our experiments,the phase mask P1 was optimized in 15 iterative steps of the Gerchberg-Saxton algorithm [30].

Strictly speaking, the fact that the spherical mirror is inclined with respect to the beam axisintroduces wavefront aberrations such as coma. Implementing a beamspitter would allow anorthogonal geometry and thus avoid these aberrations. However, this would also imply a re-duced light efficiency. Because the mirror tilt angle is small (below 1 degree), we decided toaccept the aberrations for the sake of a maximal light throughput. Furthermore, P1 and P2 arenot exactly located in conjugate planes, which is due to the SLM tilt with respect to the beamaxis. In our case, the tilt angle is in the range of 2-3 degrees and the assumption of conjugateplanes leads to an only minor quality reduction. A corresponding modification of the hologramcomputation could compensate the effect, however at the cost of higher numerical expenses.

The sensitivity of the setup to alignment errors is discussed in Ref. [26].

3. Experimental results

Figure 3 documents the practical realization of the particle-collecting reticle discussed in theformer section. The experimental situation is described by the sketch in the upper left corner.The upper right images show the reflection of the light field on a mirror, revealing good inten-sity reconstruction. The accuracy of the programmed phase was tested by direct interaction ofthe reticle with microparticles. The objective used is a Zeiss Neofluar 100× with a numericalaperture (NA) of 1.3. The movie strip of Fig. 3 shows four silica microbeads of 2 μm diame-


Int.

/arb

.units

0 s 1.5 s 5 s 11 s 12.5 s 20 s

5 µm

100xNA 1.3

silica bead

water

cover slip immersionoil

Fig. 3. Experimental realization of the “reticle” trap. Upper left image: Experimental sit-uation. Trapping is performed from below through a glass cover slip with a Zeiss Neo-fluar 100× objective, NA=1.3. Upper right image: Trap, projected at a mirror. Movie strip(frames taken out of the file reticle.avi, size 3 MB): Four trapped silica microbeads arealternately pushed in and out by changing the sign of the transverse photon momentum.The particles are three-dimensionally trapped, which is demonstrated by lifting them offthe ground of the object chamber (note the encircled untrapped particles in the lower leftcorner getting out of focus).

ter, each trapped in a different “arm” of the reticle. First, the transverse phase gradient of thestructure was programmed to point outwards with a modulus of 2π /μm. The according trans-verse radiation pressure pushed the silica beads towards the outer edges of the trapping field(first frame). Inversion of the phase gradient caused the particles to move towards the center(second and third frame). For silica particles in water, which show a relative refractive index of1.43/1.33, the axial gradient force is sufficient to overcome the opposing axial scattering force– thus the reticle is able to confine these particles three-dimensionally. This was demonstratedby lifting the trapped microspheres about 8 μm deep into the water volume. The axial shift canbe judged by some untrapped beads in the lower left edge, which appear out-of-focus in the lastthree frames. Finally, the phase gradient was once more inverted and the particles moved backto the outer edges of the reticle. A real-time video documentation of the experiment is attached(reticle.avi). The laser power output used for this experiment was 1.6 W, which corresponds toa net power of approximately 100 mW in the object plane.

A second experiment was performed with polystyrene beads located at a microscopic air-liquid interface [31], using a Zeiss Achroplan 63× air objective with a NA of 0.95. The inter-face is located within a tiny hole (300 μm diameter) – small enough to prevent leakage – inthe bottom of a stainless steel fluid container and can be accessed from below with an invertedmicroscope (see Fig. 4). Such a setup allows detailed examinations of interface properties [32].Moreover, for hydrophobic polystyrene particles, the exerted surface tension acts stabilizingagainst axial scattering forces. To demonstrate the ability of the method, we designed a “lightsquare” and an asymmetric “curvy” structure resembling Austria in shape which trap parti-


5 µm 0 s 0.5 s 1 s 1.5 s 2 s

Int.

/a

rb.

un

its

63xNA 0.95

polystyrene bead

air-waterinterfacewater

chamberwater

chamber

5 µm

0 s 2.6 s 5.2 s 7.8 s 10.5 s

Fig. 4. Upper left image: Trapping is performed at an air-liquid interface, which axiallystabilizes the hydrophobic microparticles (polystyrene beads, 3 μm diameter). Upper rightimage:“Square trap”. The microscopic light square traps particles and moves them alongits boundaries. The corresponding movie strip shows it “in action” (movie file square.avi,size 2.8 MB). The guiding performance of a asymmetric structure is demonstrated by thelower movie strip (movie file austria.avi, size 3.1 MB).

cles (about 3 μm in diameter) and push them along their boundaries by scattering forces. Thecorresponding transverse phase gradient has a modulus of 3.6×2π /μm. Again, the intensityreconstructions are almost speckle-free. The movie strips in Fig. 4 indicate that – in the caseof the square trap – the obtained particle speed (mean value 20 μm/s) is much higher than thatof the silica spheres in Fig. 3, although the programmed phase gradient is smaller. This is –besides the lower infrared absorption of the used air objective – mainly explained by the highernumber of beads (more beads gain more light momentum) and the higher refractive index ofthe particles (1.59 for polystyrene beads), which implies a higher scattering force. Because thisalso holds in axial direction, the gradient force by itself is now no longer sufficient to confinethe particles stably in the object plane. Hence for such particles another stabilizing force isrequired, which in our case is the surface tension. The laser power output used for the experi-ments with polystyrene beads was 1.4 W, which corresponds to a net power of approximately150 mW in the object plane.


4. Summary and discussion

Recently we have examined the applicability of a method for generating complex wavefrontsto liquid crystal SLMs. Here we demonstrate the usefulness of this technique for the opti-cal manipulation of microscopic dielectric particles that can be trapped by pre-defined optical“trapping patterns”. Similar to the method introduced in Refs. [17, 23], the method permitsindependent control over amplitude and phase of a light field, but with potentially higher effi-ciency. The technique makes use of two independent diffractive elements located in two distinctoptical planes: an “amplitude shaper” phase hologram in an image plane and a “phase shaper”hologram in a Fourier plane, thus the commonly used “Fourier manipulation techniques” (e.g.gratings and lenses for three-dimensional trap shifting) are extended by possibilities to alter thetrapping field via the element in the image plane (P1). For instance, lateral shifts of the trappingfield can be directly achieved by a corresponding motion of P1, which enables real-time steer-ing similar to that obtainable with a “Fresnel” setup [9]. We implemented the method using asingle user-programmable phase-only liquid crystal SLM.

Several alternative approaches for creating complex light fields with single phase masks havebeen suggested in the past [16, 17, 18]. Furthermore, phase retrieval techniques can be alsoemployed to reconstruct a complex amplitude field directly [33], however only in a restrictedarea and with unavoidable light losses.

Theoretically, the method described here would enable the generation of arbitrary light fieldswith almost 100% intensity efficiency. In our specific case, the maximal obtainable efficiencyis about 17%, which is mainly caused by the relatively high absorption by our SLM. With noabsorption, the obtained efficiency would be...

However, this technical restriction will be less severe in future generations of light modula-tors. In fact, even current systems claim to have light efficiencies of up to 80%, which wouldenable the presented “complex” holographic optical tweezers to reach up to 64% efficiency.

The method is well suited to create high contrast amplitudes in the plane of P2, which isespecially advantageous in the case of P2 being located in the Fourier plane, where field ampli-tudes often have a large dynamic range. However, high intensities may destroy the liquid crystal– hence, in some cases, P2 should rather be placed in a Fresnel than in the Fourier plane, wherethe amplitude contrast is typically smaller.

The presented “complex” holographic tweezers might be useful in different tasks concerningoptical trapping and manipulating microscopic particles or atoms, since its shows higher flexi-bility than commonly used techniques, which typically only allow controlling the amplitude oflight fields. For instance, one could use specifically tailored phase landscapes for transporting,sorting or controlled aggregation of microparticles.

Acknowledgments

This work was supported by the Austrian Science Foundation (FWF), Project No. P18051-N02and Project No. P19582.



2.5 Correcting aberrations of the SLM “double-filter” setup

Due to the manufacturing process, the surface of reflective SLMs is not ideally flat. Ac-cording to Holoeye Photonics, the surface deviation is almost spherical and does not extend3-4 waves at 633 nm in the case of the HEO 1080 P panel. However, due to our experience,there is also a significant astigmatism which can be a disturbing factor for some applica-tions. Especially in the experimental configuration introduced in chapters 2.3 and 2.4, theoptical path between the “phase shaper” (P1) and the “amplitude shaper” (P2) shouldshow as little aberrations as possible, since they strongly affect the quality of the phase cor-rection performed by P2. In this particular setup, even further aberrations arise from thetilt of the SLM and the spherical mirror with respect to the beam axis. The experimentalresults presented in chapters 2.2–2.4 were achieved with a “manual” astigmatism correc-tion, i.e., focal length and orientation of a diffractive cylindric lens superposed to P1 werealtered, until the obtained result appeared ideal. Although astigmatism is the dominantaberration, it is likely that further quality enhancement could be achieved if one wouldconsider also higher aberration modes such as coma, trefoil or spherical aberration. How-ever, this case requires automated aberration correction, since the manual “trial-and-error”method would be too time-consuming.

A method for adaptive wavefront-sensing which can be applied to our specific “double-filter” setup was introduced by Neil and co-workers (Neil et al., 2000b; Booth et al.,2002). The method allows to sense for individual aberration modes separately and hasbeen developed for aberration sensing in confocal microscopy. Requirements for the set ofaberration modes are orthogonality and a zero mean value of each mode over the wholeaperture. The so-called Zernike modes (Born and Wolf, 2002) are widely used in opticsand fulfill both requirements, thus we also utilize these modes to describe the aberrationsin our setup. The Zernike mode i is represented by the symbol Zi.

The method of Neil and co-workers is based upon subsequently introducing two arti-ficial bias aberrations ±∆aiZi for each mode i to the distorted beam and measuring thecorresponding peak intensities Ii+ and Ii− of the Airy discs which emerge when this beamis focused. It can be shown that the intensity differences ∆Ii = Ii+ − Ii− are in firstorder proportional to the magnitudes ai of the present aberration modes Zi and that thereis no crosstalk between different modes (Neil et al., 2000b). This allows to generate acompensating correction mask which can be refined in an iterative process.

Figure 2.3 sketches the optical path which was optimized with this approach. Theaim was to find a corrective phase pattern PC which compensates the present aberrations.Since the technique relies on measuring the peak intensity of the focal spot on P2, the“phase shaper” P2 was programmed to act as a pinhole, which diffracts only the centralfocal intensity onto a photo-detector. This could be achieved by displaying a fine blazedphase grating in the central region of P2. All remaining pixels were programmed to showa constant phase value. This “phase pinhole” is shown in Fig. 2.4. It has a diameter of8 pixels (64 µm) along the x- and y-axis, which corresponds in our case to 63% of the


P2, “phase-pinhole”



HEO 1080 P P1

photo diode

�6°

�4°

PC

input signal:light power within pinhole

output:corrective mask PC

reflected lightreflected light

light through pinhole

Figure 2.3: Optical path of our “double-filter” setup. In a closed-loop procedure, the fraction oflight going through the pinhole at P2 (and thus the quality of the focus) is maximized by displayingcorrective phase masks at P1.

Airy-disc diameter. It is obvious that at this small scale the pixel structure strongly affectsthe shape of the pinhole. Moreover, the grating inside consists of not even three periods(the grating period is three pixels). Nonetheless, experiments showed that the method isrobust enough to function with this phase pinhole.

Figure 2.4: Central region of P2, which is programmed as “phase-pinhole”. Gray shades correspondto phase values from 0 to 2π.

Prior to the optimization procedure, the “pinhole”-pattern P2 was shifted pixelwise atthe display in order to maximize the photo-detector signal. Our optimization accountedfor the Zernike modes from Z5 to Z15. Z1 (piston) as well as Z2,3 (tilt x,y) do not affectthe optical quality. Z4 (defocus) was corrected manually, because this can easily be donewithin a minute by continuously changing the focal length of a diffractive lens superposedto P1 and stopping at the point where the photo-detector signal is maximal. Aberrationmodes higher than Z15 were neglected to save time. Furthermore, due to their higherspatial frequencies they are expected to be of minor importance in our optical system.

The optimization procedure began with applying bias aberrations ±∆aiZi for everymode i=1..15, and deriving the corresponding intensity differences ∆Ii. Good convergencewas obtained by choosing ∆ai to be 0.5. The method fails if ∆ai is too large, since thelinear relation between ∆Ii and ai is not valid any more. Subsequently, a corrective phase


function PC was calculated as follows:

PC = −g15∑

i=5

∆Ii Zi. (2.6)

There, g represents a gain factor, which can again be chosen to optimize convergence:small gain factors lead to slow convergence while a too high gain factor causes oscillatorybehaviour. In our case, g was set to 0.5. Adding PC to P1 completed one iteration. Theprocedure was repeated until the detected aberration amplitudes (and also PC) converged.

1 2 3 4 5 6 7 8 9 10Iteration

Z1

0 1 2 3 4 5 6 7 8 9 101

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

Iteration

U/U

0

1 2 3 4 5 6 7 8 9 10 11-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

5 6 7 8 9 10 11 12 13 14 15

(a) determined aberration mode amplitudes (b) photo-detector signal

sigma interval

x... sigma interval

(c) convergence of Z (astigmatism x)5

a5

ai

Zernike mode index i

Iteration

-0.43

-0.42

-0.41

-0.4

-0.39

-0.38

-0.37

-0.36

-0.35

-0.34

0.5

1

1.5

2

2.5

3

3.5

phase

(d) final correction mask PC

Figure 2.5: Results of the closed-loop optimization. 10 iterations were performed. (a) shows thedetermined Zernike aberration amplitudes. (b),(c) Development of photo-detector signal and a5,respectively. (d) final corrective phase mask.

Figure 2.5 summarizes the experimental results, which represent mean values over 5 sub-sequently conducted independent optimization procedures. 10 Iterations were performed ineach procedure. Figure 2.5(a) indicates that the aberrations are dominated by astigmatismin x-direction (Z5) and Coma in y-direction (Z8). (b) and (c) show the development of thephoto-detector signal and the aberration mode amplitude a5. It can be seen that already


the first iteration yields the major improvement. (d) shows the final corrective phase maskPC .

105 125 145 165 185 205 225 2450

200

400

600

800

105 125 145 165 185 205 225 2450

200

400

600

800

80 100 120 140 160 180 200 220 240 2600

200

400

600

800

(a) without correction (b) astigmatism manually corrected (c) closed-loop corrected

pixel brightness

counts

Figure 2.6: Reconstruction of a light field resembling the contours of Austria with (a) no aberrationcorrection, (b) manual corrected astigmatism and (c) the correction mask PC of Fig. 2.5, whichwas obtained with the described automated aberration sensing procedure. The second row showscorresponding image brightness histograms.

Figure 2.6 shows how the aberration correction affects the quality of light fields createdwith the technique presented in chapters 2.3 and 2.4. The light fields in the first rowresemble the contour of Austria (with some smoothed edges). The second row containsthe corresponding image histograms. Pixel values below 100 were neglected for the sakeof clarity. The left image was created without optical correction. Manually pre-correctingastigmatism already shows an improved performance (see middle image). Finally, theclosed-loop optimization procedure again significantly improved the quality of light fieldreconstruction (right image).

Although the automated aberration correction proved to be beneficial for the opticalquality of the setup in Fig. 2.3, it is obvious that it is not able to correct all aberrationspresent. For instance, it only allows to correct aberrations for rays parallel to the beamaxis. Inclined rays, which appear when P1 is a general phase diffractive pattern, are notconsidered. Furthermore, errors arising from the fact that the SLMs is tilted with respectto the beam path are also not fully detected. However, the resulting errors are most likelynegligible, since the maximal beam inclination angles are in the range of 0.01 rad and theSLM inclination angle is also relatively small (see Fig. 2.3).

Chapter 3

Summary and Discussion

In this thesis, possibilities of shaping light with two cascaded user-programmable spatialphase modulators were examined. It was found that the combination of two independentSLMs in a 2-f-configuration represents a powerful instrument. To my knowledge, theprinciple was introduced for the first time in 1984 by H. O. Bartelt (Bartelt, 1984), whoperformed experiments with bleached films as phase masks.

One can see such a setup as a flexible optical Fourier processor: One SLM displaysthe phase pattern to be filtered while the other generates an arbitrary phase filter. Byexploiting this principle, we showed that it is possible to create line structures with helicalphase profiles (Jesacher et al., 2008a) by programming the filter SLM to display a phasevortex of the form exp [iθ], with θ being the angular coordinate in a polar coordinate system.Such line structures show a very specific phase profile, where the transverse phase gradientalong a line depends on the local line curvature. We further demonstrated that the linearphoton momentum which is linked with these transverse phase gradients is sufficient tomake polystyrene microspheres move along the lines of such a helical light structure. Thusthe method represents a straightforward way to create arbitrarily shaped and speckle-freeline tweezers which have the ability to exert transverse scattering forces. Nonetheless, someproperties of the method restrict its practical applicability: Due to the structure of thevortex filter, the transverse phase gradient necessarily has - independent of the trappingstructure - a mean value of 2π/L, where L is the length of the line trap. The correspondingtransverse photon momentum might bee too small for specific micromanipulation tasks.Moreover, in many cases a transverse momentum which can be adjusted independentlyfrom the trap geometry is preferable.

In this context, we demonstrated that the cascaded SLM configuration is also capableof dynamically shaping arbitrary scalar complex wavefields (Jesacher et al., 2008b,c). Thebasic principle is that the first phase element (the “amplitude shaper”) creates the ampli-tude function of the desired light field’s Fourier transform in the plane of the second phaseelement (“phase shaper”), which is programmed to “imprint’ the corresponding phasefunction. Compared to alternative techniques (Kirk and Jones, 1971; Neil et al., 2000c;Roichman and Grier, 2006; Hsieh et al., 2007), the method can in principle be almost

43

CHAPTER 3. SUMMARY AND DISCUSSION 44

lossless, regardless of the amplitude function which has to be shaped. In our experimentalconfiguration, efficiencies of 17% were achieved, which are mainly dominated by absorptionlosses caused by the SLM panel. We demonstrated the diffractive generation of differentlight fields in one and two axial planes with high purity. The obtained performance wascompared to that of a single-element phase diffractive pattern, which was optimized withthe Gerchberg-Saxton algorithm.

We also investigated the applicability of this “double-filter” principle to optical trapping.There it can be particularly useful because the ability of shaping phase and amplitude oflight independently gives full control over the acting gradient and scattering forces. In thiscontext we demonstrated controlled transport of polystyrene and silica microparticles alongpre-defined pathways, where the spatial structure as well as the particle-driving transversescattering forces could be freely determined by the user. Moreover, the method will allowto create two- and even three-dimensional trap arrays without so-called “ghost-traps” –artefact-traps which can emerge when a trap array is encoded in purely phase-diffractivepatterns. Although the current efficiency of 17% may be too low for generating large traparrays, it has to be considered that this efficiency depends quadratic on that of a singleSLM, i.e., even a small technological efficiency gain can already represent a considerableimprovement of for the “double-filter” setup. Moreover, recently offered SLM systems claimto have a diffraction efficiency of up to 80% – the corresponding double-filter efficiencywould hence be 64%.

Curriculum Vitae

Personal data

name: Alexander Franz Jesacher

date of birth: 19. September 1978

place of birth: Lienz

citizenship: Austria

marital status: unmarried

Education

Sep. 1993 – Jun. 1998 student at the technical college fortelecommunications engineering in Innsbruck

Oct. 1998 – Sep. 1999 civilian service at the Austrian Red Cross

Oct. 1999 – Sep. 2004 diploma studies in physics at theLeopold-Franzens University in Innsbruck

10. Sep. 2004 graduation with distinctiontitle of master thesis: “Anwendungen holografischerLaserpinzetten im Fresnel- und Fourier-Regime”

since Sep. 2004 postgraduate studies in physics and “Image GuidedDiagnosis and Therapy” at the Division forBiomedical Physics, Innsbruck Medical University

3. Aug. 2007 doctoral examination in physics passed with distinctiontitle of doctoral thesis: “Applications of spatiallight modulators for optical trapping and image processing”

45

Awards

Apr. 2005 DOC scholarship of the Austrian Academy of Sciences

Nov. 2005 Poster prize at the 1st meeting of doctoral students of the InnsbruckMedical University. Title: “SLM as diffractive element in applied optics”(together with Severin Furhapter)

Feb. 2007 Prize of the Furstentum Liechtenstein for scientific research at theInnsbruck universities

Publications in peer reviewed journals

1. A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte. Diffractive opticaltweezers in the Fresnel regime.Opt. Expr., 12:2243–2250, May 2004.

2. A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte. Size selective trappingwith optical “cogwheel” tweezers. Opt. Expr., 12:4129–4135, August 2004.

3. S. Furhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte. Spiral phase contrastimaging in microscopy. Opt. Expr., 13:689–694, February 2005.

4. A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte. Shadow Effects in SpiralPhase Contrast Microscopy. Phys. Rev. Lett., 94:233902, June 2005.

5. S. Furhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte. Spiral Interferometry.Opt. Lett., 30:1953–1955, August 2005.

6. S. Bernet, A. Jesacher, S. Furhapter, C. Maurer, and M. Ritsch-Marte. Quantitativeimaging of complex samples by spiral phase contrast microscopy. Opt. Expr., 14:3792-3805, May 2006.

7. A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte. Spiral InterferogramAnalysis. J. Opt. Soc. Am. A, 23:1400–1409, June 2006.

8. A. Jesacher, S. Furhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte. Holographicoptical tweezers for object manipulations at an air-liquid interface. Opt. Expr.,14:6342–6352, June 2006.

9. A. Jesacher, S. Furhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte. Reverseorbiting of microparticles in optical vortices. Opt. Lett., 31:2824–2826, October 2006.

10. C. Maurer, S. Furhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte. Tailoring ofarbitrary optical vector beams. New J. Phys., 9:78, March 2007.

46

11. A. Jesacher, A. Schwaighofer, S. Furhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte. Wavefront correction of spatial light modulators using an optical vorteximage. Opt. Expr., 15:5801–5808, April 2007.

12. A. Jesacher, A. Schwaighofer, C. Maurer, S. Bernet, and M. Ritsch-Marte. Near-perfect hologram reconstruction with a spatial light modulator. Opt. Expr., 16:2597–2603, February 2008.

13. A. Jesacher, A. Schwaighofer, C. Maurer, S. Bernet, and M. Ritsch-Marte. Full phaseand amplitude control of holographic optical tweezers with high efficiency, Opt. Expr.,16:4479–4486, March 2008.

14. A. Jesacher, A. Schwaighofer, C. Maurer, S. Furhapter, S. Bernet, and M. Ritsch-Marte. Optical tweezers of programmable shape with transverse scattering forces.Opt. Comm., 281:2207-2212, April 2008.

15. C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte. Upgradinga microscope with a spiral phase plate, J. Microsc., 230:134-142, April 2008.

Book chapters

1. S. Furhapter, A. Jesacher, C. Maurer, S. Bernet, M. Ritsch-Marte. Spiral Phase Mi-croscopy. In: Advances in Imaging and Electron Physics, Vol. 146, Peter W. Hawkes(Ed.), Academic Press Inc., April 2007.

Conference Proceedings

1. A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte. Holographic opticaltweezers in the Fresnel regime. Optical Trapping and Optical Micromanipulation.Edited by Dholakia, Kishan; Spalding, Gabriel C.. Proceedings of the SPIE, Volume5514, pp. 293-298, October 2004.

2. S. Furhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte. Interference imagingwith a spatial spiral phase filter. Building European OLED Infrastructure. Edited byPearsall, Thomas P.; Halls, Jonathan. Proceedings of the SPIE, Volume 5962, pp.415-419, September 2005.

3. A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte. Vortex filter in lightmicroscopy. Lasers and Applications. Edited by Abramski, Krzysztof M.; Lapucci,Antonio; Plinski, Edward F. Proceedings of the SPIE, Volume 5959, pp. 30-34,September 2005.

47

4. A. Jesacher, S. Furhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte. Holographicoptical tweezers: manipulations at an air-liquid interface. Optical Trapping and Op-tical Micromanipulation III. Edited by Dholakia, Kishan; Spalding, Gabriel C.. Pro-ceedings of the SPIE, Volume 6326, pp. 63261U, September 2006.

Selected talks

1. A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte. Holographic opticaltweezers in the Fresnel regime, SPIE International Symposium for Optical Scienceand Technology, Denver, USA, August 2004.

2. A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte. Vortex filter in lightmicroscopy, SPIE International Congress on Optics and Optoelectronics, Warsaw,Poland, August 2005.

3. A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte. Computer controlledspatial light modulator as phase diffractive element in light microscopy, DiffractiveOptics meeting, Warsaw, Poland, September 2005.

4. A. Jesacher, S. Furhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte. Holographicoptical tweezers: Manipulations on an air-water interface, SPIE symposium “Optics& Photonics”, San Diego, USA, August 2006.

5. A. Jesacher, S. Furhapter, C. Maurer, A. Schwaighofer, S. Bernet, and M. Ritsch-Marte. Biomedical Applications of Spatial Light Modulators, ESF conference “Trendsin Optical Micromanipulation”, Obergurgl, Austria, February 2007, (invited).

6. A. Jesacher, C. Maurer, S. Bernet, and M. Ritsch-Marte. Complex holographic op-tical tweezers, conference “Nanotech-Montreux”, Montreux, Switzerland, November2007. A. Jesacher, C. Maurer, S. Bernet, and M. Ritsch-Marte. Complex holographicoptical tweezers, Nanotech-Montreux, Montreux, Switzerland, November 2007, (in-vited).

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Advanced shaping of light fields with cascaded spatial light modulators