Spatial LC-modulatorand diffractive optics

Experimentelle Übungen IIAdvanced laboratory course in the Institute of Applied Physics

October 2015

1 Theoretical foundations1

The diffraction of light at dynamically adjustable optical elements as represented by the LCcells of a spatial light modulator can be described by the transmission through the LC material,which is characterized by its electro-optical properties, and the then following pattern formationdue to propagation of the diffracted wave. Diffractive optical elements (DOEs) are used moreand more in modern optical instruments. The optical function is caused by the diffraction andinterference of light in contrast to refractive optical components. Spatial light modulators offerthe dynamical realization of diffractive optical elements.

The usage of diffraction and interference requires tiny structures in the dimension of the opticalwavelength. The fabrication of such small structures became possible in the context of modernmethods of nanotechnology. Lithographical production technologies and replication processeshave made it possible to mass produce DOEs. Thus diffractive optical elements, which canpotentially replace lenses, prisms, or beam-splitters and can even be used to create image-likediffraction patterns, are easier to produce and more compact than their refractive counterparts,if these exist all.

A well-known visible application of DOEs in the consumer market is an optical head to bemounted on a laser pointer to create arrows, crosses and other patterns. It is less wellknown thatsome digital cameras for the consumer market make use of a DOE and a weak (and thereforeeye-safe) infrared laser diode for their autofocus system.

In more technical applications, DOEs are often used as viewfinder systems which enable tosee at which point a device or tool is working, or as spot-array generators e.g. for 3-D surfacemeasurements. Also, diffractive optical beam-splitters can create arrays of beams with the sameintensity in a geometrical grid. Such elements are used for example to measure objectives and

Figure 1: LC2002 spatial light modulator

1This experiment is based on the OptiXplorer Kid developed by the company HOLOEYE Photonics AG. Someparts of the instructions have been extracted in word and image from the corresponding manual [1].

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telescope mirrors faster and more precisely compared to measurements with one beam or withmechanical scan devices.

In the experiments a liquid crystal micro-display will be used as a spatial light modulator tocreate diffractive optical structures, for exploration of dynamical diffracting objects as well asthe investigation of the functionality and the physical properties of the device itself.

Liquid crystal displays (LCDs) with pixel sizes significantly smaller than 100µm are usednowadays in digital clocks, digital thermometers, pocket calculators, video and data projectorsand rear projection TVs. Due to the low cost, robustness, compactness and the advantage ofelectrical addressing with low power consumption, LCDs are superior to other technologies.

1.1 Electro-optical properties of liquid crystal cells

Liquid crystals (LCs) are considered a phase of matter, in which the molecule order is betweenthe crystalline solid state and the liquid state. The LCs differ from ordinary liquids due to longrange orders of their basic particles (i.e. molecules) which are typical for crystals. As a resultthey usually show anisotropy of certain properties, including dielectric and optical anisotropies.However, at the same time they show typical flow behaviour of liquids and do not have stablepositioning of single molecules.

There are different types of liquid crystals, among which are nematic and smectic liquid crystals.Nematic liquid crystals show a characteristic linear alignment of the molecules, they have anorientation order but a random distribution of the molecule centres. Smectic liquid crystalsadditionally form layers, and these layers have different linear orientation directions. Thereforesmectic liquid crystals have an orientational and a translational order.

For usage in LCD’s, liquid crystals are arranged in spatially separated cells with carefully chosendimensions. The optical properties of such cell can be manipulated by application of an external

Figure 2: Illustration of different liquid crystal phases with respect on orientation and positioning[2].

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electric field which changes the orientation of the molecules in a reversible way. Due to thelong range order of the molecules and the overall regular orientation, a single LCD elementfeatures voltage-dependent birefringent properties. Therefore an incidental light field sees differentrefraction indices for certain states of polarisation. Therefore an LCD element can change thepolarisation state of such a light field specifically by providing a well-defined voltage.

The LC cells have boundaries which are needed to firstly separate the cell and secondly toaccommodate the wires needed for addressing each cell with an independent voltage value (i.e.the electric field). Because the cells are arranged in a regular two-dimensional array, the cellboundaries act as a two-dimensional grating and produce a corresponding diffraction effect.

1.1.1 Twisted nematic LC cell

The following discussion will focus on LCD based on twisted nematic liquid crystals. In the cellsof such LCDs, the bottom and the top cover have alignment structures for the molecules whichare typically perpendicular to each other. As a result of the long range order of the LC, themolecules form a helix structure, which means that the angle of the molecular axis changes alongthe optical path of light propagating through the LC cell.

The helix structure of twisted nematic crystals can be used to change the polarisation status ofincident light. When the polarisation of the light is parallel to the molecules of the cell at theentrance facet, the polarisation follows the twist of the molecule axis. Therefore the light leavesthe LC cell with a polarisation that is perpendicular to the incident polarisation.

In order to realize a dynamic optical element, a voltage is applied to the LC cell. This voltagecauses changes of the molecular orientation, as is illustrated in Figure 4 for three voltagesVA, VB, VC . Additionally to the twist caused by the alignment layers (present already at VA = 0),the molecules experience a voltage-dependent tilt if the voltage is higher than a certain threshold(VB > Vthr). With increasing voltage (VC � Vthr), only some molecules close to the cell surface

Figure 3: Transmission of linearly polarised light through a nematic LC cell (without appliedvoltage)[1].

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Figure 4: LC cell with different applied voltages: VA = 0 with molecules in initial state, VB > Vthrwith tilted molecules in the direction of the field, VC � Vthr with parallely orientatedmolecules at the central area of the cell.

are still influenced by the alignment layers, but the majority of molecules in the centre of the cellwill get aligned parallel to the electric field direction.

If the helix arrangement of the LC cell is disturbed by the external voltage, the guidance of thelight gets less effective and eventually ceases to happen at all, so that the light leaves the cellwith unchanged linear polarisation.

It is straightforward to combine such switchable element with a polariser (referred to as analyser)to obtain a light valve for incident polarised light. For non polarised light sources, it is onlynecessary to place a second polariser in front of the LC cell to obtain the same functionality. Togain a more detailed insight, it is necessary to review the polarisation of light fields.

1.1.2 Polarisation of light

The polarisation of light is defined by orientation of its field amplitude vector. While non polarisedlight consists of contributions of all the different possible directions of the field amplitude vectors,polarised light can be characterized by either a single field component (linear polarisation) or bya superposition of field components in two directions. Partial polarised light is a mix of polarisedand non polarised light. The polarisation state of such light can be described with the Stokesparameters.

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For a completely polarised light field the state of polarisation of this light field can be expressedby a Jones vector representation. Without loss of generality, we consider a light beam propagatingin z direction

E(r) = E0(r) · ei(kz−ωt) + c.c., (1)

where E0 = (Ex, Ey, Ez)T and Ei are the components of the electrical part of the light fieldwith the linear dispersion relation ω = ck, where the wave number k = 2π/λ is given by thewavelength λ. Since Ez = 0, we restrict ourselves to the representation of the polarisation in atransverse, two-dimensional space J , created by the superposition of the orthogonal polarisationsExex und Eyey. The Jones-vectors are given by

J =(ExEy

)(2)

beschrieben, where Ex und Ey are complex numbers which tell about the relative amplitudesand phases of the two basic linear polarisations. It is convenient to normalise this vector J sothat |J| = 1 and the field strength (i.e. amplitude) is expressed in a separate variable.

A linear polarisation is given by vectors of the form

J =(

cos (α)sin (α)

)(3)

which tells that the polarisation components in x and y direction do not have a mutual phasedelay. Arbitrary states of polarisation are referred to as elliptic polarisation and are given byvectors

J =(

cos (α) exp (iΓ/2)sin (α) exp (−iΓ/2)

)(4)

where Γ denotes the phase delay between the polarisation components.

The expression of polarisation states can be used to analyse the propagation of light in anisotropicmedia like solid state matter crystals or liquid crystals.

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Figure 5: Illustration of refractive indices: ordinary no, extraordinary ne and resulting extraordi-nary refractive index neo(θ) for different orientations of the molecule.

1.1.3 Propagation in anisotropic media

Materials which on the atomic level can be described as a regular arrangement of particles ina well-defined lattice are referred to as crystals. Liquid crystals can be described by the samemodel due to the translational order of the molecules.

If the material is not isotropic with respect to its optical properties, the refractive index andcorrespondingly the speed of light becomes polarisation dependent for most propagation directions.However, the highly ordered state of matter nevertheless leads to the existence of optical axes inthe material. If a light wave propagates parallel to an optical axis, the material appears to beisotropic for that wave.

Light propagation along directions that are not parallel to an optical axis is characterized by twoindices of refraction n1 and n2 valid for two orthogonal states of polarisation, which are usuallydifferent. This effect is referred to as birefringence.

Here the discussion shall be limited to uniaxial crystals, in which the polarisation states arereferred to as ordinary (o) and extraordinary (e) polarisation, with refractive indices no and ne.Along the optical axis, the refractive index is given by no for all polarisation directions withthe velocity c/no. For all other propagation directions the ordinary polarised light propagateswith c/no, too. The velocity c/neo of the extraordinary polarised light depends on the angle ϑbetween the direction of propagation and the optical axis:

1n2

eo(θ) = cos2(θ)n2

o+ sin2(θ)

n2e

. (5)

The impact of a birefringent material on a transmitting light wave can be expressed by matriceswhich convert the Jones vector of the incident light (see previous section) to a new Jones vector,

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the so called Jones matrices. In its simplest form, the Jones-calculus is a systematic method ofcalculation to determine the effects of different, the polarisation condition affecting elements toa completely polarised light wave. When using this calculation the vector of the incident lightwave will be multiplied one after another with characteristic matrices, the Jones matrices - onefor each optical element. Finally, the result will be the vector of the electric field strength of thewave exiting the optical system.

The different refractive indices introduce a mutual phase delay between the two partials fieldscorresponding to the two linear polarisations which are propagating with the velocities c/noand c/ne. The transmitted light after a distance d (given by the thickness of the material) istherefore given by

J =(E′eoE′o

)= Wd

(EeoEo,

)(6)

where

Wd =(

exp(−ineω

c d)

00 exp

(−inoω

c d)) . (7)

1.1.4 Wave plates

An optical component with parallel entrance and exit facets made from an uniaxial birefringentmaterial with its optical axis perpendicular to the direction of light propagation is referred to asa wave plate. Such optical components can be described in different coordinate systems.

The matrix Wd of a wave plate can be expressed in the form

Wd = exp(−iφ)(

exp (−iΓ/2) 00 exp (−iΓ/2)

), (8)

where the quantity Γ, describing the relative phase delay, is given by

Γ = (ne − no) 2πλd (9)

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and the quantity φ, describing the absolute phase, is defined by

φ = 12 (ne + no) 2π

λd. (10)

The phase factor exp(−iφ) may be neglected in some cases, for example when there is nointerference phenomena considered.

A half-wave plate is a particular example of a wave plate with a thickness

d = λ

2 (ne − no) , (11)

so that the mutual phase delay is given by Γ = π. This means that the optical path between thetwo waves with orthogonal polarisations differs by half the wavelength of the light field. TheJones matrix of a half-wave plate, whose extraordinary axis meets the x axis of the laboratorysystem x-y, is obtained as

WHWP =(−i 00 i

). (12)

This means that light polarised parallel to the x direction, which is assumed to have an angleof 45◦ with respect to the direction of ordinary polarisation, will have its Jones vector changedfrom

J =(ExEy

)=(

10

)(13)

to

J′ =(−i 00 i

)(10

)=(

0i

)= exp

(−iπ2

)(01

), (14)

which corresponds to a rotation of the polarisation direction by an angle of 90◦.

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1.1.5 Jones matrix representation of a twisted nematic LC cell

A twisted nematic LC cell can be described as a succession of a high number of very thin waveplates which change the orientation of their optical axis according to the change of the directionof the molecular axis. The total Jones matrix of the cell is then obtained by matrix multiplicationof all the matrices of the assumed thin wave plates.

If a LC cell satisfies α� β, which is the case for thick cells, the Jones matrix can be significantlysimplified and permits an intuitive interpretation:

WTN-LC ≈ R(−α)(

exp(−iβ) 00 exp(iβ)

), (15)

where R is a rotation matrix, α the twist angle of the molecules through the entrance and exitfacets of the cell and the quantity β is given by

β = Γ2 = πd

λ(ne − no) . (16)

If the incident light is polarised parallel to the x- or y axis the polarisation axis will be rotatedby the twist angle α between the directions of the alignment layers, as implied by the intuitiveexplanation illustrated in 3.

1.1.6 Properties of a TN-LC cell with an applied voltage

If a voltage is applied to the cell, the molecules tend to align parallel to the electric field. Therebythe anisotropy of the liquid crystal is reduced because the angle between the direction of lightpropagation and the molecular axes gets smaller until eventually both directions are parallel,and the optical axis of the liquid crystal is parallel to the direction of light propagation.

The analysis of the intermediate cases in which the molecules are no more aligned in the helixstructure but not yet parallel to the field can be done, but are not within the framework of thisinstruction. With such analysis, the voltage-dependent optical properties can be obtained. As aresult, incident light with linear polarisation leaves the cell with an elliptic state of polarisation.

However, the voltages applied to the LC cell are not directly accessible in the LC2002 devicecontained in this kit. The voltages applied to the cells can be controlled via a customizedelectronic drive board. This drive board receives information on what voltage should be appliedto the cell as grey level values of the signal created by the VGA output of a graphics adapter ofa common PC.

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1.1.7 Amplitude and phase modulation using TN-LC cells

The voltages that are applied to the cells of the LC display are in a range that permits analmost continuous transition between the helix state which rotates the incident polarisation tothe isotropic state in which the polarisation remains unchanged. It is obvious that by insertingan analyser behind the display the LC display one can achieve an amplitude modulation of atransmitting polarised light wave.

By examination of the Jones matrices one can also deduce that the phase of the light passing theanalyser is changed dependent on the voltage-dependent parameter β. If the SLM is illuminatedby a coherent light source (e.g. a laser) various diffraction effects that are based on this phasemodulation can be observed. For the experiments it is important to note that the incidentpolarisation for obtaining a comparatively strong phase modulation with only weak amplitudemodulation is not parallel or perpendicular to the alignment layers.

At the LC2002 amplitude and phase modulation are coupled. Using the position of polariserand analyser, however, different ratios of amplitude and phase modulation can be realized. Amaximum amplitude modulation with minimal phase modulation is called an amplitude-mostlyconfiguration. A maximum phase and minimum amplitude modulation is called a phase-mostlyconfiguration

For doing experiments with the LC2002 dealing mainly with diffraction effects, it is not necessaryto review the changes of the polarisation states in detail. It is sufficient to understand that thesystem which comprises of polariser, LCD and analyser can be seen as an optical componentwhich can be used to introduce a mutual voltage-dependent phase shift between the waves passingthrough individual LC cells.

The main steps in the transition from a TN-LC cell to a LC-based micro-display are of coursethe arrangement of the cells in one-dimensional or two-dimensional arrays, and the establishmentof an interface that permits individual addressing of the cells. This results not only the creationof phase or amplitude modulation, but also the creation of a desired spatial distribution of thismodulation, resulting in the creation of a spatial light modulator.

A LCD sandwiched between polarisers can thus be used not only as an image source in aprojection system, but also (with other polariser settings) as a switchable diffractive elementwhich can be used to represent dynamically optical elements like Fresnel zone lenses, gratingsand beam splitters, which can be modified by means of electronic components.

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2 Scalar theory of light waves and diffraction – Fourier optics

2.1 Plane waves and interference

The ability of interference is a fundamental property of light, which is caused by its wave nature.Interference refers to the effects of mutual amplification and cancellation which are observedwhen two or more waves are superimposed. When monochromatic light waves of the samefrequency interfere, the amplitudes of the resulting field at any place and at any time is given bythe addition of the amplitudes of the involved waves.

In the following we will only review the interference of linearly polarised waves with amplitudevectors parallel to each other. Therefore, in the mathematical description of the interference onecan use the notation of complex amplitudes instead of the summation of the complex vector fieldamplitudes.

Unlike sound waves, light waves have certain preconditions for their ability to create interferenceeffects. These preconditions originate from the process of light emission, and are summarized inthe term and the concept of coherence.

2.1.1 Interference of plane waves

A single plane wave can be written as

E(r, t) = A0ei(kr−ωt+δ). (17)

where ω denotes the frequency of the light and k is the wave vector, while δ denotes a constantphase-offset. For two overlapping light waves at an arbitrary time t, we get spatially dependentamplitudes

E1(r) = A1ei(k1r+δ1) and E2(r) = A2e

i(k2r+δ2). (18)

For any position r the resulting complex amplitude is given by

E(r) = E1(r) + E2(r) = A1ei(k1r+δ1) +A2e

i(k2r+δ2). (19)

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The intensity of the interference is obtained as

I(r) ∝ E(r)E∗(r) = A21 +A2

2 +A1A2ei(r(k1−k2)+(δ1−δ2)) +A1A2e

−i(r(k1−k2)+(δ1−δ2)), (20)

or

I = I1 + I2 + 2√I1I2 cos(∆φ). (21)

The phase difference ∆φ of both interfering waves is

∆φ = φ1 − φ2 = r(k1 − k2) + (δ1 − δ2). (22)

Considering two waves of equal amplitude (A1 = A2 = A0), the intensity of the interferencechanges periodically between 0 and 4I0. Strict additivity of intensities applies only if the addend(the so called interference element)

2√I1I2 cos(∆φ) (23)

is zero. If this is the case, there is no interference. Interference means deviation from theadditivity of intensities. For all spatial positions which satisfy

∆φ = 2Nπ mit N = 0, 1, 2, ... (24)

a maximum intensity can be found

Imax = I1 + I2 + 2√I1I2. (25)

The positions with minimal intensity are

Imin = I1 + I2 − 2√I1I2 (26)

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and satisfy

∆φ = 2(N + 1)π. (27)

It is important to note that the intensity as a whole is neither increased nor decreased. It is thenature of interference to merely distribute the energy differently, as the total energy must bepreserved.

The intensity pattern with observable bright-dark contrasts is referred to as interference fringes.An important parameter for the characterization of their visibility is the contrast. It is defined asthe sum of the maximum and minimum intensity normalised by the difference between maximumand minimum intensity:

C = Imax − IminImax + Imin

. (28)

For the superposition of two flat monochromatic waves we obtain

C = 2√I1I2

I1 + I2. (29)

2.1.2 Coherence of light

The previous explanations deliver only a rather vague description of the true nature of theinterference of light waves and assume conditions (monochromatic waves, point light sources),which in reality are not met. As experience shows, it is in general not possible to observeinterference effects by superposition of two waves emitted by different thermal light sources orfrom two different points of an extended thermal light source.

Although light emission is often treated as a continuous process, it can be described moreaccurately as a sequence of emissions of many short wave trains. Electrons in the atom moveinto excited states by absorption of energy. The duration of the irradiation corresponding to thelimited lifetime of these states of about 10−8 s leads to the emission of short wave trains of about3m in length. Furthermore, the light emitted from different points of the thermal light source isstatistically distributed, and the phase relationship between two successive wave trains emittedby a point source changes from emission event to emission event in an unpredictable way.

In the previous discussion, it was implicitly assumed that the difference of the phase constantsδ1 and δ2 remains constant for the observation time tb. However, the waves emitted from anextended light source have neither a spatially nor temporally stable phase relationship. As a

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result, many waves with statistically different phase relationships are superimposed successivelyduring the observation time. The resulting interference pattern is not stationary but changes itsappearance in intervals of 10−8 s. Only the timeaveraged intensity

〈I〉tb = I1 + I2 + 2√I1I2

1tb

∫ tb

0cos(∆φ)dt (30)

can be measured, assuming that the amplitudes of the individual waves are constant over thetime interval tb.

If the phase relations of the superimposed waves vary in a way that during the observation timeall phase differences between 0 and 2Nπ occur for equally long time intervals, the temporalaverage of the quantity cos(∆φ) is zero, and just the sum of the individual intensities I1 and I2is measured. In this case, one can no longer speak of interference, and these light sources arereferred to as incoherent.

If in contrast the difference (δ1 − δ2) remains constant over the whole observation period, theparticipating waves are called coherent, which means that there exists is a fixed phase relationshipbetween them. In this case, the measured intensity is indeed described by equation (30). Theradiation emitted by real light sources is partially coherent, strictly coherent and incoherent light,respectively, is only obtained from the light of infinitely extended or punctual light sources.

In Section 2.1.1, the interference of monochromatic light waves was considered, emitted by idealpoint light sources. These assumptions are obviously idealizations. Real light sources are alwayslight emitting areas of finite size. The relationship between the size of the light source and theobtainable contrast in the interference fringes leads to the coherence condition, which tells thatthe product of the light source width b and the emission aperture sinα (2α is the opening oraperture angle) has to be very small compared to half the wavelength of the emitted radiation.Although the individual point sources emit waves with statistically distributed phase relations, acertain extension of the light source for producing interference pattern is allowed.

Wave trains with fixed phase relationships can be generated by splitting the light of one lightsource into two or more partial waves. The coherent division can take place using one of thefollowing two principles:

1. Division of the amplitudeAn interferometer, which is based on this method, is the Michelson interferometer.

2. Division of the wavefrontThis principle is used in the Young interferometer, for example.

As a result of the division, two waves are obtained, in which the phase changes in an unpredictableway, but in the same way. The difference (δ1 − δ2) therefore remains constant.

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After paths of different length, creating a phase difference, the partial waves will be reunited.With the help of Fourier’s integral theorems, however, it can be shown that the wave trainsare only quasi-monochromatic because of their limited length Lk. They have a finite spectralbandwidth. Only an infinitely extended wave would be monochromatic.

Bei zunehmender optischer Wegdifferenz nimmt der Kontrast der Interferenz ab, da sich diephasenmäßig korrelierten Wellen nicht mehr überlagern. Der Gangunterschied, bei dem derKontrast auf 1/e abgefallen ist, heißt Kohärenzlänge.

As the optical path difference between the two wave trains increases, the contrast of theinterference pattern decreases, since the phase relation becomes increasingly destroyed. The pathdifference, at which the contrast is reduced to 1/e is called coherence length.

The Michelson interferometer allows a very simple and rapid determination of the coherencelength. For approximately equal paths, the contrast of the interference fringes is quite high.Increasing the optical path difference will decrease the contrast. If the retardation exceeds thecoherence length, the contrast decreases to zero.

2.2 Diffraction theory

2.2.1 Fraunhofer diffraction

Consider a plane wave propagating in z direction that passes an obstacle at z = 0. This obejct isregarded as thin and is described by a complex transmission function τ(x, y). The transmittedfield reads as

Et(x, y, z = 0) = τ(x, y)Ei(x, y, z = 0). (31)

Nach dem Huygens’schen Prinzip kann die weitere Ausbreitung durch die Annahme beschriebenwerden, dass von jedem Punkt (x, y bei z = 0) der beugenden Struktur eine Kugelwelle ausgeht.Um die Feldamplitude an einem Ort (x′, y′, z) hinter dem beugenden Objekt zu erhalten, mussdaher über alle Kugelwellen summiert (integriert) werden.

The resulting light propagation can be described by applications of Huygens’ principle. Accordingto this principle, a spherical wave is created by each point (x, y at z = 0) of the diffracting object.All spherical waves must be added (i.e. integrated), to obtain the field amplitude from a point(x′, y′, z) behind the diffracting object.

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The Fraunhofer approximation applies if

(x2 + y2

) πλ� z (32)

is satisfied for (x, y) and (x′, y′) and the diffracting object is illuminated by a plane wave. In thiscase

E(x′, y′, z) = A(x′, y′, z)F [E(x, y, 0)] (νx, νy) mit A(x′, y′, z) = eikz

iλz . (33)

The quantities νx and νy are referred to as spatial frequencies analogue to the frequencies of theFourier transformation (above written as F) of temporal signals.

The far-field in the Fraunhofer approximation is given by the Fourier transformation of the fielddirectly behind the diffracting object. The spatial frequencies of the diffracting structure createwaves, which propagate with angles

α ≈ tanα = x′

z= λνx β ≈ tan β = y′

z= λνy (34)

to the optical axis, respectively. With the help of a lens the far-field of the light propagation canbe obtained in the focal plane of the lens.

This means that in optics propagation of the light field realizes a Fourier transformation in anatural way simply by propagation. The Fourier transformation of a two-dimensional object

F (νx, νy) = F [f(x, y)] (νx, νy) =∫ ∞−∞

∫ ∞−∞

f(x, y)e−2πi(νxx+νyy)dxdy (35)

can be observed directly as a function of the spatial frequencies, which can be associated withdiffraction orders. These spatial frequencies can be manipulated and filtered. The Fourier filteringis a passive parallel image processing performed at the speed of light (cf. laboratory courseoptical Fourier transformation).

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2.3 Diffraction at spatially periodic objects

Spatially periodic objects, in optics often referred to as gratings; show a discrete far-field diffractionpattern, in contrast to the spatially continuous diffraction patterns of spatially aperiodic objects.This is due to spatial frequency spectrum of periodic objects that consists of discrete frequenciesonly.

2.3.1 Diffraction orders in the Fraunhofer diffraction pattern

Each discrete spatial frequency of a one-dimensional periodical object with spatial periodicity g isa multiple of the fundamental frequency 1/g and produces by illumination with monochromaticlight a maximum in the far-field, a so-called diffraction order.

For periodic diffracting objects, the Fourier integral of the transmission function of a Fourierseries can be simplified. For a 2-dimensional object with a spatially dependent complex-valuedtransmission function τ(x, y) and spatial periodicities gx and gy , the Fourier coefficients Al,mdescribing the complex-valued amplitudes of the diffracted waves are given by

Al,m = Aingxgy

∫ gx

0

∫ gy

0τ(x, y) exp

(−2πi

(l

gxx+ m

gyy

))dxdy. (36)

The complex transmission function τ(x, y) = ρ(x, y) exp(iφ(x, y)) summarizes changes of ampli-tude and phase of the wave transmitted through the diffracting object. For a onedimensionalperiodic object the amplitude simplifies to

Al = Aing

∫ g

0τ(x) exp

(−2πi l

gx

)dx. (37)

In this equation, the diffracting object is described by a spatially resolved complex-valuedtransmission function τ(x) as a function of the position x. The given integral does not takethe second spatial coordinate into account, so that it is only valid for objects with a constanttransmission function with respect to the y direction (linear gratings).

Such transmission function τ(x) can be sinusoidal, as in the case of a grating obtained byholographic recording of a two-wave-interference (see section 2.1). The transmission functionmay in this example take any value within a certain range.

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Figure 6: Binary lattice with two transition points.

2.3.2 Fraunhofer diffraction at linear binary gratings

The values of the transmission function obtained when addressing a LCD are limited to 256values due to the addressing of the transmission function value as a single 8-bit colour channel ofthe VGA signal. The most simple example of a discretization of the transmission function isof course a signal that consists of only two different transmission values (binary elements). Forlinear gratings it is possible to describe the grating with the transition points between the areaswith the two transmission values τ1 und τ2.

We will now analyse a simple linear binary grating with only two transition points (cf. figure 6).For a grating period g there is only one free transition point called x1, the second is located at 0or (totally equivalent) at g. There is a general transmission function

τ(x) ={τ1 für 0 ≤ x ≤ x1τ2 für x1 ≤ x ≤ g

. (38)

Thus the amplitude of the zero order is

A0 = Ain ·[τ2 −

x1g

(τ2 − τ1)]

(39)

and that of the higher orders is given by

Al = iAin2πl (τ2 − τ1)

(1− exp

(−i2π l

gx1

)). (40)

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Energy-related quantities as for example light intensity and light power are proportional to thevalued square A ·A∗ of the complex-valued amplitude A. The diffraction efficiency, as the ratiobetween energy quantities, therefore is given by

ηl = Al ·A∗lAin ·A∗in

. (41)

The diffraction efficiencies depending to x1 are then

η0 =∣∣∣∣(τ2 − τ1)

2

(1− 2x1

g

)+ (τ1 + τ2)

2

∣∣∣∣2 (42)

for the zero order and

ηl = |τ2 − τ1|2

π2l2· sin2

(πlx1g

)(43)

for l 6= 0.

The diffraction efficiencies of the individual orders therefore have a characteristic envelope ofthe form sinc2(πlx1/g), which is not dependent on the individual transmission values τ1 and τ2.This means, for example, that with a transition point x1 = g/k (with an integer number k) alldiffraction orders l = nk disappear with the exception of the zero order.

For this reason a grating with a ratio of the structure widths of 1:1 (i.e. x1 = g/2), producesonly odd diffraction orders. Depending on the amplitudes ρ1, ρ2 and the phases φ1, φ2 of the twotransmittance values τ1 and τ2, the transmission function of such a grating can be written as

τ(x) ={ρ1 exp(iφ1) für 0 ≤ x ≤ g

2ρ2 exp(iφ2) für g

2 ≤ x ≤ g(44)

By evaluation of the integral in equation (37), the amplitude of the zero diffraction order A0 isobtained as

A0 = Ain2 (ρ1 exp(iφ1) + ρ2 exp(iφ2)) (45)

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and for m = 2k + 1 the amplitude in the lth diffraction order Al is

Al = iAinπl

(ρ2 exp(iφ2) + ρ1 exp(iφ1)) . (46)

With the phase difference ∆φ = φ1 − φ2, the diffraction efficiency in the diffraction orders isobtained as

η0 = 14[ρ2

1 + ρ22 + 2ρ1ρ2 cos(∆φ)

], (47)

ηl = 1π2l2

[ρ2

1 + ρ22 − 2ρ1ρ2 cos(∆φ)

]für (l 6= 0). (48)

The diffraction efficiencies in the diffraction orders are independent of the grating period g. Usingthe setting of the greyscale values of the addressed binary grating the amplitudes ρ1, ρ2 and therelative phases φ1, φ2 will be adjusted..

2.3.3 Diffraction angles of the orders

With the spatial periodicity g, often referred to as grating period, the diffraction angles αl aredetermined by the grating equation

g(sin(θ + αl)− sin θ) = l · λ (49)

where θ denotes the angle of incidence of the light. For perpendicularly incident light we haveθ = 0, and the equation is simplified to

g sinαl = l · λ. (50)

It can be seen that this equation can be written equivalently in terms of the x components ofthe wave vectors k for the incident wave and k′ for the diffracted wave, yielding

k′x = kx + l · 2πg

= kx + l · kg, (51)

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where kg denotes the absolute value of the wave vector of the grating. Introducing the spatialgrating frequency νg, a similar correlation for the spatial frequencies of the diffracted waves isobtained:

ν ′x = νx + l · 1g

= νx + l · νg. (52)

The notation with grating frequencies or wave vectors is be preferable when calculating thedirections of light propagation (or the diffraction angle) for gratings with a twodimensionalperiodicity. For example, the wave vector of the diffracted wave satisfies equation (51) in thetwo directions perpendicular to the direction light propagation. The missing vector componentof the propagation direction can then be calculated from the absolute value of the wave vector,which is determined by the wavelength.

2.4 Applications of Fourier optics

So there are many applications of the Fourier optic thinkable, only one selected example shallbe presented here. The paragraph of this theoretical introduction presents the concept of thespatial frequency filtering, which quite clearly illustrates the concept of ‘spatial frequencies’.

2.4.1 Design of diffractive elements

By solving the inverse diffraction problem for a desired diffraction pattern the required diffractingstructure can be calculated and produced with suitable micro-fabrication methods. The result isa so called diffractive optical element (DOE) which reconstructs the desired image (by diffractionand interference) in the far-field when illuminated with a coherent light source. With somerestrictions DOEs can replace classical optical elements like lenses, beam splitter, prisms andeven beam forming elements. Furthermore, even more complex elements like multi-focusinglenses can be created.

For many applications, the suppression of the zero diffraction order and of undesired higherdiffraction orders is the challenge. DOEs have considerable chromatic aberrations and thediffraction efficiency is limited. Nevertheless DOEs are already used in many applications,especially when the available space is limited or the optical function could not be realized withother optical elements.

At least, in the framework of this laboratory course we generate exemplarly arbitrary diffractiveelements and investigate them qualitatively. In particular, this demonstrates the advantages ofdynamically changeable structures.

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3 Safety instructions

The provided laser is a class 3B laser, therefore special safety measures have to be taken. Bothdirect and indirect irradiation of this laser is dangerous for the eyes. Never look in the laserbeam. Reflecting accessoires like for instance watches or rings must not be worn. During thecourses, laser safety glasses have to be worn.

4 Practical part

Equipment:

• LC-Display

– Active area: 26, 6 mm× 20, 0 mm,Pixel: 832 px× 624 px,Pixel size: 32µm

• Laser diode:

– Wavelength: 650 nm,Power: 8mW,Laser class: 3B

• Polarizer & Analyzer

• Lenses

• Camera

• Screen

• Power meter

4.1 Amplitude modulation and projection

4.1.1 Angular distribution of linearly polarised light

1. Mount the laser, so that its beam axis points along the rail. Many lasers are manufacturedto emit linear polarised light.

2. First, the beam passes an analyzer which axis is rotated at an angle ϑ with respect to thex axis (e.g. plane of the table).

3. A suitable lens focusses the beam on the power meter.

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4. Vary ϑ in steps of 10◦ and measure the transmitted intensity I.

5. Plot the transmitted intensity I in dependence of the rotation angle ϑ.

6. Fit a suitable function to the recorded data points and discuss your results with respect toMalus’ law.

7. Finally, calculate the contrast.

4.1.2 Preparations for an optimized application of the SLM

1. All optical elements except of the laser are removed. Collimate the laser beam by adjustingthe lens in front of the laser diode (screwed).

2. The SLM is in amplitude mostly configuration when the input polarization is at an angleof −45◦ and the analyzer at an angle of −135◦, respectively. Since the angle of the linearpolarized laser beam is well known from the previous experiment, the laser is rotated inits mount, so that it will have −45◦. Therefore, if you wish you can temporarily place apolarizer in the beam and observe the transmitted intensity.

3. Subsequently, the SLM is mounted, followed by the analyzer ϑ = 135◦ and the camera.

4. Using the OptiXplorer Software, a Horizontally Divided Screen is created (half black (0),half white (255)). The camera can be used to observe the image created by the SLM.

5. Determine the contrast by measuring the intensity. Only for this measurement, temporarilya lens is mounted into the beam and the power meter is used instead of the camera (in focusplane). Now choose Blank Screen with gray values 0 and 255 instead of the HorizontallyDivided Screen

6. Afterwards make sure (qualitatively, without lens, power meter, but using HorizontallyDivided Screen) that the contrast varies, if the analyzer is rotated around 45◦/90◦/135◦

and that an inverted image is shown at an angle of 90◦.Take pictures at all four angles ofthe analyzer.

7. Describe and explain the optimization of the contrast using the SLM as an amplitudemodulator and use the optimal configuration for the following experiments.

4.1.3 Pixel size of the LC display

1. The pixel size of the display can be calculated with the equations of classical optics. Thedisplay will be addressed with a rectangular object of known dimension. It will be imagedby a (thin) lens, whose focal distance is also known.

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2. A white image with a black square with the size of 200× 200 pixels in its centre will beaddressed on the display. The object size G of the black square on the display can becalculated from the image size B, the image distance b and the focal length f of the lens:

G = B

b/f − 1 (53)

3. Use a suitable lens and determine the image size B in an sufficient distance to the focalplane.

4. Estimate the inaccuracy ∆G and the resulting value of the pixel size.

4.1.4 Relation between pixel voltage and modification of the polarisation state

1. For six grey levels [250, 200, 150, 100, 50, 0] the rotation angle ϑi of the analyser for thesmallest and largest measured power values is determined. The graphical representation ofthe result yields a different ellipse for each grey level. Also note for each gray value themaximum and minimum intensity.

2. Explain the relation between gray value and rotation angle. What is the orientation of theliquid crystal in the SLM without applied voltage?

3. Image for at least 3 of the gray values (e.g. [250, 150, 0]) the eccentricity of the light inpolar coordinates: The graphical illustration is performed in parametrical form as an ellipse.The maximum intensity corresponds to the large half-axis a of the ellipse, the minimumintensity corresponds to the small half-axis b. The angle of the analyzer for the maximumintensity is the angle ϑi, at which the main axis is tilted with respect to the x axis.

4.2 Using the SLM as DOE

Illuminating a spatial light modulator with a coherent light source generates diffraction patternsbehind the display similar to those that appear behind a conventional optical grating. Onecan consider the non-addressed display already as an optical grating. This diffraction patterngenerates a diffraction pattern in the far-field when illuminated by a collimated light source. Witha so-called ‘Fourier lens’ the diffraction far-field pattern can be created in the rear focal plane ofthe lens. This diffraction pattern allows conclusions to be drawn about the characteristics of thedisplay. For example, the pixel size can be determined.

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4.2.1 Determination of the pixel size

1. The diffracting properties of the LC display are based on the pixelated layout. It acts asa grating with a period g given by the pixel size, and the created diffraction orders areobserved at angles α satisfying

g · sin(α) = m · λ with m ∈ Z. (54)

2. Experimental setup: collimated laser, unplugged SLM and analyzer in optimal configuration,lens (focal length f), screen.

3. Display the SLM (placed at the negative focal plane of the lens) on the screen at thepositive focal plane. Determine the distance d of two of the higher orders to each other andcalculate the lattice constant g. The angle α is given by tan(d/2f) ≈ d/(2f). Compare thedetermined pixel size with your results from part 4.1.3 as well as with the specificationsgiven at the materials part.

4.2.2 Intensity distribution in diffraction orders of non-addressed display

1. The intensity of the single diffraction orders in horizontal and vertical direction shall bemeasured. From the results one can calculate the fill factor of the LCD cells.

2. To keep the setup simple, the expanding optic of the laser module will be used to focusthe beam. When placing the laser right in front of the light modulator, the diffractionorders will be separated enough for measurements of individual orders to be made in adistance of about 70− 90 cm. Using the power meter, measure at the focal plane the 11intensity values of the orders 0− 10, in both horizontal and vertical direction and imagethem graphically.

3. Why is there a modulation in the imaged values? A single pixel of the LC display consistsof a transparent part and an opaque control electronic. Determine the duty cycle, i.e. theratio of transparent and non-transparent part using the measured intensity modulation.Estimate the filling factor for the transparent part.

4.2.3 Recording of different diffraction patterns with the camera

1. Setup: collimated laser, SLM and analyzer in optimal configuration, lens, camera in focusplane.

2. Image a few diffraction patterns of different DOEs: single slit, double slit, grating. Recordat least one of this DOEs two times using different structure sizes.

3. What can you observe? What changes for different structure sizes and why?

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4. For a grating of your choice: measure the intensity of the 0. and 1. diffraction maximum atdifferent gray values and calculate the diffraction efficiency. One possible definition of thediffraction efficiency is η = I1.max/I0.max.

5. Why is it possible to create diffraction patterns with the SLM?

4.3 Computer generated holograms (CGHs)

Computer generated holograms can be calculated with the OptiXplorer software. The advantagesof such dynamic structures are one of the topics of this module. The holograms will be modifiedwith lens functions included in the software. The optical parameters of these diffractive elementswill be determined.

4.3.1 Focal length of the diffractive lens

1. Setup: collimated laser, SLM, analyzer, lens, camera.

2. A DOE is created with the OptiXplorer software. In the simplest case it is a blank screen.With the toolbar at the right window edge a lens phase will be added. The focus createdby this diffractive lens should be found and the corresponding focal length be determined.

3. Familiarise yourselves with how computer generated holograms can act e.g. as lenses withdifferent focal length by creating 4 different lens phases [100, 75, 50, 25]. Note in a tablethe lens phase with the corresponding focal langth.

4.3.2 Design arbitrary diffractive optical elements as CGHs

1. Using the OptiXplorer Software, create diffractive elements from arbitrary images. Here,an iterative Fourier transformation algorithm (IFTA) is applied.

2. Create at least one own hologram using the OptiXplorer software and record this.

3. Note: The program only accepts bitmap pictures with 200 × 200 pixels. These imagesshould preferentially be white on black background, for visibility reasons. The imagingquality of the holograms is not perfect, so restrict yourselves to simple images or letterswith good contrast.

4.3.3 Demonstration of two DOEs addressed on one display

1. Finally a short experiment with a qualitative description. Two different DOEs will beaddressed side by side on the screen. For example the width of two full screen windows

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will be halved. The resulting diffraction pattern contains both images generated by thetwo DOEs but each will only have half the intensity.

2. Describe and record qualitatively how the diffraction pattern looks like. Discuss theunderlying physical effects.

3. Consider a thin statical hologram (e.g. imprinted on a glas substrate). What is thedifference of the diffraction pattern for the case, that the complete hologram is illuminated,or for the case, that the glass plate is broken in to halfs and you just illuminate one half ofit?

References

[1] HOLOEYE Photonics AG. OptiXplorer. Holoeye.com/optics-education-kit/. 2007.

[2] www.merck-performance materials.de/. Date: 12. Oktober 2015.

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