Theory and Basic Concepts of Holography
The concept of holography involves a two-step procedure, the recording and reconstruction of optical wave fronts. Recording a hologram denotes the process of storing the electric-field distribution of at least two interfering electromagnetic waves inside a photo-active medium.
Unlike conventional photography, the stored pattern has no similarity to the object from which the light is emitted. The reason is that the interference pattern does not only contain information about the amplitude of the light wave, but also about its phase. The spatial infor-mation can be restored in a subsequent reconstruction step, generating a spatial impression identical to the one of the original image.
It was D. Gabor who invented the technique in 1948[81]and suggested the term “holo-graphy”, which originates from Greek and means writing the complete information. His idea was to improve the resolution of transmission electron microscopy, but different problems arose, e. g., the missing of coherent light sources with sufficient intensity. It took until the invention of the laser in the 1960s when holography was revived by E. N. Leith and J. Upatnieks.[82]Although Gabor did not solve the problem he originally intended, he finally was awarded the Nobel Prize in 1971 “for his invention and development of the holographic method”.[83]
Half a century later, holography has developed to an extensive area of research. Differ-ent techniques have evolved, starting from Gabor’s inline holograms to the now common off-axis method. Theories have been developed to describe the diffraction off volume and surface-relief gratings. Depending on the physical properties of the storage medium, either the Raman-Nath or Bragg diffraction regime applies. A summary of important aspects is given in the following.
2.1 Plane-Wave Holography
In general, the interference of a plane wave and the coherent light emitted from an extended, three-dimensional object yields a complicated field distribution in the hologram plane. In the special case of the superposition of two coherent plane waves of equal wavelength, a simple sinusoidal interference pattern results. The latter is used for the inscription of volume or surface relief gratings. Both grating types contain all essential information about the suitability of a material as a holographic storage medium. Holographic reconstruction with a plane wave corresponds to grating diffraction in this case.
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Superimposing an object and a reference plane wave~Eobj and~Eref with equal frequencyω yields the total field
~Etot(x,z,t) =~Eobj+~Eref=~E0,objei(~kobj~r−ωt) +~E0,refei(~kref~r−ωt+∆ϕ). (2.1) If both waves are coherent, they have a fixed phase relationship and the common time-dependent phase factor exp(−iωt) can be dropped for simplicity. The phase shift ∆ϕ in Eq. (2.1) corresponds to the difference in travel time in this case. In the coordinate system depicted in Fig. 2.1 the wave vectors become
~kref=2πn0
withλwbeing the vacuum wavelength of the incident writing waves andn0 the refractive index of the medium. The vector sum of the object and reference wave vector results in the grating vector~K. The above assumptions imply that the waves already propagate inside the medium (n=n0forz<0). The boundaries can simply be taken into account by calculating the angles of the plane waves inside the medium according to Snellius’ law.
d0
Figure 2.1: Schematic diagram of the sinusoidal interference pattern generated by two interfering plane waves in a medium of thicknessd0. They-axis is perpendicular to thex-z-plane. Λ: Grating period,~K: grating vector,Φ: slant angle.
If the angles of incidence are equal (θref =θobj=θ), the grating is unslanted, i. e., Φ=90° (cf. Fig. 2.1). In this case the total electric field becomes
~Etot(x,z) =
2.1 PLANE-WAVE HOLOGRAPHY 11 Without specification of the polarization state of object and reference wave, the total field is an unknown function ofx and z. It can be seen, however, that the z dependency just describes the propagation of the wave, while thex dependency is responsible for the formation of a grating with grating periodΛ.
Basically a large number of polarization states is conceivable, but only few of them are of practical importance. Without loss of generality, the phase difference can be chosen to be∆ϕ =0. For equal amplitudes |~Eobj|=|~Eref|=E0 and some selected polarization states of the reference and object wave, the electric field in thex-y-plane atz=0 takes the form shown in Table 2.1. The oscillation of~Etot perpendicular to the plane of incidence is denoted assspolarization, while the single letters indicate the polarization states of the individual waves. Accordingly, the oscillation of~Etotin thex-z-plane is thepp-polarization state. The±45°-state is generated by using waves with polarizations rotated to an angle of
±45° with respect to this plane. The termrlcpdescribes the interference of a left and a right circularly polarized wave also denoted as “orthogonal” polarization. Further polarization configurations can be found in the literature.[62,84,85]
Table 2.1: Electric field of two interfering plane waves of equal amplitude and angle of incidence in thex-y-plane atz=0. The arrows indicate the direction and time-dependent amplitude of the field for small angles of incidence, i. e.,θ→0. Red dots indicate the direction of ~Etot att=0, while black dots mean zero intensity. The coordinate system corresponds to the one shown in Fig. 2.1. A prefactoridenotes an optical phase shift byπ/2.
z
The superposition of two waves with equal linear polarization, i. e.,ssorpp, results in a sinusoidal variation of the intensity with the same polarization. Pure polarization gratings with constant intensity are obtained by mixing different states of polarization, such as in the sp, ±45°, and rlcp configuration. It is worth to mention that onlyrlcp illumination generates a completely linearly polarized electric field with a rotating polarization vector.
Other polarization gratings change from linearly to circularly polarization and, therefore, the transition areas are exposed to elliptical polarized light. The formulas listed in Table 2.1
the plotted arrows become tilted if θ strongly deviates from zero.[86] The black arrows also indicate an oscillation of~Etot as time proceeds, while the red dots represent its initial direction. As the interaction of a photo-active material with the electric field only depends on the field amplitude but not on its sign, the periodicity of the inscribed grating is given by Λ.
2.1.2 Grating Diffraction
As mentioned above, a photo-active medium placed at the plane of interference interacts with the electric field distribution. This interaction changes the optical parameters (refrac-tive index, absorption or sample thickness) in the illuminated areas and, ideally, results in a permanent storage of the interference pattern. An off-axis hologram is generated, if ref-erence and object beam have different propagation directions while they enter the medium from the same side. If the object beam contains information about a three-dimensional ob-ject its reconstruction would appear on the other side of the sample, suggesting the term
“transmission hologram”. Since there is no object present in the case of plane-wave holo-graphy, the storage of the interference pattern of object and reference beam results in a very simple hologram, i. e., a diffraction grating.
nair≈1
n0>1
x
z y
θl, trans
θr
Λ
H θl, refl
Tl
Rl
Figure 2.2: Diffraction of a plane wave off a grating with arbitrary periodic shape. Reflected orders appear under angles θl, refl, transmitted ones underθl, trans. The maximum height of the grating isH.
Three types of diffraction gratings may be distinguished, i. e., absorption, refractive-index and surface-modulated gratings. A schematic diagram of the latter is shown in Fig. 2.2.
To provide access to the theory of grating diffraction in a more general manner, the surface of the depicted grating is assumed to be modulated periodically but the shape may, for the time being, be given by any function that fulfills d(x) =d(x+mΛ) with m∈N. Analo-gous expressions shall be valid for absorption or refractive-index gratings. Here the surface stays flat and the grating constitutes a periodic modulation of the respective parameter in the volume of the material. Figure 2.1 of the preceding chapter provides an illustration of the situation in this case. From the assumption that refractive index n, absorption α, or
2.1 PLANE-WAVE HOLOGRAPHY 13 thicknessd of the material are periodic, it follows that they can be represented as Fourier series:
The diffraction off the grating can be measured either by illumination with one of the writing beams or a plane wave at a different wavelengthλr. The latter has the advantage thatλrcan be chosen such that it does not influence the process of grating formation. If any of the three material parametersα,n, ordsatisfies Eq. (2.4), the diffracted fieldE0tot(x,z)becomes pseudo periodic,[87,88] i. e., E0tot(x+Λ,z) =E0tot(x,z)exp(ikrΛsinθr) with kr=2πnair/λr
andθrbeing the angle of incidence of the reading wave. Based on this assumption it can be shown,[88]that for the situation depicted in Fig. 2.2 the diffracted electric field of an incident s-polarized plane wave takes the form
E0tot(x,z) =
To simplify matters, the amplitude of the incident field has been set to one in Eq. (2.5).
The diffracted and the incident beam have the same polarization, therefore~E0tot(x,z) is s-polarized (~E0tot(x,z) =E0tot(x,z)eˆy). As can easily be seen, the diffracted field before and behind the grating resembles a plane-wave expansion and various diffracted orders can emerge. For the sake of clarity, only thelth transmitted and reflected order have been plot-ted in Fig. 2.2. It is an important result that arbitrary grating types (surface relief or volume gratings) of arbitrary shape (sinusoidal, rectangular, etc.) follow Eq. (2.5) as long as they do not meet the requirements of the Bragg regime, which are discussed later. The reason for this is the assumption of the pseudo-periodicity of the diffracted wave, which is affected byΛonly and not by the exact grating geometry. However, the grating type and shape will affect the amplitudesRl andTl, which need to be derived for the special physical situation.
Additionally, the famous grating equation results from Eq. (2.5) in a more general way than in the case of a geometrical consideration of the optical paths. Fromβl=krsin(θl, refl)
=2πnairsin(θl, refl)/λrforz<−Handβl=2πn0sin(θl, trans)/λrforz>0 it follows that
nairsinθl, refl=nairsinθr+lλr/Λ ifz<−H (2.6)
n0sinθl, trans=nairsinθr+lλr/Λ ifz>0. (2.7)
For simplicity, the above results were derived from the assumption that ans-polarized plane wave is diffracted. The analysis forp-polarized light is similar.
The surface relief gratings discussed in this thesis emerge due to the unique photo-physical properties of the azobenzene functionalization of the storage medium. As discussed later, these materials allow for a macroscopic, photo-induced material transport if they are exposed to an intensity or polarization gradient. The modulation of the thickness of SRG
with d0 being the initial sample thickness before the illumination. Here, the modulation amplitude d1 is related to the grating height by H=2d1. SRGs inevitably feature an in-phase absorption modulation, since the writing waves are absorbed more strongly in the hills than in the valleys of the grating. The contribution of the absorption grating to the diffracted field becomes negligible if λr of the readout beam is chosen to be outside the absorption of the photo-active medium.
Volume gratings, on the other side, are based on a local, molecular reorientation, which results in a periodic modulation of the refractive index. As long as the material response is linear, the refractive index readsn(x) =n0+n1sin(2πx/Λ)with all other Fourier com-ponents being zero. Again, the absorption stays unmodulated for the readout beam and α(x)in Eq. (2.4) reduces toα(x) =α0, being the absorption of the material at the readout wavelength. The interpretation of∆nin photo-orientable materials is slightly different for gratings generated with intensity or polarization modulated fields. In the former case,∆n simply corresponds to the difference of the refractive index between illuminated and dark regions. For the polarization gratings, however, the whole medium is homogeneously illu-minated with constant intensity. Hence,∆nis a measure of the optically induced anisotropy, which is defined as the difference between the ordinary and the extra-ordinary refractive index. Unlike in uniaxial crystals, the optical axis is not predefined here; instead it is photo-induced with its orientation depending on the polarization azimuth of the incident light.
This distinction is important, since the theoretical description differs for the various grating types.
To characterize the holographic performance of the investigated materials, the intensities of the diffracted orders need to be analyzed. A suitable quantity for this purpose is the diffraction efficiency
ηl = Il
Itot , (2.8)
defined as the ratio of the light intensity diffracted into thelth order and the total incident intensityItot. The theoretical description in the next chapters will focus on the determination of the diffraction into the 1storder. Two diffraction regimes are introduced and the difference between scalar, polarization and surface relief gratings is discussed in detail.