Writing and reading of holograms with plane waves

The simplest hologram one can think of is the symmetric interference of two plane waves with the same wavelength and the same angle of incidence, as shown in figure 2.1a. A hologram generated by two plane waves is also called the hologram of no object. In this case the object and reference are equivalent and lie together with the transmitted beams in the x-z-plane of the coordinate system shown in figure 2.1b. The standard polarization configuration for holographic experiments is ss. In this case both beams are s-polarized meaning that their electric-field vector oscillates perpendicular to the plane of incidence so also the polarization oscillates along the y-axis. For this polarization configuration, the interference of the electric fields of the reference and the object beamcauses an intensity grating I:



2.2 Holographically induced volume gratings 11 with:

ER electric field of the reference wave EO electric field of the object wave

R amplitude of the electric field of the reference beam kR wave vector of the reference beam

ω angular frequency

O amplitude of the electric field of the object beam kO wave vector of the object beam

I_R intensity of the reference beam I_O intensity of the object beam

Figure 2.1. Principles of writing and reading of holographic gratings. a) Writing of a hologram with reference and object beam propagating in the x-z-plane. b) Responding coordinate system. c) Enlarged region of interference. d) Reconstruction of the object beam by illuminating the inscribed hologram with the reference beam coming from the upper left corner.

From equation 1 one can see that the phase information of the two beams is stored in the interference term of the intensity distribution.

The resulting grating period Λ is:



 sin 2n₀



 (2)

x

y z

a) b)

c)

d)

sample

12 2 Basic theory with:

λ vacuum wavelength of the incident laser light

θ angle of incidence of the laser light inside the material n₀ refractive-index of the material

Equation 2 is valid inside and outside the material, the grating period Λ is the same in both cases. Another important parameter is the contrast V defined as:

O

Figure 2.2.Total electric-field vector distribution in the region of interference at five distinct phase differences between the writing beams. The coordinate system in the upper left corner corresponds to the one shown in figure 2.1b (viewed from a different direction).

Thus, equation 1 can be transformed to:



2.2 Holographically induced volume gratings 13 dark areas to the maximum in the bright areas. Hence, a pure intensity grating can be obtained in this ss-configuration, as shown in figure 2.1c.

Besides the s-polarized beams as discussed above, also other polarizations of the beams are possible. In the case of p-polarization, the electric-field vector oscillates in the plane of incidence. A 45° polarization corresponds to an angle of 45° with respect to this plane. Furthermore, right-circularly polarized (rcp) and left-circularly polarized (lcp) light can be used. These polarizations can be combined to achieve different polarization configurations, which are summarized in figure 2.2. If an s-polarized and a p-polarized wave interfere, the resulting so-called sp-configuration generates a pure polarization grating with spatially constant intensity but varying polarization direction. ±45° and right and left circularly polarized (rlcp) are mainly polarization gratings with only a small amount of intensity variation, whereas pp, ++45°, and right and right circularly polarized (rrcp) are mainly intensity gratings with a slight variation of the polarization direction. The most common polarization configuration for holographic experiments is ss, a pure intensity grating.

When the holographic grating is read out, as shown in figure 2.1d, there are

n₁ first spatial component of the refractive-index modulation (cf.

equation 9) Gratings are called thin if:

1

Q and  Q1 (7)

The thickness of thick holographic gratings is much larger than the grating period Λ. Thick sinusoidal gratings show Bragg diffraction meaning that the light is only diffracted into one diffraction order. In contrast, thin sinusoidal holographic gratings show Raman-Nath diffraction with many diffraction orders.

14 2 Basic theory 2.2.2 Diffraction off thin holographic gratings

To store the intensity grating, a photo-sensitive medium has to be placed in the region of interference. If ss-polarization is used, the material is expected to react to the illumination in the bright regions, whereas it stays unaffected in the dark regions. The intensity grating leads to a change of the absorptivity α and refractive-index n. These quantities can be written as Fourier series:



α₀ absorption coefficient of the unexposed material

αm amplitude of the m-th spatial Fourier component of the absorption coefficient

n_m amplitude of the m-th spatial Fourier component of the refractive-index

The holograms investigated in this thesis consist of a spatial refractive-index modulation as described in equation 9. The change of the absorption coefficient can be neglected.

The response of the material can be determined by measuring the first-order diffraction of a light beam off the inscribed grating. Only the amplitude of the first Fourier component of the refractive-index, n₁, determines the diffraction into the first order. The higher Fourier components influence the higher diffraction orders and are only needed to calculate the difference between the minimum and maximum refractive index. The diffraction efficiency η is

I₁ intensity of the light diffracted into the first order I₀ intensity of the incident light

2.2 Holographically induced volume gratings 15 The diffraction efficiency of thin volume gratings can be calculated as ^[76]:



J₁ Bessel function of the first kind of first order

With equation 11 the amplitude of the first spatial Fourier component of the refractive-index, which is the fundamental oscillation of the refractive-index change, can be calculated. In the following the higher orders are neglected and n₁ is called refractive-index modulation. According to the properties of the Bessel function of the first kind of first order, the maximum diffraction efficiency of thin gratings is 33 %.

The definition of the Bessel function by its Taylor series expansion around

For the case of small γ, equation 11 can be written with the help of the above equation as: refractive-index modulation from the measured diffraction efficiency.