Hologram Classifications

The type of the hologram depends not only on the photochemical parameters of the recording medium but also on its physical dimensions and the profile of the interference patterns. In the following, we will briefly describe the properties of thin (plane, or two dimensional), thick (volume, or three dimensional), amplitude, phase, transmission and reflection holograms. Any photoinduced effect is the result of a change in the complex index of refraction $\tilde{n}$ or the optical path length $d_{\mathrm{opt}} = d \times n$. When $\Delta \kappa \gg \Delta n$, one speaks of an amplitude recording, since a change in the absorption index is felt most strongly by the amplitude of the transmitted wave. When the phase of the transmitted wave is more strongly modulated than the amplitude, then one speaks of a phase hologram. In this case, $\Delta n \gg \Delta \kappa$, or, for the optical path length, $\Delta (d \times n) = n \Delta d + d \Delta n \approx n \Delta d$, i.e., when exposure to light produces a change in the profile of the surface of the recording medium, the optical path length is also changed by variation of $\Delta d$, thus, we again have a phase hologram. Such recordings can only be made when the thickness of the medium is large enough to allow a change $\Delta d$ without burning through the material. Thus, in this case, one speaks of a thickness modulated hologram [2].

Figure 7: Interference patterns in transmission (a,b) and reflection (c) [2]

Figure 8: Holographic grating modes (a,b) sinusoidal modulation (c) Rectangular (d) Sawtooth-like [2]

Volume (or 3-D) holograms can also be either phase or amplitude holograms, depending on whether the optical path length or the optical constants $(n, \kappa)$ of the medium are changed. The reconstruction of a volume hologram, which can be done with a parallel white light beam, produces only a single image whether it is real or imaginary (virtual) image depends only on the direction of the incident reference beam. Let us discuss the hologram properties of plane wave interference patterns. The intensity of the subject beam is $I_{S} \sim A^{2}_{S}$ and of the reference beam is $I_{R} \sim A^{2}_{R}$. For plane waves, the interference pattern is thus (Fig. 7a)

$$I(\boldsymbol{x})=A^{2}_{S} + A^{2}_{R} + 2 A^{2}_{S}A^{2}... ...(\boldsymbol{K}_R - \boldsymbol{K}_S)\boldsymbol{x}} \quad ,$$ (14)

where $\boldsymbol{x}$ is a vector, parallel to the surface of the medium, and

$$\boldsymbol{K}=\boldsymbol{K}_R -\boldsymbol{K}_S$$ (15)

is the so called grating vector. $\boldsymbol{K}$ is perpendicular to the planes of the grating and is of length $\mid\!\!\!\boldsymbol{K}\!\!\!\mid = 2\pi\!/\Lambda.$ The period of the grating $\Lambda$ is related to the angles of incidence $\theta_{S}$ and $\theta_{R}$ (Fig. 7a)

$$\Lambda = \frac{\lambda}{\sin{\theta_{S}}+\sin{\theta_{R}}} \quad ,$$ (16)

where $\lambda$ is the wavelength of the reference beam outside of the hologram. If $\theta_{S}=\theta_{R}=\theta$,

$$\Lambda = \frac{\lambda}{2\sin{\theta}} \quad .$$ (17)

Note, when $\theta\rightarrow\pi/2$, the period $\Lambda\rightarrow\lambda/2$ and when $\theta\rightarrow 0$, $\Lambda\rightarrow\infty$. The angles of incidence of the subject and reference beams determine the orientation of the holographic grating, which can be found by a simple vector model $\boldsymbol{K}=\boldsymbol{K}_S +\boldsymbol{K}_R$ (Fig. 7). If the object and reference beams are directed at the recording medium from different sides, the interference patterns (and the holographic grating) are parallel to the surface of the medium. The smallest period grating is produced when $A^{2}_{S}$ and $A^{2}_{R}$ are oriented perpendicular to the surface.Thus, minimal grating period for this case is

$$\Lambda_{\mathrm{min}} = \frac{\lambda}{2n} \quad ,$$ (18)

where $n$ stands for the refractive index of the recording medium and $\lambda$ for the wavelength outside it. Such holograms are called reflection holograms. The holographic image is produced on the same side from which the reconstructing beam is incident. The holograms can be distinguished as thick (3-D) or thin (2-D) by the $Q$-parameter criterion given by Klein [8]:

$$Q = \frac{2\pi\lambda d}{n\Lambda^{2}} \quad ,$$ (19)

where $\lambda$ is the wavelength of the illuminating reference beam in a vacuum, $n$ is the refractive index of the recording medium and $\Lambda$ is the period of the holographic grating (Fig. 7). Generally, if $Q \ge 10$ the hologram is classified as a volume hologram. Volume holograms act as spatial interference filters. This was first shown experimentally and theoretically by Denisyuk [8]. They are used in "holographic art" to create ethereal images of three dimensional objects. Benton[9] invented the rainbow phase hologram. These are recorded with a special dispersion element so that in viewing, the color changes with viewing angle. The region $1\ge Q \ge 10$ represents intermediate holograms, the properties of which are between 3-D and 2-D holograms. However, such quasi-volume holograms can be used in many applications (holographic optical elements, phase gratings, etc.). For thin holograms (2-D) is $Q\ge 1$ and these holograms can be reconstructed only with coherent laser light. Now should be explained an expression Holographic Gratings. When the fringes of a holographic grating vary sinusoidally the changes in the absorption index $\kappa(x)$ and the index of refraction $n(x)$ are given by

$$\kappa(\boldsymbol{x}) = \kappa_{0} + \kappa_{A}\cos{\boldsymbol{Kx}} \quad ,$$ (20)

$$n(\boldsymbol{x}) = n_{0} + n_{A}\cos{\boldsymbol{Kx}} \quad ,$$ (21)

Where $\kappa_{0}$ and $n_{0}$ are the average values of the absorption and refraction indices after exposure and $\boldsymbol{x}$ is parallel to the surface of the medium (Fig. 8a,b). These expressions are valid only for processes that respond linearly to the exposure $(I\times t)$ and for sinusoidal gratings. In general, however, any hologram can be expressed by the superposition of elementary sinusoidal gratings:

$$q = \sum_{i=1}^{i_{\mathrm{max}}} q_{A,i}\cos{\boldsymbol{K_{i}x}} \quad ,$$ (22)

where $q$ is either $n$ or $k$. The properties of non-sinusoidal gratings are analyzed in [8,6]. Rectangular and sawtooth gratings (Fig. 8c, d) can be produced by photon or electron beam lithography. Their behavior is extremely interesting. These gratings are very useful in some specialized applications (computer generated holograms, holographic optical elements).