Holography

 

In 1947, Dennis Gabor invented a new interferometric technique that records both the amplitude (intensity) and the phase of the light waves. This technique is called holography, and the recorded interference pattern on a transparent photographic plate is called a hologram. The word “holography” comes from the Greek words holo (complete) and gramma (writing).

Principle of Holography

In holography, a highly coherent laser beam is split into two beams of appropriate intensity: the object beam and the reference beam.

• The object beam illuminates the object.
• The light scattered by the object interferes with the reference beam.
• The resulting interference pattern is recorded on a transparent photographic plate.

Holography is thus a two-stage process:

1. Construction (recording) of the hologram – formation of the interference pattern.
2. Reconstruction of the image – the hologram acts as a diffraction grating and produces the image when illuminated by a suitable reference beam.

Construction of the Hologram

A highly coherent laser beam is incident on an optically plane beam splitter. The beam splitter divides the laser beam into two parts of suitable intensity:

• One part (object beam), after reflection from mirror, illuminates the object. The waves scattered by the object fall on the photographic plate.
• The other part (reference beam) is directed by mirror onto the same photographic plate.

The object beam and the reference beam superpose on the photographic plate and form an interference pattern. This interference pattern is the hologram. In the hologram both the amplitude and the phase of the object waves are recorded in the form of irregular interference fringes.

Reconstruction of the Image by the Hologram

The hologram now acts as a complex diffraction grating. Two diffracted waves emerge:

• One diffracted wave converges to form a real image, which can be photographed by placing a photographic plate at its location. When viewed from different angles this real image appears three-dimensional, just like the original object.
• The other diffracted wave appears to diverge from the original position of the object and forms a virtual image behind the hologram.

Theory

An object can be regarded as a collection of point scatterers. The total wave reflected from the object is the vector sum of all the scattered waves.

Let the photographic plate $P$ lie in the $𝑥𝑦$-plane at $𝑧=0$.

The object wave at a point $(𝑥,𝑦)$ on the plate may be written as

$${𝐸}_{𝑜}(𝑥,𝑦,𝑡)={𝑎}_{𝑜}(𝑥,𝑦)\cos [𝜔𝑡+𝜙(𝑥,𝑦)],$$

where ${𝑎}_{𝑜}(𝑥,𝑦)$ and $𝜙(𝑥,𝑦)$ are the amplitude and phase of the object wave, both depending on the position on the plate.

The reference wave is

$${𝐸}_{𝑟}(𝑥,𝑦,𝑡)={𝑎}_{𝑟}(𝑥,𝑦)\cos (𝜔𝑡+\overrightarrow{𝑘}\cdot \overrightarrow{𝑟})\mathrm{.}$$

Since the propagation vector $\overrightarrow{𝑘}$ lies in the $𝑥𝑧$-plane,

$$\overrightarrow{𝑘}\cdot \overrightarrow{𝑟}=𝑘𝑥\sin 𝜃+𝑘𝑧\cos 𝜃\mathrm{.}$$

At the plate ($𝑧=0$) this reduces to

$$\overrightarrow{𝑘}\cdot \overrightarrow{𝑟}=𝑘𝑥\sin 𝜃=𝛼𝑥,$$

where $𝛼=𝑘\sin 𝜃$. Thus

$${𝐸}_{𝑟}(𝑥,𝑦,𝑡)={𝑎}_{𝑟}(𝑥,𝑦)\cos (𝜔𝑡+𝛼𝑥)\mathrm{.}$$

The resultant field on the photographic plate is

$$𝐸={𝐸}_{𝑜}+{𝐸}_{𝑟}={𝑎}_{𝑜}(𝑥,𝑦)\cos [𝜔𝑡+𝜙(𝑥,𝑦)]+{𝑎}_{𝑟}(𝑥,𝑦)\cos (𝜔𝑡+𝛼𝑥)\mathrm{.}$$

The photographic emulsion responds to the time-averaged intensity of the resultant field,

$$𝐼=⟨{𝐸}^2⟩\mathrm{.}$$

The intensity recorded on the photographic plate is the time average of the square of the resultant field:

$$𝐼=⟨{𝐸}^2⟩=⟨({𝐸}_{𝑜}+{𝐸}_{𝑟}{)}^2⟩$$

$$\begin{array}{cccc}& 𝐼=\frac{1}{2}{𝑎}_{𝑜}^2+\frac{1}{2}{𝑎}_{𝑟}^2+{𝑎}_{𝑜}{𝑎}_{𝑟}\cos [𝛼𝑥-𝜙(𝑥,𝑦)]& & \text{(5)}\end{array}$$

The transmission coefficient $𝑇$ of the photographic plate is proportional to the intensity of the resultant beam. Therefore,

$$𝑇=𝐵[\frac{1}{2}{𝑎}_{𝑜}^2+\frac{1}{2}{𝑎}_{𝑟}^2+{𝑎}_{𝑜}{𝑎}_{𝑟}\cos (𝛼𝑥-𝜙)]$$

or

$$\begin{array}{cccc}& 𝑇={𝑇}_0+𝐵{𝑎}_{𝑜}{𝑎}_{𝑟}\cos (𝛼𝑥-𝜙)& & \text{(6)}\end{array}$$

where

$${𝑇}_0=\frac{\mathrm{B}}{2}({𝑎}_{𝑜}^2+{𝑎}_{𝑟}^2)$$

and $𝐵$ is a constant.

In the reconstruction process, the hologram is illuminated by a reference beam that has the same amplitude and direction as the original reference beam used during recording. The wave transmitted through the hologram is therefore

$$\begin{array}{rl}{\displaystyle {𝐸}_{𝑡}}& {\displaystyle =𝑇{𝐸}_{𝑟}}\\ {\displaystyle }& {\displaystyle =𝐵[\frac{1}{2}{𝑎}_{𝑜}^2+\frac{1}{2}{𝑎}_{𝑟}^2+{𝑎}_{𝑜}{𝑎}_{𝑟}\cos (𝛼𝑥-𝜙)]{𝑎}_{𝑟}\cos (𝜔𝑡+𝛼𝑥)}\\ {\displaystyle }& {\displaystyle =𝐵[\frac{1}{2}({𝑎}_{𝑜}^2+{𝑎}_{𝑟}^2){𝑎}_{𝑟}\cos (𝜔𝑡+𝛼𝑥)}\\ {\displaystyle }& {\displaystyle +{𝑎}_{𝑜}{𝑎}_{𝑟}^2\cos (𝛼𝑥-𝜙)\cos (𝜔𝑡+𝛼𝑥)]}\end{array}$$

Using the trigonometric identity for the product of cosines, this expands to

$$\begin{array}{rl}{\displaystyle {𝐸}_{𝑡}}& {\displaystyle =𝐵[\frac{1}{2}({𝑎}_{𝑜}^2+{𝑎}_{𝑟}^2){𝑎}_{𝑟}\cos (𝜔𝑡+𝛼𝑥) }\\ {\displaystyle }& {\displaystyle +\frac{1}{2}{𝑎}_{𝑜}{𝑎}_{𝑟}^2\cos (𝜔𝑡+𝜙)}\\ & {\displaystyle }& {\displaystyle +\frac{1}{2}{𝑎}_{𝑜}{𝑎}_{𝑟}^2\cos (𝜔𝑡+2𝛼𝑥-𝜙)]}& & \end{array}$$

Equation contains three distinct terms:

1. The first term represents the attenuated incident reference beam (zeroth-order diffracted wave).
2. The second term represents a wave identical to the original object wave. This wave forms a virtual image at the original location of the object relative to the hologram.
3. The third term represents a wave whose phase is opposite to that of the object wave. This wave converges to form a real image, which can be recorded on a photographic plate.