    # Optical Foundations of holography:-Brief Discussion  holography:-The discusses the physical basis of holography and holographic interferometry. The primary phenomena constituting holography are interference and diffraction, which occur because of light’s wave nature.

So this chapter begins with a description of the wave theory of light as far as it is required to understand the recording and reconstruction of holograms and the effect of holographic interferometry.

In holographic interferometry, the variation of a physical parameter is measured by its influence on the phase of an optical wave field. Therefore the dependence of the phase upon the geometry of the optical setup and the different parameters to be measured is outlined. holography

## Light Waves: holography

### Solutions of the Wave Equation: holography

Holography-Light is a transverse, electromagnetic wave characterized by time-varying electric and magnetic fields. Since electromagnetic waves obey the Maxwell equations, the propagation of light is described by the wave equation which follows from the Maxwell equations. The wave equation for the propagation of light in a vacuum is
∇2E − 1
c2
∂2E
∂t2 = 0 (2.1)
where E is the electric field strength.
(x, y, z) are the Cartesian spatial coordinates, t denotes the temporal coordinate, the time, and C is the propagation speed of the wave.

∇2 = ∂2
∂x2 +
∂2
∂y2 +
∂2
∂z2

The speed of light in vacuum c0 is a constant of nature C0 = 299 792 458 m s−1 or almost exactly C0 = 3 × 108 m s−1. Transverse waves vibrate at right angles to the direction of propagation and so they must be described in vector notation. The wave may vibrate horizontally, vertically, or in any direction combined of these.

Such effects are called polarization effects. Fortunately for most applications, it is not necessary to use the full vectorial description of the fields, so we can assume a wave vibrating in a single plane. Such a wave is called plane polarized. For a plane-polarized wave field propagating in the z-direction, the scalar wave equation is sufficient

∂2E
∂z2 − 1
c2
∂2E
∂t2 = 0

It is easily verified that
E(z, t) = f(z − ct) or E(z, t) = g(z + ct)

are also solutions of this equation, which means that the wave field retains its form during propagation. Due to the linearity of
E(z, t) = a f(z − ct) + b g(z + ct)
is likewise a solution to the wave equation. This superposition principle is valid for linear differential equations in general and thus for also.
The most important solution  is the harmonic wave, which in real notation is
E(z, t) = E0 cos(kz − ωt)
E0 is the real amplitude of the wave, and the term (kz − ωt) gives the phase of the wave. The wave number k is associated to the wavelength λ by
k = 2π
λ
Typical figures of λ for visible light are 514.5 nm (green line of argon-ion laser) or 632.8 nm (red light of helium-neon laser). The angular frequency ω is related to the frequency ν of the wave by
ω = 2πν
where ν is the number of periods per second, that means
ν = c
λ or νλ = c
If we have not the maximum amplitude at x = 0 and t = 0, we have to introduce the relative phase φ
E(z, t) = E0 cos(kz − ωt + φ)
With the period T, the time for a full 2π-cycle, we can write
E(z, t) = E0 cos (2π)
λ z − 2π
T t + φ
displays two aspects of this wave shows the temporal distribution of the field at two points z = 0 and z = z1 > 0, and gives the spatial distribution of two periods for time instants t = 0 and t = t1 > 0. We see that a point of constant phase moves with the so-called phase velocity, the speed c.
The use of trigonometric functions leads to cumbersome calculations, which can be circumvented by using the complex exponential which is related to the trigonometric functions, by Euler’s formula
eiα = cos α + i sin α
where i = √−1 is the imaginary unit. Since the cosine now is
cos α = 1
2
(eiα + e−iα)
The harmonic wave is
E(z, t) = 1/2.
E0 ei(kz − ωt + φ) +1/2E0 e−i(kz − ωt + φ).
The second term on the right-hand side is the complex conjugate of the first term and can be omitted as long as it is understood that only the real part of E(z, t) represents the physical wave. Thus the harmonic wave in complex notation is
E(z, t) = 1
2
E0 ei(kz − ωt + φ).
A wavefront refers to the spatial distribution of the maxima of the wave, or other surfaces of constant phase, as these surfaces propagate. The wavefronts are normal to the direction of propagation.

A plane wave is a wave that has a constant phase in all planes orthogonal to the propagation direction for a given time t. For describing the spatial distribution of the wave, we can assume t = 0 in an arbitrary time scale. Since
k · r = const
is the equation for a plane in three-dimensional space, with the wave vector k = (kx, ky, kz) and the spatial vector r = (x, y, z), a plane harmonic wave at time t = 0 is
E(r) = E0 ei(k · r + φ)

### Interference of Light-holography

holography-The interference effect which occurs if two or more coherent light waves are superposed, is the basis of holography and holographic interferometry.

So in this coherent superposition, we consider two waves, emitted by the same source, which differ in the directions k1 and k2, and the phases φ1 and φ2, but for convenience have the same amplitude E0 and frequency ω and are linearly polarized in the same direction. Then in scalar notation
E1(r, t) = E0 ei(k1 · r − ωt + φ1)
E2(r, t) = E0 ei(k2 · r − ωt + φ2). (2.29)
For determination of the superposition of these waves, we decompose the vectors k1 and k2 into components of equal and opposite directions, Fig. 2.2: k = (k1 + k2)/2 and k=(k1 − k2)/2. If θ is the angle between k1 and k2 then
|k| = 2πλ sinθ/2

#### Hologram Recording-holography

Holography optical wave field consists of an amplitude distribution as well as a phase distribution, but all detectors or recording material like photographic film only register intensities: The phase is lost in the registration process.

Now we have seen that if two waves of the same frequency interfere, the resulting intensity distribution is temporally stable and depends on the phase difference ∆φ. This is used in holography where the phase information is coded by interference into a recordable intensity. Clearly, to get a temporally stable intensity.Optical Foundations of holography

### Types of Holograms: holography

holography-The reference wave is the light passing unaffected by the particles and the object wave is the wave field scattered by the particles. We may use divergent or collimated light.

Reconstruction by the illuminating wave alone without the object gives the virtual image of the particles or the transparent object at its original position as well as the real image on the opposite side of the hologram. There is no beam splitting into reference and object beam, so the method sometimes is called single beam holography.

The main disadvantages of in-line holograms are the disturbed reconstruction due to the bright reference beam and the twin images: Virtual and real images are along the same line of sight, so while focusing on one of them we see it overlayed by an out of focus image of the
other one. These drawbacks are avoided by the off-axis arrangement.

Here the laser beam is split by wavefront division, or by amplitude division, into reference and object waves. This approach has been called split-beam holography or two-beam holography.

If the offset angle between reference and object wave is large enough, we have no overlap between the virtual and the real reconstructed images nor do we stare into the directly transmitted reference beam.

Let the complex amplitude leaving the object plane be EP (x, y), then the complex amplitude of the wave field at the holographic plate located in the back focal plane of the lens is the
Fourier transform of EP (x, y)

EP (ξ,η) = F{EP (x, y)}

The reference wave is a spherical wave emitted from a point source at (x0, y0) in the front focal plane. Without loss of generality, we can assume unit amplitude, so ER(x, y) = δ(x − x0, y − y0). δ(x, y) describes the Dirac delta impulse.

The complex amplitude of this reference wave in the hologram plane is

ER(ξ,η) = F{δ(x − x0, y − y0)} = e−i2π(ξx0 + ηy0)

according to the rules for manipulating Fourier transforms. The ⊗ denotes correlation and the sign change in the arguments is because the lens performs a direct Fourier transform instead of an inverse one. The first term of this reconstructed wave field represents a focus that is the pointwise dc-term.

The second term constitutes a halo around this focus. The third term is proportional to the original object wave field but inverted, while the fourth term is a conjugate of the original wave shifted by (−2×0, −2y0). Both images are real and sharp and can be registered by film or TV cameras in the back focal plane.

The conjugated image in the registered intensity distribution is identified only by its geometric inversion. The main advantage of such a Fourier hologram is the stationary reconstructed image even when the hologram is translated in its plane, due to the shift invariance of the Fourier transform intensity.

Collimation by the lens in Fourier transform holography means that the object points and the reference source are at infinity. Lensless Fourier transform holography is possible if the object point and reference source are at a finite but the same distance from the holographic plate. To prove this statement, let the wave leaving the object be EP (x, y).

If the coherent object and reference waves impinge onto the hologram from opposite sides we get interference layers nearly parallel to the hologram surface.

The distance between subsequent interference layers is λ/(2 sin θ2 ). For reconstruction, this thick hologram is illuminated with white light which is reflected at the layers. Dependent on the wavelength the reflected waves interfere constructively in defined directions, an effect called Bragg reflection.

Let the distance of the interference layers be d and illuminate by an angle α, then we find the n-th diffraction order for the wavelength λ in the observation direction of angle β.

d = nλ/sin α + sin β

The most intense wave is that in the first diffraction order n = 1. So if we look under the angle β onto the hologram illuminated under the angle α, we see a clear image with color corresponding to λ. The parallel layers modulated by the information about the image react like an interference filter for the specific wavelength λ. An arrangement for recording such a white light hologram.

The expanded and collimated laser beam is directed through the hologram plate onto the object. The reference wave is given by the light coming directly from the laser, the wave passing through the plate and reflected by the object is the object wave. For good results, we need strongly reflecting objects and a hologram plate close to the object.

By reversing the direction of the reference beam or by turning the hologram by 180◦ a real pseudoscopic image is reconstructed. In front of the hologram a horizontal slit aperture is now placed and a hologram of the wave field passing through this slit is recorded.

This has the first effect that the vertical parallax is lost, but this is not recognized immediately, as long as the eyes of the observer are horizontally arranged. The second effect is that the different colors reconstructed from the second hologram still overlap in space. But the colors converge to different reconstructed slits.

Although neighboring colors are overlayed in the reconstructed slits, the range from blue to red can be stretched over a broad area so that each reconstructed slit produces a sharp image. The eyes of the observer are placed in one reconstructed slit and see the object in one color.

If the head is moved in the vertical direction the object is seen in another spectral color than before. Since in this way the object can be observed in the successive colors of the rainbow, the secondary hologram is called a
rainbow hologram and we speak of rainbow holography.

#### Laser: holography

Holography-Optical holography and holographic interferometry in the visible range of the spectrum became possible with the invention of a source radiating coherent light, the laser. The basic principle behind the laser is the stimulated emission of radiation. Contrary to the ubiquitous
spontaneous emission here the emission of photons is triggered by an electromagnetic wave.

All photons generated this way have the same frequency, polarization, phase, and direction as the stimulating wave. Normally in a collection of atoms, each one tends to hold the lowest energy configuration. Therefore in thermal equilibrium or even when excited the majority of all atoms are in the ground state.

Only if one succeeds in bringing a larger part of the atoms into a higher excited state than remaining in a lower state, which may be the ground state or itself an excited state, then an impinging wave stimulates the emission of an avalanche of waves, all with equal phase and propagating in the same direction.

This stimulated emission takes place as long as the population inversion between the lower and the higher energy states is maintained. To achieve the inversion, energy must be provided to the system by a process called pumping. So the laser can be regarded as an amplifier since an impinging wave generates a manifold of waves of the same direction, frequency, and phase.

To prevent this amplifier from amplifying only noise, feedback is introduced by installing two mirrors on opposite sides of the active medium. If plane mirrors are adjusted exactly parallel, photons are reflected back and forth, and what we get is an oscillator of high quality.

Standing waves will be formed between the two mirrors. If one of the mirrors is semitransparent, some of the photons can leave the laser as the so-called laser beam of coherent

radiation. The resonant frequencies possible in a cavity of length L, the separation between the mirrors, are
νn = nc/2L with n as an integer and c as the speed of light.

The existence of several frequencies νn under the gain curve – one speaks of the longitudinal modes – results in a coherence length too short for most holographic applications.

On the other hand, short cavities would produce the desired single longitudinal mode under the gain curve, but with extremely low power. The solution to this contradiction is an intracavity etalon, which is a Fabry-Perot interferometer.  