User:Epzcaw

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$< m a t h > I n s e r t f o r m u l a h e r e$ </math>This is my user page.

This is a draft of material which I hope ot add to the holography article

1 Theory of holography
2 Lens Resolution
3 Diffraction grating theory
- 3.1 Single-slit diffraction

[edit] Theory of holography

A light wave can be modelled by a vector U which represents the electric or magnetic field. The amplitude and phase of the light are represented by the magnitude and direction of the vector. The object and reference at any point in the holographic system are given by U_O and U_R. The combined beam is given be U_O + U_R. The energy of the combined beams is proportional to the square of magnitude of the electric vector:

$|U_O + U_R|^2=U_O U_R^*+|U_r|^2+|U_O|^2+ U_O^*U_R$

If a photographic plate is exposed to the two beams, and then developed, its transmittance, T, is proportional to the light energy which was incident on the plate, and is given by

$T=k[U_O U_R^*+|U_r|^2+|U_O|^2+ U_O^*U_R]$

where k is a constant. When the developed plate is illuminated by the reference beam, the light transmitted through the plate, U_H is

$U_H=TU_R=k[U_O U_R^*+|U_r|^2+|U_O|^2+ U_O^*U_R]U_R=k[U_O+|U_r|^2U_R+|U_O|^2U_R+ U_O^*U_R^2]$

It can be seen that U_H has four terms. The first of these is kU_O, since U_RU_R^* is equal to one, and this is the re-constructed object beam. The second term represents the reference beam whose amplitude has been modifed by U_R². The third also represent the reference beam which has had its amplitude modifed by U_O²; this modification will cause the reference beam to be diffracted around its central direction. The fourth term is know as the 'conjugate object beam'. It has the reverse curvature to the object beam itself, and forms a real image of the object in the space beyond the holographic plate.

Early holograms had both the object and reference beams illuminating the recording medium normally which meant that all the four beams were superimposed on one another. The off-axis hologram, developed by Leith and Upatnieks, separated them and allowed the virtual image ot be seen clearly, (needs a lot more to be said here!) ____________

Amendments to section on optical resolution

[edit] Lens Resolution

The ability of a lens to resolve detail is usually determined by the quality of the lens but is ultimately limited by diffraction. The aperture of the lens, or the edge of the lens itself, diffracts the light coming from a point in the object so that it forms a diffraction pattern in the image which has a central spot, which is known as the Airy disk. The angular diameter of the Airy disk is given by

$\sin \theta = 2.44 \frac{\lambda}{D}$

where

θ is the angular resolution,

λ is the wavelength of light,

and D is the diameter of the lens aperture.

Two adjacent points in the object give rise to two diffraction patterns. If the angular separation of the two points is significantly less than the Airy diks angular diameter, then the two points cannot be resolved in the image, but if their angular separation is much greater than this, distinct images of the two points are formed and they can therefore be reolved in the image. Rayleigh defined the arbitrary criterion that two points whose angular separation is half the Airy disk angular diameter can be considered to be resolved. It can be seen that the greater the diameter of the lens or its aperture, the finer the resolution. Astronomical telescopes have increasingly large lenses so they can 'see' ever finer detail in the stars.

[edit] Diffraction grating theory

The relationship between the grating spacing and the angles of the incident light is know as the grating equation. For light incident normally on the grating, the diffracted beams occur at angles sinθ_n=|n|λ/d, where n is an integer.

This equation is derived here in terms of diffraction and interference using a somewhat simplistic model. Diffraction theory says that when a light wave propagates, each point on the wavefront acts as a point source of light, and the wavefront at any subsequent point can be found by adding together the contributions from each of these individual point sources. Interference theoyr says that when a set of waves are superimposed, the amplitude of the resultant wave is found by adding the adding the individal waves as vectors or complex numbers.

The grating can be considered to be made up of a set of long and infinitely narrow slits of spacing d. When a plane wave of wavelength λ, is incident normally on the grating, each of the point slits in the grating acts as a point source which propagates in all directions. The light in a given direction is made up of the interfering compenents from each slit. In a given direction, θ, the phase difference between light from adjacent slits is given by 2πdsinθ/λ. Generally, the phases of the waves from different slits will vary from one another, and will tend to cancel out. However, in directions for which dsinθ/λ=|n|, the waves will be in phase. Thus, the diffracted beams are found at angles sinθ=|n|λ/d ............

This derivation of the grating equation has used an idealised simple grating. However, the relationship between the angles of the diffracted beams, the grating spacing and the wavelength of the light applied to any regular structure of the same spacing, because the phase relationship between light scattered from adjacent elements of the grating will remain the same. The detailed distribution of the light depends on the detailed structure of the grating elements, but it will be distributed around the directions given by the grating equation,

[edit] Single-slit diffraction

Main article: Diffraction formalism

$Numerical approximation of diffraction pattern from a slit of width four wavelengths with an incident plane wave. The main central beam, nulls, and phase reversals are apparent.$

Numerical approximation of diffraction pattern from a slit of width four wavelengths with an incident plane wave. The main central beam, nulls, and phase reversals are apparent.

$Graph and image of single-slit diffraction$

Graph and image of single-slit diffraction

A long slit of infinitesmal width which is illuminated by light diffracts the light into a series of circular waves and the wavefront which emerges from the slit is a cylindrical wave of uniform intensity.

The diffraction pattern produced by a slit which is wider than a wavelength arises from the interference of a large number of simple sources spaced evenly across the width of the slit. The relative phases of the light from these sources varies with angle.

We can show that a minimum is obtained in the diffracted light by the following reasoning. The light from a source located at the top edge of the slit interferes destructively with a source located at the middle of the slit, when the path difference between them is equal to λ/2. Similarly, the source just below the top of the slit will interferes destructively with the source located just to below the middle of the slit at the same angle. We can continue this reasoning along the entire height of the slit to conclude that the condition for destructive interference for the entire slit is the same as the condition for destructive interference between two narrow slits a distance apart that is half the width of the slit. The path difference is given by (d sinθ)/2 so that the minimum intensity occurs at an angle θ_min given by

$d \sin \theta_{min} = \lambda \,$

where d is the width of the slit.

A similar argument can be used to show that if we imagine the slit to be divided into four, six eight parts, etc, minima are obtained at angles _nθ given by

$d \sin \theta_{n} = n\lambda \,$

where n is an integer greater than zero.

There is no such simple argument to enable us to to find the maxima of the diffraction pattern. The intensity profile can be shown using the Fraunhofer diffraction integral to be given by

$I(\theta)\,$