For simplicity we will consider an incoming plane wave. At the interface we will have three different waves, the incoming (k), the reflected (k') and the refracted (k").

These three wave must have the same value at the interface for all times, because at the interface they are one and the same thing.

Looking at the expression of a plane wave the condition, this implies that the scalar product of the 'k' and 'r' must have the same value.This implies that the three wave vectors are all co-planar (the three light beam are in one plane). To understand we consider a coordinate system in which the interface forms the xz-plane and the in coming wave vector 'k' lies in the xy plane. The three angles between the vectors and the interface normal are (abc). We can then write the boundary condition as (Sine equation)

Now, as k and k' are in the same medium the must have the same absolute value. The angle of incidence therefore equals the angle of reflection, an observation most should have made by the time they read this a webside.
If the propagation speed (Phase velocity) in the two media is different, otherwise there would be no refraction, the absolute value of k" differes from the absolute of k. Consequently the angle of refraction differs from the incident angle. The ratio of their sinuses is given by Snell's law of refraction.

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Reflection from Metals

Let's begin simply by considering reflection from the surface of an ideal metal (ie, a perfect mirror). When we say we have an ideal metal, we actually think of it as perfect conductor. A metal contains many free or nearly-free electrons. In a perfect conductor these electrons can move freely under the influence of an electric field. As a consequence a perfect conductor is field free. If there were an electric field the electrons would start to move and the displacement of charges will just compensate the electrical field. As a consequence there is none.

Let us consider the incoming photons as an electromagnetic wave. The experiment tells us that the light or electromagnetic wave is being reflected at the metal surface. The assumption that we have a perfect conductor implis now two different boundary conditions. First the component of the electrical field parallel to the metal surface must be zero. Secondly the component of the magnetic field perpendicular to the metal surface must also vanish.

A photon travels at the speed of light in a definite direction, carrying momentum. When it reaches the surface of the metal, the collision must involve the conservation of energy and momentum.

These electrons dominate the electromagnetic (EM) field in the metal and form an EM barrier at the surface. The incoming photon is reflected by this barrier.

Conservation of momentum requires that the reflection occurs in the plane of the incoming photon and that the incident angle equals the exit angle. The conservation of energy means the wavelength of the photon does not change on reflection.

First published on the web: 15 February 2000.

Author: Richard Payling

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Refraction of light

When light reaches an interface between two media, for example, between air and water, or air and glass, some of the light is reflected and some propagates into the second medium. If the second medium is transparent, like glass or water, then most of the light will propagate through the new medium with a change in direction and speed and only a little will be reflected. The change in direction of a light bean at the boundary of the two media is called refraction.

The details of the interactions between incident light and matter (for example, how much light is reflected, refracted or absorbed) depend on the interactions between the electromagnetic nature of light and the electric charges (free or bound, coupled or un-coupled, positive or negative) that comprise the target matter.

It is a lengthy subject and one invariably dealt with by approximations and idealisations. These approximations and idialisations may not always be satisfactory for a pure scientist, but they are actually very helpfull and usually necessary in order to get some basic and usefull understanding. Useful in the sense of enabling us to predict what is going to happen in a specific case. There are many way's for approaching this subject and many books even many good books have been written on the subject. Should you go searching in books on optics and optical physics you will find they fall, more or less, into four distinct approaches:

Two extremes of matter are ideal metals (dominated by free electrons) and perfect insulators or dielectrics (dominated by bound electrons). These are often approximated by their average properties such as conductivity (or resistivity), permittivity, permeability, etc.

Previously, we discussed reflection from an ideal metal. The metal surface forms a potential barrier that promotes reflection. But it is a feature of quantum mechanics that particles can penetrate barriers, the intensity of radiation decaying exponentially into the metal. Hence, if we make a metal film sufficiently thin, some light will penetrate the film and be detected on the other side.

Insulators, like glass, on the other hand, are dominated by bound charges. The wavefunction of the incoming photon interacts with the electron wavefunctions in the insulator and causes an oscillation of the centre of gravity of the charge density in the substance.

Reflection and refraction at the interface between to optical media.

Let us consider the interface between air and water. From experiment we know :

When a light beam is directed towards the interface between two media of different "refraction index" some of the light beam may be reflected. For example when we see the mirror image of the moon in the surface of a river, lake or the open sea. The light of the moon is directed towards the water, reflected at the boundary between the air and water and finally reaches our eye. That is reflection.

Refraction is what happens to the light beam when it actually enters the second medium. When trying to catch a fish in the aquarium one is sometimes seriously mistaken about the actual position of the fish. When you hold a stick into the aquarium and observe it from a somewhat tilt position, the stick apears bent. When you slowly move the stick towards the fish, and the fish is sufficiently cooperative, you may understand what is happening when you miss the position of the fish. Refraction changes the ligth on its way from the water (medium 1) to the air (medium 2). Our day to day experience told us that light moves in straight lines. When we see a fish in a certain direction, our brain assumes it actually is in that direction, but it is not. Let us continue the

The result is a cange in speed and direction, given by Snell's law (or in France by Descartes' law):

where v₁ and v₂ are the speeds in the first and second media, respectively, n₂₁ is the refractive index of the second medium compared with the first medium, and q₁ and q₂ are the incident and refracted angles, relative to the normal (i.e. perpendicular).

In going from air into glass, the photon speed is reduced. Hence the refractive index of glass is greater than 1.

First published on the web: 15 February 2000.

Authors: Richard Payling and Thomas Nelis

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Interference

Diffraction is a specific form of electromagnetic (EM) interference. The classic example of interference is Young's experiment.

Copyright © 1996,1997 Serge G. Vtorov

A light source is first restricted by a single pinhole, P₀, and then by two equidistant pinholes, P₁ and P₂, and the result observed on a screen. While Young used pinholes, today we normally use slits to make the interference pattern easier to see.

Several things are observable in the pattern on the screen obtained using a white light source:

• the central vertical line is white, indicating no colour (wavelength) separation there
• a single colour (eg, red) forms a series of equally spaced vertical lines on the screen
• Colours are separated so that longer wavelengths are further from the centre
• the intensity decreased away from the centre, becomes very low, then increases slightly, before dropping off again.

The overall change in intensity from the centre to the sides of the screen is consistent with a single slit diffraction pattern.

The colour separation and series of vertical coloured  lines is Young's interference. The intensity from the source can be reduced so that individual photons pass through slit P₀ and when sufficient number have been measured the same pattern is seen on the screen. The only conceivable interpretation for this phenomenon is that each photon from the source which passes through the first slit P₀ must then pass through both slits P₁ and P₂ before being detected at a point x on the screen. A particle could not do this but an EM disturbance could.

A photon entering pinhole P₀ has its spatial location severely restricted. From Heisenberg's uncertainty principle this means its momentum is very much broadened. It therefore leaves the slit, still travelling at the speed of light, but with no preferred direction. It therefore spreads out and adopts nearly spherical symmetry. The "spread-out" disturbance is then restricted in two locations simultaneously by pinholes P₁ and P₂ so that again the two disturbances exit with no preferred direction.

The two disturbances then reach the screen, but the screen is not capable of responding to part of a photon disturbance. They will therefore be detected as a whole photon at random somewhere on the screen but with a probability at any point equal to the strength of the disturbance at that point.

A photon is an undulating disturbance with a characteristic wavelength. We do not need to know its exact shape, merely that it repeats regularly once each wavelength. Any fixed point on this undulation can be labelled as a 'wavefront'. If the wavefronts of the two disturbances exiting pinholes P₁ and P₂ arrive at point x at the same time they will create a combined disturbance twice as great as their individual disturbances. There is therefore a strong likelihood that such a strong disturbance would be detected on the screen.

This combined disturbance will repeat as we look along the screen whenever the difference in path length (r₁-r₂) corresponds to one wavelength. Colours with longer wavelengths (eg, red) will therefore take a longer distance to repeat than shorter wavelengths (eg, blue).

Because of the finite size of the pinholes, the momentum is not completely broadened and so the exiting wavefronts will not be perfectly spherical. There is therefore a limit to how far the slits P₁ and P₂ can be placed apart.

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Fraunhoffer diffraction:

A way to describe how light behaves when it goes through the slits of a grating is offered by the Fraunhofer diffraction according to which a wave is diffracted into several outgoing waves when passed through an aperture, slit or opening. Fraunhofer diffraction occurs when both incident and diffracted waves are plane. To create such a situation, one must make the distance from the light source to the diffracting obstacle to the observation point large enough to neglect the curvature of incident and diffracted light.  In other words, when the light reaches the diffracting aperture, the spherical wave front should be large enough that it is virtually a planar wave front (basically a flat, vertical line).   The rays must then be parallel, or close to parallel, as they reach the diffracting object.  The point at which the diffraction pattern is observed becomes the Fraunhofer plane. The Fourier Transform, as it turns out, proves to be a powerful tool when it comes to describing and analysing diffraction patterns in the Fraunhofer plane. The Fourier transform decomposes a function into a continuous spectrum of its frequency components, and the inverse transform synthesises a function from its spectrum of frequency components. A useful analogy is the relationship between a set of notes in musical notation (the frequency components) and the sound of the musical chord represented by these notes (the function/signal itself). Using physical terminology, the Fourier transform of a signal x(t) can be thought of as a representation of a signal in the "frequency domain"; i.e. how much each frequency contributes to the signal. This is similar to the basic idea of the various other Fourier transforms including the Fourier series of a periodic function. Here are examples of some functions and their Fourier transforms.

Convolution: The convolution, is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. The Fourier transform translates between convolution and multiplication of functions. If f(t) and h(t) are integrable functions with Fourier transforms F(ω) and H(ω) respectively, and if the convolution of f and h exists and is absolutely integrable, then the Fourier transform of the convolution is given by the product of the Fourier transforms F(ω) H(ω) (possibly multiplied by a constant factor depending on the Fourier normalisation convention). In the normalisation convention we have used here, this means that if

Concerning a grating, it can be seen as n repetitions of a single feature (Rect(a)) over a total width of B (Rect(B)), which can be represented through the notation of the Fourier Transform and the convolution as follows:

We then have a diffraction image, which means repetitions of the diffraction image of B with intensity modulation by diffraction image of a single feature.

First published on the web: 8 Decembre 2007

Authors: Aranka Derzsi and Giovanni Lotito. The text is based on a lecture given by Thomas Nelis at the first Gladnet training course in Antwerp Sept. 2007

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Blazed grating 

A blazed grating combines diffraction with reflection in order to put most of the light into one diffracted order. A grating may, therefore, be described as "blazed at 250 nm" or "blazed at 1 micron" etc. by appropriate selection of groove geometry. A blazed grating can be one in which the grooves of the diffraction grating are controlled to form right triangles with a "blaze angle, q". Blazed grating groove profiles are calculated for the Littrow condition where the incident and diffracted rays are in auto collimation. The input and output rays, therefore, propagate along the same axis. In this case at the "blaze" wavelength l :

In the Littrow condition incident and refracted angle are equal to the Blaze angle. We therefore find fot the blaze wavelength:

First published on the web: 19.11.2007

Authors: Aranka Derzsi and Giovanni Lotito. The text is based on a lecture given by Thomas Nelis at the first Gladnet training course in Antwerp Sept. 2007

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Types of Reflection gratings

Initially, reflection diffraction gratings used in spectrometers were  ruled by engines, and since the 1960s are produced by holographic techniques. The latest generation is that of the echelle holographic grating, whose mirroring angle is obtained by ionic machining.

The ruled gratings are cut line by line on special machines called ruling engines. Despite the very high precision of these machines, the gratings incorporated ruling errors which caused phantom spectra (ghosts, caused by periodic defaults) and stray light (caused by random defaults).

Large holographic gratings mounted ready for installation

For holographic gratings, a polished support covered with photosensitive resin is exposed to the interference of two laser beams and processed chemically. This process allows strictly parallel and equally-spaced lines to be obtained. The grating is free of phantom spectra and stray light. However, the brightness achieved by the conventional holographic gratings is not as high as that of ruled gratings, as the groove profile is sinusoidal.

To improve brightness, this sinusoidal profile is then machined by ion bombardment, to achieve an echelle profile according to the selected mirroring angle (blaze angle). Such blazed holographic gratings are much brighter than conventional holographic gratings, as light emission is stronger into the direction of the blaze angle.

Authors: Jean Charles Lefebvre, Jobin-Yvon Emission and Th. Nelis

First published on the web: 15 December 1999.

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Concave grating:

A concave grating (also known as Rowling grating) combines the functions of optical imaging and diffraction into one optical element. Made by spacing straight grooves equally along the chord of a concave spherical or parabolic mirror surface, both collimates and disperses the light falling upon it. The grating equation is the same as for a flat grating. The important part is that the angles AMC and BMC, incidend and defraction angles respectively, is does not vary with the point 'M' choosen on the grating surface.

Traditional mountings for the concave spherical grating mounting use a slit source oriented to maximal resolution. The spectral lines obtained with a concave grating show the same aberrations as images obtained with a concave mirror, primarily astigmatism. Concave grating are typicallw used in polychromators based on the Rowland circle.

First published on the web: 8 December 2007.

Authors: Aranka Derzsi and Giovanni Lotito. The text is based on a lecture given by Thomas Nelis at the first Gladnet training course in Antwerp Sept. 2007

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Selfabsorption

When one atom emits a photon there is a chance some other atom will absorb it. To absorb the photon the energy of the photon must match a possible electron transition in the atom, normally this means the absorbing atom must be of the same type as the emitting atom, hence the term 'self-absorption'.

This means, for example, emissions from iron atoms can only be absorbed by other iron atoms, because only other iron atoms have matching electron energy levels. Also, to match energy levels, the emitting and absorbing atoms must normally be in the same excitation state, and since most atoms in analytical plasmas are in or near the ground state, self-absorption is normally only seen for electron transitions involving the ground or near-ground states. These transitions are called resonance or near-resonance transitions.

The effect of self-absorption is to rob signal from the emission line. As the number of emitting atoms increases, the likelihood of self-absorption increases and the there is no longer a linear relationship between the number of emitting atoms and the measured intensity.

In simple models of self-absorption, it possible to show that the severity of self-absorption depends on the product of the number of emitters and the effective absorption cross-section,(1) given by (2)

where M is the atomic weight of the emitting atom or ion, T is the absolute temperature of the plasma gas, l₀ the wavelength emitted, and f the oscillator strength. Self-absorption then tends to be highest when there is a large number of emitters, of high atomic weight, at lower gas temperatures, longer wavelengths, and higher oscillator strengths. Clearly the problem of self-absorption will vary greatly from one emission line to another and one emission source to another.

Spark OES is known to have severe self-absorption on some lines, ICP-OES has moderate problems at high concentrations, and GD-OES has less severe problems, though still present for some important lines in some materials, e.g. Zn I 213.8 nm and Cu I 327.3 nm in brass.

For more details on self-absorption, especially related to glow discharge, the GDOES site.

References

1. R Payling, M S Marychurch and A Dixon, in Glow Discharge Optical Emission Spectrometry, R Payling, D G Jones and A Bengtson (Eds), John Wiley & Sons (1997), pp 376-91.
2. N P Ferreira and H G C Human, Spectrochim. Acta 36B, 215 (1981).
3. Th. Nelis R. Payling, A practical Guide Glow Discharge Optical Emission Spectroscopy,RSC, 2004

First published on the web: 15 November 1999.

Author: Richard Payling

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