Optical theorem

# Optical theorem

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In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, the optical theorem is a general law of wave
Wave
In physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...

scattering theory
Scattering theory
In mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Prosaically, wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunlight scattered by rain drops to form a...

, which relates the forward scattering amplitude
Scattering amplitude
In quantum physics, the scattering amplitude is the amplitude of the outgoing spherical wave relative to the incoming plane wave in the stationary-state scattering process...

to the total cross section
Cross section (physics)
A cross section is the effective area which governs the probability of some scattering or absorption event. Together with particle density and path length, it can be used to predict the total scattering probability via the Beer-Lambert law....

of the scatterer. It is usually written in the form

where f(0) is the scattering amplitude with an angle of zero, that is, the amplitude of the wave scattered to the center of a distant screen. Because the optical theorem is derived using only conservation of energy
Conservation of energy
The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...

, or in quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

from conservation of probability, the optical theorem is widely applicable and, in quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, includes both elastic
Elasticity (physics)
In physics, elasticity is the physical property of a material that returns to its original shape after the stress that made it deform or distort is removed. The relative amount of deformation is called the strain....

and inelastic scattering. Note that the above form is for an incident plane wave
Plane wave
In the physics of wave propagation, a plane wave is a constant-frequency wave whose wavefronts are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector....

; a more general form discovered by Werner Heisenberg
Werner Heisenberg
Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...

can be written

Notice that as a natural consequence of the optical theorem, an object that scatters any light at all ought to have a nonzero forward scattering amplitude, and so a bright central spot.

## History

The optical theorem was originally discovered independently by Sellmeier and Lord Rayleigh
John Strutt, 3rd Baron Rayleigh
John William Strutt, 3rd Baron Rayleigh, OM was an English physicist who, with William Ramsay, discovered the element argon, an achievement for which he earned the Nobel Prize for Physics in 1904...

in 1871. Lord Rayleigh recognized the forward scattering amplitude
Scattering amplitude
In quantum physics, the scattering amplitude is the amplitude of the outgoing spherical wave relative to the incoming plane wave in the stationary-state scattering process...

in terms of the index of refraction as

which he used in a study of the color and polarization of the sky. The equation was later extended to quantum scattering theory by several individuals, and came to be known as the Bohr–Peierls–Placzek relation after a 1939 publication. It was first referred to as the Optical Theorem in print in 1955 by Hans Bethe
Hans Bethe
Hans Albrecht Bethe was a German-American nuclear physicist, and Nobel laureate in physics for his work on the theory of stellar nucleosynthesis. A versatile theoretical physicist, Bethe also made important contributions to quantum electrodynamics, nuclear physics, solid-state physics and...

and Frederic de Hoffmann
Frederic de Hoffmann
Frederic de Hoffmann was a nuclear physicist who worked on the Manhattan Project. He came to the United States of America in 1941 and graduated from Harvard in 1944. Upon graduating, de Hoffmann was sent to Los Alamos National Laboratory where he assisted Edward Teller in the development of the...

, after it had been known as a "well known theorem of optics" for some time.

## Derivation

The theorem can be derived rather directly from a treatment of a scalar
Scalar (physics)
In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations . This is in contrast to a vector...

wave
Wave
In physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...

. If a plane wave
Plane wave
In the physics of wave propagation, a plane wave is a constant-frequency wave whose wavefronts are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector....

is incident on an object, then the wave amplitude a great distance away from the scatterer is given approximately by

All higher terms, when squared, vanish more quickly than , and so are negligible a great distance away. Notice that for large values of z and small angles the binomial theorem
Binomial theorem
In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with , and the coefficient a of...

gives us

We would now like to use the fact that the intensity
Intensity (physics)
In physics, intensity is a measure of the energy flux, averaged over the period of the wave. The word "intensity" here is not synonymous with "strength", "amplitude", or "level", as it sometimes is in colloquial speech...

is proportional to the square of the amplitude . Approximating the r in the denominator as z, we have

If we drop the term and use the fact that we have

Now suppose we integrate
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

over a screen in the x-y plane, at a distance which is small enough for the small angle approximations to be appropriate, but large enough that we can integrate the intensity from to with negligible error. In optics
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

, this is equivalent to including many fringes of the diffraction
Diffraction
Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...

pattern. To further simplify matters, let's approximate . We quickly obtain

where A is the area of the surface integrated over. The exponentials can be treated as Gaussians and so

which is just the probability of reaching the screen if none were scattered, lessened by an amount , which is therefore the effective scattering cross section
Cross section (physics)
A cross section is the effective area which governs the probability of some scattering or absorption event. Together with particle density and path length, it can be used to predict the total scattering probability via the Beer-Lambert law....

of the scatterer.