N-slit interferometric equation
Encyclopedia
Quantum mechanics
was first applied to optics
, and interference in particular, by Paul Dirac
. Feynman, in his lectures
, uses Dirac’s notation to describe thought experiment
s on double-slit interference of electron
s. Feynman’s approach was extended to N-slit interferometers using narrow-linewidth laser
illumination, that is, illumination by indistinguishable photon
s, by researchers working on the measurement of complex interference patterns.
, as
Using a wavefunction representation for probability amplitudes, after some algebra, the corresponding probability becomes
where N is the total number of slits in the array, or transmission grating, and the term in parenthesis represents the phase that is directly related to the exact geometry of the N-slit interferometer
. The Dirac
-Duarte
interferometric equation applies to the propagation of a single photon, or the propagation of an ensemble of indistinguishable photons, and enables the accurate prediction of measured N-slit interferometric patterns continuously from the near to the far field. Interferograms generated with this equation have been shown to compare well with measured interferograms for both even (N = 2, 4, 6...) and odd (N = 3, 5, 7...) values of N from 2 to 1600.
s.
Also, the N-slit interferometric equation has been applied to describe interference, diffraction
, refraction
(Snell's law
), and reflection
, in a rational and unified approach, using quantum mechanics principles. For example, the phase term (in parenthesis) can be used to derive
which is also known as the diffraction grating equation. Here,
is the angle of incidence, 
is the angle of diffraction, 
is the wavelength, and M is the order of diffraction. The N-slit interferometric approach is one of several approaches applied to describe basic optical phenomena in a cohesive and unified manner.
Note: given the various terminologies in use, for N-slit interferometry, it should be made explicit that the N-slit interferometric equation applies to two-slit interference, three-slit interference, four-slit interference, etc.
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...
was first applied to 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...
, and interference in particular, by Paul Dirac
Paul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
. Feynman, in his lectures
The Feynman Lectures on Physics
The Feynman Lectures on Physics is a 1964 physics textbook by Richard P. Feynman, Robert B. Leighton and Matthew Sands, based upon the lectures given by Feynman to undergraduate students at the California Institute of Technology in 1961–63. It includes lectures on mathematics, electromagnetism,...
, uses Dirac’s notation to describe thought experiment
Thought experiment
A thought experiment or Gedankenexperiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences...
s on double-slit interference of electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s. Feynman’s approach was extended to N-slit interferometers using narrow-linewidth laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...
illumination, that is, illumination by indistinguishable photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
s, by researchers working on the measurement of complex interference patterns.
Probability amplitudes and the N-slit interferometric equation
In this approach the probability amplitude for propagation from a source (s) to an interference plane (x) via an array of slits (j) is given, using Dirac’s notationBra-ket notation
Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...
, as
Using a wavefunction representation for probability amplitudes, after some algebra, the corresponding probability becomes
where N is the total number of slits in the array, or transmission grating, and the term in parenthesis represents the phase that is directly related to the exact geometry of the N-slit interferometer
N-Slit interferometer
The N-slit interferometer is an extension of the double-slit interferometer also known as Young's double-slit interferometer. One of the first known uses of N-slit arrays in optics was illustrated by Newton...
. The Dirac
Paul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
-Duarte
F. J. Duarte
F. J. Duarte is a laser physicist and author/editor of several well-known books on tunable lasers. He introduced the generalized multiple-prism dispersion theory and has discovered various multiple-prism grating oscillator laser configurations...
interferometric equation applies to the propagation of a single photon, or the propagation of an ensemble of indistinguishable photons, and enables the accurate prediction of measured N-slit interferometric patterns continuously from the near to the far field. Interferograms generated with this equation have been shown to compare well with measured interferograms for both even (N = 2, 4, 6...) and odd (N = 3, 5, 7...) values of N from 2 to 1600.
Applications
At a practical level, the N-slit interferometric equation was introduced for imaging applications and is routinely applied to predict N-slit laser interferograms, both in the near and far field. Thus, it has become a valuable tool in the alignment of large, and very large, N-slit laser interferometers used in the study of clear air turbulence and the propagation of interferometric characters for secure free-space optical communicationFree-space optical communication
Free-space optical communication is an optical communication technology that uses light propagating in free space to transmit data for telecommunications or computer networking."Free space" means air, outer space, vacuum, or something similar...
s.
Also, the N-slit interferometric equation has been applied to describe interference, 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...
, refraction
Refraction
Refraction is the change in direction of a wave due to a change in its speed. It is essentially a surface phenomenon . The phenomenon is mainly in governance to the law of conservation of energy. The proper explanation would be that due to change of medium, the phase velocity of the wave is changed...
(Snell's law
Snell's law
In optics and physics, Snell's law is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water and glass...
), and reflection
Reflection (physics)
Reflection is the change in direction of a wavefront at an interface between two differentmedia so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves...
, in a rational and unified approach, using quantum mechanics principles. For example, the phase term (in parenthesis) can be used to derive
which is also known as the diffraction grating equation. Here,
is the angle of incidence, is the angle of diffraction, is the wavelength, and M is the order of diffraction. The N-slit interferometric approach is one of several approaches applied to describe basic optical phenomena in a cohesive and unified manner.Note: given the various terminologies in use, for N-slit interferometry, it should be made explicit that the N-slit interferometric equation applies to two-slit interference, three-slit interference, four-slit interference, etc.
See also
- Beam expanderBeam expanderBeam expanders are used in laser physics either as intracavity or extracavity elements. They can be telescopic in nature or prismatic. Generally prismatic beam expanders use several prisms and are known as multiple-prism beam expanders....
- Dirac's notationBra-ket notationBra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...
- Free-space optical communicationsFree-space optical communicationFree-space optical communication is an optical communication technology that uses light propagating in free space to transmit data for telecommunications or computer networking."Free space" means air, outer space, vacuum, or something similar...
- Grating equation
- N-slit interferometerN-Slit interferometerThe N-slit interferometer is an extension of the double-slit interferometer also known as Young's double-slit interferometer. One of the first known uses of N-slit arrays in optics was illustrated by Newton...