Mode-locking is a technique in optics by which a laser can be made to produce pulses of light of short duration, on the order of picoseconds or femtoseconds. A laser operated in this way is sometimes referred to as a femtosecond laser, for example in modern refractive surgery; the basis of the technique is to induce a fixed-phase relationship between the longitudinal modes of the laser's resonant cavity. Constructive interference between these modes can cause the laser light to be produced as a train of pulses; the laser is said to be'phase-locked' or'mode-locked'. Although laser light is the purest form of light, it is not of a single, pure frequency or wavelength. All lasers produce light over some natural range of frequencies. A laser's bandwidth of operation is determined by the gain medium from which the laser is constructed, the range of frequencies over which a laser may operate is known as the gain bandwidth. For example, a typical helium–neon laser has a gain bandwidth of about 1.5 GHz, whereas a titanium-doped sapphire solid-state laser has a bandwidth of about 128 THz.
The second factor to determine a laser's emission frequencies is the optical cavity of the laser. In the simplest case, this consists of two plane mirrors facing each other, surrounding the gain medium of the laser. Since light is a wave, when bouncing between the mirrors of the cavity, the light will constructively and destructively interfere with itself, leading to the formation of standing waves or modes between the mirrors; these standing waves form a discrete set of frequencies, known as the longitudinal modes of the cavity. These modes are the only frequencies of light which are self-regenerating and allowed to oscillate by the resonant cavity. For a simple plane-mirror cavity, the allowed modes are those for which the separation distance of the mirrors L is an exact multiple of half the wavelength of the light λ, such that L = qλ/2, where q is an integer known as the mode order. In practice, L is much greater than λ, so the relevant values of q are large. Of more interest is the frequency separation between any two adjacent modes q and q+1.
Using the above equation, a small laser with a mirror separation of 30 cm has a frequency separation between longitudinal modes of 0.5 GHz. Thus for the two lasers referenced above, with a 30-cm cavity, the 1.5 GHz bandwidth of the HeNe laser would support up to 3 longitudinal modes, whereas the 128 THz bandwidth of the Ti:sapphire laser could support 250,000 modes. When more than one longitudinal mode is excited, the laser is said to be in "multi-mode" operation; when only one longitudinal mode is excited, the laser is said to be in "single-mode" operation. Each individual longitudinal mode has some bandwidth or narrow range of frequencies over which it operates, but this bandwidth, determined by the Q factor of the cavity, is much smaller than the intermode frequency separation. In a simple laser, each of these modes oscillates independently, with no fixed relationship between each other, in essence like a set of independent lasers all emitting light at different frequencies; the individual phase of the light waves in each mode is not fixed, may vary randomly due to such things as thermal changes in materials of the laser.
In lasers with only a few oscillating modes, interference between the modes can cause beating effects in the laser output, leading to fluctuations in intensity. If instead of oscillating independently, each mode operates with a fixed phase between it and the other modes, the laser output behaves quite differently. Instead of a random or constant output intensity, the modes of the laser will periodically all constructively interfere with one another, producing an intense burst or pulse of light; such a laser is said to be'mode-locked' or'phase-locked'. These pulses occur separated in time by τ = 2L/c, where τ is the time taken for the light to make one round trip of the laser cavity; this time corresponds to a frequency equal to the mode spacing of the laser, Δν = 1/τ. The duration of each pulse of light is determined by the number of modes which are oscillating in phase. If there are N modes locked with a frequency separation Δν, the overall mode-locked bandwidth is NΔν, the wider this bandwidth, the shorter the pulse duration from the laser.
In practice, the actual pulse duration is determined by the shape of each pulse, in turn determined by the exact amplitude and phase relationship of each longitudinal mode. For example, for a laser producing pulses with a Gaussian temporal shape, the minimum possible pulse duration Δt is given by Δ t = 0.441 N Δ ν. The value 0.441 is known as the'time-bandwidth product' of the pulse, varies depending on the pulse shape. For ultrashort pulse lasers, a hyperbolic-secant-squared pulse shape is assumed, givi
In physics, interference is a phenomenon in which two waves superpose to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive interference result from the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have the same or nearly the same frequency. Interference effects can be observed with all types of waves, for example, radio, surface water waves, gravity waves, or matter waves; the resulting images or graphs are called interferograms. The principle of superposition of waves states that when two or more propagating waves of same type are incident on the same point, the resultant amplitude at that point is equal to the vector sum of the amplitudes of the individual waves. If a crest of a wave meets a crest of another wave of the same frequency at the same point the amplitude is the sum of the individual amplitudes—this is constructive interference. If a crest of one wave meets a trough of another wave the amplitude is equal to the difference in the individual amplitudes—this is known as destructive interference.
Constructive interference occurs when the phase difference between the waves is an multiple of π, whereas destructive interference occurs when the difference is an odd multiple of π. If the difference between the phases is intermediate between these two extremes the magnitude of the displacement of the summed waves lies between the minimum and maximum values. Consider, for example, what happens when two identical stones are dropped into a still pool of water at different locations; each stone generates a circular wave propagating outwards from the point where the stone was dropped. When the two waves overlap, the net displacement at a particular point is the sum of the displacements of the individual waves. At some points, these will be in phase, will produce a maximum displacement. In other places, the waves will be in anti-phase, there will be no net displacement at these points. Thus, parts of the surface will be stationary—these are seen in the figure above and to the right as stationary blue-green lines radiating from the centre.
Interference of light is a common phenomenon that can be explained classically by the superposition of waves, however a deeper understanding of light interference requires knowledge of wave-particle duality of light, due to quantum mechanics. Prime examples of light interference are the famous double-slit experiment, laser speckle, anti-reflective coatings and interferometers. Traditionally the classical wave model is taught as a basis for understanding optical interference, based on the Huygens–Fresnel principle; the above can be demonstrated in one dimension by deriving the formula for the sum of two waves. The equation for the amplitude of a sinusoidal wave traveling to the right along the x-axis is W 1 = A cos where A is the peak amplitude, k = 2 π / λ is the wavenumber and ω = 2 π f is the angular frequency of the wave. Suppose a second wave of the same frequency and amplitude but with a different phase is traveling to the right W 2 = A cos where φ is the phase difference between the waves in radians.
The two waves will superpose and add: the sum of the two waves is W 1 + W 2 = A. Using the trigonometric identity for the sum of two cosines: cos a + cos b = 2 cos cos , this can be written W 1 + W 2 = 2 A cos cos ; this represents a wave at the original frequency, traveling to the right like the components, whose amplitude is proportional to the cosine of φ / 2. Constructive interference: If the phase difference is an multiple of π: φ = …, − 4 π, − 2 π, 0, 2 π, 4 π, …
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is thus the inverse of the spatial frequency. Wavelength is determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. Wavelength is designated by the Greek letter lambda; the term wavelength is sometimes applied to modulated waves, to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids. Assuming a sinusoidal wave moving at a fixed wave speed, wavelength is inversely proportional to frequency of the wave: waves with higher frequencies have shorter wavelengths, lower frequencies have longer wavelengths. Wavelength depends on the medium. Examples of wave-like phenomena are sound waves, water waves and periodic electrical signals in a conductor.
A sound wave is a variation in air pressure, while in light and other electromagnetic radiation the strength of the electric and the magnetic field vary. Water waves are variations in the height of a body of water. In a crystal lattice vibration, atomic positions vary. Wavelength is a measure of the distance between repetitions of a shape feature such as peaks, valleys, or zero-crossings, not a measure of how far any given particle moves. For example, in sinusoidal waves over deep water a particle near the water's surface moves in a circle of the same diameter as the wave height, unrelated to wavelength; the range of wavelengths or frequencies for wave phenomena is called a spectrum. The name originated with the visible light spectrum but now can be applied to the entire electromagnetic spectrum as well as to a sound spectrum or vibration spectrum. In linear media, any wave pattern can be described in terms of the independent propagation of sinusoidal components; the wavelength λ of a sinusoidal waveform traveling at constant speed v is given by λ = v f, where v is called the phase speed of the wave and f is the wave's frequency.
In a dispersive medium, the phase speed itself depends upon the frequency of the wave, making the relationship between wavelength and frequency nonlinear. In the case of electromagnetic radiation—such as light—in free space, the phase speed is the speed of light, about 3×108 m/s, thus the wavelength of a 100 MHz electromagnetic wave is about: 3×108 m/s divided by 108 Hz = 3 metres. The wavelength of visible light ranges from deep red 700 nm, to violet 400 nm. For sound waves in air, the speed of sound is 343 m/s; the wavelengths of sound frequencies audible to the human ear are thus between 17 m and 17 mm, respectively. Note that the wavelengths in audible sound are much longer than those in visible light. A standing wave is an undulatory motion. A sinusoidal standing wave includes stationary points of no motion, called nodes, the wavelength is twice the distance between nodes; the upper figure shows three standing waves in a box. The walls of the box are considered to require the wave to have nodes at the walls of the box determining which wavelengths are allowed.
For example, for an electromagnetic wave, if the box has ideal metal walls, the condition for nodes at the walls results because the metal walls cannot support a tangential electric field, forcing the wave to have zero amplitude at the wall. The stationary wave can be viewed as the sum of two traveling sinusoidal waves of oppositely directed velocities. Wavelength and wave velocity are related just as for a traveling wave. For example, the speed of light can be determined from observation of standing waves in a metal box containing an ideal vacuum. Traveling sinusoidal waves are represented mathematically in terms of their velocity v, frequency f and wavelength λ as: y = A cos = A cos where y is the value of the wave at any position x and time t, A is the amplitude of the wave, they are commonly expressed in terms of wavenumber k and angular frequency ω as: y = A cos = A cos in which wavelength and wavenumber are related to velocity and frequency as: k = 2 π λ = 2 π f v = ω
In optics, the refractive index or index of refraction of a material is a dimensionless number that describes how fast light propagates through the material. It is defined as n = c v, where c is the speed of light in vacuum and v is the phase velocity of light in the medium. For example, the refractive index of water is 1.333, meaning that light travels 1.333 times as fast in vacuum as in water. The refractive index determines how much the path of light is bent, or refracted, when entering a material; this is described by Snell's law of refraction, n1 sinθ1 = n2 sinθ2, where θ1 and θ2 are the angles of incidence and refraction of a ray crossing the interface between two media with refractive indices n1 and n2. The refractive indices determine the amount of light, reflected when reaching the interface, as well as the critical angle for total internal reflection and Brewster's angle; the refractive index can be seen as the factor by which the speed and the wavelength of the radiation are reduced with respect to their vacuum values: the speed of light in a medium is v = c/n, the wavelength in that medium is λ = λ0/n, where λ0 is the wavelength of that light in vacuum.
This implies that vacuum has a refractive index of 1, that the frequency of the wave is not affected by the refractive index. As a result, the energy of the photon, therefore the perceived color of the refracted light to a human eye which depends on photon energy, is not affected by the refraction or the refractive index of the medium. While the refractive index affects wavelength, it depends on photon frequency and energy so the resulting difference in the bending angle causes white light to split into its constituent colors; this is called dispersion. It can be observed in prisms and rainbows, chromatic aberration in lenses. Light propagation in absorbing materials can be described using a complex-valued refractive index; the imaginary part handles the attenuation, while the real part accounts for refraction. The concept of refractive index applies within the full electromagnetic spectrum, from X-rays to radio waves, it can be applied to wave phenomena such as sound. In this case the speed of sound is used instead of that of light, a reference medium other than vacuum must be chosen.
The refractive index n of an optical medium is defined as the ratio of the speed of light in vacuum, c = 299792458 m/s, the phase velocity v of light in the medium, n = c v. The phase velocity is the speed at which the crests or the phase of the wave moves, which may be different from the group velocity, the speed at which the pulse of light or the envelope of the wave moves; the definition above is sometimes referred to as the absolute refractive index or the absolute index of refraction to distinguish it from definitions where the speed of light in other reference media than vacuum is used. Air at a standardized pressure and temperature has been common as a reference medium. Thomas Young was the person who first used, invented, the name "index of refraction", in 1807. At the same time he changed this value of refractive power into a single number, instead of the traditional ratio of two numbers; the ratio had the disadvantage of different appearances. Newton, who called it the "proportion of the sines of incidence and refraction", wrote it as a ratio of two numbers, like "529 to 396".
Hauksbee, who called it the "ratio of refraction", wrote it as a ratio with a fixed numerator, like "10000 to 7451.9". Hutton wrote it as a ratio with a fixed denominator, like 1.3358 to 1. Young did not use a symbol for the index of refraction, in 1807. In the next years, others started using different symbols: n, m, µ; the symbol n prevailed. For visible light most transparent media have refractive indices between 1 and 2. A few examples are given in the adjacent table; these values are measured at the yellow doublet D-line of sodium, with a wavelength of 589 nanometers, as is conventionally done. Gases at atmospheric pressure have refractive indices close to 1 because of their low density. All solids and liquids have refractive indices above 1.3, with aerogel as the clear exception. Aerogel is a low density solid that can be produced with refractive index in the range from 1.002 to 1.265. Moissanite lies at the other end of the range with a refractive index as high as 2.65. Most plastics have refractive indices in the range from 1.3 to 1.7, but some high-refractive-index polymers can have values as high as 1.76.
For infrared light refractive indices can be higher. Germanium is transparent in the wavelength region from 2 to 14 µm and has a refractive index of about 4. A type of new materials, called topological insulator, was found holding higher refractive index of up to 6 in near to mid infrared frequency range. Moreover, topological insulator material are transparent; these excellent properties make them a type of significant materials for infrared optics. According to the theory of relativity, no information can travel faster than the speed of light in vacuum, but this does not mean that the refractive index cannot be lower than 1; the refractive index measures the phase velocity of light. The phase velocity is the speed at which the crests of the wave move and can be faster than the speed of light in vacuum, thereby give a refractive index below 1; this can occur close to resonance frequencies, for absorbing media, in plasmas, for X-rays. In the X-ray regime the refractive indices are
Active laser medium
The active laser medium is the source of optical gain within a laser. The gain results from the stimulated emission of electronic or molecular transitions to a lower energy state from a higher energy state populated by a pump source. Examples of active laser media include: Certain crystals doped with rare-earth ions or transition metal ions. Liquids, in the form of dye solutions as used in dye lasers. In order to fire a laser, the active gain medium must be in a nonthermal energy distribution known as a population inversion; the preparation of this state is known as laser pumping. Pumping may be achieved with electrical currents or with light, generated by discharge lamps or by other lasers. More exotic gain media can be pumped by chemical reactions, nuclear fission, or with high-energy electron beams. A universal model valid for all laser types does not exist; the simplest model includes two systems of sub-levels: lower. Within each sub-level system, the fast transitions ensure that thermal equilibrium is reached leading to the Maxwell–Boltzmann statistics of excitations among sub-levels in each system.
The upper level is assumed to be metastable. Gain and refractive index are assumed independent of a particular way of excitation. For good performance of the gain medium, the separation between sub-levels should be larger than working temperature. In the case of amplification of optical signals, the lasing frequency is called signal frequency. However, the same term is used in the laser oscillators, when amplified radiation is used to transfer energy rather than information; the model below seems to work well for most optically-pumped solid-state lasers. The simple medium can be characterized with effective cross-sections of absorption and emission at frequencies ω p and ω s. Have N be concentration of active centers in the solid-state lasers. Have N 1 be concentration of active centers in the ground state. Have N 2 be concentration of excited centers. Have N 1 + N 2 = N; the relative concentrations can be defined as n 1 = N 1 / N and n 2 = N 2 / N. The rate of transitions of an active center from ground state to the excited state can be expressed with W u = I p σ a p ℏ ω p + I s σ a s ℏ ω s and The rate of transitions back to the ground state can be expressed with W d = I p σ e p ℏ ω p + I s σ e s ℏ ω s + 1 τ, where σ a s and σ a p are effective cross-sections of absorption at the frequencies of the signal and the pump.
Σ e s and σ e p are the same for stimulated emission. The kinetic equation for relative populations can be written as follows: d n 2 d t = W u n 1 − W d n 2, d n 1 d t = − W
A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at the fixed frequencies; these fixed frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies. A physical object, such as a building, bridge, or molecule, has a set of normal modes and their natural frequencies that depend on its structure and boundary conditions; when relating to music, normal modes of vibrating instruments are called "harmonics" or "overtones". The most general motion of a system is a superposition of its normal modes; the modes are normal in the sense that they can move independently, to say that an excitation of one mode will never cause motion of a different mode. In mathematical terms, normal modes are orthogonal to each other. In physics and engineering, for a dynamical system according to wave theory, a mode is a standing wave state of excitation, in which all the components of the system will be affected sinusoidally under a specified fixed frequency.
Because no real system can fit under the standing wave framework, the mode concept is taken as a general characterization of specific states of oscillation, thus treating the dynamic system in a linear fashion, in which linear superposition of states can be performed. As classical examples, there are: In a mechanical dynamical system, a vibrating rope is the most clear example of a mode, in which the rope is the medium, the stress on the rope is the excitation, the displacement of the rope with respect to its static state is the modal variable. In an acoustic dynamical system, a single sound pitch is a mode, in which the air is the medium, the sound pressure in the air is the excitation, the displacement of the air molecules is the modal variable. In a structural dynamical system, a high tall building oscillating under its most flexural axis is a mode, in which all the material of the building -under the proper numerical simplifications- is the medium, the seismic/wind/environmental solicitations are the excitations and the displacements are the modal variable.
In an electrical dynamical system, a resonant cavity made of thin metal walls, enclosing a hollow space, for a particle accelerator is a pure standing wave system, thus an example of a mode, in which the hollow space of the cavity is the medium, the RF source is the excitation and the electromagnetic field is the modal variable. When relating to music, normal modes of vibrating instruments are called "harmonics" or "overtones"; the concept of normal modes finds application in optics, quantum mechanics, molecular dynamics. Most dynamical system can be excited under several modes; each mode is characterized according to the modal variable field. For example, a vibrating rope in the 2D space is defined by a single-frequency, but a vibrating rope in the 3D space is defined by two frequencies. For a given amplitude on the modal variable, each mode will store a specific amount of energy, because of the sinusoidal excitation. From all the modes of a dynamical system, the normal or dominant mode of a system, will be the mode storing the minimum amount of energy, for a given amplitude of the modal variable.
Or equivalently, for a given stored amount of energy, will be the mode imposing the maximum amplitude of the modal variable. A mode of vibration is characterized by a mode shape, it is numbered according to the number of half waves in the vibration. For example, if a vibrating beam with both ends pinned displayed a mode shape of half of a sine wave it would be vibrating in mode 1. If it had a full sine wave it would be vibrating in mode 2. In a system with two or more dimensions, such as the pictured disk, each dimension is given a mode number. Using polar coordinates, we have an angular coordinate. If one measured from the center outward along the radial coordinate one would encounter a full wave, so the mode number in the radial direction is 2; the other direction is trickier, because only half of the disk is considered due to the antisymmetric nature of a disk's vibration in the angular direction. Thus, measuring 180° along the angular direction you would encounter a half wave, so the mode number in the angular direction is 1.
So the mode number of the system is 2–1 or 1–2, depending on which coordinate is considered the "first" and, considered the "second" coordinate. In linear systems each mode is independent of all other modes. In general all modes have different mode shapes. In a one-dimensional system at a given mode the vibration will have nodes, or places where the displacement is always zero; these nodes correspond to points in the mode shape. Since the vibration of a system is given by the mode shape multiplied by a time function, the displacement of the node points remain zero at all times; when expanded to a two dimensional system, these nodes become lines where the displacement is always zero. If you watch the animation above you will see two circles and a straight line bisecting the disk, where the displacement is close to zero. In an idealized system these lines equal zero as shown to the right. Consider two equ
Light is electromagnetic radiation within a certain portion of the electromagnetic spectrum. The word refers to visible light, the visible spectrum, visible to the human eye and is responsible for the sense of sight. Visible light is defined as having wavelengths in the range of 400–700 nanometres, or 4.00 × 10−7 to 7.00 × 10−7 m, between the infrared and the ultraviolet. This wavelength means a frequency range of 430–750 terahertz; the main source of light on Earth is the Sun. Sunlight provides the energy that green plants use to create sugars in the form of starches, which release energy into the living things that digest them; this process of photosynthesis provides all the energy used by living things. Another important source of light for humans has been fire, from ancient campfires to modern kerosene lamps. With the development of electric lights and power systems, electric lighting has replaced firelight; some species of animals generate their own light, a process called bioluminescence.
For example, fireflies use light to locate mates, vampire squids use it to hide themselves from prey. The primary properties of visible light are intensity, propagation direction, frequency or wavelength spectrum, polarization, while its speed in a vacuum, 299,792,458 metres per second, is one of the fundamental constants of nature. Visible light, as with all types of electromagnetic radiation, is experimentally found to always move at this speed in a vacuum. In physics, the term light sometimes refers to electromagnetic radiation of any wavelength, whether visible or not. In this sense, gamma rays, X-rays and radio waves are light. Like all types of EM radiation, visible light propagates as waves. However, the energy imparted by the waves is absorbed at single locations the way particles are absorbed; the absorbed energy of the EM waves is called a photon, represents the quanta of light. When a wave of light is transformed and absorbed as a photon, the energy of the wave collapses to a single location, this location is where the photon "arrives."
This is. This dual wave-like and particle-like nature of light is known as the wave–particle duality; the study of light, known as optics, is an important research area in modern physics. EM radiation, or EMR, is classified by wavelength into radio waves, infrared, the visible spectrum that we perceive as light, ultraviolet, X-rays, gamma rays; the behavior of EMR depends on its wavelength. Higher frequencies have shorter wavelengths, lower frequencies have longer wavelengths; when EMR interacts with single atoms and molecules, its behavior depends on the amount of energy per quantum it carries. EMR in the visible light region consists of quanta that are at the lower end of the energies that are capable of causing electronic excitation within molecules, which leads to changes in the bonding or chemistry of the molecule. At the lower end of the visible light spectrum, EMR becomes invisible to humans because its photons no longer have enough individual energy to cause a lasting molecular change in the visual molecule retinal in the human retina, which change triggers the sensation of vision.
There exist animals that are sensitive to various types of infrared, but not by means of quantum-absorption. Infrared sensing in snakes depends on a kind of natural thermal imaging, in which tiny packets of cellular water are raised in temperature by the infrared radiation. EMR in this range causes molecular vibration and heating effects, how these animals detect it. Above the range of visible light, ultraviolet light becomes invisible to humans because it is absorbed by the cornea below 360 nm and the internal lens below 400 nm. Furthermore, the rods and cones located in the retina of the human eye cannot detect the short ultraviolet wavelengths and are in fact damaged by ultraviolet. Many animals with eyes that do not require lenses are able to detect ultraviolet, by quantum photon-absorption mechanisms, in much the same chemical way that humans detect visible light. Various sources define visible light as narrowly as 420–680 nm to as broadly as 380–800 nm. Under ideal laboratory conditions, people can see infrared up to at least 1050 nm.
Plant growth is affected by the color spectrum of light, a process known as photomorphogenesis. The speed of light in a vacuum is defined to be 299,792,458 m/s; the fixed value of the speed of light in SI units results from the fact that the metre is now defined in terms of the speed of light. All forms of electromagnetic radiation move at this same speed in vacuum. Different physicists have attempted to measure the speed of light throughout history. Galileo attempted to measure the speed of light in the seventeenth century. An early experiment to measure the speed of light was conducted by Ole Rømer, a Danish physicist, in 1676. Using a telescope, Rømer observed one of its moons, Io. Noting discrepancies in the apparent period of Io's orbit, he calculated that light takes about 22 minutes to traverse the diameter of Earth's orbit. However, its size was not known at that time. If Rømer had known the diameter of the Earth's orbit, he would have calculated a speed of 227,000,000 m/s. Another, more accurate, measurement of the speed of light was performed in Europe by Hippolyte Fizeau in 1849.