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Frequency compatibility

In telecommunication, the term frequency compatibility has the following meanings:

1. Of an electronic device, the extent to which it will operate at its designed performance level in its intended operational environment (including the presence of interference) without causing interference to other devices.

2. The degree to which an electrical or electronic device or devices operating on or responding to a specified frequency or frequencies is capable of functioning with other such devices.

title: Frequency compatibility


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Frequency deviation

Frequency deviation ($$f_{\Delta}$$) is used in FM radio to describe the difference between the minimum or maximum extent of a frequency modulated signal, and the nominal center or carrier frequency. The term is sometimes mistakenly used as synonymous with frequency drift, which is an unintended offset of an oscillator from its nominal frequency.

The frequency deviation of a radio is of particular importance in relation to bandwidth, because less deviation means that more channels can fit into the same amount of frequency spectrum. The FM broadcasting range between 87.5 and 108 MHz uses a typical channel spacing of 100 or 200 kHz, with a maximum frequency deviation of +/-75 kHz, in some cases leaving a buffer above the highest and below the lowest frequency to reduce interaction with other channels.

The most common FM transmitting applications use peak deviations of +/-75 kHz (100 or 200 kHz spacing), +/-5 kHz (15–25 kHz spacing), +/-2.5 kHz (3.75-12.5 kHz spacing), and +/-2 kHz (8.33 kHz spacing, 7.5 kHz spacing, 6.25 kHz spacing or 5 kHz spacing).

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Frequency deviation


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Frequency-exchange signaling

In telegraphy, frequency-exchange signaling or two-source frequency keying is frequency-change signaling in which the change from one significant condition to another is accompanied by decay in amplitude of one or more frequencies and by buildup in amplitude of one or more other frequencies.

Frequency-exchange signaling applies to supervisory signaling and user-information transmission.

title: Frequency-exchange signaling


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Frequency frogging

In telecommunication, the term frequency frogging has the following meanings:

 * 1) The interchanging of the frequencies of carrier channels to accomplish specific purposes, such as to prevent feedback and oscillation, to reduce crosstalk, and to correct for a high frequency response slope in the transmission line.
 * 2) In microwave radio relay systems, the alternate use of two frequencies at repeater sites to prevent feedback and oscillation.

Note: Frequency frogging is accomplished by having modulators, which are integrated into specially designed repeaters, translate a low-frequency group to a high-frequency group, and vice versa. A frequency channel will appear in the low group for one repeater section and will then be translated to the high group for the next section because of frequency frogging. This results in nearly constant attenuation with frequency over two successive repeater sections, and eliminates the need for large slope equalization and adjustments. Singing and crosstalk are minimized because the high-level output of a repeater is at a different frequency than the low-level input to other repeaters. It also diminishes group delay distortion. A repeater that receives on the high band from both direction and sends on the low band is called Hi-Lo; the other kind Lo-Hi.

title: Frequency frogging


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Frequency sharing

In telecommunication, frequency sharing or channel sharing is the assignment to or use of the same radio frequency by two or more stations that are separated geographically or that use the frequency at different times. It reduces the potential for mutual interference where the assignment of different frequencies to each user is not practical or possible.

Channel sharing in digital television
U.S. mobile data usage in 2017 was 40 times that in 2010, forcing frequencies to be reallocated. The FCC's 2016 auction allowed two or more stations to share a single 6 MHz television channel while retaining their licenses and all rights.

NBC sold the spectrum of three of its stations in the 2017 FCC auction: WNBC New York, Telemundo WSNS-TV Chicago and WWSI Philadelphia. Other NBC stations in the market would begin channel sharing with those stations; for instance, Comcast moved Channel 28 WNBC onto Telemundo's Channel 35 WNJU, broadcasting both stations from WNJU's antenna. Stations had to either channel-share with another TV station in this way or go off the air by Jan. 23, 2018.

title: Frequency sharing


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Frequency shift

In the physical sciences and in telecommunication, the term frequency shift may refer to:

 * Any change in frequency
 * A Doppler shift
 * In facsimile, a frequency modulation system where one frequency represents picture black and another frequency represents picture white
 * Spectrum shifting in signal processing, see Discrete Fourier transform

title: Frequency shift


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Frequency-shift keying

Frequency-shift keying (FSK) is a frequency modulation scheme in which digital information is encoded on a carrier signal by periodically shifting the frequency of the carrier between several discrete frequencies. The technology is used for communication systems such as telemetry, weather balloon radiosondes, caller ID, garage door openers, and low frequency radio transmission in the VLF and ELF bands. The simplest FSK is binary FSK (BFSK, which is also commonly referred to as 2FSK or 2-FSK), in which the carrier is shifted between two discrete frequencies to transmit binary (0s and 1s) information.

Modulating and demodulating
Reference implementations of FSK modems exist and are documented in detail. The demodulation of a binary FSK signal can be done using the Goertzel algorithm very efficiently, even on low-power microcontrollers.

Continuous-phase frequency-shift keying
In principle FSK can be implemented by using completely independent free-running oscillators, and switching between them at the beginning of each symbol period. In general, independent oscillators will not be at the same phase and therefore the same amplitude at the switch-over instant, causing sudden discontinuities in the transmitted signal.

In practice, many FSK transmitters use only a single oscillator, and the process of switching to a different frequency at the beginning of each symbol period preserves the phase. The elimination of discontinuities in the phase (and therefore elimination of sudden changes in amplitude) reduces sideband power, reducing interference with neighboring channels.

Gaussian frequency-shift keying
Rather than directly modulating the frequency with the digital data symbols, "instantaneously" changing the frequency at the beginning of each symbol period, Gaussian frequency-shift keying (GFSK) filters the data pulses with a Gaussian filter to make the transitions smoother. This filter has the advantage of reducing sideband power, reducing interference with neighboring channels, at the cost of increasing intersymbol interference. It is used by Improved Layer 2 Protocol, DECT, Bluetooth, Cypress WirelessUSB, Nordic Semiconductor, Texas Instruments, IEEE 802.15.4, Z-Wave and Wavenis devices. For basic data rate Bluetooth the minimum deviation is 115 kHz.

A GFSK modulator differs from a simple frequency-shift keying modulator in that before the baseband waveform (with levels −1 and +1) goes into the FSK modulator, it passed through a Gaussian filter to make the transitions smoother to limit spectral width. Gaussian filtering is a standard way to reduce spectral width; it is called pulse shaping in this application.

In ordinary non-filtered FSK, at a jump from −1 to +1 or +1 to −1, the modulated waveform changes rapidly, which introduces large out-of-band spectrum. If the pulse is changed going from −1 to +1 as −1, −0.98, −0.93, ..., +0.93, +0.98, +1, and this smoother pulse is used to determine the carrier frequency, the out-of-band spectrum will be reduced.

Minimum-shift keying
Minimum frequency-shift keying or minimum-shift keying (MSK) is a particular spectrally efficient form of coherent FSK. In MSK, the difference between the higher and lower frequency is identical to half the bit rate. Consequently, the waveforms that represent a 0 and a 1 bit differ by exactly half a carrier period. The maximum frequency deviation is δ = 0.25 fm, where fm is the maximum modulating frequency. As a result, the modulation index m is 0.5. This is the smallest FSK modulation index that can be chosen such that the waveforms for 0 and 1 are orthogonal.

Gaussian minimum-shift keying
A variant of MSK called Gaussian minimum-shift keying (GMSK) is used in the GSM mobile phone standard.

Audio frequency-shift keying
Audio frequency-shift keying (AFSK) is a modulation technique by which digital data is represented by changes in the frequency (pitch) of an audio tone, yielding an encoded signal suitable for transmission via radio or telephone. Normally, the transmitted audio alternates between two tones: one, the "mark", represents a binary one; the other, the "space", represents a binary zero.

AFSK differs from regular frequency-shift keying in performing the modulation at baseband frequencies. In radio applications, the AFSK-modulated signal normally is being used to modulate an RF carrier (using a conventional technique, such as AM or FM) for transmission.

AFSK is not always used for high-speed data communications, since it is far less efficient in both power and bandwidth than most other modulation modes. In addition to its simplicity, however, AFSK has the advantage that encoded signals will pass through AC-coupled links, including most equipment originally designed to carry music or speech.

AFSK is used in the U.S.-based Emergency Alert System to notify stations of the type of emergency, locations affected, and the time of issue without actually hearing the text of the alert.

Multilevel frequency-shift keying
Phase 1 radios in the Project 25 system use 4-level frequency-shift keying (4FSK).

Applications
In 1910, Reginald Fessenden invented a two-tone method of transmitting Morse code. Dots and dashes were replaced with different tones of equal length. The intent was to minimize transmission time.

Some early Continuous Wave (CW) transmitters employed an arc converter that could not be conveniently keyed. Instead of turning the arc on and off, the key slightly changed the transmitter frequency in a technique known as the compensation-wave method. The compensation-wave was not used at the receiver. Spark transmitters used for this method consumed a lot of bandwidth and caused interference, so it was discouraged by 1921.

Most early telephone-line modems used audio frequency-shift keying (AFSK) to send and receive data at rates up to about 1200 bits per second. The Bell 103 and Bell 202 modems used this technique. Even today, North American caller ID uses 1200 baud AFSK in the form of the Bell 202 standard. Some early microcomputers used a specific form of AFSK modulation, the Kansas City standard, to store data on audio cassettes. AFSK is still widely used in amateur radio, as it allows data transmission through unmodified voiceband equipment.

AFSK is also used in the United States' Emergency Alert System to transmit warning information. It is used at higher bitrates for Weathercopy used on Weatheradio by NOAA in the U.S.

The CHU shortwave radio station in Ottawa, Ontario, Canada broadcasts an exclusive digital time signal encoded using AFSK modulation.

Caller ID and remote metering standards
Frequency-shift keying (FSK) is commonly used over telephone lines for caller ID (displaying callers' numbers) and remote metering applications. There are several variations of this technology.

European Telecommunications Standards Institute
In some countries of Europe, the European Telecommunications Standards Institute (ETSI) standards 200 778-1 and -2 – replacing 300 778-1 & -2 – allow 3 physical transport layers (Telcordia Technologies (formerly Bellcore), British Telecom (BT) and Cable Communications Association (CCA)), combined with 2 data formats Multiple Data Message Format (MDMF) & Single Data Message Format (SDMF), plus the Dual-tone multi-frequency (DTMF) system and a no-ring mode for meter-reading and the like. It's more of a recognition that the different types exist than an attempt to define a single "standard".

Telcordia Technologies
The Telcordia Technologies (formerly Bellcore) standard is used in the United States, Canada (but see below), Australia, China, Hong Kong and Singapore. It sends the data after the first ring tone and uses the 1200 bits per second Bell 202 tone modulation. The data may be sent in SDMF – which includes the date, time and number – or in MDMF, which adds a NAME field.

British Telecom
British Telecom (BT) in the United Kingdom developed their own standard, which wakes up the display with a line reversal, then sends the data as CCITT v.23 modem tones in a format similar to MDMF. It is used by BT, wireless networks like the late Ionica, and some cable companies. Details are to be found in BT Supplier Information Notes (SINs) 227 (link broken 28/7/21) and 242 (link broken 28/7/21); another useful document is Designing Caller Identification Delivery Using XR-2211 for BT from the EXAR website.

Cable Communications Association
The Cable Communications Association (CCA) of the United Kingdom developed their own standard which sends the information after a short first ring, as either Bell 202 or V.23 tones. They developed a new standard rather than change some "street boxes" (multiplexors) which couldn't cope with the BT standard. The UK cable industry use a variety of switches: most are Nortel DMS-100; some are System X; System Y; and Nokia DX220. Note that some of these use the BT standard instead of the CCA one. The data format is similar to the BT one, but the transport layer is more like Telcordia Technologies, so North American or European equipment is more likely to detect it.

title: Frequency-shift keying


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Frequency standard

A frequency standard is a stable oscillator used for frequency calibration or reference. A frequency standard generates a fundamental frequency with a high degree of accuracy and precision. Harmonics of this fundamental frequency are used to provide reference points.

Since time is the reciprocal of frequency, it is relatively easy to derive a time standard from a frequency standard. A standard clock comprises a frequency standard, a device to count off the cycles of the oscillation emitted by the frequency standard, and a means of displaying or outputting the result.

Frequency standards in a network or facility are sometimes administratively designated as primary or secondary. The terms primary and secondary, as used in this context, should not be confused with the respective technical meanings of these words in the discipline of precise time and frequency.

Frequency reference
A frequency reference is an instrument used for providing a stable frequency of some kind. There are different sorts of frequency references, acoustic ones such as tuning forks but also electrical ones that emit a signal of a certain frequency (a frequency standard).

Among the most stable frequency references in the world are caesium standards (including caesium fountains) and hydrogen masers. Caesium standards are widely recognized as having better long-term stability, whereas hydrogen masers can attain superior short-term performance; therefore, several national standards laboratories use ensembles of caesium standards and hydrogen masers in order to combine the best attributes of both.

The carrier of time signal transmitters, Loran-C transmitters and of several long wave and medium wave broadcasting stations is derived from an atomic clock and can be therefore used as frequency standard.

title: Frequency standard


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Fresnel zone

A Fresnel zone, named after physicist Augustin-Jean Fresnel, is one of a series of confocal prolate ellipsoidal regions of space between and around a transmitter and a receiver. The primary wave will travel in a relative straight line from the transmitter to the receiver. Aberrant transmitted radio, sound, or light waves which are transmitted at the same time can follow slightly different paths before reaching a receiver, especially if there are obstructions or deflecting objects between the two. The two waves can arrive at the receiver at slightly different times and the aberrant wave may arrive out of phase with the primary wave due to the different path lengths. Depending on the magnitude of the phase difference between the two waves, the waves can interfere constructively or destructively. The size of the calculated Fresnel zone at any particular distance from the transmitter and receiver can help to predict whether obstructions or discontinuities along the path will cause significant interference.

Significance
In any wave-propagated transmission between a transmitter and receiver, some amount of the radiated wave propagates off-axis (not on the line-of-sight path between transmitter and receiver). This can then deflect off objects and then radiate to the receiver. However, the direct-path wave and the deflected-path wave may arrive out of phase, leading to destructive interference when the phase difference is half an odd integer ($${(2z+1)/2, z \in \mathbb Z}$$) multiple of the period. The n-th Fresnel zone is defined as the locus of points in 3D space such that a 2-segment path from the transmitter to the receiver that deflects off a point on that surface will be between n-1 and n half-wavelengths out of phase with the straight-line path. The boundaries of these zones will be ellipsoids with foci at the transmitter and receiver. In order to ensure limited interference, such transmission paths are designed with a certain clearance distance determined by a Fresnel-zone analysis.

The dependence on the interference on clearance is the cause of the picket-fencing effect when either the radio transmitter or receiver is moving, and the high and low signal strength zones are above and below the receiver's cut-off threshold. The extreme variations of signal strength at the receiver can cause interruptions in the communications link, or even prevent a signal from being received at all.

Fresnel zones are seen in optics, radio communications, electrodynamics, seismology, acoustics, gravitational radiation, and other situations involving the radiation of waves and multipath propagation. Fresnel zone computations are used to anticipate obstacle clearances required when designing highly directive systems such as microwave parabolic antenna systems. Although intuitively, clear line-of-sight between transmitter and receiver may seem to be all that is required for a strong antenna system, but because of the complex nature of radio waves, obstructions within the first Fresnel zone can cause significant weakness, even if those obstructions are not blocking the apparent line-of-sight signal path. For this reason, it is valuable to do a calculation of the size of the 1st, or primary, Fresnel zone for a given antenna system. Doing this will enable the antenna installer to decide if an obstacle, such as a tree, is going to make a significant impact on signal strength. The rule of thumb is that the primary Fresnel zone would ideally be 80% clear of obstacles, but must be at least 60% clear.

Spatial structure
Fresnel zones are confocal prolate ellipsoidal shaped regions in space (e.g. 1, 2, 3), centered around the line of the direct transmission path (path AB on the diagram). The first region includes the ellipsoidal space which the direct line-of-sight signal passes through. If a stray component of the transmitted signal bounces off an object within this region and then arrives at the receiving antenna, the phase shift will be something less than a quarter-length wave, or less than a 90º shift (path ACB on the diagram). The effect regarding phase-shift alone will be minimal. Therefore, this bounced signal can potentially result in having a positive impact on the receiver, as it is receiving a stronger signal than it would have without the deflection, and the additional signal will potentially be mostly in-phase. However, the positive attributes of this deflection also depends on the polarization of the signal relative to the object.

The 2nd region surrounds the 1st region but excludes it. If a reflective object is located in the 2nd region, the stray sine-wave which has bounced from this object and has been captured by the receiver will be shifted more than 90º but less than 270º because of the increased path length, and will potentially be received out-of-phase. Generally this is unfavorable. But again, this depends on polarization. Use of same circular polarization (e.g. right) in both ends, will eliminate odd number of reflections (including one).

The 3rd region surrounds the 2nd region and deflected waves captured by the receiver will have the same effect as a wave in the 1st region. That is, the sine wave will have shifted more than 270º but less than 450º (ideally it would be a 360º shift) and will therefore arrive at the receiver with the same shift as a signal might arrive from the 1st region. A wave deflected from this region has the potential to be shifted precisely one wavelength so that it is exactly in sync with the line-of-sight wave when it arrives at the receiving antenna.

The 4th region surrounds the 3rd region and is similar to the 2nd region. And so on.

If unobstructed and in a perfect environment, radio waves will travel in a relatively straight line from the transmitter to the receiver. But if there are reflective surfaces that interact with a stray transmitted wave, such as bodies of water, smooth terrain, roof tops, sides of buildings, etc., the radio waves deflecting off those surfaces may arrive either out-of-phase or in-phase with the signals that travel directly to the receiver. Sometimes this results in the counter-intuitive finding that reducing the height of an antenna increases the signal-to-noise ratio at the receiver.

Although radio waves generally travel in a relative straight line, fog and even humidity can cause some of the signal in certain frequencies to scatter or bend before reaching the receiver. This means objects which are clear of the line of sight path will still potentially block parts of the signal. To maximize signal strength, one needs to minimize the effect of obstruction loss by removing obstacles from both the direct radio frequency line of sight (RF LoS) line and also the area around it within the primary Fresnel zone. The strongest signals are on the direct line between transmitter and receiver and always lie in the first Fresnel zone.

In the early 19th century, French scientist Augustin-Jean Fresnel created a method to calculate where the zones are — that is, whether a given obstacle will cause mostly in-phase or mostly out-of-phase deflections between the transmitter and the receiver.

Clearance calculation

The concept of Fresnel zone clearance may be used to analyze interference by obstacles near the path of a radio beam. The first zone must be kept largely free from obstructions to avoid interfering with the radio reception. However, some obstruction of the Fresnel zones can often be tolerated. As a rule of thumb the maximum obstruction allowable is 40%, but the recommended obstruction is 20% or less.

For establishing Fresnel zones, first determine the RF line of sight (RF LoS), which in simple terms is a straight line between the transmitting and receiving antennas. Now the zone surrounding the RF LoS is said to be the Fresnel zone. The cross sectional radius of each Fresnel zone is the longest at the midpoint of the RF LoS, shrinking to a point at each vertex, behind the antennas.

Formulation
Consider an arbitrary point P in the LoS, at a distance $$d_1$$ and $$d_2$$ with respect to each of the two antennas. To obtain the radius $$r_n$$ of zone $$n$$, note that the volume of the zone is delimited by all points for which the difference in distances, between the reflected wave ($$\overline{AP} + \overline{PB}$$) and the direct wave ($$D=d_1+d_2$$) is the constant $$n\frac{\lambda}{2}$$ (multiples of half a wavelength). This effectively defines an ellipsoid with the major axis along $$\overline{AB}$$ and foci at the antennas (points A and B). So:

 * $$\overline{AP} + \overline{PB} - D = n\frac{\lambda}{2}$$

Re-writing the expression with the coordinates of point $$P$$ and the distance between antennas $$D$$, it gives:

 * $$\sqrt{d_1^2+r_n^2}+\sqrt{d_2^2+r_n^2}-(d_1+d_2)=n\frac{\lambda}{2}$$
 * $$d_1\left(\sqrt{1+r_n^2/d_1^2}-1\right)+d_2\left(\sqrt{1+r_n^2/d_2^2}-1\right)=n\frac{\lambda}{2}$$

Assuming the distances between the antennas and the point $$P$$ are much larger than the radius and applying the binomial approximation for the square root, $$\sqrt{1+x} \approx 1+x/2$$ (for x≪1), the expression simplifies to:

 * $$\frac{r_n^2}{2}\left(\frac{1}{d_1}+\frac{1}{d_2}\right)\approx n\frac{\lambda}{2}$$

which can be solved for $$r_n$$:

 * $$r_n\approx\sqrt{n\frac{d_1\ d_2}{D}\lambda},\quad d_1, d_2 \gg n\lambda,$$

For a satellite-to-Earth link, it further simplifies to:

 * $$r_n\approx \sqrt{n d_1 \lambda},\quad d_1 \gg n\lambda,\quad d_2\approx D$$

Note that when $$d_1=0$$ or $$d_2=0\implies r_n=0$$, which implies that the foci seem to coincide with the vertices of the ellipsoid. This is not correct and it's a consequence of the approximation made.

Setting the point $$P$$ to one of the vertices (behind an antenna), it's possible to obtain the error $$\epsilon$$ of this approximation:

 * $$\epsilon + \left(\epsilon+D\right) - D = n\frac{\lambda}{2}\implies\epsilon=n\frac{\lambda}{4}$$

Since the distance between antennas is generally tens of km and $$\lambda$$ of the order of cm, the error is negligible for a graphical representation.

On the other hand, considering the clearance at the left-hand antenna, with $$d_1=0, d_2=D$$, and applying the binomial approximation only at the right-hand antenna, we find:
 * $$\left(\sqrt{d_1^2+r_n^2}-d_1\right)+0.5 r_n^2/d_2=0.5 n \lambda$$
 * $$r_n +0.5 r_n^2/D=0.5 n \lambda$$

The quadratic polynomial roots are:
 * $$r_n=D\left(-1 \pm \sqrt{1+n\lambda/D}\right)$$

Applying the binomial approximation one last time, we finally find:
 * $$r_n=0.5n\lambda,\quad d_1=0$$

So, there should be at least half a wavelength of clearance at the antenna in the direction perpendicular to the line of sight. The vertical clearance at the antenna in a slant direction inclined at an altitude angle a would be:
 * $$v_n=r_n \sec(a).$$

Maximum clearance
For practical applications, it is often useful to know the maximum radius of the first Fresnel zone. Using $$n = 1$$, $$d_1 = d_2 = D/2$$, and $$\lambda = c/f$$ in the above formula gives

 * $$F_1 = {1 \over 2} \sqrt{\lambda D} = {1 \over 2} \sqrt{c D \over f},$$

where
 * $$D$$ is the distance between the two antennas,
 * $$f$$ is the frequency of the transmitted signal,
 * $$c$$ ≈ $299,700,000 m/s$ is the speed of light in the air.

Substitution of the numeric value for $$c$$ followed by a unit conversion results in an easy way to calculate the radius of the first Fresnel zone $$F_1$$, knowing the distance between the two antennas $$D$$ and the frequency of the transmitted signal $$f$$:

 * $$F_1 \mathrm{[m]} = 8.656 \sqrt{D \mathrm{[km]} \over f \mathrm{[GHz]}}$$
 * $$F_1 \mathrm{[ft]}= 36.03 \sqrt{D \mathrm{[mi]} \over f \mathrm{[GHz]}}$$

title: Fresnel zone


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Front-to-back ratio

In telecommunication, the term front-to-back ratio (also known as front-to-rear ratio) can mean:
 * 1) The ratio of power gain between the front and rear of a directional antenna.
 * 2) Ratio of signal strength transmitted in a forward direction to that transmitted in a backward direction. For receiving antennas, the ratio of received-signal strength when the antenna is rotated 180°.

The ratio compares the antenna gain in a specified direction, i.e., azimuth, usually that of maximum gain, to the gain in a direction 180° from the specified azimuth. A front-to-back ratio is usually expressed in decibels (dB).

In point-to-point microwave antennas, a "high performance" antenna usually has a higher front to back ratio than other antennas. For example, an unshrouded 38 GHz microwave dish may have a front to back ratio of 64 dB, while the same size reflector equipped with a shroud would have a front to back ratio of 70 dB. Other factors affecting the front to back ratio of a parabolic microwave antenna include the material of the dish and the precision with which the reflector itself was formed.

In other electrical engineering the front to back ratio is a ratio of parameters used to characterize rectifiers or other devices, in which electric current, signal strength, resistance, or other parameters, in one direction is compared with that in the opposite direction.

title: Front-to-back ratio


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FTS2000

Federal Telecommunications System 2000 (FTS2000) is a long distance telecommunications service for the United States federal government, including services such as switched voice service for voice or data up to 4.8 kbit/s, switched data at 56 kbit/s and 64 kbit/s, switched digital integrated service for voice, data, image, and video up to 1.544 Mbit/s, packet switched service for data in packet form, video transmission for both compressed and wideband video, and dedicated point-to-point private line for voice and data.

Note 1: Use of FTS2000 contract services is mandatory for use by U.S. Government agencies for all acquisitions subject to 40 U.S.C. 759.

Note 2: No U.S. Government information processing equipment or customer premises equipment other than that which are required to provide an FTS2000 service are furnished.

Note 3: The FTS2000 contractors will be required to provide service directly to an agency's terminal equipment interface. For example, the FTS2000 contractor might provide a terminal adapter to an agency location in order to connect FTS2000 ISDN services to the agency's terminal equipment.

Note 4: GSA awarded two 10-year, fixed-price contracts covering FTS2000 services on December 7, 1988.

Note 5: The Warner Amendment excludes the mandatory use of FTS2000 in instances related to maximum security. FTS2000 was completed in 2000, then replaced by FTS2001, and thereafter, in 2008, by Networx.

title: FTS2000


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Full width at half maximum

In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve measured between those points on the y-axis which are half the maximum amplitude. Half width at half maximum (HWHM) is half of the FWHM if the function is symmetric. The term full duration at half maximum (FDHM) is preferred when the independent variable is time.

FWHM is applied to such phenomena as the duration of pulse waveforms and the spectral width of sources used for optical communications and the resolution of spectrometers. The convention of "width" meaning "half maximum" is also widely used in signal processing to define bandwidth as "width of frequency range where less than half the signal's power is attenuated", i.e., the power is at least half the maximum. In signal processing terms, this is at most −3 dB of attenuation, called half-power point or, more specifically, half-power bandwidth. When half-power point is applied to antenna beam width, it is called half-power beam width.

Normal distribution
If the considered function is the density of a normal distribution of the form $$f(x) = \frac{1}{\sigma \sqrt{2 \pi} } \exp \left[ -\frac{(x-x_0)^2}{2 \sigma^2} \right]$$ where σ is the standard deviation and x0 is the expected value, then the relationship between FWHM and the standard deviation is $$ \mathrm{FWHM} =  2\sqrt{2 \ln 2 } \; \sigma \approx 2.355 \; \sigma.$$ The FWHM does not depend on the expected value x0; it is invariant under translations. The area within this FWHM is approximately 76% of the total area under the function.

Other distributions
In spectroscopy half the width at half maximum (here γ), HWHM, is in common use. For example, a Lorentzian/Cauchy distribution of height $1⁄πγ$ can be defined by $$f(x) = \frac{1}{\pi\gamma \left[1 + \left(\frac{x - x_0}{\gamma}\right)^2\right]} \quad \text{ and } \quad \mathrm{FWHM} = 2 \gamma. $$

Another important distribution function, related to solitons in optics, is the hyperbolic secant: $$f(x) = \operatorname{sech} \left( \frac{x}{X} \right).$$ Any translating element was omitted, since it does not affect the FWHM. For this impulse we have: $$\mathrm{FWHM} = 2 \operatorname{arcsch} \left(\tfrac{1}{2}\right) X = 2 \ln (2 + \sqrt{3}) \; X \approx 2.634 \; X $$ where $arcsch$ is the inverse hyperbolic secant.

title: Full width at half maximum


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Functional profile

In telecommunication, a functional profile is a standardization document that characterizes the requirements of a standard or group of standards, and specifies how the options and ambiguities in the standard(s) should be interpreted or implemented to (a) provide a particular information technology function, (b) provide for the development of uniform, recognized tests, and (c) promote interoperability among different network elements and terminal equipment that implement a specific profile.

title: Functional profile


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Fuse

Fuse or FUSE may refer to:

Devices

 * Fuse (electrical), a device used in electrical systems to protect against excessive current
 * Fuse (automotive), a class of fuses for vehicles
 * Fuse (hydraulic), a device used in hydraulic systems to protect against sudden loss of fluid pressure
 * Fuse (explosives) or fuze, the part of the device that initiates function
 * Fuze or fuse, a mechanism for exploding military munitions such as bombs, shells, and mines

Computing

 * Fuse ESB, an open-source integration platform based on Apache Camel
 * Filesystem in Userspace, a virtual file system interface for Unix-like operating systems
 * Fuse (emulator), the Free Unix Spectrum Emulator of the ZX Spectrum
 * Fuse Internet Service, a former Cincinnati Bell Internet service provider based in Cincinnati, Ohio, United States
 * Fuse Universal, a learning platform
 * Adobe Fuse CC, formerly Fuse Character Creator, 3D computer graphics program, originally developed by Mixamo, used to create 3D characters.
 * e-fuse

Science

 * Far Ultraviolet Spectroscopic Explorer, a space-based ultraviolet telescope and spectroscope
 * Intramembranous ossification, the fusing of bones of the skeletal system
 * Cell fusion, several biological cells combining
 * Fuse (thermonuclear)
 * Fused compound, a class of multi-ring chemical structures

Entertainment

 * Fuse (film), a 2003 film by Pjer Žalica, original Bosnian title Gori vatra
 * Fuse Teppō Musume no Torimonochō, a 2012 Japanese animated film
 * Fuse (magazine), a Canadian arts and culture magazine

Music

 * Fuse (band), an American rock group
 * Fuse (Fuse album), 1970
 * Fuse (Joe Henry album), 1999
 * Fuse (Colin James album), 2000
 * Fuse (Everything but the Girl album), 2023
 * Fuse (Keith Urban album), 2013
 * The Fuse (band), a band from the East of Scotland
 * The Fuse (album), by Pennywise, 2005
 * Fused (album), a 2005 album by Tony Iommi
 * F.U.S.E., an alias for electronic artist Richie Hawtin
 * Fuse Festival, an Australian contemporary music event
 * Fuse (club), an electronic dance music club in Brussels
 * "Fuse", an instrumental by Chaz Jankel from his 1980 album Chas Jankel
 * "The Fuse", a song by Bruce Springsteen from his 2002 album The Rising
 * "The Fuse", a song by Jackson Browne from his 1976 album The Pretender
 * "The Fuse", a song by Heaven 17 from their 1984 album How Men Are

Radio and television

 * Fuse (TV channel), an American cable television station
 * Fuse (radio program), a musical radio program on CBC
 * The Fuse (game show), a UK game show broadcast on ITV1

Games

 * Fuse Games, a game developer
 * Fuse (video game), a video game by Insomniac Games
 * Fuse, a playable character in the game Apex Legends

Other uses

 * Fuse, a consumer electronics company started by Tony Fadell
 * Fuse (chocolate bar), a brand of chocolate bar made by Cadbury
 * Fuse (surname), a Japanese surname
 * Fuse, Shimane, a village located in Oki District, Shimane, Japan
 * Fuse Station, a station in Osaka, Japan
 * Scion Fuse, a concept car

title: Fuse


text:
Garble

Garble may refer to:
 * Garble (My Little Pony), a character in My Little Pony
 * Garble, a character in Freddy and the Men from Mars

title: Garble


text:
Gateway

Gateway often refers to:

 * A gate or portal

Gateway or The Gateway may also refer to:

Films

 * Gateway (film), a 1938 drama
 * The Gateway (2015 film), a horror film
 * The Gateway (2017 film), a science-fiction film
 * The Gateway (2021 film), a crime thriller

Music

 * Gateway (band), a jazz trio featuring John Abercrombie, Jack DeJohnette, and Dave Holland
 * Gateway (Gateway album) (1976)
 * Gateway (Bongzilla album)
 * Gateway, an album by Erik Wøllo

Literature

 * Gateway (novel), a 1977 science-fiction novel by Frederik Pohl
 * Gateway (comics), a supporting character in Marvel's X-Men series
 * The Gateway (student magazine), the student magazine at the University of Alberta, Canada
 * The Gateway (student newspaper), a UK student business and careers newspaper
 * SF Gateway, a science-fiction imprint of British book publisher Victor Gollancz Ltd
 * Gateway Editions, a philosophy imprint of American book publisher Regnery Publishing
 * Gateway, a 2014 four-issue comic book limited series published by IDW Publishing

Gaming

 * Gateway (video game), a 1992 interactive fiction game based on Frederik Pohl's novel
 * Gateway II: Homeworld, the 1993 sequel
 * Gateway Galaxy, the opening level in the video game Super Mario Galaxy
 * Gateway 3DS, a flash cartridge for the Nintendo 3DS; see Nintendo 3DS storage devices

Buildings

 * The Gateway, Nantwich, England
 * The Gateway, Hong Kong, Tsim Sha Tsui, Kowloon, Hong Kong, a shopping centre
 * The Gateway, Singapore, a twin-building office complex
 * Gateway Theatre of Shopping, a shopping centre in uMhlanga, South Africa
 * Westfield Gateway, an enclosed shopping mall in Lincoln, Nebraska, United States
 * The Gateway (New Brunswick, New Jersey), New Jersey, United States
 * The Gateway (Salt Lake City), Salt Lake City, Utah, United States, a residential, retail and office complex
 * Gateway Sports and Entertainment Complex, a stadium complex in Cleveland, Ohio, United States
 * Gateway Arch, St. Louis, Missouri, United States
 * Gateway Center (disambiguation)
 * Gateway Mall (disambiguation)
 * Gateway Theatre (disambiguation)
 * Gateway Tower (disambiguation)

Companies

 * Gateway, Inc., a computer manufacturer
 * Gateway Newstands, a Canadian convenience store chain
 * Gateway Supermarket, a former UK supermarket chain rebranded as Somerfield in the 1990s
 * Gateway Television, a defunct American-owned Pan-African subscription television network

Places in Antarctica

 * The Gateway (Antarctica)

Places in the United States

 * Gateway, Alaska, a census-designated place
 * Gateway, Arkansas, a town
 * Gateway, Nevada County, California, an unincorporated community
 * Gateway, San Diego, California, a neighborhood of San Diego
 * Gateway, Colorado, an unincorporated community
 * Gateway, Florida, a census-designated place
 * Gateway, Pinellas County, Florida, a business district in Florida
 * Gateway, Camden, a neighborhood of Camden, New Jersey
 * Gateway, Oregon, an unincorporated community
 * Gateway National Recreation Area, parks and beaches in New York City and New Jersey
 * Gateway Region, the most urbanized part of northeastern New Jersey

Schools in the United States

 * Gateway Regional High School (Massachusetts)
 * Gateway Academy (Laredo, Texas), a charter high school in Laredo, Texas
 * Gateway Academy, Scottsdale, a private special school in Scottsdale, Arizona
 * Gateway Technical College, a public technical college in southeastern Wisconsin

Sports

 * Gateway International Raceway, a racetrack in Madison, Illinois, United States
 * Gateway Collegiate Athletic Conference, a women's sports conference that operated from 1983 to 1992 before being absorbed by the Missouri Valley Conference
 * Gateway Football Conference, a college football league now known as the Missouri Valley Football Conference

Technology

 * Gateway (telecommunications), a network node equipped for interfacing with another network that uses different communication protocols
 * Gateway (web page), a web page designed to attract visitors and search engines to a particular website
 * Gateway Technology, a cloning system in molecular biology
 * Lunar Gateway, a planned moon orbit space station

Transportation

 * Gateway Bridge (disambiguation)
 * Gateway Motorway, a motorway in Brisbane
 * Gateway Boulevard, a major street in Edmonton, Alberta, Canada
 * Gateway Transportation Center, a rail and bus terminal station in downtown St. Louis, Missouri
 * Gateway station (disambiguation)
 * Gateway Project, a rail expansion project between New York City and Newark, New Jersey
 * Greater Vancouver Gateway Program, an infrastructure project for Greater Vancouver
 * Takanawa Gateway Station, a train station in Tokyo, Japan

title: Gateway


text:
Gating

Gating may refer to:

Neurobiology

 * Gating (electrophysiology), the opening (activation) or closing (deactivation) of ion channels
 * Sensory gating, an automatic process by which the brain adjusts to stimuli
 * Synaptic gating, neural circuits suppressing inputs through synapses

Technology

 * Gating (telecommunication), a process of selectively modifying signals
 * Gating system metalwork, a process in casting
 * Gating signal, a signal that provides a time window
 * Clock gating, a power-saving techniques used in synchronous circuits
 * Power gating, a power-saving technique for circuits
 * Noise gate, a term in audio signal processing
 * Frequency-resolved optical gating, a term related to auto correlation in optics
 * Gating (cytometry), manually selecting a population during flow cytometry analysis

title: Gating


text:
Gaussian beam

In optics, a Gaussian beam is an idealized beam of electromagnetic radiation whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This fundamental (or TEM00) transverse Gaussian mode describes the intended output of many lasers, as such a beam diverges less and can be focused better than any other. When a Gaussian beam is refocused by an ideal lens, a new Gaussian beam is produced. The electric and magnetic field amplitude profiles along a circular Gaussian beam of a given wavelength and polarization are determined by two parameters: the waist $w_{0}$, which is a measure of the width of the beam at its narrowest point, and the position $z$ relative to the waist.

Since the Gaussian function is infinite in extent, perfect Gaussian beams do not exist in nature, and the edges of any such beam would be cut off by any finite lens or mirror. However, the Gaussian is a useful approximation to a real-world beam for cases where lenses or mirrors in the beam are significantly larger than the spot size w(z) of the beam

Fundamentally, the Gaussian is a solution of the axial Helmholtz equation, the wave equation for an electromagnetic field. Although there exist other solutions, the Gaussian families of solutions are useful for problems involving compact beams.

Mathematical form
The equations below assume a beam with a circular cross-section at all values of $z$; this can be seen by noting that a single transverse dimension, $r$, appears. Beams with elliptical cross-sections, or with waists at different positions in $z$ for the two transverse dimensions (astigmatic beams) can also be described as Gaussian beams, but with distinct values of $w_{0}$ and of the $z = 0$ location for the two transverse dimensions $x$ and $y$.

The Gaussian beam is a transverse electromagnetic (TEM) mode. The mathematical expression for the electric field amplitude is a solution to the paraxial Helmholtz equation. Assuming polarization in the $x$ direction and propagation in the $w_{0} = 2λ$ direction, the electric field in phasor (complex) notation is given by:

$${\mathbf E(r,z)} = E_0 \, \hat{\mathbf x} \, \frac{w_0}{w(z)} \exp \left( \frac{-r^2}{w(z)^2}\right ) \exp \left(\! -i \left(kz +k \frac{r^2}{2R(z)} - \psi(z) \right) \!\right)$$

where

 * $r$ is the radial distance from the center axis of the beam,
 * $z$ is the axial distance from the beam's focus (or "waist"),
 * $i$ is the imaginary unit,
 * $+z$ is the wave number (in radians per meter) for a free-space wavelength $λ$, and $n$ is the index of refraction of the medium in which the beam propagates,
 * $k = 2πn/λ$, the electric field amplitude at the origin ($E_{0} = E(0, 0)$, $r = 0$),
 * $z = 0$ is the radius at which the field amplitudes fall to $w(z)$ of their axial values (i.e., where the intensity values fall to $1/e$ of their axial values), at the plane $z$ along the beam,
 * $1/e^{2}$ is the waist radius,
 * $w_{0} = w(0)$ is the radius of curvature of the beam's wavefronts at $z$, and
 * $R(z)$ is the Gouy phase at $z$, an extra phase term beyond that attributable to the phase velocity of light.

The physical electric field is obtained from the phasor field amplitude given above by taking the real part of the amplitude times a time factor: $$\mathbf E_\text{phys}(r,z,t) = \operatorname{Re}(\mathbf E(r,z) \cdot e^{i\omega t}),$$ where $\omega$ is the angular frequency of the light and $t$ is time. The time factor involves an arbitrary sign convention, as discussed at.

Since this solution relies on the paraxial approximation, it is not accurate for very strongly diverging beams. The above form is valid in most practical cases, where $ψ(z)$.

The corresponding intensity (or irradiance) distribution is given by

$$ I(r,z) = { |E(r,z)|^2 \over  2 \eta } = I_0 \left( \frac{w_0}{w(z)} \right)^2 \exp \left( \frac{-2r^2}{w(z)^2}\right),$$

where the constant $η$ is the wave impedance of the medium in which the beam is propagating. For free space, $w_{0} ≫ λ/n$ ≈ 377 Ω. $η = η_{0}$ is the intensity at the center of the beam at its waist.

If $I_{0} = |E_{0}|^{2}/2η$ is the total power of the beam, $$I_0 = {2P_0 \over \pi w_0^2}.$$

Evolving beam width
At a position $z$ along the beam (measured from the focus), the spot size parameter $w$ is given by a hyperbolic relation: $$w(z) = w_0 \, \sqrt{ 1+ {\left( \frac{z}{z_\mathrm{R}} \right)}^2 },$$ where $$z_\mathrm{R} = \frac{\pi w_0^2 n}{\lambda}$$ is called the Rayleigh range as further discussed below, and $$n$$ is the refractive index of the medium.

The radius of the beam $P_{0}$, at any position $z$ along the beam, is related to the full width at half maximum (FWHM) of the intensity distribution at that position according to: $$w(z)={\frac {\text{FWHM}(z)}{\sqrt {2\ln2}}}.$$

Wavefront curvature
The curvature of the wavefronts is largest at the Rayleigh distance, $1/e^{2}$, on either side of the waist, crossing zero at the waist itself. Beyond the Rayleigh distance, $2w$, it again decreases in magnitude, approaching zero as $w(z)$. The curvature is often expressed in terms of its reciprocal, $R$, the radius of curvature; for a fundamental Gaussian beam the curvature at position $z$ is given by:

$$\frac{1}{R(z)} = \frac{z} {z^2 + z_\mathrm{R}^2} ,$$

so the radius of curvature $z = ±z_{R}$ is $$R(z) = z \left[{ 1+ {\left( \frac{z_\mathrm{R}}{z} \right)}^2 } \right].$$ Being the reciprocal of the curvature, the radius of curvature reverses sign and is infinite at the beam waist where the curvature goes through zero.

Elliptical and astigmatic beams
Many laser beams have an elliptical cross-section. Also common are beams with waist positions which are different for the two transverse dimensions, called astigmatic beams. These beams can be dealt with using the above two evolution equations, but with distinct values of each parameter for $x$ and $y$ and distinct definitions of the $|z| > z_{R}$ point. The Gouy phase is a single value calculated correctly by summing the contribution from each dimension, with a Gouy phase within the range $z → ±∞$ contributed by each dimension.

An elliptical beam will invert its ellipticity ratio as it propagates from the far field to the waist. The dimension which was the larger far from the waist, will be the smaller near the waist.

Gaussian as a decomposition into modes
Arbitrary solutions of the paraxial Helmholtz equation can be decomposed as the sum of Hermite–Gaussian modes (whose amplitude profiles are separable in $x$ and $y$ using Cartesian coordinates), Laguerre–Gaussian modes (whose amplitude profiles are separable in $r$ and $θ$ using cylindrical coordinates) or similarly as combinations of Ince–Gaussian modes (whose amplitude profiles are separable in $ξ$ and $η$ using elliptical coordinates). At any point along the beam $z$ these modes include the same Gaussian factor as the fundamental Gaussian mode multiplying the additional geometrical factors for the specified mode. However different modes propagate with a different Gouy phase which is why the net transverse profile due to a superposition of modes evolves in $z$, whereas the propagation of any single Hermite–Gaussian (or Laguerre–Gaussian) mode retains the same form along a beam.

Although there are other modal decompositions, Gaussians are useful for problems involving compact beams, that is, where the optical power is rather closely confined along an axis. Even when a laser is not operating in the fundamental Gaussian mode, its power will generally be found among the lowest-order modes using these decompositions, as the spatial extent of higher order modes will tend to exceed the bounds of a laser's resonator (cavity). "Gaussian beam" normally implies radiation confined to the fundamental (TEM00) Gaussian mode.

Beam parameters
The geometric dependence of the fields of a Gaussian beam are governed by the light's wavelength $λ$ (in the dielectric medium, if not free space) and the following beam parameters, all of which are connected as detailed in the following sections.

Beam waist

The shape of a Gaussian beam of a given wavelength $z$ is governed solely by one parameter, the beam waist $R(z)$. This is a measure of the beam size at the point of its focus ($z = 0$ in the above equations) where the beam width $±π/4$ (as defined above) is the smallest (and likewise where the intensity on-axis ($w(z)$) is the largest). From this parameter the other parameters describing the beam geometry are determined. This includes the Rayleigh range $w_{0}$ and asymptotic beam divergence $b$, as detailed below.

Rayleigh range and confocal parameter
The Rayleigh distance or Rayleigh range $z_{R}$ is determined given a Gaussian beam's waist size:

$$z_\mathrm{R} = \frac{\pi w_0^2 n}{\lambda}.$$

Here $Θ$ is the wavelength of the light, $λ$ is the index of refraction. At a distance from the waist equal to the Rayleigh range $w_{0}$, the width $θ$ of the beam is $z = 0$ larger than it is at the focus where $w(z)$, the beam waist. That also implies that the on-axis ($r = 0$) intensity there is one half of the peak intensity (at $z_{R}$). That point along the beam also happens to be where the wavefront curvature ($z_{R}$) is greatest.

The distance between the two points $z_{R}$ is called the confocal parameter or depth of focus of the beam.

Beam divergence
Although the tails of a Gaussian function never actually reach zero, for the purposes of the following discussion the "edge" of a beam is considered to be the radius where $√2$. That is where the intensity has dropped to $w = w_{0}$ of its on-axis value. Now, for $r = 0$ the parameter $z = 0$ increases linearly with $λ$. This means that far from the waist, the beam "edge" (in the above sense) is cone-shaped. The angle between that cone (whose $1/R$) and the beam axis ($z = ±z_{R}$) defines the divergence of the beam: $$\theta = \lim_{z\to\infty} \arctan\left(\frac{w(z)}{z}\right).$$

In the paraxial case, as we have been considering, $n$ (in radians) is then approximately $$\theta = \frac{\lambda}{\pi n w_0}$$

where $w$ is the refractive index of the medium the beam propagates through, and $z$ is the free-space wavelength. The total angular spread of the diverging beam, or apex angle of the above-described cone, is then given by $$\Theta = 2 \theta\, .$$

That cone then contains 86% of the Gaussian beam's total power.

Because the divergence is inversely proportional to the spot size, for a given wavelength $θ$, a Gaussian beam that is focused to a small spot diverges rapidly as it propagates away from the focus. Conversely, to minimize the divergence of a laser beam in the far field (and increase its peak intensity at large distances) it must have a large cross-section ($r = w(z)$) at the waist (and thus a large diameter where it is launched, since $1/e^{2}$ is never less than $z ≫ z_{R}$). This relationship between beam width and divergence is a fundamental characteristic of diffraction, and of the Fourier transform which describes Fraunhofer diffraction. A beam with any specified amplitude profile also obeys this inverse relationship, but the fundamental Gaussian mode is a special case where the product of beam size at focus and far-field divergence is smaller than for any other case.

Since the Gaussian beam model uses the paraxial approximation, it fails when wavefronts are tilted by more than about 30° from the axis of the beam. From the above expression for divergence, this means the Gaussian beam model is only accurate for beams with waists larger than about $w(z)$.

Laser beam quality is quantified by the beam parameter product (BPP). For a Gaussian beam, the BPP is the product of the beam's divergence and waist size $r = w(z)$. The BPP of a real beam is obtained by measuring the beam's minimum diameter and far-field divergence, and taking their product. The ratio of the BPP of the real beam to that of an ideal Gaussian beam at the same wavelength is known as $r = 0$ ("M squared"). The $w_{0}$ for a Gaussian beam is one. All real laser beams have $w(z)$ values greater than one, although very high quality beams can have values very close to one.

The numerical aperture of a Gaussian beam is defined to be $w_{0}$, where $n$ is the index of refraction of the medium through which the beam propagates. This means that the Rayleigh range is related to the numerical aperture by $$z_\mathrm{R} = \frac{n w_0}{\mathrm{NA}} .$$

Gouy phase
The Gouy phase is a phase shift gradually acquired by a beam around the focal region. At position $λ$ the Gouy phase of a fundamental Gaussian beam is given by $$\psi(z) = \arctan \left( \frac{z}{z_\mathrm{R}} \right).$$

The Gouy phase results in an increase in the apparent wavelength near the waist ($2λ/π$). Thus the phase velocity in that region formally exceeds the speed of light. That paradoxical behavior must be understood as a near-field phenomenon where the departure from the phase velocity of light (as would apply exactly to a plane wave) is very small except in the case of a beam with large numerical aperture, in which case the wavefronts' curvature (see previous section) changes substantially over the distance of a single wavelength. In all cases the wave equation is satisfied at every position.

The sign of the Gouy phase depends on the sign convention chosen for the electric field phasor. With $w_{0}$ dependence, the Gouy phase changes from $M^{2}$ to $M^{2}$, while with $M^{2}$ dependence it changes from $NA = n sin θ$ to $z ≈ 0$ along the axis.

For a fundamental Gaussian beam, the Gouy phase results in a net phase discrepancy with respect to the speed of light amounting to $λ$ radians (thus a phase reversal) as one moves from the far field on one side of the waist to the far field on the other side. This phase variation is not observable in most experiments. It is, however, of theoretical importance and takes on a greater range for higher-order Gaussian modes.

Power through an aperture
With a beam centered on an aperture, the power $n$ passing through a circle of radius $z$ in the transverse plane at position $π$ is $$P(r,z) = P_0 \left[ 1 - e^{-2r^2 / w^2(z)} \right],$$ where $$P_0 = \frac{ 1 }{ 2 } \pi I_0 w_0^2$$ is the total power transmitted by the beam.

For a circle of radius $e^{iωt}$, the fraction of power transmitted through the circle is $$\frac{P(z)}{P_0} = 1 - e^{-2} \approx 0.865.$$

Similarly, about 90% of the beam's power will flow through a circle of radius $-π/2$, 95% through a circle of radius $+π/2$, and 99% through a circle of radius $e^{-iωt}$.

Peak intensity
The peak intensity at an axial distance $P$ from the beam waist can be calculated as the limit of the enclosed power within a circle of radius $r$, divided by the area of the circle $+π/2$ as the circle shrinks: $$I(0,z) = \lim_{r\to 0} \frac {P_0 \left[ 1 - e^{-2r^2 / w^2(z)} \right]} {\pi r^2} .$$

The limit can be evaluated using L'Hôpital's rule: $$I(0,z) = \frac{P_0}{\pi} \lim_{r\to 0} \frac { \left[ -(-2)(2r) e^{-2r^2 / w^2(z)} \right]} {w^2(z)(2r)} = {2P_0 \over \pi w^2(z)} .$$

Complex beam parameter
The spot size and curvature of a Gaussian beam as a function of $z$ along the beam can also be encoded in the complex beam parameter $-π/2$ given by: $$ q(z) = z + iz_\mathrm{R} .$$

The reciprocal of $r = w(z)$ contains the wavefront curvature and relative on-axis intensity in its real and imaginary parts, respectively:

$${1 \over q(z)} = {1 \over R(z)} - i {\lambda \over n \pi w^2(z)} .$$

The complex beam parameter simplifies the mathematical analysis of Gaussian beam propagation, and especially in the analysis of optical resonator cavities using ray transfer matrices.

Then using this form, the earlier equation for the electric (or magnetic) field is greatly simplified. If we call $z$ the relative field strength of an elliptical Gaussian beam (with the elliptical axes in the $r$ and $z$ directions) then it can be separated in $u$ and $x$ according to: $$u(x,y,z) = u_x(x,z)\, u_y(y,z) ,$$

where $$\begin{align} u_x(x,z) &= \frac{1}{\sqrt{{q}_x(z)}} \exp\left(-i k \frac{x^2}{2 {q}_x(z)}\right), \\ u_y(y,z) &= \frac{1}{\sqrt{{q}_y(z)}} \exp\left(-i k \frac{y^2}{2 {q}_y(z)}\right), \end{align}$$

where $r = 1.07 × w(z)$ and $r = 1.224 × w(z)$ are the complex beam parameters in the $y$ and $x$ directions.

For the common case of a circular beam profile, $r = 1.52 × w(z)$ and $πr^{2}$, which yields $$u(r,z) = \frac{1}{q(z)}\exp\left( -i k\frac{r^2}{2 q(z)}\right) .$$

Beam optics

When a gaussian beam propagates through a thin lens, the outgoing beam is also a (different) gaussian beam, provided that the beam travels along the cylindrical symmetry axis of the lens, and that the lens is larger than the width of the beam. The focal length of the lens $$f$$, the beam waist radius $$w_0$$, and beam waist position $$z_0$$ of the incoming beam can be used to determine the beam waist radius $$w_0'$$ and position $$z_0'$$ of the outgoing beam.

Lens equation
As derived by Saleh and Teich, the relationship between the ingoing and outgoing beams can be found by considering the phase that is added to each point $$(x,y)$$ of the gaussian beam as it travels through the lens. An alternative approach due to Self is to consider the effect of a thin lens on the gaussian beam wavefronts.

The exact solution to the above problem is expressed simply in terms of the magnification $$M$$

\begin{align} w_0' &= Mw_0\\[1.2ex] (z_0'-f) &= M^2(z_0-f). \end{align} $$

The magnification, which depends on $$w_0$$ and $$z_0$$, is given by

M = \frac{M_r}{\sqrt{1+r^2}} $$

where

r = \frac{z_R}{z_0-f}, \quad M_r = \left|\frac{f}{z_0-f}\right|. $$

An equivalent expression for the beam position $$z_0'$$ is

\frac{1}{z_0+\frac{z_R^2}{(z_0-f)}}+\frac{1}{z_0'} = \frac{1}{f}. $$

This last expression makes clear that the ray optics thin lens equation is recovered in the limit that $$\left|\left(\tfrac{z_R}{z_0}\right)\left(\tfrac{z_R}{z_0-f}\right)\right|\ll 1$$. It can also be noted that if $$\left|z_0+\frac{z_R^2}{z_0-f}\right|\gg f$$ then the incoming beam is "well collimated" so that $$z_0'\approx f$$.

Beam focusing
In some applications it is desirable to use a converging lens to focus a laser beam to a very small spot. Mathematically, this implies minimization of the magnification $$M$$. If the beam size is constrained by the size of available optics, this is typically best achieved by sending the largest possible collimated beam through a small focal length lens, i.e. by maximizing $$z_R$$ and minimizing $$f$$. In this situation, it is justifiable to make the approximation $$z_R^2/(z_0-f)^2\gg 1$$, implying that $$M\approx f/z_R$$ and yielding the result $$w_0'\approx fw_0/z_R$$. This result is often presented in the form

\begin{align} 2w_0' &\approx \frac{4}{\pi}\lambda F_\# \\[1.2ex] z_0' &\approx f \end{align} $$

where

F_\# = \frac{f}{2w_0}, $$

which is found after assuming that the medium has index of refraction $$n\approx 1$$ and substituting $$z_R=\pi w_0^2/\lambda$$. The factors of 2 are introduced because of a common preference to represent beam size by the beam waist diameters $$2w_0'$$ and $$2w_0$$, rather than the waist radii $$w_0'$$ and $$w_0$$.

Wave equation
As a special case of electromagnetic radiation, Gaussian beams (and the higher-order Gaussian modes detailed below) are solutions to the wave equation for an electromagnetic field in free space or in a homogeneous dielectric medium, obtained by combining Maxwell's equations for the curl of $y$ and the curl of $x$, resulting in: $$ \nabla^2 U = \frac{1}{c^2} \frac{\partial^2 U}{\partial t^2},$$ where $y$ is the speed of light in the medium, and $E$ could either refer to the electric or magnetic field vector, as any specific solution for either determines the other. The Gaussian beam solution is valid only in the paraxial approximation, that is, where wave propagation is limited to directions within a small angle of an axis. Without loss of generality let us take that direction to be the $q(z)$ direction in which case the solution $H$ can generally be written in terms of $c$ which has no time dependence and varies relatively smoothly in space, with the main variation spatially corresponding to the wavenumber $U$ in the $U$ direction: $$ U(x, y, z, t) = u(x, y, z) e^{-i(kz-\omega t)} \, \hat{\mathbf x} \, .$$

Using this form along with the paraxial approximation, $q(z)$ can then be essentially neglected. Since solutions of the electromagnetic wave equation only hold for polarizations which are orthogonal to the direction of propagation ($u$), we have without loss of generality considered the polarization to be in the $k$ direction so that we now solve a scalar equation for $q_{x}(z)$.

Substituting this solution into the wave equation above yields the paraxial approximation to the scalar wave equation: $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 2ik \frac{\partial u}{\partial z}.$$ Writing the wave equations in the light-cone coordinates returns this equation without utilizing any approximation. Gaussian beams of any beam waist $q_{y}(z)$ satisfy the paraxial approximation to the scalar wave equation; this is most easily verified by expressing the wave at $z$ in terms of the complex beam parameter $q_{x}(z) = q_{y}(z) = q(z)$ as defined above. There are many other solutions. As solutions to a linear system, any combination of solutions (using addition or multiplication by a constant) is also a solution. The fundamental Gaussian happens to be the one that minimizes the product of minimum spot size and far-field divergence, as noted above. In seeking paraxial solutions, and in particular ones that would describe laser radiation that is not in the fundamental Gaussian mode, we will look for families of solutions with gradually increasing products of their divergences and minimum spot sizes. Two important orthogonal decompositions of this sort are the Hermite–Gaussian or Laguerre-Gaussian modes, corresponding to rectangular and circular symmetry respectively, as detailed in the next section. With both of these, the fundamental Gaussian beam we have been considering is the lowest order mode.

Hermite-Gaussian modes
It is possible to decompose a coherent paraxial beam using the orthogonal set of so-called Hermite-Gaussian modes, any of which are given by the product of a factor in $z$ and a factor in $x$. Such a solution is possible due to the separability in $z$ and $x$ in the paraxial Helmholtz equation as written in Cartesian coordinates. Thus given a mode of order $x^{2} + y^{2} = r^{2}$ referring to the $y$ and $x$ directions, the electric field amplitude at $+z$ may be given by: $$ E(x,y,z) = u_l(x,z) \, u_m(y,z) \, \exp(-ikz), $$ where the factors for the $y$ and $x$ dependence are each given by: $$ u_J(x,z) = \left(\frac{\sqrt{2/\pi}}{ 2^J \, J! \; w_0}\right)^{\!\!1/2} \!\! \left( \frac{{q}_0}{{q}(z)}\right)^{\!\!1/2} \!\! \left(- \frac{{q}^\ast(z)}{{q}(z)}\right)^{\!\! J/2} \!\! H_J\!\left(\frac{\sqrt{2}x}{w(z)}\right) \, \exp \left(\! -i \frac{k x^2}{2 {q}(z)}\right) , $$ where we have employed the complex beam parameter $∂^{2}u/∂z^{2}$ (as defined above) for a beam of waist $u(x, y, z)$ at $y$ from the focus. In this form, the first factor is just a normalizing constant to make the set of $w_{0}$ orthonormal. The second factor is an additional normalization dependent on $x$ which compensates for the expansion of the spatial extent of the mode according to $q(z)$ (due to the last two factors). It also contains part of the Gouy phase. The third factor is a pure phase which enhances the Gouy phase shift for higher orders $y$.

The final two factors account for the spatial variation over $z$ (or $z$). The fourth factor is the Hermite polynomial of order $J$ ("physicists' form", i.e. $(l, m)$), while the fifth accounts for the Gaussian amplitude fall-off $x, y, z$, although this isn't obvious using the complex $x$ in the exponent. Expansion of that exponential also produces a phase factor in $y$ which accounts for the wavefront curvature ($q(z)$) at $J$ along the beam.

Hermite-Gaussian modes are typically designated "TEMlm"; the fundamental Gaussian beam may thus be referred to as TEM00 (where TEM is transverse electro-magnetic). Multiplying $w_{0}$ and $u_{J}$ to get the 2-D mode profile, and removing the normalization so that the leading factor is just called $w(z)/w_{0}$, we can write the $H_{1}(x) = 2x$ mode in the more accessible form:

$$\begin{align} E_{l, m}(x, y, z) ={} & E_0 \frac{w_0}{w(z)}\, H_l \!\Bigg(\frac{\sqrt{2} \,x}{w(z)}\Bigg)\, H_m \!\Bigg(\frac{\sqrt{2} \,y}{w(z)}\Bigg) \times {} \\ & \exp \left( {-\frac{x^2+y^2}{w^2(z)}} \right) \exp \left( {-i\frac{k(x^2 + y^2)}{2R(z)}} \right) \times {} \\ & \exp \big(i \psi(z)\big) \exp(-ikz). \end{align}$$

In this form, the parameter $exp(−x^{2}/w(z)^{2})$, as before, determines the family of modes, in particular scaling the spatial extent of the fundamental mode's waist and all other mode patterns at $1/R(z)$. Given that $u_{l}(x, z)$, $u_{m}(y, z)$ and $E_{0}$ have the same definitions as for the fundamental Gaussian beam described above. It can be seen that with $(l, m)$ we obtain the fundamental Gaussian beam described earlier (since $w_{0}$). The only specific difference in the $q$ and $x$ profiles at any $z$ are due to the Hermite polynomial factors for the order numbers $x$ and $y$. However, there is a change in the evolution of the modes' Gouy phase over $z$: $$ \psi(z) = (N+1) \, \arctan \left( \frac{z}{z_\mathrm{R}} \right), $$

where the combined order of the mode $l$ is defined as $z = 0$. While the Gouy phase shift for the fundamental (0,0) Gaussian mode only changes by $w_{0}$ radians over all of $m$ (and only by $w(z)$ radians between $R(z)$), this is increased by the factor $l = m = 0$ for the higher order modes.

Hermite Gaussian modes, with their rectangular symmetry, are especially suited for the modal analysis of radiation from lasers whose cavity design is asymmetric in a rectangular fashion. On the other hand, lasers and systems with circular symmetry can better be handled using the set of Laguerre-Gaussian modes introduced in the next section.

Laguerre-Gaussian modes
Beam profiles which are circularly symmetric (or lasers with cavities that are cylindrically symmetric) are often best solved using the  Laguerre-Gaussian modal decomposition. These functions are written in cylindrical coordinates using generalized Laguerre polynomials. Each transverse mode is again labelled using two integers, in this case the radial index $H_{0} = 1$ and the azimuthal index $z$ which can be positive or negative (or zero): $$\begin{align} u(r, \phi, z) ={} &C^{LG}_{lp}\frac{1}{w(z)}\left(\frac{r \sqrt{2}}{w(z)}\right)^{\! |l|} \exp\! \left(\! -\frac{r^2}{w^2(z)}\right)L_p^{|l|} \! \left(\frac{2r^2}{w^2(z)}\right) \times {} \\ &\exp \! \left(\! - i k \frac{r^2}{2 R(z)}\right) \exp(-i l \phi) \, \exp(i \psi(z)) , \end{align}$$

where $N = l + m$ are the generalized Laguerre polynomials. $±π/2$ is a required normalization constant: $$C^{LG}_{lp} = \sqrt{\frac{2 p!}{\pi(p+|l|)!}} \Rightarrow \int_0^{2\pi}d\phi\int_0^\infty dr\; r \,|u(r,\phi,z)|^2=1,$$.

$±π/4$ and $±z_{R}$ have the same definitions as above. As with the higher-order Hermite-Gaussian modes the magnitude of the Laguerre-Gaussian modes' Gouy phase shift is exaggerated by the factor $N + 1$: $$\psi(z) = (N+1) \, \arctan \left( \frac{z}{z_\mathrm{R}} \right) ,$$ where in this case the combined mode number $p ≥ 0$. As before, the transverse amplitude variations are contained in the last two factors on the upper line of the equation, which again includes the basic Gaussian drop off in $N$ but now multiplied by a Laguerre polynomial. The effect of the rotational mode number $z$, in addition to affecting the Laguerre polynomial, is mainly contained in the phase factor $L_{p}^{l}$, in which the beam profile is advanced (or retarded) by $l$ complete $CLG lp$ phases in one rotation around the beam (in $r$). This is an example of an optical vortex of topological charge $l$, and can be associated with the orbital angular momentum of light in that mode.

Ince-Gaussian modes
In elliptic coordinates, one can write the higher-order modes using Ince polynomials. The even and odd Ince-Gaussian modes are given by

$$ u_\varepsilon \left( \xi ,\eta ,z\right) = \frac{w_{0}}{w\left( z\right) }\mathrm{C}_{p}^{m}\left( i\xi ,\varepsilon \right) \mathrm{C} _{p}^{m}\left( \eta ,\varepsilon \right) \exp \left[ -ik\frac{r^{2}}{ 2q\left( z\right) }-\left( p+1\right) \zeta\left( z\right) \right] , $$ where $l$ and $φ$ are the radial and angular elliptic coordinates defined by $$\begin{align} x &= \sqrt{\varepsilon /2}\;w(z) \cosh \xi \cos \eta ,\\ y &= \sqrt{\varepsilon /2}\;w(z) \sinh \xi \sin \eta. \end{align}$$ $z$ are the even Ince polynomials of order $l$ and degree $ξ$ where $η$ is the ellipticity parameter. The Hermite-Gaussian and Laguerre-Gaussian modes are a special case of the Ince-Gaussian modes for $l = p = 0$ and $w(z)$ respectively.

Hypergeometric-Gaussian modes
There is another important class of paraxial wave modes in cylindrical coordinates in which the complex amplitude is proportional to a confluent hypergeometric function.

These modes have a singular phase profile and are eigenfunctions of the photon orbital angular momentum. Their intensity profiles are characterized by a single brilliant ring; like Laguerre–Gaussian modes, their intensities fall to zero at the center (on the optical axis) except for the fundamental (0,0) mode. A mode's complex amplitude can be written in terms of the normalized (dimensionless) radial coordinate $R(z)$ and the normalized longitudinal coordinate $N + 1$ as follows:

$$\begin{align} u_{\mathsf{p}m}(\rho, \phi, \Zeta) {}={} &\sqrt{\frac{2^{\mathsf{p} + |m| + 1}}{\pi\Gamma(\mathsf{p} + |m| + 1)}}\; \frac{\Gamma\left(\frac{\mathsf{p}}{2} + |m| + 1\right)}{\Gamma(|m| + 1)}\, i^{|m|+1} \times{} \\ &\Zeta^{\frac{\mathsf{p}}{2}}\, (\Zeta + i)^{-\left(\frac{\mathsf{p}}{2} + |m| + 1\right)}\, \rho^{|m|} \times{} \\ &\exp\left(-\frac{i\rho^2}{\Zeta + i}\right)\, e^{im\phi}\, {}_1F_1 \left(-\frac{\mathsf{p}}{2}, |m| + 1; \frac{\rho^2}{\Zeta(\Zeta + i)}\right) \end{align}$$

where the rotational index $p$ is an integer, and $$ {\mathsf p}\ge-|m| $$ is real-valued, $N = |l| + 2p$ is the gamma function and $exp(−ilφ)$ is a confluent hypergeometric function.

Some subfamilies of hypergeometric-Gaussian (HyGG) modes can be listed as the modified Bessel-Gaussian modes, the modified exponential Gaussian modes, and the modified Laguerre–Gaussian modes.

The set of hypergeometric-Gaussian modes is overcomplete and is not an orthogonal set of modes. In spite of its complicated field profile, HyGG modes have a very simple profile at the beam waist ($2π$): $$u(\rho, \phi, 0) \propto \rho^{\mathsf{p} + |m|}e^{-\rho^2 + im\phi}.$$

title: Gaussian beam


text:
Gel

A gel is a semi-solid that can have properties ranging from soft and weak to hard and tough. Gels are defined as a substantially dilute cross-linked system, which exhibits no flow when in the steady state, although the liquid phase may still diffuse through this system.

Gels are mostly liquid by mass, yet they behave like solids because of a three-dimensional cross-linked network within the liquid. It is the cross-linking within the fluid that gives a gel its structure (hardness) and contributes to the adhesive stick (tack). In this way, gels are a dispersion of molecules of a liquid within a solid medium. The word gel was coined by 19th-century Scottish chemist Thomas Graham by clipping from gelatine.

The process of forming a gel is called gelation.

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Definitions
The International Union of Pure and Applied Chemistry (IUPAC) defines a gel and related terms as:

Gel: Nonfluid colloidal network or polymer network that is expanded throughout its whole volume by a fluid. Note 1: A gel has a finite, usually rather small, yield stress. Note 2: A gel can contain:1. a covalent polymer network, e.g., a network formed by crosslinking polymer chains or by nonlinear polymerization;

2. a polymer network formed through the physical aggregation of polymer chains, caused by hydrogen bonds, crystallization, helix formation, complexation, etc., that results in regions of local order acting as the network junction points. The resulting swollen network may be termed a "thermoreversible gel" if the regions of local order are thermally reversible;

3. a polymer network formed through glassy junction points, e.g., one based on block copolymers. If the junction points are thermally reversible glassy domains, the resulting swollen network may also be termed a thermoreversible gel;

4. lamellar structures including mesophases, e.g., soap gels, phospholipids, and clays;

5. particulate disordered structures, e.g., a flocculent precipitate usually consisting of particles with large geometrical anisotropy, such as in V2O5 gels and globular or fibrillar protein gels. Hydrogel: Gel in which the swelling agent is water. Note 1: The network component of a hydrogel is usually a polymer network. Note 2: A hydrogel in which the network component is a colloidal network may be referred to as an aquagel. Xerogel: Open network formed by the removal of all swelling agents from a gel. Note 1: Examples of xerogels include silica gel and dried out, compact macromolecular structures such as gelatin or rubber.

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Composition
Gels consist of a solid three-dimensional network that spans the volume of a liquid medium and ensnares it through surface tension effects. This internal network structure may result from physical bonds such as polymer chain entanglements (see polymers) (physical gels) or chemical bonds such as disulfide bonds (see thiomers) (chemical gels), as well as crystallites or other junctions that remain intact within the extending fluid. Virtually any fluid can be used as an extender including water (hydrogels), oil, and air (aerogel). Both by weight and volume, gels are mostly fluid in composition and thus exhibit densities similar to those of their constituent liquids. Edible jelly is a common example of a hydrogel and has approximately the density of water.

Polyionic polymers
Polyionic polymers are polymers with an ionic functional group. The ionic charges prevent the formation of tightly coiled polymer chains. This allows them to contribute more to viscosity in their stretched state, because the stretched-out polymer takes up more space. This is also the reason gel hardens. See polyelectrolyte for more information.

Colloidal gels
A colloidal gel consists of a percolated network of particles in a fluid medium, providing mechanical properties in particular the rise of elastic behaviour. The particles can show attractive interactions through osmotic depletion or through polymeric links.

Colloidal gels have three phases in their lifespan: gelation, aging and collapse. The gel is initially formed by the assembly of particles into a space-spanning network, leading to a phase arrest. In the aging phase, the particles slowly rearrange to form thicker strands, increasing the elasticity of the material. Gels can also be collapsed and separated by external fields such as gravity. Colloidal gels show linear response rheology at low amplitudes. These materials have been explored as candidates for a drug release matrix.

Hydrogels

A hydrogel is a network of polymer chains that are hydrophilic, sometimes found as a colloidal gel in which water is the dispersion medium. A three-dimensional solid results from the hydrophilic polymer chains being held together by cross-links. Because of the inherent cross-links, the structural integrity of the hydrogel network does not dissolve from the high concentration of water. Hydrogels are highly absorbent (they can contain over 90% water) natural or synthetic polymeric networks. Hydrogels also possess a degree of flexibility very similar to natural tissue, due to their significant water content. As responsive "smart materials," hydrogels can encapsulate chemical systems which upon stimulation by external factors such as a change of pH may cause specific compounds such as glucose to be liberated to the environment, in most cases by a gel-sol transition to the liquid state. Chemomechanical polymers are mostly also hydrogels, which upon stimulation change their volume and can serve as actuators or sensors. The first appearance of the term 'hydrogel' in the literature was in 1894.

Organogels
An organogel is a non-crystalline, non-glassy thermoreversible (thermoplastic) solid material composed of a liquid organic phase entrapped in a three-dimensionally cross-linked network. The liquid can be, for example, an organic solvent, mineral oil, or vegetable oil. The solubility and particle dimensions of the structurant are important characteristics for the elastic properties and firmness of the organogel. Often, these systems are based on self-assembly of the structurant molecules. (An example of formation of an undesired thermoreversible network is the occurrence of wax crystallization in petroleum. )

Organogels have potential for use in a number of applications, such as in pharmaceuticals, cosmetics, art conservation, and food.

Xerogels

A xerogel is a solid formed from a gel by drying with unhindered shrinkage. Xerogels usually retain high porosity (15–50%) and enormous surface area (150–900 m2/g), along with very small pore size (1–10 nm). When solvent removal occurs under supercritical conditions, the network does not shrink and a highly porous, low-density material known as an aerogel is produced. Heat treatment of a xerogel at elevated temperature produces viscous sintering (shrinkage of the xerogel due to a small amount of viscous flow) which results in a denser and more robust solid, the density and porosity achieved depend on the sintering conditions.

Nanocomposite hydrogels
Nanocomposite hydrogels or hybrid hydrogels, are highly hydrated polymeric networks, either physically or covalently crosslinked with each other and/or with nanoparticles or nanostructures. Nanocomposite hydrogels can mimic native tissue properties, structure and microenvironment due to their hydrated and interconnected porous structure. A wide range of nanoparticles, such as carbon-based, polymeric, ceramic, and metallic nanomaterials can be incorporated within the hydrogel structure to obtain nanocomposites with tailored functionality. Nanocomposite hydrogels can be engineered to possess superior physical, chemical, electrical, thermal, and biological properties.

Properties
Many gels display thixotropy – they become fluid when agitated, but resolidify when resting. In general, gels are apparently solid, jelly-like materials. It is a type of non-Newtonian fluid. By replacing the liquid with gas it is possible to prepare aerogels, materials with exceptional properties including very low density, high specific surface areas, and excellent thermal insulation properties.

Thermodynamics of gel deformation
A gel is in essence the mixture of a polymer network and a solvent phase. Upon stretching, the network crosslinks are moved further apart from each other. Due to the polymer strands between crosslinks act as entropic springs, gels demonstrate elasticity like rubber (which is just a polymer network, without solvent). This is so because the free energy penalty to stretch an ideal polymer segment $$N$$ monomers of size $$b$$ between crosslinks to an end-to-end distance $$R$$ is approximately given by

 * $$F_\text{ela} \sim kT \frac{R^2}{Nb^2}.$$

This is the origin of both gel and rubber elasticity. But one key difference is that gel contains an additional solvent phase and hence is capable of having significant volume changes under deformation by taking in and out solvent. For example, a gel could swell to several times its initial volume after being immersed in a solvent after equilibrium is reached. This is the phenomenon of gel swelling. On the contrary, if we take the swelled gel out and allow the solvent to evaporate, the gel would shrink to roughly its original size. This gel volume change can alternatively be introduced by applying external forces. If a uniaxial compressive stress is applied to a gel, some solvent contained in the gel would be squeezed out and the gel shrinks in the applied-stress direction. To study the gel mechanical state in equilibrium, a good starting point is to consider a cubic gel of volume $$V_{0}$$ that is stretched by factors $$\lambda_1$$, $$\lambda_2$$ and $$\lambda_3$$ in the three orthogonal directions during swelling after being immersed in a solvent phase of initial volume $$V_{s0}$$. The final deformed volume of gel is then $$\lambda_1\lambda_2\lambda_3V_{0}$$ and the total volume of the system is $$V_{0}+V_{s0}$$ that is assumed constant during the swelling process for simplicity of treatment. The swelled state of the gel is now completely characterized by stretch factors $$\lambda_1$$, $$\lambda_2$$ and $$\lambda_3$$ and hence it is of interest to derive the deformation free energy as a function of them, denoted as $$f_\text{gel}(\lambda_1,\lambda_2,\lambda_3)$$. For analogy to the historical treatment of rubber elasticity and mixing free energy, $$f_\text{gel}(\lambda_1,\lambda_2,\lambda_3)$$ is most often defined as the free energy difference after and before the swelling normalized by the initial gel volume $$V_{0}$$, that is, a free energy difference density. The form of $$f_\text{gel}(\lambda_1,\lambda_2,\lambda_3)$$ naturally assumes two contributions of radically different physical origins, one is associated with the elastic deformation of the polymer network, and the other with the mixing of the network with the solvent. Hence, we write

 * $$f_\text{gel}(\lambda_1, \lambda_2, \lambda_3) = f_\text{net}(\lambda_1, \lambda_2, \lambda_3) + f_\text{mix}(\lambda_1, \lambda_2, \lambda_3).$$

We now consider the two contributions separately. The polymer elastic deformation term is independent of the solvent phase and has the same expression as a rubber, as derived in the Kuhn's theory of rubber elasticity:

 * $$f_\text{net}(\lambda_1,\lambda_2,\lambda_3) = \frac{G_0}{2} (\lambda_1^2 + \lambda_2^2 + \lambda_3^2 - 3),$$

where $$G_0$$ denotes the shear modulus of the initial state. On the other hand, the mixing term $$f_\text{mix}(\lambda_1,\lambda_2,\lambda_3)$$ is usually treated by the Flory-Huggins free energy of concentrated polymer solutions $$f(\phi)$$, where $$\phi$$ is polymer volume fraction. Suppose the initial gel has a polymer volume fraction of $$\phi_0$$, the polymer volume fraction after swelling would be $$\phi=\phi_0/\lambda_1\lambda_2\lambda_3$$ since the number of monomers remains the same while the gel volume has increased by a factor of $$\lambda_1\lambda_2\lambda_3$$. As the polymer volume fraction decreases from $$\phi_0$$ to $$\phi$$, a polymer solution of concentration $$\phi_0$$ and volume $$V_{0}$$ is mixed with a pure solvent of volume $$(\lambda_1\lambda_2\lambda_3-1)V_{0}$$ to become a solution with polymer concentration $$\phi$$ and volume $$\lambda_1\lambda_2\lambda_3V_{0}$$. The free energy density change in this mixing step is given as

 * $$V_{g0} f_\text{mix}(\lambda_1 \lambda_2 \lambda_3) = \lambda_1 \lambda_2 \lambda_3 f(\phi) - [V_0 f(\phi_0) + (\lambda_1 \lambda_2\lambda_3 - 1) f(0)],$$

where on the right-hand side, the first term is the Flory–Huggins energy density of the final swelled gel, the second is associated with the initial gel and the third is of the pure solvent prior to mixing. Substitution of $$\phi = \phi_0/\lambda_1\lambda_2\lambda_3$$ leads to

 * $$f_\text{mix}(\lambda_1, \lambda_2, \lambda_3) = \frac{\phi_0}{\phi} [f(\phi) - f(0)] - [f(\phi_0) - f(0)].$$

Note that the second term is independent of the stretching factors $$\lambda_1$$, $$\lambda_2$$ and $$\lambda_3$$ and hence can be dropped in subsequent analysis. Now we make use of the Flory-Huggins free energy for a polymer-solvent solution that reads

 * $$f(\phi) = \frac{kT}{v_c} [\frac{\phi}{N} \ln\phi + (1 - \phi) \ln(1 - \phi) + \chi \phi (1 - \phi)],$$

where $$v_c$$ is monomer volume, $$N$$ is polymer strand length and $$\chi$$ is the Flory-Huggins energy parameter. Because in a network, the polymer length is effectively infinite, we can take the limit $$N\to\infty$$ and $$f(\phi)$$ reduces to

 * $$f(\phi) = \frac{kT}{v_c} [(1 - \phi) \ln(1 - \phi) + \chi \phi(1 - \phi)].$$

Substitution of this expression into $$f_\text{mix}(\lambda_1,\lambda_2,\lambda_3)$$ and addition of the network contribution leads to

 * $$f_\text{gel}(\lambda_1, \lambda_2, \lambda_3) = \frac{G_0}{2} (\lambda_1^2 + \lambda_2^2 + \lambda_3^2) + \frac{\phi_0}{\phi} f(\phi).$$

This provides the starting point to examining the swelling equilibrium of a gel network immersed in solvent. It can be shown that gel swelling is the competition between two forces, one is the osmotic pressure of the polymer solution that favors the take in of solvent and expansion, the other is the restoring force of the polymer network elasticity that favors shrinkage. At equilibrium, the two effects exactly cancel each other in principle and the associated $$\lambda_1$$, $$\lambda_2$$ and $$\lambda_3$$ define the equilibrium gel volume. In solving the force balance equation, graphical solutions are often preferred.

In an alternative, scaling approach, suppose an isotropic gel is stretch by a factor of $$\lambda$$ in all three directions. Under the affine network approximation, the mean-square end-to-end distance in the gel increases from initial $$R_0^2$$ to $$(\lambda R_0)^2$$ and the elastic energy of one stand can be written as

 * $$F_\text{ela} \sim kT \frac{(\lambda R_0)^2}{R_\text{ref}^2},$$

where $$R_\text{ref}$$ is the mean-square fluctuation in end-to-end distance of one strand. The modulus of the gel is then this single-strand elastic energy multiplied by strand number density $$\nu=\phi/Nb^3$$ to give

 * $$G(\phi) \sim \frac{kT}{b^3} \frac{\phi}{N} \frac{(\lambda R_0)^2}{R_\text{ref}^2}.$$

This modulus can then be equated to osmotic pressure (through differentiation of the free energy) to give the same equation as we found above.

Modified Donnan equilibrium of polyelectrolyte gels
Consider a hydrogel made of polyelectrolytes decorated with weak acid groups that can ionize according to the reaction

 * $$\text{HA} \rightleftharpoons \text{A}^- + \text{H}^+$$

is immersed in a salt solution of physiological concentration. The degree of ionization of the polyelectrolytes is then controlled by the $$\text{pH}$$ and due to the charged nature of $$\text{H}^+$$ and $$\text{A}^-$$, electrostatic interactions with other ions in the systems. This is effectively a reacting system governed by acid-base equilibrium modulated by electrostatic effects, and is relevant in drug delivery, sea water desalination and dialysis technologies. Due to the elastic nature of the gel, the dispersion of $$\text{A}^-$$ in the system is constrained and hence, there will be a partitioning of salts ions and $$\text{H}^+$$ inside and outside the gel, which is intimately coupled to the polyelectrolyte degree of ionization. This ion partitioning inside and outside the gel is analogous to the partitioning of ions across a semipemerable membrane in classical Donnan theory, but a membrane is not needed here because the gel volume constraint imposed by network elasticity effectively acts its role, in preventing the macroions to pass through the fictitious membrane while allowing ions to pass.

The coupling between the ion partitioning and polyelectrolyte ionization degree is only partially by the classical Donnan theory. As a starting point we can neglect the electrostatic interactions among ions. Then at equilibrium, some of the weak acid sites in the gel would dissociate to form $$\text{A}^-$$that electrostatically attracts positive charged $$\text{H}^+$$ and salt cations leading to a relatively high concentration of $$\text{H}^+$$ and salt cations inside the gel. But because the concentration of $$\text{H}^+$$ is locally higher, it suppresses the further ionization of the acid sites. This phenomenon is the prediction of the classical Donnan theory. However, with electrostatic interactions, there are further complications to the picture. Consider the case of two adjacent, initially uncharged acid sites $$\text{HA}$$ are both dissociated to form $$\text{A}^-$$. Since the two sites are both negatively charged, there will be a charge-charge repulsion along the backbone of the polymer than tends to stretch the chain. This energy cost is high both elastically and electrostatically and hence suppress ionization. Even though this ionization suppression is qualitatively similar to that of Donnan prediction, it is absent without electrostatic consideration and present irrespective of ion partitioning. The combination of both effects as well as gel elasticity determines the volume of the gel at equilibrium. Due to the complexity of the coupled acid-base equilibrium, electrostatics and network elasticity, only recently has such system been correctly recreated in computer simulations.

Animal-produced gels
Some species secrete gels that are effective in parasite control. For example, the long-finned pilot whale secretes an enzymatic gel that rests on the outer surface of this animal and helps prevent other organisms from establishing colonies on the surface of these whales' bodies.

Hydrogels existing naturally in the body include mucus, the vitreous humor of the eye, cartilage, tendons and blood clots. Their viscoelastic nature results in the soft tissue component of the body, disparate from the mineral-based hard tissue of the skeletal system. Researchers are actively developing synthetically derived tissue replacement technologies derived from hydrogels, for both temporary implants (degradable) and permanent implants (non-degradable). A review article on the subject discusses the use of hydrogels for nucleus pulposus replacement, cartilage replacement, and synthetic tissue models.

Applications
Many substances can form gels when a suitable thickener or gelling agent is added to their formula. This approach is common in manufacture of wide range of products, from foods to paints and adhesives.

In fiber optic communications, a soft gel resembling hair gel in viscosity is used to fill the plastic tubes containing the fibers. The main purpose of the gel is to prevent water intrusion if the buffer tube is breached, but the gel also buffers the fibers against mechanical damage when the tube is bent around corners during installation, or flexed. Additionally, the gel acts as a processing aid when the cable is being constructed, keeping the fibers central whilst the tube material is extruded around it.

title: Gel