Fourier Transform

The Fourier transform, named after Joseph Fourier, is a mathematical transformation employed to transform signals between time (or spatial) domain andfrequency domain, which has many applications in physics and engineering. It is reversible, being able to transform from either domain to the other. The term itself refers to both the transform operation and to the function it produces.

This is the basic mathematical tool used for the calculation of difficult mathematical problems, specially, when the problem is related to any domain. like time domain, frequency domain, space domain and others.

Fourier transform converts every domain to s domain which has real and negative part. The poles and zeros are on either real part or on imaginary part. If they are on real axis and at negative side then the system is stable, If the are on positive side then system is unstable. If they are on imaginary axis then system is marginal stable.

Fourier transform gives the relation between time domain to frequency domain, so we can easily convert a signal in other domains.  In frequency domain it is very easier to solve and complex problem. but in time domain it very easier to design the system. So for system designing and its performance characterization we need Fourier Transform.

Fourier transform is the natural mechanism used by the nature to solve the optical problem as the lens give the Fourier transform of any signal at its focus, when signal is originated from its other focus. It can be so simple in the solving a differentiation or integral problem. Since it is in the form of exponential and differentiation and integration of the exponential is exponential.

The Fourier Transform is extensively used in the field of Signal Processing. In fact, the Fourier Transform is probably the most important tool for analyzing signals in that entire field. So what exactly is signal processing? I'll try to give a one paragraph high level overview. A signal is any waveform (function of time). This could be anything in the real world - an electromagnetic wave, the voltage across a resistor versus time, the air pressure variance due to your speech (i.e. a sound wave), or the value of Apple Stock versus time. Signal Processing then, is the act of processing a signal to obtain more useful information, or to make the signal more useful. How can a signal be made better? Suppose that you are listening to a recording, and there is a low-pitched hum in the background. By applying a low-frequency filter, we can eliminate the hum. Or suppose you have a digital photograph, and it is very noisy (that is, there are random specs of light everywhere). We can use signal processing and fourier transforms to filter out this undesirable "noise".

Fiber Bragg Grating

"FBG is a simple device used for the separation of wavelength from a group of wavelengths."  Rayc


A fiber Bragg grating (FBG) is a type of distributed Bragg reflector constructed in a short segment of optical fiber that reflects particular wavelengths of light and transmits all others. This is achieved by creating a periodic variation in the refractive index of the fiber core, which generates a wavelength specific dielectric mirror. A fiber Bragg grating can therefore be used as an inline optical filter to block certain wavelengths, or as a wavelength-specific reflector.

The first in-fiber Bragg grating was demonstrated by Ken Hill in 1978. Initially, the gratings were fabricated using a visible laser propagating along the fiber core. In 1989, Gerald Meltz and colleagues demonstrated the much more flexible transverse holographic inscription technique where the laser illumination came from the side of the fiber. This technique uses the interference pattern of ultraviolet laser light to create the periodic structure of the fiber Bragg grating.

Fiber Bragg gratings are created by "inscribing" or "writing" systematic (periodic or aperiodic) variation of refractive index into the core of a special type of optical fiber using an intense ultraviolet(UV) source such as a UV laser. Two main processes are used: interference and masking. The method that is preferable depends on the type of grating to be manufactured. Normally agermanium-doped silica fiber is used in the manufacture of fiber Bragg gratings. The germanium-doped fiber is photosensitive, which means that the refractive index of the core changes with exposure to UV light. The amount of the change depends on the intensity and duration of the exposure as well as the photosensitivity of the fibre. To write a high reflectivity fiber Bragg grating directly in the fiber the level of doping with germanium needs to be high. However, standard fibers can be used if the photosensitivity is enhanced by pre-soaking the fiber in hydrogen. More recently, fiber Bragg gratings have also been written in polymer fibers, this is described in the PHOSFOS entry

Fiber bragg grating can easily designed by using a simple fiber in presence of the interference of two ultravoilet light beam. Which give constructive and destructive interferences. At constructive interference refractive index decrease and at destructive interference refractive index remains same so we get two different refractive indices. Which periodically makes FBG.

FBG  are of two types:

1. Simple FBG 

2. Chirped FBG

 Simple FBG  in this refractive index of the dense medium is fixed. In this reflected light is the function of length of a period of refractive index.

 Chirped FBG in this refractive index of the dense medium is getting denser.In this refracted light is combination of different wavelengths are refracted at different time intervals.

Fiber Bragg Gratings are made by laterally exposing the core of a single-mode fiber to a periodic pattern of intense ultraviolet light. The exposure produces a permanent increase in the refractive index of the fiber's core, creating a fixed index modulation according to the exposure pattern. This fixed index modulation is called a grating.

At each periodic refraction change a small amount of light is reflected. All the reflected light signals combine coherently to one large reflection at a particular wavelength when the grating period is approximately half the input light's wavelength. This is referred to as the Bragg condition, and the wavelength at which this reflection occurs is called the Bragg wavelength. Light signals at wavelengths other than the Bragg wavelength, which are not phase matched, are essentially transparent.

Therefore, light propagates through the grating with negligible attenuation or signal variation. Only those wavelengths that satisfy the Bragg condition are affected and strongly back-reflected. The ability to accurately preset and maintain the grating wavelength is a fundamental feature and advantage of fiber Bragg gratings.

The central wavelength of the reflected component satisfies the Bragg relation: λrefl=2nΛ, with n the index of refraction and Λ the period of the index of refraction variation of the FBG. Due to the temperature and strain dependence of the parameters n and Λ, the wavelength of the reflected component will also change as function of temperature and/or strain, see Figure 2. This dependency is well known what allows determining the temperature or strain from the reflected FBG wavelength.

There have been many advances in the methods used to perform strain measurements. The three most prevalent technologies today are electrical foil gages, electrical vibrating wire, and fiber Bragg grating (FBG) optical sensors. 

For most standard strain sensing applications, electrical sensing has been and will continue to be the best and most effective solution. However, optical sensors can offer an important alternative in traditionally challenging applications. Harsh environments, distributed systems, or long-term deployments are a few examples where the characteristics of an optical sensing system can make it a more effective solution compared to conventional electrical sensors. 

Applications often require measurement solutions that encompass the benefits and attributes of multiple sensing technologies. It is therefore important to consider a hybrid approach. This paper provides a brief overview of each technology and weighs their benefits and drawbacks.

Electrical Sensing: Metal Foil Gages

Foil strain gages use the relationship between electrical resistance and conductor length to measure changes in strain. As the foil is stretched, its length is increased, which translates into a minute increase in resistance. To accurately measure these small changes in resistance, additional signal conditioning is necessary, often in the form of a Wheatstone bridge resistance network. A constant voltage is applied across the resistance network, and the varying proportional drop in voltage across the foil can be translated to strain.

Gradient Force

"Gradient" refers to how rapidly a quantity (such as pressure or temperature) changes in a given distance. It can be thought of as measure of "steepness", like the topography on a contour plot.

The pressure gradient force is the force which results when there is a difference in pressure across a surface. In general, a pressure is a force per unit area, across a surface. A difference in pressure across a surface then implies a difference in force, which can result in an acceleration according to Newton's second law, if there is no additional force to balance it. The resulting force is always directed from the region of higher-pressure to the region of lower-pressure. When a fluid is in an equilibrium state (i.e. there are no net forces, and no acceleration), the system is referred to as being in hydrostatic equilibrium. In the case of atmospheres, the pressure gradient force is balanced by the gravitational force, maintaining hydrostatic equilibrium. In the Earth's atmosphere, for example, air pressure decreases at increasing altitudes above the Earth's surface, thus providing a pressure gradient force which counteracts the force of gravity on the atmosphere

In optical tweezer Gradient force is used to compensate Scattering force.

In cases where the diameter of a trapped particle is significantly greater than the wavelength of light, the trapping phenomenon can be explained using ray optics. As shown in the figure, individual rays of light emitted from the laser will be refracted as it enters and exits the dielectric bead. As a result, the ray will exit in a direction different from which it originated. Since light has a momentum associated with it, this change in direction indicates that its momentum has changed. Due to Newton's third law, there should be an equal and opposite momentum change on the particle.

Most optical traps operate with a Gaussian beam (TEM₀₀ mode) profile intensity. In this case, if the particle is displaced from the center of the beam, as in the right part of the figure, the particle has a net force returning it to the center of the trap because more intense beams impart a larger momentum change towards the center of the trap than less intense beams, which impart a smaller momentum change away from the trap center. The net momentum change, or force, returns the particle to the trap center.

If the particle is located at the center of the beam, then individual rays of light are refracting through the particle symmetrically, resulting in no net lateral force. The net force in this case is along the axial direction of the trap, which cancels out the scattering force of the laser light. The cancellation of this axial gradient force with the scattering force is what causes the bead to be stably trapped slightly downstream of the beam waist.

This is the output when beam is linear.

If the particle is located at the center of the beam, then individual rays of light are refracting through the particle symmetrically, resulting in no net lateral force. The net force in this case is along the axial direction of the trap, which cancels out the scattering force of the laser light. The cancellation of this axial gradient force with the scattering force is what causes the bead to be stably trapped slightly downstream of the beam waist.

This is the output, when beam is focused.

The standard tweezers works with the trapping laser propagated in the direction of gravity and the inverted tweezers works against gravity.

Wind is simply air in motion relative to the earth's surface. We normally think of the wind as the horizontal motion of the air, although air actually moves in three dimensions. The vertical component of the wind is generally quite small, except in thunderstorm updrafts. The vertical motion of air, however, is very important in determining our weather. Air that is rising cools, which may cause it to reach saturation and form clouds and precipitation. Conversely, air that is sinking warms, which causes clouds to evaporate and produce clear weather.

Surface maps usually have H's and L's at various locations. The H's and L's represent high and low pressure systems. On weather maps highs and lows are surrounded by lines called isobars. Isobars are lines of constant pressure; they connect every location that has the same value of pressure. When isobars are packed close together, the pressure is changing rapidly over a small distance. The closer the isobars are packed together, the stronger thepressure gradient (the rate of pressure change over a given distance.) Also, notice that (in the Northern Hemisphere) the wind blows clockwise around a high pressure system and also slightly outward from its center. Around a low pressure system, the wind blows counterclockwise and slightly in towards its center.

Diffraction Limit

An ideal optical system would image an object point perfectly as a point. However, due to the wave nature of radiation, diffraction occurs, caused by the limiting edges of the system’s aperture stop. The result is that the image of a point is a blur, no matter how well the lens is corrected. This is the diffraction blur or Airy disk, named in honor of Lord George Biddel Airy, a British mathematician (1801–1892). Its cross section and its appearance are shown in the figure below.

Airy disk, energy distribution and appearance.

If an image is made through a small aperture, there is a point at which the resolution of the image is limited by the aperture diffraction. As a matter of general practice in photographic optics, the use of a smaller aperture (larger f-number) will give greater depth of field and a generally sharper image. But if the aperture is made too small , the effects of the diffraction will be large enough to begin to reduce that sharpness, and you have reached the point of diffraction-limited imaging.

If you are imaging two points of light, then the smallest separation at which you could discern that there are two could reasonably be used as the limit of resolution of the imaging process. Presuming that diffraction is the determining factor, then the generally accepted criterion for the minimum resolvable detail is the Rayleigh criterion.

This shows the intensity curves for the radial distribution of the diffracted light for different separations. Your eye sees the characteristic bulls eye distribution of light as illustrated below.

For modern digital photography where the images are projected onto a CCD, the information is collected on pixels of the digital detector. At left is an attempt to show the effect of diffraction on such imaging in cases where the diffraction is the phenomenon that limits the resolution. If the image is in focus and free of visible affects of lens aberrations, then it may be that it will fit on one pixel. But if the aperture is small enough, then diffraction can spread the image onto neighboring pixels and constitute the limit on the resolution of the image.

Diffraction of any image reduces its sharpness and losses the frequency component inside it. Since there is lot much noise comes in even scent modes.