where \(\pmb{\mathcal{E}}\) is a constant vector independent of space and time. This represents a sinusoidal wave whose phase varies with z and t as

\[\tag{163}\begin{align}\pmb{E}&=\pmb{\mathcal{E}}\exp(\text{i}k_1z-\text{i}\omega_1t)+\text{c.c.}+\pmb{\mathcal{E}}\exp(\text{i}k_2z-\text{i}\omega_2t)+\text{c.c.}\\&=2\pmb{\mathcal{E}}\lbrace\cos[(k_0+\text{d}k)z-(\omega_0+\text{d}\omega)t]+\cos[(k_0-\text{d}k)z-(\omega_0-\text{d}\omega)t]\rbrace\\&=4\pmb{\mathcal{E}}\cos(\text{d}kz-\text{d}\omega{t})\cos(k_0z-\omega_0t)\end{align}\]

\[\tag{162}\begin{align}\omega_1=\omega_0+\text{d}\omega,\;k_1=k_0+\text{d}k\\\omega_2=\omega_0-\text{d}\omega,\;k_2=k_0-\text{d}k \end{align}\]

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The constant-phase wavefront travels at the phase velocity, but the group velocity is the velocity at which energy and information travel.

Two major lens components—the objective lens and the ocular lens, or eyepiece—work together to project the image of the specimen onto a sensor. This may be the ...

Imec provides hyperspectral imaging with filters on chip. This eliminates the complex and delicate optical design of traditional hyperspectral systems.

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tau; inf; nan. Previous topic. numbers — Numeric ... π = 3.141592… e. e = 2.718281… tau. τ = 2π ... On some non-Windows builds, the underlying C library ...

The index of refraction of a certain type of glass as a function of optical wavelength around λ = 1.3 μm can be approximated as n = 1.465 - 0.0114(λ - 1.3) - 0.004(λ - 1.3)3, where λ is measured in micrometers. Therefore,

Teen substance abuse is a crisis in our culture. As parents scramble to try to protect their teens from addiction, the use of at-home drug tests is a popular solution to ensure that they are honest about their sobriety. While drug tests for teens may seem like the perfect way to monitor substance use, there are ways teens can alter the results of these tests, which could lure parents into a false sense of security about their teen’s substance use. Before you drug test your teen at home, find out the methods people have used to cheat the test.

We find that the resultant wave packet has a carrier, which has a frequency ω0 and a propagation constant k0, and an envelope, which varies in space and time as cos(dkz - dωt). This is illustrated in figure 20 above. Therefore, a fixed point on the envelope is defined by dkz - dωt = constant, and it travels with a velocity of

We find that, in this spectral region, dn/dλ < 0 for any wavelength but D > 0 for λ < 1.3 μm and D < 0 for λ > 1.3 μm. Clearly, this glass has normal phase-velocity dispersion in the entire spectral region around λ < 1.3 μm, but it has positive group-velocity dispersion for λ < 1.3 μm and negative group-velocity dispersion for λ > 1.3 μm. As an example, we find that n ≈ 1.469, N ≈ 1.481, and D ≈ 0.0072 at λ = 1 μm. We also find that n ≈ 1.463, N ≈ 1.481, and D ≈ -0.0108 at λ = 1.5 μm. Because of normal phase-velocity dispersion, the group index is always larger than the refractive index, N > n, in this spectral region.

Group-velocity dispersion and phase-velocity dispersion have different meanings. They should not be confused with each other.

When measuring the transmission delay or the broadening of optical pulses due to dispersion in optical fibers, another dispersion coefficient defined as

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Group-velocity dispersion is an important consideration in the propagation of optical pulses. It can cause broadening of an individual pulse, as well as changes in the time delay between pulses of different frequencies.

As long as there have been urine screens for drug testing, there have been people trying todilutetheir specimen in an attempt to fake their results. A diluted urine drug test is usually done by consuming excessive amounts of water before testing or adding water to the urine sample to dilute the concentration of substances in it. A diluted urine test can sometimes be detected more easily than other types of cheating, based on the dilution of other prime compounds in the urine that are detected on the test.

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The sign of the group-velocity dispersion can be either positive or negative. In the case of positive group-velocity dispersion, d2k/dω2 >0 and D > 0, a long-wavelength, or low-frequency, pulse travels faster than a short-wavelength, or high-frequency, pulse.

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In real circumstances, a propagating optical wave rarely contains only one frequency. It usually consists of many frequency components grouped around some center frequency ω0. For the simplicity of illustration, we consider a wave packet traveling in the z direction that is composed of two plane waves of equal real amplitude \(\pmb{\mathcal{E}}\). The frequencies and propagation constants of the two component plane waves are

Therefore, d2k/dω2 represents group-velocity dispersion. A dimensionless coefficient for group-velocity dispersion can be defined as

This is called the phase velocity of the wave. Note that the phase velocity is a function of optical frequency because the refractive index of a medium is a function of frequency. There is phase-velocity dispersion due to the fact that dn/dω ≠ 0. In the case of normal dispersion, dn/dω > 0 and dn/dλ < 0; in the case of anomalous dispersion, dn/dω < 0 and dn/dλ > 0.

This is the velocity of the wave packet and is called the group velocity. Because the energy of a harmonic wave is proportional to the square of its field amplitude, the energy carried by a wave packet that is composed of many frequency components is concentrated in regions where the amplitude of the envelope is large. Therefore, the energy in a wave packet is transported at group velocity vg.

The condenser serves two purposes; it regulates the amount of light reaching the specimen and it focuses the light coming from the light source. As the ...

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is usually used. This coefficient is generally expressed as a function of wavelength in units of picoseconds per kilometer per nanometer. It is a direct measure of the chromatic pulse transmission delay over a unit transmission length.

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\[\tag{166}D=c\omega\frac{\text{d}^2k}{\text{d}\omega^2}=\frac{2\pi{c^2}}{\lambda}\frac{\text{d}^2k}{\text{d}\omega^2}\]

For a point of constant phase on the space- and time-varying field, φ = constant and thus kdz - ωdt = 0. If we track this point of constant phase, we find that it is moving with a velocity of

A false negative drug test is an attempt to introduce a compound into the body or the urine directly that influences test results. Taking aspirin before the drug test is a common method of falsifying results. The chemical compounds of aspirin are known to have properties that can mask drugs within the urine, making this a popular method. Drinking bleach to pass a drug test is another common method used to fake a drug test that is highly dangerous and ineffective. Teens are putting themselves at risk not only by using substances but by employing dangerous tactics to fake their drug tests.

The Recovery Village aims to improve the quality of life for people struggling with substance use or mental health disorder with fact-based content about the nature of behavioral health conditions, treatment options and their related outcomes. We publish material that is researched, cited, edited and reviewed by licensed medical professionals. The information we provide is not intended to be a substitute for professional medical advice, diagnosis or treatment. It should not be used in place of the advice of your physician or other qualified healthcare providers.

Paula Holmes is a licensed clinical social worker, psychotherapist and freelance writer who lives and works in midcoast Maine. She received her master's degree in Social Work in 2008 from the University of Maine. Read more

Our state-specific resource guides offer a comprehensive overview of drug and alcohol addiction treatment options available in your area.

With teen drug use on the rise, parents want to safeguard against their kids’ use of substances. Often parents will purchase drug tests as a means of ensuring sobriety, particularly if their teen’s drug abuse has been ongoing. Unfortunately, these tests are fallible. Cheating a drug test is possible; there are many ways to do it. If your teen is motivated to have a clean result, they may research how to pass an at home drug test and use these methods.

The Recovery Village aims to improve the quality of life for people struggling with a substance use or mental health disorder with fact-based content about the nature of behavioral health conditions, treatment options and their related outcomes. We publish material that is researched, cited, edited and reviewed by licensed medical professionals. The information we provide is not intended to be a substitute for professional medical advice, diagnosis or treatment. It should not be used in place of the advice of your physician or other qualified healthcare provider.

Sometimes, teens using drugs will resort to using synthetic urine, or fake pee, for drug tests. Synthetic urine is not actual urine. Instead, it is a manufactured composite that mimics the look and chemical compounds of urine. It is passable in lab screens as a clean urine sample. Synthetic urine is sold in the form of a powder that is then mixed with water. Heating pads are often part of the purchase to keep the urine at a passable temperature for the test. People purchase synthetic urine online, and although it is being used to fraudulently pass drug tests, the product itself is legal.

In contrast, a short-wavelength pulse travels faster than a long-wavelength pulse in the case of negative group-velocity dispersion, d2k/dω2 < 0 and D < 0. In a given material, the sign of D generally depends on the spectral region of concern.