For multimodal distributions, the D4σ width is a better choice.

Mathematically, the D4σ beam width in the x-dimension for the beam profile is expressed as

When a beam is measured with a laser beam profiler, the wings of the beam profile influence the D4σ value more than the center of the profile since the wings are weighted by the square of its distance, x < sup > 2 </ sup >, from the center of the beam.

If the baseline value is large or if it is not subtracted out of the image, then the computed D4σ value will be larger than the actual value because the baseline value near the edges of the sensor are weighted in the D4σ integral by x < sup > 2 </ sup >.

Therefore, baseline subtraction is necessary for accurate D4σ measurements.

The D4σ width, unlike the FWHM and 1 / e < sup > 2 </ sup > widths, is meaningful for multimodal marginal distributions — that is, beam profiles with multiple peaks — but requires careful subtraction of the baseline for accurate results.

The D4σ is the ISO international standard definition for beam width.

The D4σ beam width is the ISO standard definition and the measurement of the M² beam quality parameter requires the measurement of the D4σ widths.

Measurements of the 1 / e < sup > 2 </ sup > width only depend on three points on the marginal distribution, unlike D4σ and knife-edge widths that depend on the integral of the marginal distribution.

Five definitions of the beam width are in common use: D4σ, 10 / 90 or 20 / 80 knife-edge, 1 / e < sup > 2 </ sup >, FWHM, and D86.

1 / e < sup > 2 </ sup > width measurements are noisier than D4σ width measurements.

For an ideal single-mode Gaussian beam, the D4σ, D86 and 1 / e < sup > 2 </ sup > width measurements would give the same value.

On the other hand, the D4σ, 1 / e < sup > 2 </ sup >, and FWHM widths encompass fractions of power that are beam-shape dependent.

The other definitions provide complementary information to the D4σ.

The D4σ and knife-edge widths are sensitive to the baseline value, whereas the 1 / e < sup > 2 </ sup > and FWHM widths are not.

He knew instinctively that next to voice and face an actor's hands are his most useful possession -- that in fiction as in the theatre, gesture is an indispensable shorthand for individualizing character and dramatizing action and response.

The term ' the 10th of August ' is widely used by historians as a shorthand for the Storming of the Tuileries Palace on the 10th of August, 1792, the effective end of the French monarchy until it was restored in 1814.

Thus, for example, V ← V + 1 is commonly understood to be a shorthand for ( V ,( V ) + 1 ).

( In modern times, the use of is regarded as a shorthand for

However, as an exception, the American shorthand " 9 / 11 " is widely understood as referring to the 11 September 2001 terrorist attacks.

The term " democracy " is sometimes used as shorthand for liberal democracy, which is a variant of representative democracy that may include elements such as political pluralism ; equality before the law ; the right to petition elected officials for redress of grievances ; due process ; civil liberties ; human rights ; and elements of civil society outside the government.

A defendant or defender ( Δ in legal shorthand ) is any party required to answer a plaintiff's complaint in a civil lawsuit, or any party that has been formally charged or accused of violating a criminal statute.

A film maker is colluding with his audience to provide a shorthand learned through a common cinematic cultural background.

The purpose is to create safe handling habits, and to discourage reasoning along the lines of, " I know my gun is unloaded so certain unsafe practices are OK ." The proposition " the gun is always loaded " is used as a shorthand, even though it may be assumed — or even positively known — that this is not true of a particular firearm.

For this reason, however, ' narcotic ' has come to mean any illegally used drug, but it is useful as a shorthand for referring to a controlled drug in a context where its legal status is more important than its physiological effects.

That said, many individualists are willing to use " we " in reference to government or society as a convenient shorthand as long as the fact that these entities are composed of individuals is kept in mind.

If V is a formal language, then V < sub > i </ sub >, the i-th power of the set V, is a shorthand for the concatenation of set V with itself i times.

A shorthand version of Klingonaase, and later with the same term adopted by tlhIngan Hol itself, is called " battle language ", or " Clipped Klingon ".

It is situated in Holland ( then as now used by English speakers as shorthand for the Netherlands ) but by treaty is garrisoned by Prussians and 5, 000 of their troops occupy it under a Prince of Hesse.

The sound of linear timecode is a jarring and distinctive noise and has been used as a sound-effects shorthand to imply ' telemetry ' or ' computers '.

Advocates of phylogenetic nomenclature, a more nominalistic view, oppose this reading ; in their opinion, calling an animal a mammal is a shorthand way of saying that it is descended from the last common ancestor of, say, humans and platypuses.

It is possible, although highly doubtful, that he killed none at all but merely let his reputation work for him by privately claiming every unsolved murder in the state.

( The best evidence is that he received a monthly wage of about $125, very good money in an era when top hands worked for $30 and found.

this is not so, for education offers all kinds of dividends, including how to pull the wool over a husband's eyes while you are having an affair with his wife.

`` What is the scaffolding for, Brassnose ''??

He speaks your language too, for he is the grandson of a chieftain on Taui who made much magic and was strong and cunning.

This is a paradise for hunters.

`` And if the dive goes OK he has the exclusive import rights to your line for this country, is that right ''??

There is nothing for you '', Matsuo said.

It is almost time for and calinda to begin ''.

I want the room in the attic prepared for him He is a most unusual lad, quite precocious in many ways.

-- liberal considers that the need for a national economy with controls that will assure his conception of social justice is so great that individual and local liberties as well as democratic processes may have to yield before it.

In fact it has caused us to give serious thought to moving our residence south, because it is not easy for the most objective Southerner to sit calmly by when his host is telling a roomful of people that the only way to deal with Southerners who oppose integration is to send in troops and shoot the bastards down.

but for this discussion the most important division is between those who have been reconstructed and those who haven't.

Had the situation been reversed, had, for instance, England been the enemy in 1898 because of issues of concern chiefly to New England, there is little doubt that large numbers of Southerners would have happily put on their old Confederate uniforms to fight as allies of Britain.

Of greater importance, however, is the content of those programs, which have had and are having enormous consequences for the American people.

The general acceptance of the idea of governmental ( i.e., societal ) responsibility for the economic well-being of the American people is surely one of the two most significant watersheds in American constitutional history.

Reduced to its simplest terms, it is an assumption of a collective duty to compensate for the inability of individuals to cope with the rigors of the era.

National responsibility for individual welfare is a concept not limited to the United States or even to the Western nations.

For better or for worse, we all now live in welfare states, the organizing principle of which is collective responsibility for individual well-being.

( Since the time-span of the nation-state coincides roughly with the separate existence of the United States as an independent entity, it is perhaps natural for Americans to think of the nation as representative of the highest form of order, something permanent and unchanging.

There is little time for the men in the command centers to reflect about the implications of these clocks.

Only recently new `` holes '' were discovered in our safety measures, and a search is now on for more.