The intensity of a laser beam is given by the ratio of power...

The intensity of a laser beam is given by the ratio of power to area. A particular laser beam has an intensity function given by $I=e^{-r^{2}} \mathrm{mW} / \mathrm{cm}^{2},$ where $r$ is the radius off the center axis given in centimeters. Where is the beam brightest (largest intensity)?

Key Concepts

Gaussian Distribution

A Gaussian distribution is a common function that describes how a quantity decreases exponentially from a central peak. In many laser beams, the intensity profile follows a Gaussian shape, which means the highest intensity is at the center and falls off as one moves outward.

Function Optimization

Optimization involves finding the maximum or minimum value of a function. In the analysis of intensity profiles, determining where the beam is brightest entails finding the value of the radial coordinate at which the intensity function reaches its maximum value, typically using calculus or by examining the behavior of the function.

Intensity

Intensity is a measure of power per unit area, which indicates how much energy is transmitted per unit time through a given surface. In the context of beams, it determines how concentrated the energy is over the area the beam irradiates.

Radial Symmetry

Radial symmetry refers to functions or distributions that depend only on the distance from a central point rather than the direction. This symmetry simplifies problems by reducing multidimensional analysis to a single variable, typically the radius.

Transcript

00:01 So a laser beam has this intensity expression.

00:06 So the intensity as a function of r, where r is a distance from the off center.

00:13 Okay, so if a radius, if our laser beam was somewhat circular, really exaggerated beam, we find it all circle here, i assume this is a circle, okay? and so that's your horizontal axis and you could plot you right here.

00:34 This has the intensity so this unit axis has the intensity and this would be the x like the distance from the center and this is the right in the center of the beam the question is how does where is the maximum intensity and so you can just plug in our x or actually r not x so we can just plug in this this expression at r equal to zero you know that e to the 0 is 1 so intensity is intensity is one i guess this is a normalized expression obviously has some magnitude other than one but it's it's taken as a ratio of that quantity okay so it is one at r equal to zero and then when when r is greater than zero you can see that this r squared is become larger so i another way of writing that is really is i equals 1 over exponential e to the r square and so we can see that as r if r increases this thing becomes this quantity here becomes greater than one and so one over that this whole quantity here will become less than one so this is one at r equal to zero that's one everywhere else as we step away from r as r increases it's going to drop and as r tends to infinity, the value actually will drop to 1 over e to the infinity is zero.

02:24 So it's a really sharp drop.

02:25 And you can examine this curve more carefully because we know that 1 over r squared itself looks like this.

02:32 So we can use some, let's say, just for experimenting to understand this better.

02:38 You know that, so for example, for any given, let's say r just qualitatively.

02:53 Without punching and using any values.

02:55 We know that 1 over r squared goes something like this...

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