Very short pulses of high-intensity laser beams are used to...

Very short pulses of high-intensity laser beams are used to repair detached portions of the retina of the eye. The brief pulses of energy absorbed by the retina weld the detached portions back into place. In one such procedure, a laser beam has a wavelength of 810 nm and delivers 250 mW of power spread over a circular spot 510 $\mu$m in diameter. The vitreous humor (the transparent fluid that fills most of the eye) has an index of refraction of 1.34. (a) If the laser pulses are each 1.50 ms long, how much energy is delivered to the retina with each pulse? (b) What average pressure would the pulse of the laser beam exert at normal incidence on a surface in air if the beam is fully absorbed? (c) What are the wavelength and frequency of the laser light inside the vitreous humor of the eye? (d) What are the maximum values of the electric and magnetic fields in the laser beam?

Key Concepts

Electromagnetic Wave Field-Amplitude Relationships

This concept relates the intensity of an electromagnetic wave to the maximum values of its electric and magnetic fields. The intensity can be expressed in terms of the electric field amplitude using the relation I = ½ ?? c E², where ?? is the permittivity of free space and c is the speed of light in a vacuum. Additionally, the magnetic field amplitude is related to the electric field via B = E/c. These relationships are fundamental in analyzing the properties and effects of electromagnetic waves in various scientific and engineering contexts.

Pulse Energy Calculation

This concept involves determining the energy delivered by a pulsed laser beam by multiplying its power by the duration of the pulse. In general, the energy (in Joules) is found by the equation E = P × t, where P is the power in watts and t is the time in seconds. This calculation is essential when working with pulsed sources of energy in various applications, including medical procedures and material processing.

Radiation Pressure

Radiation pressure is the pressure exerted by electromagnetic radiation upon any surface due to the momentum carried by the photons. It is calculated differently depending on whether the light is fully absorbed or reflected. For a fully absorbed beam, the pressure is given by the intensity divided by the speed of light (p = I/c). This concept is important in understanding the mechanical effects of light on surfaces in fields like optical manipulation and laser surgery.

Light Propagation in Media

This concept describes how light behaves when it moves from one medium to another with a different refractive index. The wavelength of light decreases in a medium by a factor equal to the index of refraction (?_medium = ?_vacuum/n), while its frequency remains unchanged. This understanding is crucial for applications involving light transmission through various materials, such as in optical fibers, microscopy, and biomedical imaging.

Transcript

00:01 All right, so for this problem, we have a laser beam.

00:06 So we already know that the laser beam has power p equal.

00:11 So this is the part a.

00:14 The power is 250 milliwatt.

00:19 And we also know the time for this laser beam, i mean for this pulse, the total time is 1 .5 milliseconds.

00:31 So in part a, we want to find out the energy for this pulse.

00:36 So we know that the energy equal power times time, right? so this gives the total energy of this pulse.

00:44 So it's simply 250 minute watt times 1 .5 milliseconds.

00:51 And this gives you 0 .375 mili jails.

01:00 In part b, we want to, let me see, we want to find out the pressure.

01:09 Induced by this pulse.

01:13 So in a textbook, you can find the relation between the pressure of the laser beam and the intensity of the beam.

01:21 So the relation is the pressure, the radiation pressure, equal i over c.

01:30 So where i is the intensity of this beam, and c is the speed of light.

01:35 So the intensity is defined as p over x.

01:41 Where p is the power, which is the power, which is the one we found over here, and the s is the effective area of this laser beam.

01:55 Because in this problem, we're considering a situation that the beam is fully absorbed by this normal surface.

02:02 So this is the relation, and the divide by c.

02:06 So we already know that the p is 250 watt.

02:10 Watt so the area is pi r squared right because it's it's just the circle circle area pi r squared and divided by c's seize the speed of light which is a constant so the r is not directly given the problem but we already know the diameter d so you can utilize the diameter and because r the radius equal d over two right so you can find out the radius of this beam and just plug in the number and it would see that the final we're not the final result for this radiation pressure is 4 .08 midepaska.

02:52 And in part c, so suppose the laser beam enters into the human eye, we want to find out the, basically, we want to find out the frequency and the speed and the wavelengths of the light inside the eye.

03:12 So we already know the reflection, in fact, the refraction, the refraction factor inside the eye and it's 1 .34.

03:21 So it's just the ideal case.

03:25 So given this information, we also know the wavelengths of this light is, so this wavelength is the wavelength in the vacuum, okay? so let's say this is a lambda v.

03:38 So v labels the vacuum.

03:41 So this is 8, 10 nanometers.

03:45 And the speed of light, i mean the speed of the beam is just the speed of light.

03:50 Which is a constant c.

03:52 So let's see this is the cv, okay? so also the labels the vacuum, which is three times 10 to the 10th meters per second, okay? and the frequency is, the frequency can be calculated by these two variables.

04:10 So the frequency is cv, sorry, cv diwa, lambda, okay? so just to use these two values and the calculate this frequency.

04:20 And the value you can obtain is, 3 .704 times 10 to the 14th.

04:29 Sorry, we just remove it times 10 to the 14th hertz.

04:42 So remember, these are the situation in the vacuum because we wanna find out the information inside the eye.

04:48 So we need to do some more calculations.

04:51 Because we already know the reflection factor is 1 .34.

04:56 So basically if the beam entered the eye, the frequency will remain unchanged...