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Laser Surgery. 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\(\mathrm { nm }\) and delivers 250\(\mathrm { mW }\) of power spread over a circular spot 510\(\mu \mathrm { m }\) 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 does the pulse of the laser beam exert on the retina as it is fully absorbed by the circular spot? (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?

Step by step solution

01

Calculate Energy Delivered per Pulse

Power delivered by the laser is given as 250 mW, which is 0.250 W. Since energy (E) is power (P) multiplied by time (t), the energy delivered per pulse is \[ E = P \times t = 0.250 \text{ W} \times 1.50 \times 10^{-3} \text{ s} = 0.375 \text{ J} \]

02

Compute Average Pressure Exerted on Retina

To find the average pressure, we use the formula for radiation pressure when light is fully absorbed: \(P_r = \frac{I}{c}\), where \(I\) is the intensity of the laser beam, and \(c\) is the speed of light. First, we calculate the intensity \(I\) using \(I = \frac{P}{A}\) where \(A = \pi r^2\), with \(r\) (radius) being half the diameter of the spot:\[ r = \frac{510 \times 10^{-6}}{2} = 255 \times 10^{-6} \text{ m} \]\[ A = \pi (255 \times 10^{-6})^2 \approx 2.042 \times 10^{-7} \text{ m}^2 \]\[ I = \frac{0.250}{2.042 \times 10^{-7}} \approx 1.223 \times 10^6 \text{ W/m}^2 \]Finally, since the index of refraction is 1.34, the speed of light \( c' = \frac{c}{n} = \frac{3 \times 10^8}{1.34} \approx 2.239 \times 10^8 \text{ m/s}\). The average pressure is\[ P_r = \frac{1.223 \times 10^6}{2.239 \times 10^8} \approx 5.46 \times 10^{-3} \text{ N/m}^2 \]

03

Determine Wavelength and Frequency in Vitreous Humor

The speed of light in the vitreous humor \( c' = \frac{c}{n} \). The wavelength \( \lambda' \) in the vitreous humor can be determined using\[ \lambda' = \frac{\lambda}{n} = \frac{810}{1.34} \approx 604.5 \text{ nm} \]The frequency \( f \) is unchanged:\[ f = \frac{c}{\lambda} = \frac{3 \times 10^8}{810 \times 10^{-9}} \approx 3.70 \times 10^{14} \text{ Hz} \]

04

Find Maximum Electric and Magnetic Fields

The intensity \( I \) is related to the maximum electric field \( E_0 \) and magnetic field \( B_0 \) by the formulas:\[ I = \frac{c\epsilon_0}{2} E_0^2 \quad \text{and} \quad I = \frac{c B_0^2}{2 \mu_0} \]where \( \epsilon_0 = 8.85 \times 10^{-12} \text{ F/m} \) and \( \mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A} \). Solving for \( E_0 \):\[ E_0 = \sqrt{\frac{2I}{c\epsilon_0}} \approx \sqrt{\frac{1.223 \times 10^6}{2.239 \times 10^8 \times 8.85 \times 10^{-12}}} \approx 1086 \text{ V/m} \]Using \( B_0 = \frac{E_0}{c'} \):\[ B_0 = \frac{1086}{2.239 \times 10^8} \approx 4.85 \times 10^{-6} \text{ T} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Physics of Laser Beams

Laser beams are highly concentrated beams of light that consist of electromagnetic waves. This concentrated nature allows lasers to be extremely precise, which makes them ideal for tasks requiring accuracy, like surgical procedures. Laser beams are characterized by their wavelength and power. The wavelength determines the type of electromagnetic wave, and the power determines how much energy the beam carries. In laser surgery, pulses of this energy are used to perform specific functions such as repairing tissues. In this way, the physical properties of lasers make them uniquely suitable for medical purposes.

Retina Repair

Retina repair is a delicate procedure that can be assisted significantly by using laser surgery. The retina is a layer of cells at the back of the eye responsible for capturing light and enabling vision. Sometimes, parts of the retina can become detached, requiring precise and targeted intervention. Laser beams can be used to deliver very fine and controlled energy bursts to weld the detached retina back into place. This is achieved by focusing the laser beam through the eye, targeting only the affected area, and applying just enough energy to fuse the tissues without causing damage to adjacent parts. These precision capabilities make lasers incredibly valuable in ophthalmology.

Radiation Pressure

Radiation pressure is the pressure exerted by light when it strikes a surface and is absorbed. When a laser beam hits the retina, it exerts pressure, which is a result of the beam's momentum. In the context of laser surgery, this radiation pressure is calculated based on the intensity of the laser light and the speed of light, adjusted for the medium through which it passes, such as the eye’s vitreous humor. The formula used is Intensity divided by the speed of light in that medium. This pressure must be understood and controlled, as it affects how the laser interacts with the tissue, and ensures that the beam does not cause unintended damage.

Electromagnetic Waves

Lasers emit electromagnetic waves, which are synchronized and concentrated beams of light. These waves travel in the form of photons at high speeds. The properties of electromagnetic waves, such as wavelength and frequency, determine the type of light the laser emits. When applied to surgery, these waves allow for highly focused energy transfer. In mediums like the vitreous humor of the eye, the speed and wavelength change due to the medium's refractive index, but the frequency remains constant. Understanding these properties is crucial for tailoring the laser for different surgical tasks, such as adjusting the energy for precise retina repair procedures.