Photorefractive keratectomy (PRK) is a laser-based surgical procedure that corrects near- and farsightedness by removing part of the lens of the eye to change its curvature and hence focal length. This procedure can remove layers $0.25$ mm thick using pulses lasting $12.0$ ns from a laser beam of wavelength $193$ nm. Low-intensity beams can be used because each individual photon has enough energy to break the covalent bonds of the tissue. If a $1.50$-mW beam is used, how many photons are delivered to the lens in each pulse?

Verified step by step guidance

1. Step 1: Start by calculating the energy of a single photon using the formula \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant \( (6.626 \times 10^{-34} \, \text{J·s}) \), \( c \) is the speed of light \( (3.00 \times 10^8 \; \text{m/s}) \), and \( \lambda \) is the wavelength of the laser beam \( (193 \; \text{nm} = 193 \times 10^{-9} \; \text{m}) \).
2. Step 2: Determine the total energy delivered by the laser beam during each pulse. Use the formula \( E_{\text{pulse}} = P \cdot t \), where \( P \) is the power of the laser beam \( (1.50 \; \text{mW} = 1.50 \times 10^{-3} \; \text{W}) \) and \( t \) is the duration of the pulse \( (12.0 \; \text{ns} = 12.0 \times 10^{-9} \; \text{s}) \).
3. Step 3: Calculate the number of photons delivered in each pulse by dividing the total energy of the pulse \( E_{\text{pulse}} \) by the energy of a single photon \( E \). Use the formula \( N = \frac{E_{\text{pulse}}}{E} \).
4. Step 4: Substitute the values for \( E \) and \( E_{\text{pulse}} \) into the formula \( N = \frac{E_{\text{pulse}}}{E} \) to find the number of photons delivered in each pulse.
5. Step 5: Ensure all units are consistent throughout the calculations (e.g., converting nanometers to meters, milliwatts to watts, and nanoseconds to seconds) to avoid errors in the final result.

Key Concepts

Photon Energy

The energy of a photon is determined by its wavelength and can be calculated using the formula E = hc/λ, where E is energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. In this context, the wavelength of the laser beam (193 nm) is crucial for determining the energy of each photon, which is necessary to understand how many photons are involved in the procedure.

Power and Energy Relationship

Power is defined as the rate at which energy is transferred or converted, measured in watts (W). In this case, the power of the laser beam (1.50 mW) indicates how much energy is delivered per second. To find the total energy delivered in a pulse, we multiply the power by the duration of the pulse, allowing us to calculate the total energy available for photon delivery.

Photon Count Calculation

To determine the number of photons delivered in each pulse, we can use the relationship between energy and the number of photons. By dividing the total energy delivered in a pulse by the energy of a single photon, we can find the total number of photons. This calculation is essential for understanding the effectiveness of the laser in breaking covalent bonds in the eye tissue during the PRK procedure.