Question

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 \(\mu\)m 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. (a) In what part of the electromagnetic spectrum does this light lie? (b) What is the energy of a single photon? (c) If a 1.50-mW beam is used, how many photons are delivered to the lens in each pulse?

Short answer

(a) UV spectrum, (b) \(1.03 \times 10^{-18} \text{ J}\), (c) \(1.75 \times 10^7\) photons per pulse.

Step-by-step solution

Part of the Electromagnetic Spectrum

Find the part of the electromagnetic spectrum the wavelength corresponds to. Given the wavelength is 193 nm, convert it to meters: \(193 \text{ nm} = 193 \times 10^{-9} \text{ m}\). Since this wavelength is between 100 nm and 400 nm, it falls within the ultraviolet (UV) spectrum.

Calculate Energy of a Photon

Use the formula \(E = \frac{hc}{\lambda}\) to calculate the energy of a photon, where \(h = 6.626 \times 10^{-34} \text{ J⋅s}\) (Planck's constant) and \(c = 3.00 \times 10^8 \text{ m/s}\) (speed of light). Substitute \(\lambda = 193 \times 10^{-9} \text{ m}\) into the formula: \[E = \frac{(6.626 \times 10^{-34})(3.00 \times 10^8)}{193 \times 10^{-9}} = 1.03 \times 10^{-18} \text{ J}\] Thus, the energy of a single photon is approximately \(1.03 \times 10^{-18} \text{ J}\).

Calculate Number of Photons per Pulse

First, find the energy delivered by the beam in each pulse. The power of the laser is given as 1.50 mW, which is \(1.50 \times 10^{-3} \text{ W}\). The time duration of each pulse is \(12.0 \text{ ns} = 12.0 \times 10^{-9} \text{ s}\). The energy per pulse is calculated by: \[ E_{\text{pulse}} = (1.50 \times 10^{-3}) \times (12.0 \times 10^{-9}) = 1.80 \times 10^{-11} \text{ J}\] To find the number of photons, divide the total energy per pulse by the energy of a single photon: \[N = \frac{E_{\text{pulse}}}{E} = \frac{1.80 \times 10^{-11}}{1.03 \times 10^{-18}} \approx 1.75 \times 10^7\] Thus, approximately \(1.75 \times 10^7\) photons are delivered to the lens in each pulse.

Key concepts

Ultraviolet Spectrum

The ultraviolet (UV) spectrum is a part of the electromagnetic spectrum that spans wavelengths from around 10 nm to 400 nm. This range is just beyond the violet portion of visible light, which is why it's called "ultraviolet." The UV spectrum is divided into three main categories:

• UV-A (320-400 nm): Also known as long-wave UV or "black light," commonly encountered in tanning beds and fluorescent lamps.
• UV-B (280-320 nm): Has a medium wavelength and is known for causing sunburns. It's partially absorbed by the ozone layer.
• UV-C (100-280 nm): Short-wave UV that is mostly absorbed by the Earth's atmosphere and is used in germicidal lamps.

In the exercise, the laser light used in photorefractive keratectomy (PRK) has a wavelength of 193 nm, which places it well within the UV-C category of the ultraviolet spectrum. This region has sufficiently high energy to break covalent bonds, making it ideal for delicate medical procedures like reshaping the cornea.

Electromagnetic Spectrum

The electromagnetic spectrum is a broad range of all possible frequencies of electromagnetic radiation. It ranges from low-energy, long-wavelength radio waves to high-energy, short-wavelength gamma rays. The main segments of the electromagnetic spectrum include:

• Radio Waves: Longest wavelength, used for broadcasting audio and visual signals.
• Microwaves: Slightly shorter than radio waves, used in cooking and radar technology.
• Infrared: Just below visible light, utilized in thermal imaging and remote controls.
• Visible Light: The range that is visible to the human eye, spanning from violet (shortest wavelength) to red (longest wavelength).
• Ultraviolet: Shorter than visible light, useful for sterilization and in scientific research.
• X-Rays: Even shorter; used in medical imaging to view the internal structure of organisms.
• Gamma Rays: Shortest wavelength and highest energy, originating from nuclear reactions.

When considering parts of the spectrum like the UV used in the exercise, it is important to recognize how energy increases as the wavelength decreases. This principle underlines why the UV-C spectrum is capable of disruption at the molecular level, such as breaking covalent bonds in tissue during eye surgeries.

Planck's Constant

Planck's constant is a fundamental constant in physics denoted by the letter "h." Its value is approximately 6.626 x 10^-34 Js (joules per second). It plays a crucial role in the field of quantum mechanics, particularly in calculations involving the energy of photons. The famous formula, which links Planck's constant to photon energy, is:
\[ E = \frac{hc}{\lambda} \]
where \(E\) is the energy of a photon, \(h\) is Planck's constant, \(c\) is the speed of light (approximately 3.00 x 10^8 m/s), and \(\lambda\) is the wavelength of the electromagnetic wave.
Planck's constant fundamentally connects the wave and particle nature of light. This duality is central to understanding how photons can carry energy, and it explains how lasers can be safe for delicate operations yet effective in applications like PRK. By understanding this balance, one can appreciate how small-scale quantum activities manifest in large-scale applications, directly impacting fields ranging from physics to medical technology.