Problem 31 - College Physics

Question

Photorefractive keratectomy (PRK) is a laser-based surgery process 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 \(0.25 \mu \mathrm{m}\) thick in pulses lasting \(12.0 \mathrm{~ns}\) with a laser beam of wavelength \(193 \mathrm{nm}\). 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 \(1.50 \mathrm{~mW}\) beam is used, how many photons are delivered to the lens in each pulse?

Step-by-Step Solution

Answer

(a) Ultraviolet light; (b) \(1.03\times10^{-18}\) J; (c) \(1.75\times10^{7}\) photons per pulse.

Step 1: Determine the electromagnetic spectrum region

To identify the part of the electromagnetic spectrum this light lies in, we start with the given wavelength of the laser light, which is \(193\ nm\). The electromagnetic spectrum's ultraviolet (UV) range typically spans from about \(10\ nm\) to \(400\ nm\). Since \(193\ nm\) falls within this range, the light is categorized as ultraviolet (UV) light.

Step 2: Calculate the energy of a single photon

To find the energy of a single photon, we use the formula \(E = \frac{hc}{\lambda}\), where \(E\) is the energy, \(h\) is Planck's constant \((6.63\times10^{-34}\ J\cdot s)\), \(c\) is the speed of light \((3.00\times10^{8}\ m/s)\), and \(\lambda\) is the wavelength of the light in meters \((193\times10^{-9}\ m)\). The calculation is:\[E = \frac{(6.63\times10^{-34}) \times (3.00\times10^{8})}{193\times10^{-9}} = 1.03\times10^{-18}\ J.\]Thus, the energy of a single photon is approximately \(1.03\times10^{-18} J\).

Step 3: Calculate the number of photons per pulse

First, convert the power of the beam from milliwatts to watts: \(1.50\ mW = 1.50\times10^{-3}\ W\). The energy delivered per pulse \((E_{pulse})\) is given by \(E_{pulse} = \text{Power} \times \text{pulse duration}\). Replace the values to find:\[E_{pulse} = 1.50\times10^{-3}\ W \times 12.0\times10^{-9}\ s = 1.80\times10^{-11}\ J.\]Next, to find how many photons are delivered, we divide the energy per pulse by the energy per photon:\[\text{Number of photons} = \frac{1.80\times10^{-11}\ J}{1.03\times10^{-18}\ J/photon} \approx 1.75\times10^{7}\ photons.\] Therefore, approximately \(1.75\times10^{7}\) photons are delivered in each pulse.

Key Concepts

Ultraviolet Light

Ultraviolet (UV) light is a type of electromagnetic radiation that is not visible to the human eye. It has shorter wavelengths than visible light, ranging from about 10 nm to 400 nm. The light produced in photorefractive keratectomy falls within this range, specifically at a wavelength of 193 nm.
This identifies it as UV light.
UV light is important in laser surgeries like PRK because its high energy allows it to effectively remove thin layers of tissue.

• UV light is classified into three main types: UVA, UVB, and UVC.
• UVC, which includes wavelengths around 193 nm, is the most energetic and can break molecular bonds energy-efficiently.

Unfortunately, exposure to UV light can also be harmful, which is why protective measures are essential when working with UV lasers.
In medical applications, UV light's high energy is exploited to precisely ablate tissues without causing excessive heat damage.

Photon Energy

Photon energy corresponds to the amount of energy carried by a single photon, which is the elementary particle of light. This energy can be calculated using the formula:\[ E = \frac{hc}{\lambda} \] Where:

• \( E \) is the energy of a photon,
• \( h \) is Planck's constant \( (6.63 \times 10^{-34} \, J \cdot s) \),
• \( c \) is the speed of light \( (3.00 \times 10^{8} \, m/s) \),
• \( \lambda \) is the wavelength of light.

Using the given wavelength of 193 nm, photon energy is approximately \( 1.03 \times 10^{-18} \, J \).
This high energy is enough to break covalent bonds in tissues, making it effective for such precision applications as photorefractive keratectomy.

Electromagnetic Spectrum

The electromagnetic spectrum is a complete range of electromagnetic waves, characterized by their differing wavelengths and frequencies. UV light, utilized in PRK, is one part of this spectrum.
Here's a brief overview of its structure:

• Radio Waves: Longest wavelength, used in communication.
• Microwaves: Shorter than radio waves, used in radar and cooking.
• Infrared: Felt as heat, used in thermal imaging.
• Visible Light: The only part human eyes can see.
• Ultraviolet: Just beyond visible light, used in sterilization and surgeries.
• X-Rays: Penetrates objects, used in medical imaging.
• Gamma Rays: Shortest wavelength and highest energy, used in cancer treatments.

The UV section of the spectrum is important in medical and industrial applications due to its high energy ability to interact with materials at an atomic level.

Optics

Optics is the branch of physics concerned with light, its properties, and its interactions with matter. In the context of PRK, optics is crucial because it deals with the behavior and characteristics of light waves as they interface with biological tissues.

• Reflection: This occurs when light bounces off a surface. In PRK, controlling reflection is important to ensure laser precision.
• Refraction: Light bending when passing through substances. Lasers must account for refraction to maintain accurate focus on target tissues.
• Diffraction: The slight bending of light around the edges of objects. Understanding diffraction helps in the design of optical systems to minimize errors.

In laser surgery, optics guides the development of precise instruments that can alter or correct vision by reshaping the corneal curvature to adjust focusing power for better vision.

Laser Surgery

Laser surgery utilizes focused light beams to perform medical procedures, including photorefractive keratectomy. This technique offers a high level of precision and minimal invasiveness, achieving results beyond traditional surgical methods.

• Types of Lasers: Different lasers are used for various procedures, each selected based on wavelength needs that affect tissue interaction.
• Benefits: Lasers provide precise targeting, reduced bleeding, lower infection risks, and faster recovery times.
• Applications: Beyond vision correction, laser surgery is used in dermatology, oncology, and dentistry.

In PRK, the use of UV lasers allows for the accurate removal of microscopic layers of the cornea to correct vision by reshaping its surface.