Comprehensive Study Notes on Lasers and Optical Fibers

LASER: Definition and Fundamental Characteristics

• LASER is an acronym that stands for Light Amplification by Stimulated Emission of Radiation.
• A laser device produces a beam of light with the following specific characteristics:
  • Coherence: The waves of the laser beam move in phase with each other.
  • Monochromaticity: The light consists of a single wavelength or frequency.
  • Intensity: The light is highly concentrated and bright.
  • Directionality: The light travels in a narrow path in a single direction.
• Comparison with Ordinary Light:
  • Ordinary light is incoherent, meaning its waves do not move in phase.
  • Ordinary light spreads out and travels in many different directions.
  • Laser light is considered highly organized compared to ordinary light.
• General Applications of Lasers:
  • Lasers are versatile tools used in Engineering, Medicine, Defence, Entertainment, and Communication.
  • Common specific uses include reading barcodes, cutting and welding metals, light show displays, playing music, printing documents, and guiding missiles to targets.

Basic Principles: Interaction of Radiation with Matter

• Production of laser light is a consequence of the interaction of radiation with matter under specific conditions.
• This interaction causes transitions of a quantum system (atoms or molecules) from one quantum energy state to another.
• Material media are composed of identical atoms or molecules with discrete allowed energy levels.
• An atom moves between energy states through a "quantum jump" or transition by receiving or releasing energy equal to the difference between the two states.
• Consider a two-level energy system:
  • $E_1$ is the energy of the lower (ground) state.
  • $E_2$ is the energy of the excited state.
  • Radiation is viewed as a stream of photons with energy calculated as $(E_2 - E_1) = h\nu$.

Types of Interaction Between Radiation and Matter

• Induced Absorption:

  • Definition: The excitation of atoms from a lower energy state to a higher energy state by absorbing incident photons.
  • An atom in state $E_1$ absorbs a photon of energy $(E_2 - E_1)$ and moves to state $E_2$.
  • For every transition, one photon disappears from the incident beam.
  • Expression: $A + h\nu \rightarrow A^*$ (where $A^*$ is the excited state).
  • The rate of induced absorption is proportional to the number of atoms in the ground state ($N_1$) and the energy density of incident radiation ($U_{\nu}$).
  • $\text{Rate of induced absorption} = B_{12}U_{\nu}N_1$.
  • $B_{12}$ is the Einstein coefficient of absorption (a proportionality constant for absorption probability).

• Spontaneous Emission:

  • Definition: An unstable atom at the higher energy state $E_2$ returns to the lower state $E_1$ on its own, emitting a single photon of energy $(E_2 - E_1) = h\nu$.
  • Expression: $A^* \rightarrow A + h\nu$.
  • The transition happens without outside control; variables like timing, direction, phase, and polarization are random.
  • The resulting light is incoherent.
  • The rate depends on the number of atoms in the excited state ($N_2$).
  • $\text{Rate of spontaneous emission} = A_{21}N_2$.
  • $A_{21}$ is the Einstein coefficient of spontaneous emission.

• Stimulated Emission:

  • Definition: Photons are emitted by an atomic system under external influence.
  • First predicted by Einstein in 1916, where an incident photon of energy $h\nu = (E_2 - E_1)$ induces an excited atom to drop to the ground state.
  • This results in the emission of a second photon identical to the incident one.
  • Expression: $A^* + h\nu \rightarrow A + 2h\nu$.
  • The emitted photon has the same frequency, phase, direction, and polarization as the incident photon.
  • The rate depends on the number of atoms in the excited state ($N_2$) and the energy density ($U_{\nu}$).
  • $\text{Rate of stimulated emission} = B_{21}U_{\nu}N_2$.
  • $B_{21}$ is the Einstein coefficient of stimulated emission. This process is the basis for laser action.

Key Definitions and Parameters

• Atomic System: A system of atoms or molecules with discrete energy levels.
• Active Medium: A material medium supporting the interaction of radiation with matter in thermal equilibrium.
• Energy Density ($U_{\nu}$): Total radiation energy per unit volume per unit frequency. It follows Planck’s distribution law:
  • $U_{\nu} = \frac{8\pi h\nu^3}{c^3} \times \frac{1}{e^{\frac{h\nu}{kT}} - 1}$
• Population: The number of atoms per unit volume ($N$) in a given energy state.
• Boltzmann Factor: The ratio of populations in thermal equilibrium:
  • $\frac{N_2}{N_1} = e^{-\frac{h\nu}{kT}}$
  • In equilibrium, $N_1 > N_2$.
• Population Inversion: A non-equilibrium state where the number of atoms in the higher energy state ($N_2$) exceeds the number in the ground state ($N_1$) ($N_2 > N_1$).

Einstein Coefficients and Energy Density Expression

• At thermal equilibrium, the rate of absorption must equal the total rate of emission (spontaneous + stimulated):
  • $B_{12}N_1U_{\nu} = A_{21}N_2 + B_{21}N_2U_{\nu}$
  • Rearranging for energy density: $U_{\nu} = \frac{A_{21}N_2}{B_{12}N_1 - B_{21}N_2}$
  • Using the Boltzmann factor $\frac{N_1}{N_2} = e^{\frac{h\nu}{kT}}$, the equation becomes:
    • $U_{\nu} = \frac{A_{21}}{B_{21}} \times \frac{1}{\frac{B_{12}}{B_{21}}e^{\frac{h\nu}{kT}} - 1}$
• By comparing this to Planck's law, it is determined that:
  • $B_{12} = B_{21}$ (The probability of induced absorption equals the probability of stimulated emission).
  • $\frac{A_{21}}{B_{21}} = \frac{8\pi h\nu^3}{c^3}$
• Final energy density expression at equilibrium:
  • $U_{\gamma} = \frac{A}{B} \times \frac{1}{e^{\frac{h\gamma}{kT}} - 1}$

Conditions for Light Amplification

• The ratio of stimulated emission to spontaneous emission is proportional to the radiation density ($U_{\nu}$). High energy density enhances stimulated transitions.
• Amplification occurs only if stimulated emission dominates over absorption, which requires Population Inversion ($N_2 > N_1$).
• If $N_2 < N_1$, the medium will absorb more energy than it emits.

Requisites of a Laser System

• Active Medium: The material (solid, liquid, or gas) where laser action occurs. Specific "active centers" (atoms/ions) are responsible for stimulated emission.
• Pumping Mechanism: The process of supplying energy to transport atoms to higher energy states to achieve population inversion.
  • Optical Pumping: Using light for excitation (e.g., Ruby and Nd:YAG lasers).
  • Electric Discharge: Using an electric field to ionize and excite atoms (e.g., Argon ion laser).
  • Inelastic Atom-Atom Collision: Gas molecules (e.g., Helium) are excited via discharge and collide with another species (e.g., Neon) to transfer energy (e.g., He-Ne laser).
  • Direct Conversion: Converting electrical energy directly into light (e.g., GaAs semiconductor laser).
• Metastable State: An intermediate state with a longer lifetime ($10^{-6}$ to $10^{-3}\,s$) compared to the excited state ($10^{-9}\,s$). This helps achieve population inversion.
• Optical Resonator/Cavity: Consists of two parallel mirrors (one 100% reflective, one semi-transparent). It provides positive feedback and selects the direction and frequency of light.
  • Resonance condition: $L = n \times \frac{\lambda}{2}$, or $\lambda = \frac{2L}{n}$.

Carbon Dioxide (CO2) Laser

• Background: Developed by Prof. C.K.N. Patel in 1963. It is a four-level molecular gas laser.
• Output: Operates at $10.6\,\mu m$ (primary) and $9.6\,\mu m$ in the far infrared (IR) region. Efficiency is up to 30%.
• Vibrational Modes of CO2:
  • Symmetric Stretching (100): Oxygen atoms oscillate along the axis relative to the stationary Carbon atom. Intermediate energy.
  • Asymmetric Stretching (001): Oxygen atoms move in one direction while Carbon moves in the other. Highest energy level.
  • Bending Mode (010, 020): Atoms oscillate normal to the molecular axis. State (010) has the least energy.
• Construction:
  • Quartz discharge tube (5 m long, 2.5 cm diameter).
  • Gaseous mixture: $CO_2:N_2:He$ in a ratio of $1:2:3$ at 6–17 torr pressure.
  • Equipped with NaCl Brewster windows for polarization and external mirrors.
• Working Mechanism:
  • Electric discharge excites $N_2$ molecules to a metastable vibrational state ($v=1$) through collisions with electrons: $e_1 + N_2 \rightarrow e_2 + N_2^*$.
  • $N_2^*$ transfers energy to $CO_2$ via resonance: $N_2^* + CO_2 \rightarrow N_2 + CO_2^*$.
  • $CO_2$ is raised to the (001) state ($E_5$).
  • Transitions from $E_5$ to (100) ($E_4$) emit $10.6\,\mu m$ photons.
  • Transitions from $E_5$ to (020) ($E_3$) emit $9.6\,\mu m$ photons.
  • Helium serves to cool the mixture and helps depopulate lower energy levels through collisions, maintaining population inversion.

Medical Applications of Lasers

• Ophthalmology (Eye Surgery):
  • LASIK: Reshapes the cornea to correct myopia, hyperopia, or astigmatism.
  • Cataract Treatment: Use of femtosecond lasers for photodisruption and removal of the lens.
  • Retinal Coagulation: Treating diabetic retinopathy and macular edema using pulsed diode lasers.
• Dermatology (Skin Surgery):
  • Ablative Resurfacing: The laser destroys the epidermis while heating the dermis to stimulate collagen production.
  • CO2 Lasers: Used for wrinkles, scars, and warts. Fractionated CO2 uses very short pulses (ultrapulse).
  • Erbium Lasers: Suited for surface-level wrinkles and darker skin tones; has fewer side effects and faster recovery (typically one week compared to two for CO2).

Optical Fibers: Structure and Principles

• Physical Structure:
  • Core: Inner cylindrical layer made of glass or plastic.
  • Cladding: Surrounds the core; made of material with a lower refractive index ($n_2 < n_1$).
  • Sheath/Jacket: Polyurethane layer protecting the fiber from chemical or mechanical damage.
• Propagation Mechanism: Based on Total Internal Reflection (TIR). Light entering at an angle greater than the critical angle remains confined to the core.
• Numerical Aperture (NA): Measures light-gathering ability.
  • $NA = \sin(\theta_0) = \sqrt{n_1^2 - n_2^2}$ (where $n_0 = 1$ for air).
• Fractional Index Change ($\Delta$):
  • $\Delta = \frac{n_1 - n_2}{n_1}$
  • Relation to NA: $NA = n_1 \sqrt{2\Delta}$.
• Modes of Propagation (V-number):
  • $V = \frac{\pi d}{\lambda} \sqrt{n_1^2 - n_2^2}$
  • For step-index fibers, number of modes $\approx \frac{V^2}{2}$.
  • For graded-index fibers, number of modes $\approx \frac{V^2}{4}$.

Types of Optical Fibers

• Step Index Single Mode Fiber (SMF):
  • Uniform core RI with a sudden step at the cladding interface.
  • Narrow core (8–10 $\mu m$), cladding (60–70 $\mu m$).
  • Supports only one mode; zero intermodal dispersion. Ideal for submarine cables.
• Step Index Multimode Fiber (MMF):
  • Larger core diameter supporting many modes.
  • Maximum intermodal dispersion; used for low bandwidth, short-distance data links.
• Graded Index Multimode Fiber (GRIN):
  • Core RI varies radially (highest at axis, decreasing toward cladding).
  • Rays follow sinusoidal paths. Differential speeds result in almost equal travel time for modes, minimizing intermodal dispersion.

Attenuation and Losses in Optical Fibers

• Defined as the energy loss per unit length, expressed as:
  • $\alpha = -\frac{10}{L} \log_{10}\left(\frac{P_{out}}{P_{in}}\right) \text{dB/km}$
• Absorption Loss: Caused by impurities (transition metals like iron, copper) or hydroxyl ($OH^-$) ions, and intrinsic material absorption.
• Scattering loss: Rayleigh scattering occurs due to molecular-sized imperfections. Proportional to $\frac{1}{\lambda^4}$.
• Bending losses:
  • Macroscopic: Caused by wrapping fiber on spools or turning corners.
  • Microscopic: Repetitive small-scale fluctuations in fiber axis linearity due to manufacturing stresses.
• Coupling losses: Occur at fiber junctions due to misalignment or air gaps.

Optical Fiber Communication System

• Components:
  • Transmitter: LED or semiconductor laser (more efficient due to monochromaticity).
  • Channel: Optical fiber.
  • Receiver: Photodiode (reverse-biased junction) and decoder.
  • Repeater: Used at regular intervals to amplify weak signals and correct delay distortion.
• Advantages:
  • Bandwidth up to $10^{14}\,bps$.
  • Immunity to Electromagnetic Interference (EMI).
  • Low loss ($0.1$ to $0.5\,dB/km$).
  • Security (no signal radiation) and absence of electrical hazards.

Fiber Optic Sensors

• Used as transducers to measure pressure, temperature, strain, etc.
• Intensity Modulated Temperature Sensor: Uses a silicon layer at the fiber tip. The light absorbed varies with temperature, changing the reflected light intensity.
• Phase Modulated Temperature Sensor: Uses a Mach-Zehnder arrangement.
  • Light is split into a sensing fiber and a reference fiber.
  • Heating the sensing fiber changes its refractive index, creating a phase difference.
  • The resulting interference fringe displacement is measured to determine temperature or pressure.

Questions & Discussion

1. Define the terms: Spontaneous emission, Stimulated Emission, Active medium, Population inversion.
2. List the characteristic properties of laser.
3. Give any two differences between the laser light and ordinary light.
4. Explain the requisites of a laser system.
5. With the energy level diagram explain the construction and working of $CO_2$ laser.
6. Discuss the conditions required for laser action.
7. Discuss the application of laser in eye surgery.
8. Explain the three processes which take place when radiation interacts with matter.
9. Explain the terms stimulated emission and population inversion. Obtain an expression for energy density of photons in terms of Einstein’s co-efficient.