Unit 5: Lasers and MASERs

1. Fundamentals of Lasers

What is a LASER?

The acronym LASER stands for Light Amplification by Stimulated Emission of Radiation. A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation.

Definition: A Laser is a device that produces an intense, highly monochromatic, coherent, and highly directional beam of light via stimulated emission of radiation.

Key Characteristics of Laser Light

• Monochromaticity: Laser light consists of a single wavelength or an extremely narrow band of wavelengths.
• Coherence: The emitted light waves share a constant phase relationship in both space (spatial coherence) and time (temporal coherence).
• Directionality: The beam has exceptionally low divergence and travels as a parallel beam over long distances.
• High Intensity: Laser light concentrates massive energy into a very small cross-sectional area.

2. Interaction of Radiation with Matter

When electromagnetic radiation interacts with an atomic system possessing energy levels E₁ (ground state) and E₂ (excited state), where E₂ > E₁, three primary processes occur:

1. Induced / Stimulated Absorption

An atom in the lower energy state E₁ absorbs an incident photon of frequency ν = (E₂ - E₁) / h and jumps to the higher energy state E₂.

Rate of Absorption: R₁₂ = B₁₂ × N₁ × u(ν)

Where N₁ is the number of atoms per unit volume in state E₁, u(ν) is the energy density of incident radiation, and B₁₂ is Einstein's coefficient of stimulated absorption.

2. Spontaneous Emission

An atom in the excited state E₂ naturally transitions to the ground state E₁ without external influence after its natural lifetime (~10^-8 seconds), emitting a photon of frequency ν = (E₂ - E₁) / h. Photons emitted spontaneously have random phases and directions.

Rate of Spontaneous Emission: R_21(spont) = A₂₁ × N₂

Where N₂ is the population of state E₂, and A₂₁ is Einstein's coefficient of spontaneous emission.

3. Stimulated Emission

An incident photon of energy hν = E₂ - E₁ interacts with an excited atom at level E₂, triggering it to return to level E₁ immediately. This process releases a second photon that is identical in frequency, phase, direction, and polarization to the triggering photon.

Rate of Stimulated Emission: R_21(stim) = B₂₁ × N₂ × u(ν)

Where B₂₁ is Einstein's coefficient of stimulated emission.

Comparison: Spontaneous vs. Stimulated Emission

3. Population Inversion, Optical Pumping, and Metastable States

Thermal Equilibrium vs. Population Inversion

According to Boltzmann's distribution law, at thermal equilibrium temperature T:

N₂ = N₁ × e^{-(E₂ - E₁) / (k × T)}

Since E₂ > E₁, N₁ > N₂ always holds under normal thermal equilibrium. This means absorption dominates over stimulated emission.

Population Inversion: A non-equilibrium state of matter in which the population of an upper energy level (N₂) exceeds the population of a lower energy level (N₁), i.e., N₂ > N₁.

Metastable States

Normally, excited atomic states have very short lifetimes (~10^-8 s). Under these conditions, atoms return to the ground state before a sufficient population can accumulate. A metastable state is an excited state with a significantly longer lifetime (~10^-3 s to 10^-6 s).

• Metastable states allow atoms to accumulate in the upper level.
• They serve as a storage reservoir of energy needed to establish N₂ > N₁.

Optical Pumping and Other Pumping Mechanisms

Pumping is the external process of supplying energy to excite atoms from a lower state to a higher state to achieve population inversion.

• Optical Pumping: Uses light from external sources (e.g., xenon flash lamps, secondary lasers). Used in solid-state lasers like Ruby lasers.
• Electric Discharge: Uses electric fields to accelerate electrons, which collide with gas atoms and excite them. Used in gas lasers like He-Ne.
• Direct Current Injection: Recombination of charge carriers across a forward-biased p-n junction. Used in semiconductor lasers.
• Chemical Reactions: Energy released in exothermic chemical reactions pumps the system.

4. Three-Level and Four-Level Laser Systems

Three-Level Laser System

Consists of Ground State E₁, Metastable State E₂, and Upper Pump Level E₃.

• Atoms are pumped from E₁ to E₃ via optical pumping.
• Atoms quickly decay non-radiatively from E₃ to the metastable state E₂ (~10^-8 s).
• Population inversion is built between E₂ and the ground state E₁.
• Laser action occurs between E₂ → E₁.

Drawback: More than 50% of the total ground-state population must be pumped to level E₂ to achieve N₂ > N₁, requiring very high pump energy.

Four-Level Laser System

Consists of Ground State E₁, Lower Laser Level E₂, Upper Metastable State E₃, and Pump Level E₄.

• Atoms are pumped from E₁ to E₄.
• Rapid non-radiative transition transfers atoms from E₄ to metastable state E₃.
• Lasing transition occurs between E₃ and lower laser level E₂.
• Atoms at level E₂ quickly decay non-radiatively to E₁, keeping level E₂ nearly empty.

Advantage: Since level E₂ is naturally empty, population inversion (N₃ > N₂) is achieved with very low pumping energy.

Comparison of 3-Level and 4-Level Lasers

5. Einstein’s Coefficients and Relations

Albert Einstein derived the mathematical conditions for radiative balance in thermal equilibrium using statistical mechanics.

Derivation of Einstein's Relations

At thermal equilibrium, the total rate of upward transitions equals the total rate of downward transitions:

Rate of Absorption = Rate of Spontaneous Emission + Rate of Stimulated Emission

B₁₂ × N₁ × u(ν) = A₂₁ × N₂ + B₂₁ × N₂ × u(ν)

Rearranging for u(ν):

u(ν) × [B₁₂ × N₁ - B₂₁ × N₂] = A₂₁ × N₂

u(ν) = (A₂₁ × N₂) / [B₁₂ × N₁ - B₂₁ × N₂] = (A₂₁ / B₂₁) / [ (B₁₂ / B₂₁) × (N₁ / N₂) - 1 ]

From Boltzmann's law, N₁ / N₂ = e^{hν / (k × T)}. Substituting this gives:

u(ν) = (A₂₁ / B₂₁) / [ (B₁₂ / B₂₁) × e^{hν / (k × T)} - 1 ]

Comparing this with Planck's Radiation Law:

u(ν) = (8π × h × ν³ / c³) / [ e^{hν / (k × T)} - 1 ]

By comparing corresponding coefficients, we obtain the two Einstein Relations:

First Relation: B₁₂ = B₂₁

The probability of stimulated absorption is equal to the probability of stimulated emission.

Second Relation: A₂₁ / B₂₁ = 8π × h × ν³ / c³

Important Observation

The ratio of spontaneous to stimulated emission rate is directly proportional to ν³. Thus, at higher frequencies (such as optical wavelengths), spontaneous emission dominates unless population inversion is artificially forced.

6. Requisites for Producing Laser Light

To produce sustained laser output, three basic components are essential:

1. Active Medium: A collection of atoms, molecules, or ions that can undergo electronic transitions and possess a metastable state to support population inversion.
2. Pumping Source: An external energy supply system (optical, electrical, or chemical) to transfer energy into the active medium and establish N₂ > N₁.
3. Optical Resonator Cavity: A pair of mirrors positioned at opposite ends of the active medium (one totally reflecting mirror and one partially reflecting mirror) providing positive optical feedback and wavelength selection.

7. Laser Rate Equations

Laser rate equations model the time dynamics of atomic population densities and photon density inside the laser cavity.

For a simplified two-level system with optical pumping rate R and spontaneous emission lifetime τ:

dN₂ / dt = R - (N₂ / τ) - B₂₁ × (N₂ - N₁) × u(ν)

dN₁ / dt = -R + (N₂ / τ) + B₂₁ × (N₂ - N₁) × u(ν)

Under steady-state conditions (dN₂ / dt = 0 and dN₁ / dt = 0), these equations determine the threshold pump strength required to sustain laser oscillations.

8. Optical Resonators

An optical resonator supplies positive feedback to amplify the laser beam and selects specific longitudinal modes.

Working Mechanism

• Photons emitted parallel to the cavity axis bounce back and forth between mirrors.
• As photons pass through the population-inverted active medium, they cause exponential stimulated emission, creating a cascade of coherent photons.
• Photons traveling at oblique angles escape through the side walls of the cavity.

Resonant Cavity Condition

For constructive interference inside a Fabry-Pérot cavity of length L:

L = m × (λ / 2)   or   ν = m × c / (2 × L)

Where m is an integer representing the longitudinal mode number, λ is the wavelength in the medium, and c is the speed of light.

9. Specific Laser Systems

1. He-Ne Laser (Gas Laser)

• Type: Four-level gas laser emitting continuous wave (CW) light.
• Active Medium: Mixture of Helium (He) and Neon (Ne) gases in a ratio of approximately 10:1 inside a glass tube at low pressure (~1 mm Hg).
• Pumping Method: Electric discharge.
• Working Mechanism:
  1. Electrons in electric discharge collide with He atoms, exciting them to metastable states (2¹S and 2³S).
  2. excited He atoms transfer energy to ground-state Ne atoms via resonant inelastic collisions.
  3. Ne atoms are excited to higher levels (3s and 2s).
  4. Laser action occurs during transitions of Ne atoms from 3s → 2p (emitting 632.8 nm red light).
  5. Ne atoms decay non-radiatively from 2p to 1s, and then return to the ground state via collision with the cavity walls.

2. Solid-State Laser (e.g., Ruby Laser)

• Type: Three-level solid-state laser emitting pulsed light.
• Active Medium: Synthetic Al₂O₃ (Corundum) crystal doped with ~0.05% Trivalent Chromium ions (Cr³⁺).
• Pumping Method: Optical pumping using a helical Xenon flash lamp.
• Working Output: Emits deep red light at a wavelength of λ = 694.3 nm.

3. Gas Lasers (General Characteristics)

• Use gaseous active media (pure gas, molecular gas, or gas mixtures).
• High spectral purity and spatial coherence.
• Examples: Carbon Dioxide (CO₂) laser (infrared laser with high efficiency), Argon-ion laser, He-Ne laser.

4. Semiconductor Lasers (Diode Laser)

• Type: Compact solid-state semiconductor p-n junction diode.
• Active Medium: Heavily doped direct bandgap semiconductors (e.g., Gallium Arsenide - GaAs, AlGaAs).
• Pumping Method: Direct electrical current bias (Forward Biasing).
• Mechanism: Forward bias forces electrons from the n-region and holes from the p-region into the active junction region. Electron-hole recombination emits coherent photons with wavelength corresponding to the bandgap energy E_g:

λ = h × c / E_g

Summary Comparison of Laser Types

10. Applications of Lasers

• Medical Applications: Eye surgery (LASIK for vision correction, retinal detachment repair), bloodless laser surgery, cancer photodynamic therapy, dermatology.
• Industrial Applications: High-precision cutting, drilling, welding, heat treatment of metals, 3D printing.
• Telecommunications: Carrier signals in optical fiber communications over long distances with minimal losses.
• Defense & Industry Measurements: Laser range finders, target designators, LIDAR (Light Detection and Ranging), alignment systems in construction.
• Scientific Research & Data Storage: Holography (3D image recording), optical disks (CD, DVD, Blu-Ray), interferometry, laser-induced fusion research.

11. Basic Idea of MASER

What is a MASER?

MASER stands for Microwave Amplification by Stimulated Emission of Radiation. Developed prior to the optical laser, a MASER operates on the exact same physical principle of stimulated emission, but operates in the microwave region of the electromagnetic spectrum.

Key Characteristics of MASER

• Emits coherent electromagnetic radiation at microwave frequencies (typically 1 GHz to 300 GHz).
• Uses transitions between atomic or molecular energy levels separated by small energy gaps corresponding to microwave photons.
• Examples: Ammonia MASER (uses inversion transitions of NH₃ molecules) and Hydrogen/Ruby MASER.
• Applications: Highly precise atomic clocks, ultra-low-noise microwave amplifiers in deep-space communications and radio astronomy.