Lasers, fibre optics and modern physics applications notes — Unit 5

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

⇒Characteristics of Laser Light:

• Highly Monochromatic: Emits light of a single, highly specific wavelength.
• Highly Coherent: All photons are exactly in phase with each other.
• Highly Directional: Emits a narrow beam that spreads very little over long distances.
• Extremely Intense: Massive energy is concentrated in a tiny area.

According to quantum mechanics, atoms exist in discrete energy levels (e.g., Ground state $E_1$, Excited state $E_2$). When electromagnetic radiation interacts with matter, three fundamental processes can occur:

• 1. Induced (Stimulated) Absorption
• 2. Spontaneous Emission
• 3. Stimulated Emission

All three processes involve the absorption or emission of a photon with energy $E = h\nu = E_2 - E_1$.

An atom is initially in the lower energy state ($E_1$). An incident photon of exact energy $h\nu = E_2 - E_1$ strikes the atom. The atom absorbs the photon and transitions to the excited state ($E_2$).

The rate of absorption is proportional to the number of atoms in $E_1$ (let's call this $N_1$) and the energy density of the incident radiation ($\rho(\nu)$).

Rate of Absorption = B₁₂ N₁ ρ(ν)

Where $B_{12}$ is Einstein's coefficient of induced absorption.

An atom in the excited state ($E_2$) is inherently unstable. It can only stay there for a very short lifetime (typically $10^{-8}$ seconds).

Without any external trigger, the atom spontaneously drops back down to $E_1$, emitting a photon of energy $h\nu$ in a completely random direction and random phase.

The rate of spontaneous emission depends ONLY on the number of atoms currently in the excited state ($N_2$).

Rate of Spontaneous Emission = A₂₁ N₂

Where $A_{21}$ is Einstein's coefficient of spontaneous emission. This is the source of light in ordinary bulbs.

This is the magic behind the laser. If an atom is already in the excited state ($E_2$), an incident photon of energy $h\nu$ can force (stimulate) the atom to drop to $E_1$ before its natural lifetime expires.

The atom emits a photon, and the original incident photon is also released. Result: One photon in, Two photons out.

Rate of Stimulated Emission = B₂₁ N₂ ρ(ν)

Where $B_{21}$ is Einstein's coefficient of stimulated emission.

Under thermal equilibrium, the rate of upward transitions (absorption) must equal the rate of downward transitions (spontaneous + stimulated emission).

B₁₂ N₁ ρ(ν) = A₂₁ N₂ + B₂₁ N₂ ρ(ν)

Solving for $\rho(\nu)$:

ρ(ν) = A₂₁ / [ (N₁/N₂)B₁₂ - B₂₁ ]

From Boltzmann statistics, $N_1/N_2 = e^{h\nu/kT}$. Comparing this equation with Planck's radiation law, Einstein proved:

1. B₁₂ = B₂₁ (Probabilities of induced absorption & stimulated emission are equal)
2. A₂₁ / B₂₁ = (8πhν³) / c³

Usually, $N_1 \gg N_2$ (more atoms in the ground state than excited state). In this condition, absorption dominates over stimulated emission, so light is attenuated, not amplified.

To achieve Light Amplification, we must force a situation where $N_2 > N_1$. This unnatural state is called Population Inversion.

⇒Metastable States

Population inversion is impossible without a Metastable State. This is a special excited state where atoms live much longer than usual ($\sim 10^{-3}$ seconds instead of $10^{-8}$ seconds). This "traffic jam" allows atoms to accumulate in the excited state, achieving $N_2 > N_1$.

Every laser system requires three fundamental components:

• 1. Active Medium: The material (gas, liquid, crystal, or semiconductor) containing the atoms that will undergo population inversion and emit the laser light.
• 2. Pumping Source: The external energy source used to excite atoms from $E_1$ to $E_2$ to achieve population inversion (e.g., optical flash tubes, electrical discharge).
• 3. Optical Resonator (Cavity): A pair of parallel mirrors (one 100% reflective, one 99% reflective) placed at the ends of the active medium. It bounces photons back and forth to trigger an avalanche of stimulated emissions.

The first working laser, built by Theodore Maiman in 1960. It is a three-level solid-state laser.

• Active Medium: A synthetic ruby crystal ($Al_2O_3$ doped with 0.05% $Cr^{3+}$ ions). The Chromium ions are the active centers.
• Pumping: Optical pumping using a helical Xenon flash lamp wrapped around the ruby rod.
• Output: Emits pulses of deep red light at wavelength 694.3 nm.

Because it is a 3-level system, it requires massive pumping energy to achieve population inversion, making it inefficient and limited to pulsed (non-continuous) output.

A highly successful four-level gas laser that produces a continuous wave (CW) output.

• Active Medium: A mixture of Helium and Neon gases in a 10:1 ratio inside a quartz tube.
• Pumping: Electrical discharge (high voltage).
• Output: Continuous red light at 632.8 nm.

⇒Why the mixture?

Neon is the actual active atom that emits the laser light. Helium acts as a "pumping assistant." Helium is easily excited by electrons, and its excited energy levels perfectly match the metastable states of Neon. Helium transfers its energy to Neon via resonant collisions, easily achieving population inversion in Neon.

The most common laser in the world (used in barcode scanners, fiber optics, CD players). It uses a heavily doped p-n junction (typically Gallium Arsenide - GaAs).

• Pumping: Direct electrical pumping (forward biasing the diode).
• Mechanism: Forward bias pushes electrons and holes into the junction (active region). Population inversion is achieved between the conduction band (electrons) and valence band (holes).
• Emission: When electrons recombine with holes, they emit stimulated photons. The polished ends of the crystal act as the optical resonator.

Advantages: Extremely small, highly efficient, and easily modulated (turned on/off rapidly for communication).

An optical fibre is a thin, flexible, transparent strand of very pure glass (silica) or plastic that acts as a waveguide, transmitting light between the two ends.

⇒Principle: Total Internal Reflection (TIR)

Light travels inside the fibre by bouncing repeatedly off the inner walls. This is governed by TIR, which occurs when:

• 1. Light travels from a denser medium to a rarer medium.
• 2. The angle of incidence is greater than the critical angle ($\theta_c = \sin^{-1}(n_{rare}/n_{dense})$).

A standard optical fibre consists of three concentric cylindrical layers:

• 1. Core: The innermost solid cylinder where light travels. It is made of high-quality silica with a high refractive index ($n_1$).
• 2. Cladding: The outer layer surrounding the core. It is made of silica with a slightly lower refractive index ($n_2 < n_1$). The core-cladding boundary is where TIR occurs.
• 3. Buffer/Jacket: An outer plastic coating that protects the fragile glass from moisture, physical damage, and micro-bending.

Light must be injected into the core at a specific angle to ensure it undergoes TIR at the core-cladding boundary. If injected too steeply, it will escape into the cladding.

The Acceptance Angle is the maximum angle (with respect to the fibre axis) at which light can enter the fibre and propagate through it via TIR.

Rotating this angle 360 degrees around the axis forms the Acceptance Cone. Only light rays entering within this cone are "accepted" and transmitted.

Numerical Aperture is a dimensionless parameter that represents the light-gathering capability of an optical fibre. It is simply the sine of the acceptance angle.

NA = sin(θ_a)
Using Snell's law at the air-core boundary and TIR conditions:
NA = √(n₁² - n₂²)

Where:
n₁ = refractive index of core
n₂ = refractive index of cladding

A larger NA means the fibre can gather more light, but it also increases signal distortion (dispersion) over long distances.

The fractional change in refractive index between the core and cladding is denoted by $\Delta$.

Δ = (n₁ - n₂) / n₁

Since $n_1 \approx n_2$, we can approximate $n_1 + n_2 \approx 2n_1$. This gives a very useful relation between NA and $\Delta$:

NA = √(n₁² - n₂²) = √[(n₁ - n₂)(n₁ + n₂)]
NA ≈ √[ (Δ·n₁) (2n₁) ]
NA ≈ n₁ √(2Δ)

⇒1. Step-Index Fibre

The refractive index is uniform throughout the core ($n_1$) and drops abruptly (in a "step") to $n_2$ at the cladding boundary. Light travels in zig-zag straight lines.

⇒2. Graded-Index Fibre (GRIN)

The refractive index of the core is maximum at the center and decreases gradually in a parabolic manner towards the cladding. Instead of sharp zig-zags, light travels in smooth sinusoidal curves. This drastically reduces intermodal dispersion (pulse spreading).

A "mode" is a specific path that a light ray can take through the fibre.

• Single-Mode Fibre (SMF): Has a very thin core (8-10 $\mu$m). It allows only one mode (path) of light to propagate (straight down the center). Zero intermodal dispersion, incredibly high bandwidth. Used for long-distance telecommunications.
• Multi-Mode Fibre (MMF): Has a thicker core (50-100 $\mu$m). Allows hundreds of paths. Suffers from dispersion (pulses smear together over distance). Used for short-distance LANs.

V = (2πa / λ) × NA
(Where 'a' is core radius)
If V < 2.405, the fibre is Single-Mode.

As light travels, it loses intensity. This attenuation is measured in dB/km.

Attenuation (α) = (10/L) * log₁₀(P_in / P_out) dB/km

⇒Causes of Loss:

• Absorption: By the silica glass itself, or by impurities like OH⁻ (water) ions.
• Rayleigh Scattering: Microscopic density fluctuations scatter light in all directions (loss $\propto 1/\lambda^4$).
• Bending Losses: Macrobending (sharp turns in the cable) and Microbending (tiny bumps on the core surface) cause light to strike the boundary below the critical angle and escape.

⇒Unit 5 Mastery

• Explain the difference between spontaneous and stimulated emission.
• Derive the relation between Einstein's A and B coefficients.
• What is population inversion and why is a metastable state required?
• Explain the construction and working of Ruby and He-Ne lasers.
• Derive the formula for Acceptance Angle and Numerical Aperture.
• Distinguish between Step-Index and Graded-Index fibres.
• What are the main causes of attenuation in optical fibres?