Applied Physics for Engineers: Lasers, Fibre Optics, and Quantum Physics

LASERS AND FIBRE OPTICS

Introduction and Characteristics

Laser light is distinguished from ordinary light by four unique characteristics:

• High Monochromaticity: The light consists of a single wavelength or color.

• High Coherence: All photons are in phase with one another.

• High Directionality: The beam undergoes minimal divergence over long distances.

• Sharp Focus: The beam can be focused into extremely small spots.

Interaction of Radiation with Matter

There are three fundamental ways radiation interacts with atomic systems:

1. Absorption

• An atom in the lower energy state $E_1$ absorbs a photon of frequency $f$ where $E_2 - E_1 = hf$.

• The photon disappears, and the system transitions to the upper energy state $E_2$.

2. Spontaneous Emission

• An excited atom in state $E_2$ transits to state $E_1$ without any external influence.

• The average lifetime of an excited state is approximately $10^{-8}\,s$.

• The emitted photon has energy $hf = E_2 - E_1$ but possesses random phase and direction, resulting in incoherent radiation.

3. Stimulated Emission

• An external stimulating photon of frequency $f = (E_2 - E_1)/h$ interacts with an excited atom.

• The atom descends to the ground state before its lifetime ends, emitting a second photon.

• Properties: Both the stimulating and stimulated photons have the same frequency, same phase, same state of polarization, and same direction. This process is the basis for laser action.

Population Inversion

• Boltzmann Statistics: Under thermal equilibrium at temperature $T$, the ratio of population densities is   $\frac{N_2}{N_1} = e^{-(E_2 - E_1)/kT}$

• Normally, N_2 < N_1 (Normal Population).

• For stimulated emission to exceed absorption, a non-equilibrium state is required where N_2 > N_1 (Population Inversion).

• This is facilitated by Metastable States, which have a relatively long lifetime ($10^{-3}\,s$) compared to ordinary excited states ($10^{-8}\,s$).

Einstein’s Coefficients

Einstein coefficients quantify the probabilities of radiation processes. Let $I_f$ be the energy density of frequency $f$.

Mathematical Formulation

1. Rate of Induced Absorption: $B_{12} N_1 I_f$

2. Rate of Spontaneous Emission: $A_{21} N_2$

3. Rate of Stimulated Emission: $B_{21} N_2 I_f$

At thermal equilibrium: $\text{Rate of Absorption} = \text{Rate of Spontaneous Emission} + \text{Rate of Stimulated Emission}$ $B_{12} N_1 I_f = A_{21} N_2 + B_{21} N_2 I_f$

Solving for energy density $I_f$ and comparing with Planck's Law ($I_f = \frac{8\pi hf^3}{c^3} \frac{1}{e^{hf/kT}-1}$), we find:

• $B_{12} = B_{21}$ (Probabilities of induced absorption and stimulated emission are equal).

• $\frac{A_{21}}{B_{21}} = \frac{8\pi hf^3}{c^3}$

Physical Interpretations

• $\frac{A_{21}}{B_{21}} \propto f^3$: Spontaneous emission dominates at high frequencies (e.g., visible light).

• Case 1 (hf >> kT): Spontaneous emission is significantly more likely than stimulated emission.

• Case 2 (hf << kT): Stimulated emission dominates (observed in microwave transitions at room temperature).

Construction and Components of Laser Systems

Every laser requires three essential elements:

1. Active Medium: Material (gas, liquid, crystal, semiconductor) with special energy levels for lasing.

2. Pumping System: Energy source to excite atoms and achieve population inversion.

3. Resonant Cavity: A pair of mirrors (one fully reflective, one partially reflective) to provide optical feedback and amplify light.

Specific Laser Types

• He-Ne Laser:

  • Medium: 80% Helium, 20% Neon gas mixture.

  • Process: Electrons collide with He atoms, exciting them to a metastable state ($E_3$). Energy is transferred to Ne atoms via resonant energy transfer ($E_2$). Rapid decay from $E_1$ to $E_0$ maintains inversion.

  • Output: Red laser light ($632.8\,nm$).

• Ruby Laser:

  • Medium: $Al_2O_3$ crystal doped with $Cr^{3+}$ ions.

  • Operation: Three-level system. Chromium ions are active centers. Optical pumping transitions them to $E_3$, followed by non-radiative decay to metastable state $E_2$. Stimulated emission occurs between $E_2$ and $E_1$.

  • Output: Intense pulses of coherent light ($694.4\,nm$).

• Semiconductor (Injection) Laser:

  • Medium: Heavily doped p-n junction (e.g., GaAs).

  • Principle: Forward bias and high current density ($\approx 20\,kA/cm^2$) inject charge carriers into the depletion region, causing population inversion and radiative recombination.

Applications of Lasers

• Bar Code Scanner: Uses a narrow, directional beam to scan Universal Product Code (UPC). Black stripes absorb light; white spaces reflect. A photodiode/CCD converts light variations into electrical signals.

• Laser Printer (Electrophotography):

  1. Charging: The drum is given a uniform negative charge.

  2. Exposure: Laser writing creates an electrostatic latent image.

  3. Developing: Negatively charged toner sticks to laser-exposed areas.

  4. Transfer: Toner moves to positively charged paper.

  5. Fusing: Heat and pressure fix the toner.

• Laser Cooling: Dissipative light forces reduce particle motion (momentum $p = h/\lambda$). Methods include Doppler Cooling and Sisyphus Cooling, achieving temperatures in the nano-kelvin range.

Optical Fibres

Optical fibres are thin strands of dielectric material (glass or plastic) that guide light using Total Internal Reflection (TIR).

Structure and Propagation

• Core: High refractive index $n_1$.

• Cladding: Slightly lower index $n_2$ (n_1 > n_2).

• Protective Jacket: Polyurethane coating.

• Acceptance Angle ($\theta_0$): The maximum angle at which light can enter to undergo TIR.

• Numerical Aperture (NA): Light gathering power.     $NA = n_0 \sin(\theta_0) = \sqrt{n_1^2 - n_2^2}$

• Fractional Refractive Index Change ($\Delta$):     $\Delta = \frac{n_1 - n_2}{n_1} \implies NA \approx n_1 \sqrt{2\Delta}$

• Skip Distance ($L_s$): Distance between two successive reflections within the fibre.     $L_s = d \cot(\theta_1)$

Types of Fibres

1. Single-mode Step Index: Narrow core ($5-10\,\mu m$); supports only one propagation path.

2. Multi-mode Step Index: Larger core ($50-200\,\mu m$); supports many modes but suffers from intermodal dispersion.

3. Graded-Index (GRIN) Multimode: Refractive index decreases parabolically from the center. Higher signal quality due to reduced intermodal dispersion.

Attenuation and Distortion

• Loss Sources: Absorption (transition metals, hydroxyl ions), Scattering (Rayleigh scattering $\propto 1/\lambda^4$), and Bending losses (microscopic and macroscopic).

• Distortion Types:

  • Material Dispersion: Speed of light depends on wavelength ($\lambda$).

  • Waveguide Dispersion: Light travels at different angles within the same mode.

  • Intermodal Dispersion: Different paths (modes) have different travel times (absent in single-mode fibres).

QUANTUM PHYSICS

Blackbody Radiation

A blackbody absorbs all incident radiation. To maintain thermal equilibrium, it radiates energy according to its temperature.

• Stefan’s Law: $P = \sigma A e T^4$ where $\sigma = 5.670 \times 10^{-8}\,W/m^2 K^4$.

• Wien’s Displacement Law: $\lambda_m T = 2.898 \times 10^{-3}\,m \cdot K$.

• Rayleigh-Jeans Law: Predicted intensity $I \propto \lambda^{-4}$. At short wavelengths, it predicted infinite energy, known as the Ultraviolet Catastrophe.

• Planck’s Hypothesis:

  1. Oscillators have quantized energy: $E_n = n h f$.

  2. Transitions emit/absorb energy in discrete packets ($hf$).     $I(\lambda, T) = \frac{2\pi hc^2}{\lambda^5} \frac{1}{\exp(hc/\lambda kT) - 1}$

Photoelectric Effect

Ejection of electrons when light hits a metal surface.

• Einstein's Interpretation: Electromagnetic waves are made of photons with energy $E = hf$.

• Equation: $K_{max} = hf - \phi$ where $\phi$ is the work function.

• Key Discoveries: Emission depends on frequency, not intensity; emission is instantaneous; existence of a cutoff frequency ($f_c = \phi/h$).

Compton Effect

The shift in wavelength when X-rays scatter from free electrons.

• Shift Formula: $\lambda' - \lambda = \frac{h}{mc} (1 - \cos(\theta))$

• The term $\frac{h}{mc}$ is the Compton wavelength ($\approx 2.43\,pm$ for electrons).

• This confirms the particle nature (momentum) of light photons.

Wave Properties of Particles

• De Broglie Hypothesis: Matter exhibits wave-like properties.     $\lambda = \frac{h}{p} = \frac{h}{mv}$

• Davisson-Germer Experiment: Confirmed wave nature by diffracting electrons through a nickel crystal. A peak was observed at $54\,V$ and and angle of $50^{\circ}$, yielding $\lambda \approx 0.165\,nm$, matching theoretical predictions.

The Quantum Particle

• Wave Packet: A localized entity formed by the superposition of multiple waves of slightly different frequencies.

• Phase Speed ($v_p$): Speed of individual wave crests: $v_p = \omega / k$.

• Group Speed ($v_g$): Speed of the wave packet (envelope). It is equal to the particle speed $u$.     $v_g = \frac{d\omega}{dk} = u$

Heisenberg Uncertainty Principle

It is impossible to simultaneously measure position ($x$) and momentum ($p$) with infinite precision. $\Delta x \cdot \Delta p_x \ge \frac{h}{4\pi}$ $\Delta E \cdot \Delta t \ge \frac{h}{4\pi}$

QUANTUM MECHANICS

The Wave Function ($\Psi$)

• Contains all possible information about a system.

• Born Interpretation: $|\Psi|^2$ is the probability density (probability per unit volume of finding a particle).

• Normalization: $\int_{-\infty}^{+\infty} |\psi|^2 dx = 1$.

• Expectation Value: Average position: $\langle x \rangle = \int_{-\infty}^{+\infty} \psi^* x \psi dx$.

Schrödinger Equation

• Time-Dependent: $\text{i}\hbar \frac{d\Psi}{dt} = H\Psi$

• Momemtum Operator: $p \to -\text{i}\hbar \frac{d}{dx}$

• Hamiltonian (Energy Operator): $H = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + U(x)$

• Time-Independent: Solutions for stable potentials where $\Psi(x,t) = \psi(x) e^{-\text{i}Et/\hbar}$.     $\frac{-\hbar^2}{2m} \frac{d^2\psi}{dx^2} + U(x)\psi = E\psi$

Applications of Schrödinger Equation

1. Particle in a Box (Infinite Potential Well)

• Potentials: $U = 0$ for 0 < x < L, and $U = \infty$ elsewhere.

• Energy Levels: $E_n = \frac{n^2 h^2}{8mL^2}$ for $n = 1, 2, 3...$

• Wave Function: $\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)$.

• Zero-point Energy: Energy at $n=1$ ($E_1 \neq 0$).

2. Quantum Tunneling

• When total energy $E$ is less than barrier height $U$, the wave function decays exponentially but remains non-zero on the far side.

• Transmission Coefficient: $T \approx e^{-2CL}$, where $C = \frac{\sqrt{2m(U-E)}}{\hbar}$.

3. Quantum Harmonic Oscillator (QHO)

• Potential: $U(x) = \frac{1}{2} m \omega^2 x^2$.

• Energy: $E_n = (n + 1/2) \hbar \omega$ for $n = 0, 1, 2...$

• Zero-point Energy: $E_0 = \frac{1}{2} \hbar \omega$. Even at the lowest state, the particle oscillates.

INTRODUCTION TO QUANTUM COMPUTING

Moore's Law and Its Potential End

• Moore's Law: Transistor density on integrated circuits doubles roughly every two years.

• Physical Limitations: Miniaturization leads to electrons tunneling through narrow channels (Heisenberg Uncertainty), causing functional failure.

• Computational Bottlenecks: Difficulties in simulating molecular interactions, optimizing supply chains, and breaking RSA-style cryptography.

The Quantum Bit (Qubit)

• Analogous to a classical bit (0 or 1).

• Dirac Notation: States are denoted as kets ($|0\rangle$, $|1\rangle$).

• Superposition: A qubit can exist in a linear combination of states:   $|\psi\rangle = \alpha |0\rangle + \beta |1\rangle$

• Measurement: Observing a qubit collapses it to $|0\rangle$ (prob $|\alpha|^2$) or $|1\rangle$ (prob $|\beta|^2$), where $|\alpha|^2 + |\beta|^2 = 1$.

Mathematical Formalism

• Hilbert Space: A complex linear vector space with an inner product.

• Ket ($|v\rangle$): Column matrix.

• Bra ($\langle v|$): Row matrix, Hermitian conjugate of the ket ($\langle v| = (|v\rangle)^{\dagger}$).

• Inner Product ($\langle w|v\rangle$): Produces a complex number.

• Operators: Square matrices ($n \times n$).

  • Heritage: $A = A^{\dagger}$ (Eigenvalues are real).

  • Unitary: $U^{\dagger} U = I$ (Preserves probability/norms). Quantum gates must be unitary.

Pauli Matrices

• X gate: $[0, 1; 1, 0]$ (Bit-flip / NOT).

• Y gate: $[0, -i; i, 0]$.

• Z gate: $[1, 0; 0, -1]$ (Phase-flip).

• Hadamard (H): $1/\sqrt{2} [1, 1; 1, -1]$ (Creates superposition).

Postulates of Quantum Mechanics

1. The state of a system is specified by a state vector $|\psi\rangle$ in Hilbert space.

2. Observables are represented by Hermitian operators.

3. Measurements yield eigenvalues of the operators.

4. The probability of a result is given by the square of the amplitude.

5. Evolution of a closed system is unitary ($|\psi'\rangle = U|\psi\rangle$).

Multiple Qubits and Entanglement

• Multi-qubit state: Needs $2^n$ amplitudes (e.g., $|00\rangle, |01\rangle, |10\rangle, |11\rangle$).

• Entanglement (Bell States): States like $\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$. Measurement of the first qubit immediately determines the state of the second.

• CNOT Gate: Flips the target bit if the control bit is 1.

• Toffoli Gate: CCNOT; flips the target bit if both control bits are 1.

Qubit Realizations

• Photon Polarization: Horizontal/Vertical states.

• Trapped Ions: Energy states of charged atoms.

• Superconducting Loops: Resistance-free current oscillations (Google, IBM).

• Silicon Quantum Dots: Electron spin within silicon chips (Intel).

• Diamond Vacancies: Nitrogen-vacancy centers in a diamond lattice.

DEMONSTRATION EXPERIMENTS

Numerical Aperture of Optical Fiber

• Aim: Measure acceptance angle $\theta_0$ and NA.

• Method: Laser light emerging from a cable forms a cone. The diameter $D$ of the spot is measured at distance $L$.

• Calculation: $NA = \sin[\tan^{-1}(D/2L)]$.

Photoelectric Effect (Planck's Constant Determination)

• Aim: Determine Planck's constant ($h$) and work function ($\phi$).

• Method: Measure stopping potential ($V_o$) for different frequencies ($f$).

• Calculation: Plot $V_o$ vs $f$. Slope = $h/e$. The $f$-intercept is threshold frequency $f_o$. $\phi = hf_o$.

Uncertainty Principle using Single Slit

• Aim: Verify $\Delta x \cdot \Delta p \ge h / 4\pi$.

• Method: Slit width $a = \Delta x$. Diffraction spread on screen yields angular spread $\theta$, leading to $\Delta p = (h/\lambda) \sin(\theta)$.

• Result: The product exhibits consistency with the order of magnitude of $h$.

Quantum Circuit Simulation (Quirk-E)

• X Gate: Simulates bit-flip (Prob $|0\rangle \to 100\% |1\rangle$).

• H Gate: Creates equal superposition (50/50 probability).

• Bell State: Combines H and CNOT gates to show correlated measurement outcomes in two qubits.

• Interference: Sequence $H \to X \to H$ returns a superposition state to a definite deterministic state through quantum interference.