Lasers and Fibre Optics

Objectives

• Explain basic interactions of radiation with matter (absorption, spontaneous emission, stimulated emission)

• Understand basic principle, construction and applications of laser systems

• Comprehend working principle of optical fibres and their applications

• Classify types of optical fibres with merits & demerits

Characteristics of Laser Light

• Highly monochromatic (single wavelength ➔ high spectral purity)

• Highly coherent (fixed phase relationship across space/time ➔ interference & holography)

• Highly directional (very small divergence ➔ long-distance propagation, drilling, targetting)

• Sharply focusable to extremely high power densities (micromachining, surgery)

Interaction of Radiation with Matter

Absorption

• Photon of frequency $f$ is absorbed when E2-E1 = hf

• Atom promoted from ground level E1 to excited level E2; photon disappears

Spontaneous Emission

• Excited atom (average lifetime $\approx 10^{-8}\,\text{s}$) returns to lower level without external trigger

• Emits photon of energy $hf = E<em>2-E</em>1$

• Photons random in phase, direction ➔ ordinary light = incoherent

Stimulated Emission

• Incident “stimulating” photon of proper $f$ forces excited atom to de-excite prematurely

• Two output photons:

  • Same frequency, phase, polarization, direction (perfectly coherent)

  • Basis for laser amplification

Population Inversion

• At thermal equilibrium (Boltzmann):

  

• k is Boltzmann constant,

• N1 is density of atoms with energy

• E1 & N2 is density of atoms with energy E2

• Population inversion: N2>N1 (non-equilibrium); achieved via pumping mechanisms

• Requires metastable state (lifetime $\approx 10^{-3}\,\text{s}$) to “store” atoms in upper level

Einstein Coefficients (A & B)

• $B_{12}$: probability/coeff. of induced absorption

• $A_{21}$: probability/coeff. of spontaneous emission

• $B_{21}$: probability/coeff. of stimulated emission

Rate Equations

• Absorption: $R{abs}=B{12}N1If$

• Spontaneous emission: $R{sp}=A{21}N_2$

• Stimulated emission: $R{st}=B{21}N2If$

Thermal Equilibrium Condition

• $R{abs} = R{sp} + R_{st}$

• Derivation using Planck’s law yields:

  • $\dfrac{A{21}}{B{21}} = \dfrac{8\pi hf^3}{c^3}$

  • $B{12}=B{21}$ (absorption probability equals stimulated-emission probability)

Dependence on Frequency & Temperature

• For $hf\gg kT$ : $A{21}\gg B{21}I_f$ ➔ spontaneous dominates (visible/UV)

• For $hf\approx kT$ : comparable ➔ stimulated significant (IR)

• For $hf\ll kT$ : $B{21}If\gg A_{21}$ ➔ stimulated dominates (microwave; MASER)

Amplification Under Non-Equilibrium

• With population inversion, \dfrac{R{emission}}{R{abs}}=\dfrac{N2}{N1}>1 (Eq. $1.13$)

• Optical cavity feeds back photons, exponentially growing coherent wave ➔ laser output

Essential Laser Components

• Pumping system: energy source (flash-lamp, electrical discharge, current injection)

• Active (lasing) medium: atoms/ions/molecules/semiconductor junction where population inversion occurs

• Resonant (optical) cavity: pair of mirrors (one fully, one partially reflecting) providing feedback & mode selection

Specific Laser Systems

He–Ne Laser (Gas, Four-level)

• Mixture: He 80 %, Ne 20 % at low pressure in glass discharge tube

• Pumping: electric discharge excites $\text{He}$ to metastable $E3$; resonant energy transfer pumps $\text{Ne}$ to $E2$

• Population inversion between $E2$ (Ne) and $E1$

• Output: red light $\lambda =632.8\,\text{nm}$ through partially reflecting mirror

• Brewster windows ensure linear polarization & minimize losses

Ruby Laser (Solid-state, Three-level)

• Medium: $\text{Al}2\text{O}3$ doped with $\text{Cr}^{3+}$ (ruby rod)

• Pumping: intense flash-lamp excites ions to $E3$; fast non-radiative decay to metastable $E2$

• Population inversion between $E2$ and ground $E1$

• Optical cavity: silvered end faces of rod (or external mirrors)

• Output: deep red $\lambda =694.3\,\text{nm}$ pulses (Q-switching common)

Semiconductor (Injection) Laser

• p–n junction in GaAs or GaAlAs; dimensions (1\,\text{mm}\times1\,\text{mm}\times1\,\text{mm})

• Heavy doping + forward bias ((V\approx E_g/e)) ➔ carrier overflow, population inversion in depletion region

• Mirrors: cleaved, polished crystal ends (one partial)

• High current density ((\sim20\,\text{kA/cm}^2)) for continuous wave (CW) emission

• Advantages: compact, efficient, directly modulatable; basis of CDs, telecom transmitters

Applications of Lasers (Selected)

• Scientific research: spectroscopy, nonlinear optics, plasma diagnostics

• Engineering: optical fibre communication, micro-welding, precision cutting, alignment

• Medicine: bloodless surgery (ophthalmic retina repair, dermatology, dentistry)

Bar-Code Scanner

• Laser beam scans UPC stripes; black absorbs, white reflects ➔ photodiode converts intensity variations to digital code

• Narrow, directional beam ensures readability at various angles/distances

Laser Printer

1. Charging: drum positively charged

2. Writing: modulated laser discharges pattern corresponding to image

3. Developing: toner adheres to discharged areas

4. Transfer: negatively charged paper attracts toner

5. Fusing: heat + pressure melt toner into paper

Laser Cooling

• Utilizes photon momentum ($p = h/\lambda$) to slow atoms

• Two main schemes:

  • Doppler cooling

  • Sisyphus cooling

• Achievable temperatures: mK → µK → nK; enables Bose-Einstein condensation, precision clocks

Optical Fibres

Construction

• Core: glass/plastic, refractive index $n_1$

• Cladding: lower index $n2$ (<$n1$); maintains TIR

• Protective jacket: polyurethane; multiple fibres form cable

• Typical diameters: $d{core}=10\text{–}200\,\mu\text{m},\; d{clad}=50\text{–}250\,\mu\text{m}$

Principle – Total Internal Reflection (TIR)

• For \thetai>\thetac at core–clad interface, light is totally reflected, propagating through curved paths without appreciable loss

• Critical angle: \thetac = \sin^{-1}\big(\tfrac{n2}{n_1}\big)

Acceptance Angle, Cone & Numerical Aperture (NA)

• For external medium index $n0$; incident angle $\theta0$ must satisfy:
  $\sin\theta0 = n0^{-1}\sqrt{n1^2-n2^2}$

• Numerical aperture: $\text{NA}=n0\sin\theta0=\sqrt{n1^2-n2^2}$ (for $n_0=1$)

• Fractional index difference: $\Delta =\dfrac{n1-n2}{n1}$; relation $\text{NA}=n1\sqrt{2\Delta}$

Skip Distance (inter-reflection spacing)

• $Ls = d\cot\theta1 = d\big(\csc\theta1-\cot\theta1\big)$ with $\sin\theta1 =\tfrac{n0}{n1}\sin\theta0$

Modes & Normalized Frequency

• Normalized frequency $V = \dfrac{\pi d}{\lambda}\,\text{NA}$

• Number of guided modes (multimode): $\approx \dfrac{V^2}{2}$

Fibre Types

1. Single-mode Step-Index: narrow core (5–10 µm); V<2.405 ➔ only fundamental mode; minimal dispersion, highest bandwidth

2. Multimode Step-Index: large core (50–200 µm); many modes; easy coupling but suffers intermodal dispersion

3. Multimode Graded-Index (GRIN): core index decreases parabolically from center; equalizes path lengths ➔ lower dispersion than step-index multimode

Attenuation Mechanisms

1. Absorption

   • Impurities (transition metals, OH⁻) absorb photons ➔ heat or re-emit incoherently

   • Intrinsic absorption in pure silica (IR vibrational bands, UV electronic bands)

2. Scattering (Rayleigh $\propto1/\lambda^4$)

   • Micro-density fluctuations, compositional variations, bubbles

3. Other losses

   • Micro-bending: random tiny bends along fibre

   • Macro-bending: tight curves (<10 cm radius) during installation; power radiates out of core

   • Connectors/splices: Fresnel reflections, misalignment

Dispersion (Pulse Broadening)

1. Material dispersion ($D_m$)

   • Wavelength dependence of refractive index $n(\lambda)$

   • $D_m = \dfrac{\lambda\,\Delta\lambda\,L}{c}\,\dfrac{d^2n}{d\lambda^2}$

2. Waveguide dispersion

   • Different propagation angles for different (\lambda) within same mode

3. Intermodal (modal) dispersion

   • Multiple modes travel different path lengths ➔ only in multimode fibres

Applications of Optical Fibres

Fibre-Optic Networking

• Nodes interconnected by fibre links; dedicated wavelength “lightpaths” $\lambda1,\lambda2,\dots$

• Advantages: huge bandwidth, low loss, immunity to EMI, small size/weight

Fibre-Optic Communication System

• Transmitter (laser/LED) ➔ fibre channel ➔ receiver (photodiode)

• Digital data encoded via intensity, phase or wavelength division multiplexing (WDM)

Other Uses

• Sensors (temperature, strain, chemical), gyroscopes

• Flexible fibrescope/endoscope for medical diagnostics

• Industrial inspection in hazardous environments

Ethical and Practical Implications

• Eye safety: high-power lasers demand stringent standards (ANSI Z136)

• Data privacy: enormous capacity raises security concerns; encryption essential in fibre networks

• Environmental impact: fibre cables less resource-intensive than copper; but rare-earth elements in lasers require responsible sourcing

Numerical & Statistical References (from transcript’s problem set)

• Example: three-level laser at $550\,\text{nm}$, Boltzmann ratio at $300\,\text{K}$ $\Rightarrow N2/N1 \approx1.77\times10^{-38}$

• Ruby laser: rod $6\,\text{cm}$ × $1\,\text{cm}$, pulse energy computed $16.9\,\text{J}$ when all $\text{Cr}^{3+}$ ions de-excite

• He–Ne laser 2.3 mW emits $4.4\times10^{17}$ photons/min

• Step-index fibre (core 63.5 µm, $n1=1.53, n2=1.39$): $\text{NA}=0.639,\;\theta_{accept}=39.7^{\circ}$, etc.

Connections to Foundational Principles

• Stimulated emission predicted by Einstein ➔ unifies quantum transitions with Planck radiation law

• TIR and Snell’s law foundational for waveguides; analogous to microwave waveguide reflections

• Laser cooling exploits conservation of momentum & Doppler effect, bridging quantum optics and thermodynamics

Real-World Relevance

• Internet back-bone capacity depends on DWDM (Dense WDM) using single-mode fibres and diode lasers

• Supermarkets, logistics rely on laser scanners for inventory management

• Semiconductor lasers enable optical storage (CD/DVD/Blu-ray) and LiDAR in autonomous vehicles

Summary of Merits & Demerits

• Single-mode fibre: +highest bandwidth, −difficult coupling

• Multimode step: +easy alignment, −high dispersion

• GRIN fibre: compromise, −fabrication complexity

• Ruby laser: +robust, −high threshold, pulsed only

• He–Ne: +stable CW, −bulky/low power

• Semiconductor laser: +compact, efficient, −temperature sensitive, beam divergence