Engineering Physics II: Optoelectronics and Optical Fiber Communications

Fundamentals of Optoelectronics and Light

• Definition of Optoelectronic Devices:

  • Optoelectronic devices are specialized electronic components that emit, detect, or control light using semiconductor materials.
  • They serve as an essential bridge between optics (the physical study of light) and electronics (the manipulation of electrical charge carriers), facilitating direct conversion between optical and electrical signals.
  • These devices form the technological foundation for optical communications, visual display units, advanced sensing, and photovoltaic energy conversion.

• Major Categories of Optoelectronic Devices:

  • Emitters: Devices that transform electrical current into optical radiation (e.g., Light Emitting Diodes [LEDs], Semiconductor Laser Diodes).
  • Detectors: Devices that absorb light photons to produce electrical current (e.g., Photodiodes, Phototransistors).
  • Transducers: Components that convert light energy directly into electrical power via photovoltaic action (e.g., Solar Cells).
  • Modulators: Components designed to manipulate the phase, amplitude, or polarization of light beams via electrical inputs (e.g., Electro-optic modulators).
  • Sensors: Light-sensitive elements utilized for environmental detection or image capture (e.g., Photoresistors [Light Dependent Resistors / LDRs], solid-state imaging arrays).

• Physical Nature and Properties of Light:

  • Light is a dynamic form of electromagnetic radiation visible to the human eye, propagating through space as transverse electromagnetic waves.
  • Speed of Light: In a vacuum, light travels at a constant velocity denoted by $c = 3.00 \times 10^8\,\text{m/s}$, representing a fundamental physical constant.
  • Electromagnetic Spectrum: The complete spectrum encompasses radio waves, microwaves, infrared (IR) radiation, visible light, ultraviolet (UV) radiation, X-rays, and gamma rays.

• The Visible Spectrum and Energy Quantization:

  • The visible portion of the electromagnetic spectrum is segmented by wavelength (measured in nanometers, $\text{nm}$):
  • Red: $740 - 625\,\text{nm}$
  • Orange: $625 - 590\,\text{nm}$
  • Yellow: $590 - 565\,\text{nm}$
  • Green: $565 - 520\,\text{nm}$
  • Cyan / Blue-Green: $520 - 500\,\text{nm}$
  • Blue: $500 - 435\,\text{nm}$
  • Violet: $435 - 380\,\text{nm}$
  • Photons carry quantized energy directly proportional to wave frequency $\nu$ and inversely proportional to wavelength $\lambda$:     \n    E = h\nu = \frac{hc}{\lambda}\n         where $E$ is energy, $h$ is Planck's constant ($6.626 \times 10^{-34}\,\text{J}\cdot\text{s}$), and $\nu$ is frequency.

Quantum Interaction of Radiation with Matter

• Radiation interacts with quantum state energy levels of material systems through three primary physical processes:

• 1. Stimulated Absorption:

  • Mechanism: An incident photon carrying energy $\Delta E = E_2 - E_1 = h\nu$ strikes an atom in its ground or lower energy state $E_1$. The electron absorbs the photon energy completely and transitions to an excited energy state $E_2$.
  • Consequence: Generates free charge carriers or excited atomic states. This physical principle underlies the operation of photodiodes and photovoltaic cells.

• 2. Spontaneous Emission:

  • Mechanism: An electron residing in an elevated energy state $E_2$ decays naturally to lower state $E_1$ without external physical prompts after its upper-state lifetime expires.
  • Characteristics: The emitted photon possesses energy $\Delta E = E_2 - E_1$, but its direction, phase, and polarization are entirely random.
  • Consequence: Produces incoherent, omnidirectional light, forming the fundamental operating principle of LEDs.

• 3. Stimulated Emission:

  • Mechanism: An electron occupying excited state $E_2$ is perturbed by an incoming trigger photon whose energy matches the transition gap $\Delta E = E_2 - E_1$. The transition is stimulated prematurely, prompting the electron to drop to $E_1$
  • Characteristics: The emitted secondary photon is identical in frequency, phase, direction of propagation, and polarization state to the triggering photon.
  • Consequence: Produces coherent, monochromatic, highly directional, and amplified light, providing the foundational basis for LASER operation.

LASER Physics and Light Amplification

• Definition of LASER:

  • LASER is an acronym for Light Amplification by Stimulated Emission of Radiation.
  • It is an optoelectronic device that forces stimulated emission to achieve optical amplification, yielding a high-power beam with specific physical properties.

• Key Physical Characteristics of LASER Light:

  • Monochromaticity: Emitted radiation consists of a single discrete wavelength or an extremely tight wavelength band, unlike polychromatic white light sources.

• High Coherence: Photons exhibit spatial and temporal phase alignment. Spatial coherence ensures uniform phase across the wavefront; temporal coherence maintains phase predictability over time.

• High Directionality: Light rays propagate along a parallel path with minimal angular divergence over extensive transmission distances.

• High Intensity: Energy density per unit cross-sectional area is exceptionally high due to constructive interference and beam concentration.

  • Population Inversion:

• Under thermal equilibrium at temperature $T$, atomic populations in energy levels $E_1$ and $E_2$ follow Boltzmann statistics, where ground state population $N_1$ strictly exceeds excited state population $N_2$ ($N_1 > N_2$).

• Population Inversion is an artificial non-equilibrium state generated within a material where the population of higher energy state $N_2$ substantially exceeds lower state population $N_1$:     \n    N_2 \gg N_1\n    

• Population inversion is mandatory for light amplification, ensuring that stimulated emission events outnumber stimulated absorption events.

  • Essential Components of a Laser System:

• Active Medium (Gain Medium): The physical medium containing atoms, ions, or molecules capable of undergoing population inversion and stimulated emission. Types include:

  • Solid-State: Ruby crystal ($\text{Cr}^{3+}:\text{Al}_2\text{O}_3$), Nd:YAG crystal.
  • Liquid: Organic dye solutions dissolved in solvent.
  • Gas: Helium-Neon (He-Ne), Carbon Dioxide ($\text{CO}_2$).
  • Semiconductor: Gallium Arsenide (GaAs) compounds.

• Pumping Scheme: External energy delivery mechanism used to elevate ground-state carriers to excited energy bands to create population inversion. Common excitation methods:

  • Optical Pumping: High-intensity flash lamps or auxiliary laser sources.
  • Electrical Discharge: High-voltage potential applied across gas media.
  • Electrical Current Injection: Direct forward current biasing across semiconductor junctions.
  • Chemical Energy: Exothermic chemical reaction energy.

• Optical Resonator (Cavity):

  • An optical feedback enclosure consisting of two parallel mirrors placed at opposing ends of the active medium.
  • One mirror is $100\%$ reflective (totally reflecting rear mirror), while the opposing mirror is partially reflective ($90 - 99\%$ reflectivity, functioning as the output coupler).
  • Traps photons along the central axis, bouncing them repeatedly through the gain medium to build up intensity via sustained stimulated emission before a fraction escapes as the laser output beam.

Semiconductor Lasers (Laser Diodes)

• Structural Construction of Laser Diodes:
  • Constructed from direct bandgap compound semiconductor alloys, primarily Gallium Arsenide (GaAs), Aluminium Gallium Arsenide (AlGaAs), or Indium Gallium Arsenide Phosphide (InGaAsP).
  • P-Type and N-Type Layers: Heavily doped semiconductor regions providing high concentrations of mobile holes and free electrons.
  • Active Region (Intrinsic Layer): A thin undoped or lightly doped semiconductor layer sandwiched directly at the P-N junction interface where charge carriers accumulate.
  • Optical Cavity Boundaries: The front and rear faces of the semiconductor crystal are cleaved along natural crystallographic planes or polished smooth. One face is coated for full reflectivity ($100\%$), while the opposite face is partially polished/coated to serve as the optical output aperture.
  • Ohmic Contacts: Top and bottom metallic contact plates facilitate direct current flow from an external DC power supply.

• Electronic Transition and Lasing Operation:

  • Forward Bias Injection: Applying an external DC voltage exceeding the junction barrier potential injects minority carriers (electrons from N-side, holes from P-side) into the active recombination layer at high density.
  • Absorption and Carrier Generation: Electrical energy pumps valence electrons ($E_1$) across energy band gap $E_g$ into the conduction band ($E_2$), leaving behind empty valence states (holes).
  • Spontaneous Recombination: Initial carrier recombinations spontaneously emit random photons carrying energy matching the semiconductor band gap:     \n    E = E_2 - E_1 = E_g\n    
  • Stimulated Multiplication: Spontaneous photons traveling parallel to the junction plane induce stimulated recombination of remaining electron-hole pairs, creating clone photons in exact phase and trajectory.
  • Resonant Feedback: Photons bounce repeatedly between the cleaved end mirrors within the active waveguide layer. When optical gain exceeds internal absorption and scatter losses, threshold current is reached, producing a narrow, highly intense, coherent laser beam output.

• Applications of Laser Diodes:

  • Telecommunications: Primary optical transmitters for long-haul and metro fiber-optic communication lines.
  • Consumer Electronics: Optical pickup heads in CD, DVD, and Blu-ray readers/writers.
  • Printing and Scanning: High-resolution raster engines in laser printers, barcode units, and QR scanners.
  • Medical Technologies: Precision laser surgery, ophthalmology (retinal repair), dermatological procedures, photodynamic cancer therapy.
  • Industrial Processing: Precision micromachining, laser cutting, engraving, micro-welding.
  • Sensing and Defense: Optical rangefinding, LiDAR distance measurement, target designation, high-precision spectroscopy.

Light Emitting Diodes (LEDs)

• Fundamental Operating Principle:

  • LEDs operate via electroluminescence, a solid-state phenomenon where electrical energy supplied to a forward-biased P-N junction is directly converted into optical radiation.
  • The semiconductor material used must feature a direct band gap, ensuring that electron-hole recombination momentum is conserved without phononic lattice loss, releasing energy directly as photons.

• Construction and Layered Geometry:

  • Formed by growing three distinct layers onto a supporting crystal substrate:
  1. N-Type Semiconductor Layer: Doped with donor impurities providing abundant free electrons.
  2. Active Recombination Layer: Narrow middle boundary zone where radiative recombination occurs.
  3. P-Type Semiconductor Layer: Doped with acceptor impurities rich in hole states.
  • Assembly Structure: Attached to lead frames with an Anode terminal (positive, longer lead) and Cathode terminal (negative, shorter lead), encapsulated inside a clear or tinted protective plastic/epoxy dome that acts as a focusing lens.

• Detailed Recombination Process:

  • Forward bias lowers the internal depletion barrier potential.
  • Electrons cross from the N-region and holes cross from the P-region into the active junction zone.
  • High-energy conduction band electrons drop into lower-energy valence band holes.
  • The energy differential lost during transition is released as a photon of light with energy:     \n    E = E_g = h\nu = \frac{hc}{\lambda}\n    
  • Emitted wavelength $\lambda$ is strictly defined by the semiconductor material's band gap energy $E_g$:     \n    \lambda = \frac{hc}{E_g}\n    

• Semiconductor Material Compositions and Emission Bands:

  • Gallium Arsenide (GaAs): Infrared emission ($\lambda > 880\,\text{nm}$).
  • Gallium Arsenide Phosphide (GaAsP): Red to infrared, orange emission.
  • Aluminium Gallium Arsenide Phosphide (AlGaAsP): High-brightness red, orange-red, orange, and yellow.
  • Gallium Phosphide (GaP): Red, yellow, and green emission.
  • Aluminium Gallium Phosphide (AlGaP): Pure green emission.
  • Gallium Nitride (GaN): Green, emerald green emission.
  • Gallium Indium Nitride (GaInN): Near-ultraviolet, bluish-green, and bright blue.
  • Silicon Carbide (SiC): Substrate material for blue emissions.
  • Zinc Selenide (ZnSe): Blue emission.
  • Aluminium Gallium Nitride (AlGaN): Ultraviolet emission.

• Engineering Advantages of LEDs:

  • High energy efficiency with minimal thermal loss.
  • Long operating lifespan (often exceeding $50,000\,\text{hours}$).
  • Fast switching dynamic response times (nanosecond scale).
  • Solid-state durability, compact form factors, and low operating drive voltages.

• Applications:

  • General residential and industrial solid-state lighting.
  • Automotive headlights, optical displays, traffic signal matrices.
  • Security burglar alarms, optocouplers, consumer digital displays, watches, and instrumentation.

Semiconductor Photodetectors (Photodiodes)

• General Operating Principle:

  • A photodiode is a solid-state sensor that converts incident photons into electrical current.
  • Operates under reverse bias conditions to maximize depletion region width, decrease internal junction capacitance, and minimize dark current.
  • Incident photons carrying energy greater than or equal to energy band gap ($h\nu \ge E_g$) excite valence electrons to the conduction band, producing electron-hole pairs (EHPs).
  • The strong reverse-bias electric field swept across the depletion region rapidly separates EHPs, pulling electrons to the N-side and holes to the P-side, creating a proportional photocurrent ($I_{ph}$).

• Types of Photodiodes:

• 1. Standard PN Junction Photodiode:

  • Construction: Simple P-type and N-type silicon junction with a thin surface depletion zone covered by a transparent optical window.
  • Disadvantages: Thin depletion layer limits photon capture volume resulting in low quantum efficiency, higher junction capacitance, and slow response speed.

• 2. PIN Photodiode:

  • Construction: Incorporates a thick, high-resistivity, undoped Intrinsic (I) layer sandwiched between heavily doped $P^+$ and $N^+$ regions.

• Working Mechanism: Reverse bias drops almost entirely across the wide intrinsic layer, creating a vast depletion volume. Photons entering the intrinsic layer generate EHPs that are quickly swept out by the field.

• Advantages: Significantly higher quantum efficiency, expanded optical absorption depth, reduced junction capacitance, and broad frequency bandwidth suitable for gigabit optical links.

  • 3. Avalanche Photodiode (APD):

• Construction: Layered $p^+ - i - p - n^+$ structure incorporating a specialized high-field multiplication region.

• Working Mechanism: Operated under extremely high reverse bias voltage near electrical breakdown. Primary photo-generated carriers accelerate through the high-field region gaining extreme kinetic energy.
• Impact Ionization: Energetic carriers collide with lattice atoms, knocking bound electrons into the conduction band to generate secondary EHPs. These secondary carriers accelerate and create tertiary EHPs in a cascading avalanche multiplication process.
• Advantage: Provides internal photocurrent signal gain ($M = 10 - 1000$), enabling detection of ultra-weak optical signals in long-distance communication links and LiDAR systems.

Optical Fiber Waveguides

• Concept and Structural Geometry:

  • An optical fiber is a highly transparent, flexible dielectric waveguide constructed from pure silica glass ($\text{SiO}_2$) or plastic, designed to guide light along its length.

• Concentric Layer Architecture:

  • Core: Central light-guiding cylindrical medium made of silica doped with germanium ($\text{Ge}$) or aluminium ($\text{Al}$) to elevate its refractive index $n_1$.
  • Single-mode core diameter ($2a$): $8 - 12\,\mu\text{m}$ (typically $9\,\mu\text{m}$).
  • Multimode core diameter ($2a$): $50 - 62.5\,\mu\text{m}$.
  • Typical Core Refractive Index: $n_1 = 1.461$.
  • Cladding: Surrounding layer of pure, lower-index silica glass ($n_2$) engineered such that $n_1 > n_2$
  • Outer Cladding Diameter ($2b$): Standardized at $125\,\mu\text{m}$.
  • Typical Cladding Refractive Index: $n_2 = 1.460$
  • Buffer Coating: Primary mechanical polymer coating ($250\,\mu\text{m} - 900\,\mu\text{m}$ outer diameter) protecting the glass surface against micro-abrasions and ambient humidity.
  • Strength Members: Layer of woven Aramid Yarn (Kevlar) surrounding the buffered fiber to absorb tensile loads.
  • Outer Cable Jacket: Tough outer PVC or polyethylene sheath shielding against physical stress and chemical exposure.

Waveguide Ray Optics and Parameters

• Total Internal Reflection (TIR) Mechanism:
  • Light guiding relies on Total Internal Reflection occurring at the boundary between core ($n_1$) and cladding ($n_2$).

• Derivation of Critical Angle ($\theta_c$):

  • Applying Snell's Law at the core-cladding interface:       \n      n_1 \sin(\phi) = n_2 \sin(r)\n      
  • At the critical condition, the angle of refraction $r = 90^\circ$, and angle of incidence $\phi = \theta_c$:       \n      n_1 \sin(\theta_c) = n_2 \sin(90^\circ) = n_2\n             \n      \sin(\theta_c) = \frac{n_2}{n_1} \implies \theta_c = \arcsin\left(\frac{n_2}{n_1}\right)\n      

• Mandatory Conditions for Total Internal Reflection:

  1. The core material must be optically denser than the cladding ($n_1 > n_2$).
  2. The internal ray angle of incidence at the interface must strictly exceed the critical angle ($\phi > \theta_c$).
  • Acceptance Angle ($i_A$) and Acceptance Cone:

• The Acceptance Angle ($i_A$) is the maximum launch angle relative to the central fiber axis at which light rays can enter the core face and undergo continuous total internal reflection.

• Rotating $i_A$ around the fiber axis defines a three-dimensional coaxial Acceptance Cone. Rays launched inside this cone are trapped and guided; rays launched outside leak into the cladding.

  • Derivation of Numerical Aperture (NA):

• Consider a light ray launched from an external medium (air, $n_0 = 1$) into the core ($n_1$) at acceptance angle $i_A$. The ray refracts into the core at angle $\theta$.

• From geometry at the entrance face:     \n    \theta + \phi = 90^\circ \implies \theta = 90^\circ - \phi\n    

• Applying Snell's Law at the launch face:     \n    n_0 \sin(i_A) = n_1 \sin(\theta) = n_1 \sin(90^\circ - \phi_c) = n_1 \cos(\phi_c)\n    

• Using the trigonometric identity $\cos(\phi_c) = \sqrt{1 - \sin^2(\phi_c)}$:     \n    n_0 \sin(i_A) = n_1 \sqrt{1 - \sin^2(\phi_c)}\n    

• Substituting $\sin(\phi_c) = \frac{n_2}{n_1}$ from Snell's law at the critical condition:     \n    n_0 \sin(i_A) = n_1 \sqrt{1 - \left(\frac{n_2}{n_1}\right)^2} = n_1 \sqrt{\frac{n_1^2 - n_2^2}{n_1^2}} = \sqrt{n_1^2 - n_2^2}\n    

• For light entering from air ($n_0 = 1$), the Numerical Aperture is:     \n    \text{NA} = \sin(i_A) = \sqrt{n_1^2 - n_2^2}\n    

• The acceptance angle is calculated as:     \n    i_A = \arcsin(\text{NA}) = \arcsin\left(\sqrt{n_1^2 - n_2^2}\right)\n    

  • Fractional Refractive Index Difference ($\Delta$):

• Measures the relative index drop between core and cladding:     \n    \Delta = \frac{n_1 - n_2}{n_1}\n    

• Rewriting NA in terms of $\Delta$:     \n    \text{NA} = \sqrt{(n_1 - n_2)(n_1 + n_2)} = \sqrt{(n_1 \Delta)(2n_1)} = n_1 \sqrt{2\Delta}\n    

Classification of Optical Fibers

• Optical fibers are categorized according to three distinct classification schemes:

• 1. Classification by Materials Used:

  • Glass Fibers: Core and cladding made of ultra-pure silica glass. Features extremely low loss ($< 0.2\,\text{dB/km}$ at $1550\,\text{nm}$).
  • Plastic Fibers: Core and cladding made of optical polymers (e.g., PMMA). Highly flexible and robust, but exhibits higher attenuation; used for short-range links.

• 2. Classification by Refractive Index Profile:

  • Step-Index Fibers: Core refractive index $n_1$ remains uniform across its entire radius, then drops abruptly to $n_2$ at the core-cladding interface.
  • Graded-Index (GRIN) Fibers: Core refractive index varies continuously as a parabolic function of radial distance $r$ from the center axis:     \n    n(r) = n_{\text{max}} \sqrt{1 - 2\Delta \left(\frac{r}{a}\right)^g}\n         where $n_{\text{max}}$ is the peak central index, $a$ is core radius, and $g \approx 2$ is the parabolic profile index.

• 3. Classification by Propagation Modes:

  • Single-Mode Step-Index Fiber:
  • Core Diameter: $8 - 12\,\mu\text{m}$.
  • Light Propagation: Supports only one fundamental spatial mode ($\text{HE}_{11}$).
  • Performance: Completely eliminates intermodal dispersion. Offers ultra-wide bandwidth over tens of kilometers.
  • Applications: Telecom backbones, undersea cables.
  • Multimode Step-Index Fiber:
  • Core Diameter: $50 - 200\,\mu\text{m}$.
  • Light Propagation: Supports hundreds of distinct spatial ray paths (modes).
  • Performance: High intermodal dispersion because higher-order zig-zag paths travel longer physical distances than axial paths, causing pulse broadening.
  • Applications: Short-distance local area networks (LANs).
  • Multimode Graded-Index (GRIN) Fiber:
  • Core Diameter: $50 - 100\,\mu\text{m}$.
  • Light Propagation: Light rays follow smooth sinusoidal trajectories. Rays traveling further from the axis speed up due to lower local refractive index ($v = c/n$).
  • Performance: Equalizes propagation delay across all modes, significantly reducing modal dispersion compared to multimode step-index fiber.
  • Applications: Medium-distance enterprise networks.

Signal Attenuation and Transmission Losses

• Attenuation Definition:

  • Attenuation represents the exponential reduction of optical power as light propagates through the fiber length.
  • Caused by material absorption (OH- ion resonance and intrinsic silica absorption), Rayleigh scattering (microscopic density fluctuations), and bending radiation losses.

• Logarithmic Attenuation Coefficient Formula:

  • If optical power $P_i$ is launched into a fiber of length $L$ (in kilometers), the emerging output power $P_o$ is given by:     \n    P_o = P_i 10^{-\frac{\alpha L}{10}}\n    
  • Rearranging to solve for the attenuation coefficient $\alpha$ in decibels per kilometer ($\text{dB/km}$):     \n    \alpha = \frac{10}{L} \log_{10}\left(\frac{P_i}{P_o}\right)\n    
  • For an ideal lossless fiber, $P_o = P_i$ and $\alpha = 0\,\text{dB/km}$.

Practice and Numerical Problems

• Problem 1: A step-index fiber has a core refractive index of $1.55$ and a cladding refractive index of $1.53$. Find the numerical aperture and the acceptance angle of the fiber.

  • Solution Outline:     \n    \text{NA} = \sqrt{n_1^2 - n_2^2} = \sqrt{1.55^2 - 1.53^2} = \sqrt{2.4025 - 2.3409} = \sqrt{0.0616} \approx 0.248\n         \n    i_A = \arcsin(\text{NA}) = \arcsin(0.248) \approx 14.36^\circ\n    

• Problem 2: A step-index fiber has a core of refractive index $1.50$ and a cladding of refractive index $1.40$. If the fiber is used in a water environment ($n_0 = 1.33$), find its numerical aperture and acceptance angle.

  • Solution Outline:     \n    \text{NA} = \frac{\sqrt{n_1^2 - n_2^2}}{n_0} = \frac{\sqrt{1.50^2 - 1.40^2}}{1.33} = \frac{\sqrt{2.25 - 1.96}}{1.33} = \frac{\sqrt{0.29}}{1.33} \approx \frac{0.5385}{1.33} \approx 0.405\n         \n    i_A = \arcsin(0.405) \approx 23.89^\circ\n    

• Problem 3: Optical power of $1\,\text{mW}$ is launched into an optical fiber of length $100\,\text{m}$ ($0.1\,\text{km}$). If the power emerging from the other end is $0.3\,\text{mW}$, calculate the fiber attenuation.

  • Solution Outline:     \n    \alpha = \frac{10}{0.1} \log_{10}\left(\frac{1.0}{0.3}\right) = 100 \times \log_{10}(3.333) = 100 \times 0.5228 = 52.28\,\text{dB/km}\n    

• Problem 4: What is the attenuation in $\text{dB/km}$, if $15\%$ of the power fed at the launching end of a $0.5\,\text{km}$ fiber is lost during propagation?

  • Solution Outline:
  • Power remaining $P_o = 85\%$ of $P_i = 0.85 P_i$.     \n    \alpha = \frac{10}{0.5} \log_{10}\left(\frac{P_i}{0.85 P_i}\right) = 20 \times \log_{10}(1.1765) = 20 \times 0.0706 = 1.412\,\text{dB/km}\n    

• Problem 5: An optical fiber has cladding of refractive index $1.50$ and NA of $0.39$. Find the refractive index of the core and the acceptance angle in air.

  • Solution Outline:     \n    \text{NA} = \sqrt{n_1^2 - n_2^2} \implies n_1 = \sqrt{\text{NA}^2 + n_2^2} = \sqrt{0.39^2 + 1.50^2} = \sqrt{0.1521 + 2.25} = \sqrt{2.4021} \approx 1.55\n         \n    i_A = \arcsin(0.39) \approx 22.95^\circ\n    

• Problem 6: The power of a signal after propagating through an optical fiber of $1.5\,\text{km}$ length is reduced to $25\%$ of its original value. Compute the fiber loss in $\text{dB/km}$.

  • Solution Outline:
  • $P_o / P_i = 0.25 \implies P_i / P_o = 4$.     \n    \alpha = \frac{10}{1.5} \log_{10}(4) = \frac{10}{1.5} \times 0.60206 \approx 4.013\,\text{dB/km}\n    

• Problem 7: Find the attenuation in an optical fiber of length $500\,\text{m}$ ($0.5\,\text{km}$), when a light signal of power $100\,\text{mW}$ emerges out of the fiber with a power of $90\,\text{mW}$.

  • Solution Outline:     \n    \alpha = \frac{10}{0.5} \log_{10}\left(\frac{100}{90}\right) = 20 \times \log_{10}(1.1111) = 20 \times 0.04575 = 0.915\,\text{dB/km}\n    

• Problem 8: The attenuation of light in an optical fiber is estimated at $2.2\,\text{dB/km}$. What fractional initial intensity remains after $2\,\text{km}$ and $6\,\text{km}$?

  • Solution Outline:
  • For $L = 2\,\text{km}$:       \n      \alpha L = 2.2 \times 2 = 4.4\,\text{dB} \implies \log_{10}\left(\frac{P_i}{P_o}\right) = 0.44 \implies \frac{P_o}{P_i} = 10^{-0.44} \approx 0.363\,\text{(or } 36.3\%\text{)}\n      
  • For $L = 6\,\text{km}$:       \n      \alpha L = 2.2 \times 6 = 13.2\,\text{dB} \implies \log_{10}\left(\frac{P_i}{P_o}\right) = 1.32 \implies \frac{P_o}{P_i} = 10^{-1.32} \approx 0.0479\,\text{(or } 4.79\%\text{)}\n      

• Problem 9: Calculate (a) the cladding index $n_2$ and (b) the numerical aperture of a glass-clad fiber made with core glass of refractive index $n_1 = 1.5$ and cladding doped to give a fractional index difference \Delta = 0.0005$.\n - *Solution Outline*:\n - (a) Cladding index n_2:\n             \Delta = \frac{n_1 - n_2}{n_1} \implies n_1 - n_2 = n_1 \Delta = 1.5 \times 0.0005 = 0.00075       \n             n_2 = 1.5 - 0.00075 = 1.49925       \n - (b) Numerical Aperture:\n             \text{NA} = n_1 \sqrt{2\Delta} = 1.5 \times \sqrt{2 \times 0.0005} = 1.5 \times \sqrt{0.001} = 1.5 \times 0.03162 \approx 0.0474       $$