Unit 3 - Notes

PHY109

Unit 3: Fiber Optics

1. Introduction to Fiber Optics

Fiber optics is a technology associated with the transmission of information as light pulses along a glass or plastic strand or fiber. A fiber optic cable can contain a varying number of these glass fibers—from a few up to several hundred. Another glass layer, called cladding, surrounds the glass fiber core.

1.1 Optical Fiber as a Dielectric Waveguide

An optical fiber is essentially a cylindrical dielectric waveguide. It is "dielectric" because it is made of non-conducting materials (glass or plastic), and "waveguide" because it guides electromagnetic waves (light) through it.

Structure of an Optical Fiber:

Core: The central cylindrical region made of glass or plastic with a refractive index $n_1$ . Light propagates through this region.
Cladding: The layer surrounding the core with a refractive index $n_2$ . Crucially, $n_2 < n_1$ (refractive index of cladding is less than that of the core).
Buffer Coating/Jacket: A protective plastic layer that protects the fiber from moisture and physical damage.

2. Principle of Propagation: Total Internal Reflection (TIR)

Light propagation inside an optical fiber is based on the principle of Total Internal Reflection.

2.1 Conditions for TIR

When a ray of light travels from a denser medium (Core, $n_1$ ) to a rarer medium (Cladding, $n_2$ ), it bends away from the normal. As the angle of incidence increases, the angle of refraction increases.

Condition 1: Light must travel from a high refractive index medium to a low refractive index medium ( $n_1 > n_2$ ).
Condition 2: The angle of incidence ( $\theta_i$ ) at the core-cladding interface must be greater than the Critical Angle ( $\theta_c$ ).

2.2 Critical Angle ( $\theta_c$ )

The critical angle is the angle of incidence for which the angle of refraction is $90^\circ$ (grazing emergence).
According to Snell's Law:

n_1 \sin \theta_c = n_2 \sin 90^\circ

Since

\sin 90^\circ = 1

\theta_c = \sin^{-1} \left( \frac{n_2}{n_1} \right)

3. Fiber Parameters

3.1 Acceptance Angle ( $\theta_a$ )

The acceptance angle is the maximum angle made by a light ray with the fiber axis at the entrance face of the fiber such that the ray undergoes total internal reflection at the core-cladding interface.

If the incident angle is greater than $\theta_a$ , the ray refracts into the cladding and is lost (clad mode).
The cone defined by the acceptance angle is called the Acceptance Cone.

Formula:
Assuming the medium outside the fiber is air ( $n_0 \approx 1$ ):

\theta_a = \sin^{-1} \sqrt{n_1^2 - n_2^2}

3.2 Numerical Aperture (NA)

Numerical Aperture is a dimensionless quantity that represents the light-gathering ability of the optical fiber. It is defined as the sine of the acceptance angle.

Formula:

\text{NA} = \sin \theta_a = \sqrt{n_1^2 - n_2^2}

3.3 Relative Refractive Index Difference ( $\Delta$ )

This parameter relates the refractive indices of the core and the cladding. It is defined as the ratio of the difference in refractive indices to the refractive index of the core.

Formula:

\Delta = \frac{n_1 - n_2}{n_1}

(Since $n_1 \approx n_2$ , sometimes written as $\Delta \approx \frac{n_1^2 - n_2^2}{2n_1^2}$ )

Relationship between NA and $\Delta$ :

\text{NA} = n_1 \sqrt{2\Delta}

4. V-Number (Normalized Frequency)

The V-number is a dimensionless parameter that determines the number of modes (paths) a fiber can support. It relates the fiber geometry and optical parameters to the wavelength of light.

Formula:

V = \frac{2\pi a}{\lambda} \text{NA} = \frac{2\pi a}{\lambda} \sqrt{n_1^2 - n_2^2}

Where:

$a$ = Radius of the core
$\lambda$ = Wavelength of light used

4.1 Significance of V-Number

Single Mode Operation: If $V < 2.405$ , the fiber supports only one mode (Single Mode Fiber).
Multimode Operation: If $V > 2.405$ , the fiber supports multiple modes.

4.2 Number of Modes ( $N$ )

For a multimode fiber with a large V-number:

Step Index Fiber: $N \approx \frac{V^2}{2}$
Graded Index Fiber: $N \approx \frac{V^2}{4}$

5. Classification of Optical Fibers

Fibers are classified based on the refractive index profile of the core and the number of modes they propagate.

5.1 Step Index Fiber

In a step-index fiber, the refractive index of the core is uniform throughout and undergoes an abrupt change (step) at the cladding boundary.

Structure: $n(r) = n_1$ for $r < a$ (core); $n(r) = n_2$ for $r \ge a$ (cladding).
Path of Light: Light travels in zigzag paths (meridional rays).
Disadvantages: High Intermodal Dispersion. Rays travelling at different angles travel different path lengths. The ray travelling along the axis arrives sooner than the ray reflecting off the walls, causing pulse broadening.

5.2 Graded Index (GRIN) Fiber

In a GRIN fiber, the refractive index of the core is non-uniform. It is maximum at the axis and decreases parabolically towards the cladding.

Structure:
- $n(r) = n_1 [1 - 2\Delta(r/a)^2]^{1/2}$ for $r < a$
- $n(r) = n_2$ for $r \ge a$
Path of Light: Light travels in helical/sinusoidal paths. It does not hit the cladding boundary abruptly but curves back toward the axis.
Self-Focusing Effect: Rays travelling further from the axis (longer path) move through a lower refractive index medium (higher speed). Rays on the axis (shorter path) move through a higher refractive index (lower speed). This compensates for time differences, significantly reducing intermodal dispersion.

Feature	Step Index	Graded Index
RI Profile	Uniform core	Parabolic (decreasing from center)
Light Path	Zigzag	Sinusoidal (Curved)
Dispersion	High	Low
Bandwidth	Lower	Higher
Cost	Lower	Higher

6. Losses in Optical Fibers (Attenuation)

Attenuation is the reduction in signal strength (optical power) as light travels down the fiber. It is measured in decibels per kilometer (dB/km).

Formula:

\alpha_{dB} = \frac{10}{L} \log_{10} \left( \frac{P_{in}}{P_{out}} \right)

6.1 Absorption Losses

Intrinsic Absorption: Caused by the interaction of light with the fundamental material properties of glass (SiO $_2$ ). Includes UV absorption (electronic transitions) and IR absorption (molecular vibrations).
Extrinsic Absorption: Caused by impurities in the glass, primarily:
- Hydroxyl ions (OH⁻): Water peaks occur at specific wavelengths (e.g., 1.38 µm), causing high loss.
- Transition metals (Iron, Copper, Chromium).

6.2 Scattering Losses

Rayleigh Scattering: Occurs due to microscopic density fluctuations in the glass that are smaller than the wavelength of light.
- Loss is proportional to $1/\lambda^4$ . (Shorter wavelengths scatter much more).
Mie Scattering: Caused by non-perfect cylindrical structure or imperfections larger than the wavelength of light.

6.3 Bending Losses

Macrobending: Occurs when the fiber is bent with a large radius (visible to the eye). Light strikes the interface at an angle less than the critical angle and leaks out.
Microbending: Occurs due to microscopic kinks or bumps in the fiber geometry during manufacturing or cabling. Causes mode coupling and leakage.

7. Applications of Optical Fibers

7.1 Telecommunications

Backbone Networks: Connecting cities and countries due to high bandwidth and low attenuation.
Fiber to the Home (FTTH): High-speed internet directly to consumers.
Submarine Cables: Global internet traffic.

7.2 Medical Field

Endoscopy: Uses coherent bundles (maintain spatial relationship) to view internal organs.
Laser Surgery: Delivering high-power laser beams to precise targets inside the body (e.g., clearing blocked arteries, eye surgery).
Biomedical Sensors: Measuring blood flow, pH, or oxygen levels inside the body.

7.3 Sensors

Intrinsic Sensors: The fiber itself is the sensing element (e.g., Fiber Optic Gyroscope for navigation).
Extrinsic Sensors: The fiber transmits light to an external sensor. Used for measuring temperature, pressure, strain, and displacement in industrial environments.

7.4 Industrial and Military

Noise Immunity: Fiber is immune to Electromagnetic Interference (EMI) and Radio Frequency Interference (RFI), making it ideal for noisy electrical environments.
Security: Extremely difficult to tap without detection (no radiated EM field).
Hazardous Environments: Since no electricity flows through the fiber, there is no spark hazard, making it safe for oil rigs and chemical plants.

Unit 3 - Notes

Table of Contents

Unit 3: Fiber Optics

1. Introduction to Fiber Optics

1.1 Optical Fiber as a Dielectric Waveguide

2. Principle of Propagation: Total Internal Reflection (TIR)

2.1 Conditions for TIR

2.2 Critical Angle ()

3. Fiber Parameters

3.1 Acceptance Angle ()

3.2 Numerical Aperture (NA)

3.3 Relative Refractive Index Difference ()

4. V-Number (Normalized Frequency)

4.1 Significance of V-Number

4.2 Number of Modes ()

5. Classification of Optical Fibers

5.1 Step Index Fiber

5.2 Graded Index (GRIN) Fiber

6. Losses in Optical Fibers (Attenuation)

6.1 Absorption Losses

6.2 Scattering Losses

6.3 Bending Losses

7. Applications of Optical Fibers

7.1 Telecommunications

7.2 Medical Field

7.3 Sensors

7.4 Industrial and Military

2.2 Critical Angle ( $\theta_c$ )

3.1 Acceptance Angle ( $\theta_a$ )

3.3 Relative Refractive Index Difference ( $\Delta$ )

4.2 Number of Modes ( $N$ )