Unit 3 - Notes

PHY110

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. An optical fiber is a cylindrical dielectric waveguide (non-conducting waveguide) that transmits light along its axis through the process of Total Internal Reflection.

1.1 Structure of an Optical Fiber

An optical fiber consists of three concentric cylindrical sections:

The Core: The innermost section made of glass or plastic with a high refractive index ( $n_1$ ). This is the physical medium that transports optical signals.
The Cladding: Surrounds the core and is made of a material with a slightly lower refractive index ( $n_2$ ) than the core ( $n_1 > n_2$ ). It confines the light within the core and prevents light from leaking out.
The Buffer Coating/Jacket: The outermost layer made of plastic/polymer. It protects the fiber from physical damage, moisture, and environmental hazards. It does not play a role in light propagation.

2. Optical Fiber as a Dielectric Waveguide

Unlike metallic waveguides (microwaves) or copper wires (electricity), optical fibers are dielectric waveguides. They are made of transparent dielectric (electrically non-conducting) materials.

Electromagnetic Nature: Light is an electromagnetic wave. The fiber guides these waves.
Confinement: The wave is confined to the core because the core has a higher refractive index than the cladding.
Propagation: Light propagates through the fiber in the form of "modes" (distinct electromagnetic field patterns).

3. Total Internal Reflection (TIR)

The fundamental principle behind fiber optic communication is Total Internal Reflection.

3.1 The Principle

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.

Critical Angle ( $\phi_c$ ): The angle of incidence in the denser medium for which the angle of refraction in the rarer medium becomes $90^\circ$ .
$\sin \phi_c = \frac{n_2}{n_1}$
Total Internal Reflection: If the angle of incidence is greater than the critical angle ( $\phi > \phi_c$ ), the ray does not refract into the cladding but reflects back into the core entirely.

Conditions for TIR:

Light must travel from a denser medium to a rarer medium ( $n_1 > n_2$ ).
The angle of incidence at the core-cladding interface must be greater than the critical angle.

4. Mathematical Parameters of Optical Fibers

4.1 Acceptance Angle ( $\theta_a$ )

The acceptance angle is the maximum angle of incidence at the launching end (air-core interface) of an optical fiber for which the light ray can undergo total internal reflection inside the core and propagate along the fiber.

If light enters at an angle greater than $\theta_a$ , it strikes the core-cladding interface at an angle less than the critical angle, refracts into the cladding, and is lost.
Formula:
$\theta_a = \sin^{-1} \sqrt{n_1^2 - n_2^2}$
(Assuming light is launched from air, where $n_0 \approx 1$ )

4.2 Numerical Aperture (NA)

The Numerical Aperture is a dimensionless quantity that represents the light-gathering ability of an optical fiber. It determines how much light the fiber can accept from a source.

It is defined as the sine of the acceptance angle.
$NA = \sin \theta_a = \sqrt{n_1^2 - n_2^2}$
Significance:
- High NA: Captures more light (good for LEDs), but causes higher modal dispersion (signal spreading).
- Low NA: Captures less light (requires Lasers), but supports higher bandwidth (less dispersion).

4.3 Relative Refractive Index Difference ( $\Delta$ )

This parameter is the ratio of the difference in refractive indices of the core and cladding to the refractive index of the core. It is usually less than 1 (often expressed as a percentage).

Formula:

$\Delta = \frac{n_1 - n_2}{n_1}$
(Sometimes approximated as $\Delta \approx \frac{n_1 - n_2}{n_2}$ when $n_1 \approx n_2$ )
Relationship between NA and $\Delta$ :

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

5. The 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's geometric and optical parameters to the wavelength of light.

5.1 Formula

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

Where:

$a$ = Radius of the fiber core
$\lambda$ = Wavelength of light utilized
$NA$ = Numerical Aperture

5.2 Significance of V-Number

The value of $V$ determines if the fiber is Single Mode or Multi Mode.

Single Mode Fiber (SMF): If $V \leq 2.405$ , the fiber supports only one mode (the fundamental mode).
Multi Mode Fiber (MMF): If $V > 2.405$ , the fiber supports multiple modes.

5.3 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: $N \approx \frac{V^2}{4}$

6. Classification of Optical Fibers

Fibers are classified based on the Refractive Index Profile and the Number of Modes.

6.1 Step Index Fiber

The refractive index of the core is constant throughout and undergoes an abrupt (step) change at the core-cladding boundary.

Single Mode Step Index Fiber:
- Very small core diameter ($5$ to $10 \mu m$ ).
- Supports only one path of light along the axis.
- Pros: Very low dispersion, extremely high bandwidth, suitable for long-distance.
- Cons: Difficult to couple light into the tiny core.
Multi Mode Step Index Fiber:
- Larger core diameter ( $\sim 50$ to $200 \mu m$ ).
- Supports many paths (zig-zag).
- Pros: Easy to couple light, cheaper.
- Cons: High Intermodal Dispersion. Rays taking different paths arrive at different times, causing pulse broadening.

6.2 Graded Index Fiber (GRIN)

The refractive index of the core is not constant. It is maximum at the center axis and decreases gradually (parabolically) towards the cladding interface.

Mechanism: Light rays travel in sinusoidal (helical) paths rather than zig-zag lines. Rays traveling near the edge (longer path) travel in a lower refractive index medium (faster speed), while rays at the center (shorter path) travel in a higher refractive index medium (slower speed).
Result: All rays arrive at the output end at approximately the same time.
Advantage: Drastically reduces intermodal dispersion compared to multimode step-index fibers.

7. Losses in Optical Fibers (Attenuation)

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

Formula:

\alpha (dB/km) = -\frac{10}{L} \log_{10} \left( \frac{P_{out}}{P_{in}} \right)

Where

L

is length in km,

P_{in}

is input power, and

P_{out}

is output power.

7.1 Absorption Losses

Caused by the material composition of the fiber absorbing light energy and converting it to heat.

Intrinsic Absorption: Caused by the interaction of light with the basic silica material (UV absorption by electron transitions, IR absorption by atomic vibrations).
Extrinsic Absorption: Caused by impurities, primarily hydroxyl ions ( $OH^-$ ) from water dissolved in the glass, and transition metals (Fe, Cu, Cr).

7.2 Scattering Losses

Caused by the interaction of light with density fluctuations within the fiber.

Rayleigh Scattering: Occurs due to microscopic variations in material density and refractive index that are smaller than the wavelength of light. This is the dominant loss mechanism at lower wavelengths.
- Scattering loss $\propto \frac{1}{\lambda^4}$ .
Mie Scattering: Caused by structural imperfections (bubbles, cracks) comparable in size to the wavelength.

7.3 Bending Losses

Occurs when the fiber is bent, disrupting the condition for Total Internal Reflection.

Macrobending: Occurs when the fiber is bent with a radius visible to the human eye (e.g., wrapping fiber around a spool). Higher order modes strike the interface at angles less than the critical angle and leak out.
Microbending: Caused by microscopic bumps or kinks in the fiber axis, often introduced during the manufacturing or cabling process (e.g., uneven coating pressure). This causes mode coupling and leakage.