Step-Index Optical Fiber Principles

Vocabulary flashcards covering the geometrical optics principles, formulas, derivations, dispersion, and numerical application of step-index optical fibers.

Total Internal Reflection Condition at Core/Cladding Diopter

In a step-index fiber with core index $n_1$ and cladding index $n_2 < n_1$, total internal reflection occurs at the core/cladding interface when the angle of incidence $i_1$ satisfies $i_1 > i_{1\text{lim}}$, where $i_{1\text{lim}} = \arcsin\left(\frac{n_2}{n_1}\right)$.

Entry Refraction Angle and Core/Cladding Incidence Angle Relation

For a step-index fiber with an entry face perpendicular to the optical axis, the refraction angle $r$ at entry and the incidence angle $i_1$ on the core/cladding diopter are related by $r + i_1 = \frac{\pi}{2}$ (from right triangle OIK), which gives $r = \frac{\pi}{2} - i_1$.

Guided Ray Condition on Entry Refraction Angle

In a step-index fiber, a ray is guided if the entry refraction angle $r$ at the air/core interface satisfies $r < r_{1\text{lim}} = \frac{\pi}{2} - i_{1\text{lim}}$, where $i_{1\text{lim}} = \arcsin\left(\frac{n_2}{n_1}\right)$.

Acceptance Cone Half-Angle Sine Formula

For a step-index fiber with core index $n_1$, cladding index $n_2$, and entry in air, the sine of the acceptance cone half-angle $i_{\text{acc}}$ is given by $\sin(i_{\text{acc}}) = \sqrt{n_1^2 - n_2^2}$.

Derivation Steps of Acceptance Cone Half-Angle Formula

At the limiting case $i_1 = i_{1\text{lim}}$, refraction at entry point $O$ gives $\sin(i_{\text{acc}}) = n_1 \sin(r_{1\text{lim}}) = n_1 \sin\left(\frac{\pi}{2} - i_{1\text{lim}}\right) = n_1 \cos(i_{1\text{lim}})$; using $\sin(i_{1\text{lim}}) = \frac{n_2}{n_1}$, we obtain $\cos(i_{1\text{lim}}) = \sqrt{1 - \left(\frac{n_2}{n_1}\right)^2}$, yielding $\sin(i_{\text{acc}}) = \sqrt{n_1^2 - n_2^2}$.

Entry Incidence Angle Condition for Ray Guidance

A ray entering a step-index fiber is guided if its entry incidence angle satisfies $i < i_{\text{acc}}$ (ray within the acceptance cone), because a small $i$ leads to a small $r$ via $\sin(i) = n_1 \sin(r)$, which results in a large $i_1 = \frac{\pi}{2} - r$ such that $i_1 > i_{1\text{lim}}$, ensuring total internal reflection.

Acceptance Cone Half-Angle ($i_{\text{acc}}$) Definition

In a step-index fiber, $i_{\text{acc}}$ represents the half-angle at the apex of the acceptance cone, measured relative to the optical axis $(Ox)$.

Axial Ray Minimum Travel Time ($t_{\text{min}}$)

The ray that arrives first in a step-index fiber of length $L$ and core index $n_1$ is the axial ray ($i = 0$), traveling a path of length $L$ at velocity $v = \frac{c}{n_1}$, resulting in $t_{\text{min}} = \frac{n_1 L}{c}$.

Propagation Velocity in Fiber Core ($v$)

The speed of light inside the fiber core of refractive index $n_1$ is $v = \frac{c}{n_1}$ (and not $c$).

Ray Path with Maximum Travel Time

In a step-index fiber, the ray with the longest propagation duration is the most inclined ray ($i = i_{\text{acc}}$), corresponding to limiting incidence $i_1 = i_{1\text{lim}}$ at the core/cladding diopter.

Unfolding Method for Longest Path Length ($L_{\text{max}}$)

The zigzag path of the most inclined ray ($i_1 = i_{1\text{lim}}$) is unfolded into a right triangle of adjacent side $L$ and hypotenuse $L_{\text{max}}$, yielding $\sin(i_{1\text{lim}}) = \frac{L}{L_{\text{max}}} = \frac{n_2}{n_1}$, which gives $L_{\text{max}} = \frac{n_1 L}{n_2}$.

Maximum Propagation Duration ($t_{\text{max}}$)

For a step-index fiber of length $L$ with core index $n_1$ and cladding index $n_2$, the maximum travel time corresponding to limiting incidence $i_1 = i_{1\text{lim}}$ is $t_{\text{max}} = \frac{n_1 L_{\text{max}}}{c} = \frac{n_1^2 L}{n_2 c}$.

Intermodal Dispersion Pulse Broadening Shape

For a very short input pulse at $t = 0$, the output pulse in a step-index fiber spreads out between $t_{\text{min}}$ and $t_{\text{max}}$, resulting in a pulse duration of $\Delta t = t_{\text{max}} - t_{\text{min}}$.

Pulse Broadening Duration Formula ($\Delta t$)

In a step-index fiber of length $L$, core index $n_1$, and cladding index $n_2$, the pulse duration caused by intermodal dispersion is $\Delta t = t_{\text{max}} - t_{\text{min}} = \left(\frac{n_1 L}{c}\right) \left(\frac{n_1}{n_2} - 1\right)$.

Non-Overlap Condition for Transmitted Optical Pulses

For optical pulses transmitted at frequency $f$ with negligible duration relative to period $T$, the condition to prevent overlap at the fiber output is $T > \Delta t$, which implies $f < \frac{1}{\Delta t}$, where $\Delta t$ is the output pulse broadening duration.

Step-Index Fiber Numerical Example ($n_1 = 1.456$, $n_2 = 1.410$, $L = 1\,\text{km}$)

For a step-index fiber with $n_1 = 1.456$, $n_2 = 1.410$, and length $L = 1\,\text{km}$, the pulse broadening order of magnitude is $\Delta t \approx 1.58 \times 10^{-8}\,\text{s}$, giving a maximum pulse frequency $f < \frac{1}{\Delta t} \approx 6.3\,\text{MHz}$ (maximum bit rate $\approx 6.3\,\text{Mbit/s}$).