Optical fiber

An optical fiber is a thin, flexible electromagnetic waveguide that carries light along its length. In a conventional step-index optical fiber, an optical fiber core with refractive index $n_1$ is surrounded by an optical fiber cladding with lower refractive index $n_2$. Suitable light rays remain confined by total internal reflection at this interface. The cladding is an optical part of the guide, distinct from its protective coating.^[ref]

The useful surprise is that a transparent strand can deliver light around a route that a free beam would leave. This makes optical fibers useful for communications and for carrying collected light to a fiber-fed spectrograph. Guidance, successful launch, and surviving optical power are separate questions: the optical fiber acceptance angle constrains launch geometry, while optical fiber attenuation describes loss along the route.

Guiding light and retaining useful optical power are different achievements. Charles Kuen Kao and George Alfred Hockham argued that impurities need not impose the measured losses of existing glasses on every future optical fiber. Their 1966 paper treats wave absorption, scattering of light and guidance separately; reducing impurity absorption leaves intrinsic material losses and geometric losses to address. This was a route toward practical communication, building on existing optical fibers, rather than the invention of every use of guided light.^[ref]

The launch experiment used clad glass optical fibers fixed in capillary tubes with epoxy for end polishing. Laser illumination gave transverse normal mode patterns; input-spot displacement and polarization of an electromagnetic wave controlled excitation. These observations tested guidance, not kilometre-scale optical fiber attenuation.^[ref]

Charles Kuen Kao's lecture describes the complementary material experiment: short, polished glass samples, including plasma-deposited fused silicon dioxide, were examined with purpose-built single-beam and later double-beam spectrophotometers. A separate ellipsometry measurement assessed end-surface specular reflection, a check against confusing interface loss with bulk wave absorption. The readout was loss versus wavelength. Comparing glass samples supported impurity reduction as a route to lower loss. Surface contamination and the tiny bulk loss of short samples limited inference; the result did not itself demonstrate a long drawn optical fiber. The lecture also credits subsequent manufacturing work at Corning.^[ref]

An original scale comparison makes the material problem tangible. In the uniform optical fiber attenuation model, a stipulated $20dB/km$ leaves $10^{-2}=0.01$ of launched optical power after $1\mathrm{km}$; a stipulated $0.2dB/km$ leaves $10^{-0.02}\simeq 0.95499$. Both guides could satisfy the same total internal reflection condition. A successful spot pattern at the output therefore cannot by itself establish a useful communication distance.

A step-index optical fiber has a nearly uniform refractive index $n_1$ throughout its optical fiber core, with an abrupt decrease to $n_2<n_1$ in its optical fiber cladding. In a straight, ideal guide, meridional rays stay in a plane containing the optical axis and trace a zigzag through repeated total internal reflection. This is a geometrical optics approximation; it is particularly useful for a multimode optical fiber whose core diameter is large compared with the wavelength.

The upper panel shows an accepted meridional ray; the lower panel enlarges the limiting geometry. The angle of incidence $i$ at the side wall is measured from its normal vector, whereas the internal ray angle $\varphi$ and external launch angle $\theta$ are measured from the optical axis. Thus $i+\varphi =90^{\circ }$. The limiting ray is a boundary of the ideal model, rather than a complete normal mode solution.

The dimensionless waveguide parameter $V$ combines radius, vacuum wavelength and the numerical aperture factor. In the weak-guidance scalar model the first additional spatial family appears at $V=j_{0,1}=2.404825\dots$, the first positive zero of a Bessel function of the first kind. The fundamental family has no positive cutoff. Its two polarizations are not two distinct spatial profiles.

For a light ray launched on the centreline in the plane of a circular optical fiber bend, let $R$ be centreline radius, $r$ core radius and $\beta$ the internal launch angle from the tangent. The straight segment's distance from the bend centre is $R\cos \beta$. Thus its outer-wall angle of incidence satisfies $\sin i=R\cos \beta /(R+r)$. Equating this to $n_2/n_1$ gives the limiting total internal reflection threshold in the title, provided $n_1\cos \beta >n_2$. Radially displaced or nonplanar launches need different geometry; this is not a universal wave-optical loss cutoff.

The optical fiber acceptance angle is the largest launch half-angle admitted by the specified guide model. For a flat entrance face perpendicular to the optical axis, external refractive index $n_0$, and a straight step-index optical fiber, Snell's law at the entrance givesAt the core–cladding interface, the critical angle for total internal reflection $i_c$ satisfiesFor a meridional ray, strict total internal reflection requires $i>i_c$. At the limiting geometry $i=i_c$, the refracted light ray is tangent to the interface, soThis identifies the numerical aperture with an entrance cone under these assumptions. With $n_0=1$ for air, the conventional inclusive boundary condition isfor a nonnegative launch angle $\theta$; a practical launch stays inside the boundary. If $\mathrm{NA}>n_0$, this ideal model imposes no further restriction within the forward hemisphere, and $\arcsin (\mathrm{NA}/n_0)$ must not be used. The light ray must also intersect the optical fiber core: a suitable angle alone does not correct a misplaced beam.

For an original example, take $n_1=1.50$, $n_2=1.48$ and air at the entrance. ThenThe internal limiting angle is only ${\varphi }_{\max }\simeq 9.37^{\circ }$, while the critical angle for total internal reflection is $i_c\simeq 80.63^{\circ }$. A $10^{\circ }$ launch has $\varphi \simeq 6.65^{\circ }$ and $i\simeq 83.35^{\circ }$, so it satisfies total internal reflection. A $20^{\circ }$ launch has $\varphi \simeq 13.18^{\circ }$ and $i\simeq 76.82^{\circ }$, so it fails this test.

This cone is a geometrical optics result for meridional rays. A single-mode optical fiber requires the spatial overlap of the incident electromagnetic wave with its guided normal mode; its numerical aperture does not impose an exact angular cutoff on arbitrary beams. Diffraction, finite cladding, and bends require a physical optics description.^[ref]

A multimode optical fiber supports several guided normal modes at the operating wavelength. Different mode populations can change its output cone, so launch conditions matter when measuring numerical aperture or interpreting focal-ratio degradation.

A single-mode optical fiber supports only its fundamental spatial guided normal mode at the operating wavelength, with possible distinct states of polarization of an electromagnetic wave. The designation must include wavelength: MIT's fibre laboratory explicitly notes that a fibre sold for single-mode operation can support several normal modes at its helium–neon laser wavelength.^[ref]

A graded-index optical fiber has an optical fiber core whose refractive index varies smoothly with distance from the optical axis. Refraction bends light rays through the varying medium; the straight zigzag of a step-index optical fiber no longer describes their paths. The local launch cone can depend on entry position.^[ref]

MIT's Optical Waveguides and Fibers, Laboratory Exercise No. 7 (Fall 2005), describes a teaching apparatus. A cleaned, flat-ended multimode optical fiber receives helium–neon laser light through a microscope objective optical lens; a micropositioner aligns the input. The launch deliberately overfills the fibre's cone. A beam splitter sends a sample to a reference photodetector to track laser fluctuations. The fixed output faces a screen or a photodetector on a rotation stage with angular measurement calibration. Readings every two degrees, divided by simultaneous reference readings, give an angular profile. Repeating with an underfilled launch is a control: too few excited normal modes underestimate numerical aperture. Optional index-matching fluid removes cladding-carried light. Bending the fibre reveals leakage and reduced output on an oscilloscope. The handout supplies no measured trace or absolute detector measurement calibration, and this angular scan does not determine loss per kilometre.^[ref]

Optical fiber attenuation is the reduction of transmitted optical power along an optical fiber. Define a positive loss in decibels by comparing optical power $P(0)$ already launched into the guide with surviving optical power $P(z)$:The factor ten applies to a power ratio; the factor twenty applies to a wave amplitude ratio only when power is proportional to its square under the same conditions.^[ref]

For a uniform illustrative model with constant loss coefficient ${\alpha }_{\mathrm{dB}}$ in $dB/km$ and distance $z$ in kilometres,For ${\alpha }_{\mathrm{dB}}=0.2dB/km$ over $10\mathrm{km}$, the loss is $2\mathrm{dB}$ andThus about $63.1\%$ of the launched optical power survives and $36.9\%$ is lost. This is an illustrative coefficient, not a property of all optical fibers, and is independent of the earlier illustrative refractive indices.

The exponential decay follows from assuming each equal distance interval transmits the same fraction. For two such intervals, power ratios multiply; their logarithms add, so the losses in decibels add. Two $5\mathrm{km}$ sections each transmit $10^{-0.1}\simeq 0.794328$; together they transmit $(0.794328{)}^2\simeq 0.630957$. The loss curve is straight in decibels and curved on a linear power scale.

Wave absorption and scattering of light can remove guided optical power even in a straight fibre, while bends can let it escape. Entrance losses, joints and connectors contribute additional losses; they belong separately from a uniform distance coefficient. This is why a measured end-to-end power ratio need not equal the straight-fibre model.^[ref]

An optical fiber cutback measurement estimates distributed optical fiber attenuation by comparing a long guide with the short input portion left after cutting the output end back. Keep the launch untouched. For the following original functional model, fix wavelength, source output, input alignment and collected output cone; subtract a dark reading and keep the photodetector in its linear range. Renewed end-face quality and changed normal mode populations can still spoil the comparison. The diagram describes a measurement principle, not Kao's historical short-glass apparatus.

Let $L_{\ell }>L_s$ be lengths in kilometres, $P_{\ell },P_s$ the corresponding positive dark-subtracted readings, and $C$ the same launch and detection factor. A uniform guide obeysHere ${\alpha }_{\mathrm{dB}}$ and its estimate are numerical values in $dB/km$. Dividing eliminates $C$; taking a logarithm proves the formula. A changed $C$ instead adds an error $10{\log }_{10}(C_s/C_{\ell })/(L_{\ell }-L_s)$. Reconnecting the launch between readings defeats this cancellation.

For original chosen values $L_s=0.1\mathrm{km}$, $L_{\ell }=2.1\mathrm{km}$, $P_s=100\mu \mathrm{W}$ and ${\alpha }_{\mathrm{dB}}=0.5dB/km$, the predicted long reading is $P_{\ell }=10010^{-0.1}\simeq 79.4328\mu \mathrm{W}$. The ratio gives $1\mathrm{dB}$ for the removed $2\mathrm{km}$ and hence $0.5dB/km$.

Suppose an unremoved additive detector background contributes $b=5\mu \mathrm{W}$ to both readings. Then the apparent coefficient isThis is a bias of an estimator: a common additive background pushes the ratio toward one, whereas a common multiplicative factor cancels. For independent small zero-mean errors with standard deviations ${\sigma }_s,{\sigma }_{\ell }$ on the dark-subtracted readings, a first-order Taylor expansion givesChoosing both standard deviations as $1\mu \mathrm{W}$ gives ${\sigma }_{\hat{\alpha }}\simeq 0.03491dB/km$. A shared estimated dark offset introduces covariance, so it requires a different uncertainty calculation. Near zero signal, the logarithm and this linear approximation become unreliable.

Focal-ratio degradation is the broadening of the output angular cone of an optical fiber relative to its input cone. It can make a spectrograph collimator lose transmitted light or require larger optics. Stress, bends, and mode mixing in the fiber can contribute.

The central guiding region of a conventional optical fiber; in a step-index optical fiber, its refractive index exceeds that of the surrounding optical fiber cladding.

The optical material surrounding the optical fiber core. In a conventional step-index optical fiber, its lower refractive index permits total internal reflection; it is distinct from the protective coating.