Part 1: Fiber Optics Fundamentals (a prerequisite to understanding side fire fibers)
I’ve been told a half dozen tales that purport to describe the origin of fiber optics and likely as many claiming to describe the first side fire fiber, but there is little or no evidence for any stories I have heard over my 30+ years in the field…except the tale of “the Hitachi patent”. Anyone working in side fire fiber design has likely heard of this infamous patent, but I'd venture few have read it. The Hitachi fiber is the earliest patented side fire fiber that I have found, but I prefer using the moniker of US District Courts: “the Abe fiber”, for the first named inventor. (US Patent 4,740,047, Abe, et al., was originally filed in Japan, in 1985.)
Some basic understanding of how large core, multimode optical fibers actually work is necessary in order to appreciate the issues that affect side fire fiber design and performance -- why some side fire fibers work better than others. What follows is far from a comprehensive treatise on optical fibers, but rather is the bare bones needed I as a prerequisite to understanding the true complexity of side fire fiber design issues (excluding obsolete metallic reflector models, that is).
Optical fibers emitting energy lateral to the fiber’s axis existed long before Abe. In fact, the fundamental principle of modern side fire fibers are based, has been known for more than a millennium and is commonly called ‘Snell’s Law’, for the sixteenth century Dutch astronomer (aka “the law of refraction”), even though the earliest description of the phenomenon dates to the Abbasid Caliphate circa AD 984. (I suppose the luminaries of Snellius' time might not have been fluent in Arabic.)
Snell (Willebrord Snellius) described the bending of light where it transitions from one material to another: a stick in a pond appearing to bend or break at the surface, for example. The pond’s surface represents a boundary from a low refractive index material (the air) to a higher refractive index material (the water). Light does not bend if it enters the pond perpendicular to the pond surface, at an incidence angle of zero, normal to the plane of the pond surface, but it will bend at all other angles of incidence, with the angle of the bend being proportional to the angle of incidence. This is refraction:
n1 x sin(θ1) = n2 x sin(θ2)
where n1 and n2 are the refractive indices of the first and second medium and the angles θ1 and θ2 are the angle of incidence for the light ray in medium 1 relative to the interface surface normal, and angle of refraction within medium 2, also relative to normal, respectively. Note there are four distinct cases for refraction in the figure: at left, light traveling from the air into water and from water into air again, and right, light traveling from air into silica glass and from silica glass into air again.
The larger the difference in refractive indices, the more the beam is refracted (where the angle of incidence is the same). It is subtle, but in the figure you can see that the ray hits bottom closer to the right edge of the “block” of water than the block of silica because the light has not been turned as much in the lower refractive index water. It’s also worthy of pointing out that the refraction is completely reversible; the ray returns to the same direction upon exiting the higher refractive index material, albeit shifted in space.
I find the standard nomenclature used for referencing these phenomena unfortunately inconvenient for optical fibers and for side fibers in particular. As seen in the figure above, the angles are referenced to the normal line to the interface plane such that the angle of refraction in the higher refractive index material (silica) is 36.4° where it is larger (40.6°) in the lower refractive index material (water), even though the magnitude of the bending effect within the silica is larger than that in water. This convention is even more confusing inside of an optical fiber where we refer to angles of propagation as angles relative to the fiber axis, but the laws of refraction describing the propagation itself reference angles perpendicular to the fiber axis (normal to the cylindrical surface of the fiber). This standard convention is largely abandoned within the confines of side fire patents, where angles are referenced to the fiber longitudinal axis instead.
Complicating refraction issues, partial reflections also occur at refractive index boundaries; these images reflected by calm water or clean glass window panes, familiar to us all, are “Fresnel reflections”. Briefly, and neglecting plane polarized light, the larger the angle of incidence (relative to normal) becomes, the stronger these Fresnel reflections become and the less light crosses the interface into the second medium. Where the second medium has a lower refractive index than the first, as is the case for light traveling within a glass fiber surrounded by air (see figure below), there is a critical angle (θC) where none of the light passes into the air at all; instead, 100% of the light is reflected back into the higher refractive index medium (glass). This is Total Internal Reflection or TIR.
In the left side of the illustration, an incident ray encounters the surface of the silica glass at angle θ1 where the angle is lower than the critical angle. The bulk of that light is refracted at angle θ2, but a small portion of the light is also reflected, as depicted by the pink ray. This partial reflection is the Fresnel reflection. Where the angle of incidence is high enough, as depicted at right side for angle θC, 100% of the light ray is reflected back into the silica. This is the principle that captures light within an optical fiber: total internal reflection.
While we are on the subject of basic optical fiber theory, it may be worthwhile to dig a bit deeper into this TIR phenomenon because many readers will have the same questions here. If TIR works in a simple glass rod or sheet of glass, why aren’t optical fibers just simple glass filaments? Why do we need the complication of a cladding material?
The answer is simple enough; there are practical problems with bare glass that render most applications thereof impractical, namely dirt, dust and oils. Simply put, since the critical angle θC is calculated from
θC = arcsin (n2/n1)
(my apologies for equation format, Word equations do not copy to Shopify)
where n1 is the originating medium (silica glass in the case of side fire fibers where n1 ≈ 1.457) and n2 is the second medium (air, for the example illustrated above where n2 ≈ 1.000), the critical angle θC is 43.3°. For clad fiber of the type used in side fire fibers, where the cladding is fluorine-doped silica and n2 ≈ 1.44, the critical angle θC is much larger at 81.3°. Keep in mind that 90° is parallel to the glass:air interface so just angles that are parallel and up to 8.7° off of parallel will be totally internally reflected. This is part of the confusing nomenclature I mentioned. For optical fibers we typically see critical angles manifest as Numerical Aperture (NA), and where fluorine-doped silica clad silica fiber is notoriously low NA, reflecting this low acceptance angle, yet the critical angle according to classical nomenclature is high. We have to convert to the complementary angle to describe the optical fiber TIR condition in terms that agree with fiber optic specifications.
From the equation, we see that the critical angle is proportional to the magnitude of the difference in refractive indices between the glass and the air. (Side note: high school geometry also taught us that you can’t have a critical angle if the first medium is a lower refractive index that the second, as an arcsine that is greater than 1 is undefined. This is a real phenomenon; TIR exists where light travels from glass to air but not from air to glass.) In the case of fused silica, in air the difference is roughly 0.457 (roughly because refractive indices are dependent upon wavelength). Dirt, dust, oils and water have higher refractive indices than air so the difference between contaminants’ refractive indices and that of the fiber is smaller. In other words, light that propagates by TIR within a clean, bare fiber will leak at every fingerprint and dust mote because the lower angles for TIR that are supported by clean glass to air are not supported by glass to dirt. (Nomenclature rears its ugly confusion once more, requiring clarification: the lower angles according to Snell’s law nomenclature leak but these are the higher angles according to fiber optics terminology – sometimes I think my head is going to explode….)
Light leakage at dirt on unclad surfaces is actually the basis for some colorful signage that was very popular in restaurants and bars a couple of decades ago. Edge lit sheets of glass were used to display daily specials and such with fluorescent crayons or markers. The clean sheet of glass totally reflects the light internally but, where the crayons contaminate the surface, the light leaks, illuminating the script.
Bare glass is also easily damaged, particularly high purity fused silica glass, so optical fibers are typically coated with a plastic or plastics that may serve as a cladding and/or a buffer or jacket. These terms are often interchangeable where multiple functions are supported by a single coating, giving rise to another source of confusion in this field. In brief, a cladding is a typically thin coating that has a lower refractive index than the core and is transparent to the wavelengths of light that the fiber is intended for. A buffer is a typically thin coating that is bound to the glass tightly, has a higher refractive index or is opaque to the wavelengths the fiber is intended for and serves to physically and/or chemically protect the underlying glass. The jacket is a typically thicker, extruded coating that serves a physical protective function, only, and is not bound to the glass or the buffer/cladding, and it may be removed (stripped) to expose the underlying layer(s).
Recapping, the cladding has a lower refractive index than the fiber core so TIR is supported, but because cladding materials have refractive indices that are significantly higher than air (closer to the core) -- that is 1.00 << n < 1.457 -- the critical angle θC for TIR is much larger than it would be for an “air clad” fiber. As shown in the clad fiber above, θC angles that are large relative to the normal are low angles in their complementary angle θP. The low complementary angles relative to the fiber axis are what give rise to the low angles of acceptance, or NAs, that are typical for optical fibers of the type used for side firing devices. In fact, it is the low NA that makes these types of fibers suitable for building side fire fibers.
The maximum angle that is accepted by a fiber (or the maximum angle light diverges from the fiber) becomes a bit more complicated by the fact that light refracts (to lower angles) upon entering the fiber face (from air), as we’d expect from our initial discussion of Snell’s law. While this complication does not come into play in any meaningful way with respect to side fire fiber design, it does merit a brief discussion, if for no other reason than to clear up NA versus θC in fibers.
NA can be calculated from the core and cladding refractive indices as:
NA = (ncore - ncladding)^1/2 (apologies, again: NA is the square root of the difference in the core and cladding refractive indices)
Plugging in the numbers we’ve been using where ncore ≈ 1.457 and ncladding ≈ 1.44 yields 0.22 NA, which is the NA of most fluorine-doped silica clad silica fiber or Si(F):Si fiber. But there is a tolerance to the NA due to variations in the concentration of the fluorine dopant that is trapped in the silica cladding matrix; the NA of these fiber materials is really 0.22 ± 0.02.
There is a relationship between NA and the maximum acceptance and propagation angle θMAX of a fiber that is also useful:
θMAX = arcsin(NA)
For 0.22 NA fiber, θMAX is 12.7° as depicted in the figure below (where the fiber input face is positioned facing upward to match the interface drawings we’ve used before). This maximum acceptance angle is, or course, refracted to a lower angle at the glass to air interface in keeping with the description used at the beginning of this work. Once inside the fiber the refractive index interface is rotated 90 degrees and is no longer silica glass to air, but silica glass to doped silica glass. The math still works; just make certain that you orient yourself correctly, or as I like to say, “Be the Photon”.
The incident angle θI is 12.7 degrees (angles not to scale) relative to the surface normal (blue) and the refracted angle θR is 8.7°, which is the same as the angle of propagation θP relative to the gray normal line to the core:cladding interface which is the complement of the critical angle θC or 81.3°.
It’s simple, really…
Now all we have to do is polish the fiber to an angle so that all the rays propagating in the fiber are reflected through the side of the fiber. This can be a bit confusing to some, but if we break it down into a couple of components it is simple enough. The first problem is that there are a range of angles propagating in the fiber so there is no single critical angle for all rays. There is, however, a worst case ray and a worst case critical angle where all the other rays will be incident at larger angles than their critical angles. You may determine what the worst case is and work from there, but I find it easier to simply recognize that the worst case ray angle will be the maximum angle of propagation θP more challenging than an axial ray. By calculating the critical angle for the axial ray and adding the angle of propagation, we arrive at the critical angle for the worst case ray.
First we need to get some nomenclature straight (and admit that the drawing is not to scale). The angle-polished tip is often called a bevel tip, but more commonly it is referred to as “the TIR” surface or angle. The gray normal line to the plane of the bevel and the gray arc define the critical angle for fused silica to air: it is, after all, air on the other side. We’ve calculated this before: 43.3°. The worst case ray reduces the effective critical angle by the angle of propagation, or 9.5° (not 8.7° because that is the maximum angle of propagation for the nominal NA and the worst case NA of 0.24) as indicated by the lower angle inscribed by the blue arc to the blue normal line). We must add 9.5° to the axial ray critical angle to accommodate the worst case ray, yielding 52.8°, but this is the complement to the bevel angle that produces this condition, as depicted by the green arc. A 37.2° polish angle is our result (90 – 52.8).
All rays imparting the 37.2° (relative to the fiber axis) TIR angle will be totally internally reflected through the side of the fiber, regardless of the fiber raw material lot. If you actually polish the fiber to this angle you will see that there are no axial rays – no light propagates to the right side of the page – but the output that you do achieve is a complete disaster; about 30% of the light goes in the wrong direction (generally toward the bottom of the page) and the light that does exit the desired side of the fiber is horribly distorted. We’ll examine why this happens next time and take a close look at Abe’s (partial) solution as we begin our journey through the history of side fire fiber evolution.
Thanks for reading,
Stephen Griffin
@doctorsilica
#LDD85 #Duet Fiber
© InnovaQuartz 2016