To better understand the basic properties of optical fibres, it is useful to have a look to the definition and propagation of the meridian and skew rays in cylindrical rods. A meridian plane is a plane which intersects the optical axis of the fibre. Meridian rays are those who travel on these planes. Since the meridian plane is perpendicular to the tangent at the edge of the cylinder, the meridian rays propagate always on this plane all along the fibre.
Figure 6 shows the meridian plane and the red ray travelling on this plane. The skew rays are all those rays which travel out of the meridian planes. In the same Figure 6, the green ray enters parallel to the meridian ray but out of the meridian plane. Since the incidence of the skew rays is not perpendicular to the surface of the cylinder, we can see that the skew rays “turn or rotate” along the fibre. This is the reason why we see a circle projected on a screen when we launch a laser beam inclined with respect to the optical axis (Figure 7). We can also notice that most of the rays entering an optical fibre are skew rays. However, we use the meridian rays to describe the propagation of the light along the fibres because they are easy to visualize and they remain always on the meridian plane.
Figure 6. Meridian and skew rays. The meridian rays (red) stay always on the meridian plane while the skew rays(green) enter into the fibre out of the meridian plane.
Figure 7. Projected circle pattern created by skew rays coming from an input parallel beam
4.1. Pupil position
Another interesting property of the fibres is the location of the pupil. If a light spot with a diameter smaller than the core of the fibre is launched anywhere on the fibre input end, the spot tends to expand all along the fibre. When the length of the fibre is much bigger than the diameter, the whole surface of the output fibre end appears evenly illuminated. This is the so-called scrambling power of the fibre. At the fibre end, each point on the surface of the fibre core will emit a beam with the same F/#. A multimode fibre behaves then like a telecentric system. In other words, the pupil is at infinity.
4.2. Numerical aperture
The numerical aperture (NA) in an optical fibre is a parameter that defines the maximal angle of the cone of light accepted by the fibre. Light beams entering with angles bigger than the NA are not transmitted because there is not more total internal reflection (Figure 8).
Equation below shows the relation of the NA with the refraction index and with the aperture of the input focal beam.
where no, nco and ncl are the refraction index of the air, core and cladding respectively. F/#in is the aperture of the input beam. For most practical examples, the refraction index of the air can be approximated to 1.0 (1.000293 at λ 589 nm).
As an example, the refraction index of the core and cladding of an all silica fibre are 1.4571 and 1.440 respectively at 633 nm. The NA is therefore 0.22. This is the standard value for most of step index fibres. The maximum angle of a ray can still be transmitted is θm = 12.9 deg. This corresponds to a F/# of F/2.22.
4.3. Transmission, attenuation, scattering and polarization
The internal transmission along an optical fibre is measured typically in percentage of the incoming light. It decreases exponentially with the length according to the Beer-Lambert law. If the internal transmission (T) is known for a given length (L) and a given wavelength, the internal transmission (T’) for another length L’ is given by equation
where 0 < T (or T’) <1
To get the total fibre transmission, one must not only add the reflection losses at the ends of the fibre (Fresnel losses, typically 4% per surface) but also the losses by polishing quality.
Example 1: Internal transmission in a step index silica fibre of 20m at 400 and 600 nm is typically 84 and 96 % respectively. What will be the total transmission for 5 and 40m? We have:
and at 400 nm
and at 600 nm
Including the reflection losses (8%) and neglecting the polishing errors, we have:
|Fibre length (m)||Total Transmission (%)|
|at 400 nm||at 600 nm|
Fibre manufacturers specify the fibre losses in another convenient parameter: the attenuation. It is measured in decibels per kilometre (dB/km). The relation between the internal transmission and the attenuation is:
where A is the attenuation, L the fibre length in km and T the internal transmission (0<T<1).
Example 2: What is the attenuation for the fibre described in Example 1?
Taking L = 20 m and T = 85 %, we have:
For 600 nm T = 96% and therefore
at 600 nm
The fibre transmission drops very fast in the near UV region limiting their applicability to few tens of meters. The Rayleigh scattering of light by silica is the theoretical limit of fibres transmission in the blue spectral region. Most of the commercial fibres optimized in the near UV are very close to this limit. Glasses with smaller Rayleigh scattering like sapphire may increase the transmission of fibres but the present technology is not able to make fibres in such materials or the manufacturing processes are very expensive. One hope to overcome this limit might be future improvements of the hollow core photonics fibres (Section 3.3).
About polarization properties, multi-mode step-index fibres do not maintain the state of polarization of the incoming beam. In fact, they are one of the best de-polarizing devices!
4.4. Focal Ratio Degradation (FRD)
4.4.1 Definition and origin
The focal ratio degradation or FRD is a parameter of prime importance in astronomical instrumentation. It is the main cause of flux losses linking telescopes to spectrographs. In practical terms is the gradual increase of beam aperture along the fibre. If for example a sharp F/15 telescope beam is injected into a fibre, the shape of the output beam will not be a well defined F/15 cone but fuzzy. Only a portion of the incoming flux will leave at F/15. The sharpness of the outer cone beam will depend basically of the manufacture quality and the physical conditions at which the fibre is submitted. An optical fibre will never preserve the original aperture of the incoming beam. In Optics, the FRD is the equivalent of the entropy of the geometrical étendue A𝛺.
The main cause of the FRD is the fibre bend, although twists contribute substantially to the FRD as well. Figure 9 shows the transmission of a ray travelling along the optical axis of the fibre (simulating a beam with high F/#). As soon as the fibre is bent, this ray is deviated from its initial direction and therefore it quits the fibre with an angle depending of the radius of curvature of the fibre. Therefore the emerging ray defines a beam with lower F/#.
This behaviour is not the same for rays entering with high angles with respect to the optical axis (still inside the NA!). This can be simulated by the blue ray which defines an F/3 aperture. After the fibre bend, the beam leaves the fibre with an ever bigger angle but the angle difference between this ray and the incoming one is not so large than for the case with the red beams. In other words, the FRD is smaller for faster beams. This is the reason why using fibres for astronomical applications we must work with beams with low F/#. It is not, however, advisable working at the numerical aperture of the fibre (F/2.22). The rays travelling close to the numerical aperture may go out of the fibre (no more total internal reflection) when they encounter a fibre bend! In the next section we will show what would be the best aperture – throughput compromise.
Figure 10 shows a practical example of FRD. An F/10 telescope beam is launched into the fibre. The fibre output end is distributed in cones with apertures F/2.2, F/6 and F/10. In this particular case, only 60% of the total output flux will leave the fibre in an F/10 cone, 40 % is lost by FRD! At bigger apertures, like F/6, 85% of the flux will be collected. Theoretically the 100 % of the flux should be inside the numerical aperture of the fibre (0.22 or F/2.22 for most fibres). However some flux may be still emitted beyond the NA. These rays are produced basically by surface diffusion due to polishing defects and by core/cladding interface imperfections along the fibre. These defects are easily seen when a power laser beam is launched into the fibre.
4.4.2 FRD experimental results and discussion
A way to measure the FRD is to project a well defined beam into the fibre and study the flux distribution of the output beam. Figure 11 shows one possible set up. An integrating sphere is illuminated by a monochromatic LED. The sphere is equipped with a pinhole simulating the star. The emerging beam has a Lambertian distribution and it is launched into the fibre by means of two doublets. The input aperture is defined by a number of stops mounted on a wheel. The typical apertures are F/2.22 (NA), F/3, F/4, F/5 and F/6. Smaller apertures are useless since the FRD is very high and therefore the flux losses are not acceptable.
The plane of the wheel is placed on the focal plane of the second doublet. At this location, the image of the pupil will be at infinity, i.e. at the same location that the pupil of the fibre. The size of the image of the pinhole projected on the fibre end is defined by the ratio of the focal distances of the doublets. The output fibre beam is projected directly on the detector of a CCD camera. The F/# of the output beam is defined by the ratio between the distance of the fibre to the detector and the diameter of predefined circles on the detector (computed along the data reduction).
The FRD can be relative or absolute. In the first case, the internal transmission of the fibre and the reflection losses at the surfaces of the fibre are not taken into account. Once a beam is projected on the CCD detector, the flux recorded in a circle opened at the numerical aperture defines the reference flux of 100 %. The FRD is therefore measured relative to the output flux. If the absolute FRD is needed, the total flux entering into the fibre has to be first recorded. For that, the image of the pinhole must be smaller than the fibre core diameter. The flux of the output beam projected on the CCD on a given circle (output F/#) is measured and the absolute FRD is the ratio of this flux divided by the total flux at the input of the fibre. In practice, these measurements are difficult to carry out, the optical aberrations of the projection optics must be low enough to be sure to inject all the input flux into the fibre. On the other hand, the measurement of the direct flux is easily contaminated by background illumination.
Figure 12 shows the FRD for 4 input beams (F/4, F/5, F/6 and F/8). The measured flux is absolute (it includes the reflection losses at the ends of the fibre) in the 400 to 700 nm spectral range. The fibre is 10 m long and the core has a diameter of 200 µm. Two cases are analyzed, when the fibre is free of bends (upper figure) and when is forced to follow a loop of 30 mm diameter. This loop is located in the middle of the fibre.
Figure 12. FRD in a 10m length, 200μm core bare fibre. The input beam ranges from 400 to 700nm. Above: the fibre is free of bends. Below: the fibre is submitted to a loop of 30 mm diameter
As an example, if an F/8 beam is launched into the fibre (blue curve in upper Figure), 74% of the total flux will be collected inside a cone opened at F/8 at the output fibre end. When the fibre is submitted to a bend of 30 mm diameter, the collected flux decreases down to 46% ! However, this degradation is less for faster beams. In the case of an input F/4 beam and when the fibre is free, 85% of the light will be recovered by an output cone opened to F/4. When the fibre is bent, the flux will drop to 79% only.
We have mentioned in Section 3.1.2 that gradient index fibres may be used for amateur astronomical spectroscopy. Their transmission is acceptable and their core diameter is comparable (50 and 62.5 μm) to the smallest step index fibres (usually 50 μm). Unfortunately the FRD shows in most cases unacceptable results even for fast apertures. Figure 13 shows the FRD (relative) of a 62.5 μm core, 3 m long gradient index cabled fibre. As expected, the non constant refraction index in the core scrambles all information of the ray directions along the fibre. If for instance, one wants to use such a fibre at F/5, only 38% will be recovered by a collimator opened to F/5. To that we have to add the internal ad surface reflection losses.
The only possibility is to work at apertures close to the NA (F/1.8): for an F/2.3 input beam we can recover 80% at F/2.3 at the output. In this case an optical reducer in front of the fibre will be needed to convert the telescope aperture to around F/2.3 in the fibre. We will see in Section 5.1 how to achieve this aperture coupling.
In ESO’s Fibre Laboratory we have collected the following experience:
- The FRD is the main cause of light losses when using fibres to link telescopes to spectrographs. It is higher than internal transmission, reflection losses at the fibre ends and flux differences between slit or round apertures
- The FRD depends highly of manufacturing process. Micro bends and deformations on the interface between the core and cladding may alter the direction of the reflected rays. Additional bends on the cladding may be produced by stresses generated during the extruding of the jacket (acrylate, silicon, polyimide, nylon, etc.). The FRD may be quite different in fibres with the same geometry but made by different manufacturers. More surprising is to find substantial differences in FRD in similar fibres but coming from different batches of the same manufacturer!
- The FRD does not depend of the core diameter. However, small fibres are more vulnerable of micro bends along the manufacturing and therefore the FRD may be higher
- The FRD slightly increases with the length of the fibre. However, beyond a given length, the FRD is quite constant. We have found empirically that for 50 μm fibres the length is around 2 to 4 m. For 600 μm fibres, the length should be larger than 10 m
- A bare and free fibre can have intrinsically a very low FRD, but when this fibre is prepared inside a cable and ended with ferrules or connectors, the FRD may be unacceptable! In practice, the process to assemble a fibre cable is the mayor cause of FRD.
a. Standard fibre cables include a Kevlar braid to avoid breakage of the fibre when the cable is pulled. For that, the fibre is a bit longer than the protection Kevlar braid and when the cable is pulled, the Kevlar absorbs the strength. The problem is that the fibre adopts a serpent shape with high bends and therefore the FRD increases.
b. An additional issue is the adhesive to glue the fibre inside the ferrule. Very hard glues or those who polymerize at high temperatures create strong stresses on the fibre (micro bends) affecting enormously the FRD
c. Residual twist of the fibre when is slide inside the protective cable may substantially contribute to the FRD
- A “good” fibre cable shows an FRD smaller than 15% for an F/3 input and output beam. This means a relative transmission of 85%. Including the Fresnel losses at the fibre ends (4% per surface), the transmission is reduced to 78%. To that one must add the internal transmission of the fibre. As seeing in Section 4.2, this transmission can be for instance 96% at 600 nm for a 20 m fibre. Therefore the total transmission of the final cable is 75%. Actual professional telescopes provide beams usually opened to F/8 or slower (except in especial cases when they work in prime focus configuration where the aperture can be precisely F/3). A focal reducer is then needed in front of the fibre input end to convert the given telescope aperture to around F/3. This optics introduces additional losses: aberrations, misalignment and transmission losses. Typical good full fibre links have a total throughput around 70% (in the red side of the spectrum)
4.4 How to estimate the FRD
As described in Section 4.3.2, precise measurements of the FRD in fibres require some laboratory equipment: an optical bench to define a light beam with the proper aperture, a stable light source and a detector. However, there is a rather simple method to evaluate the FRD and decide to use a given fibre or refuse it. What you need it is a low power visible laser beam. A pen laser pointer is an excellent tool. First, launch the laser beam in front of the fibre end. Of course, the fibre must be polished on both ends to perform the test! The beam has to be parallel to the optical axis of the fibre (or perpendicular to the surface of the fibre end). Send the exit beam to a white surface placed to 20 or 30 cm from the fibre end. Inspect the spot on the screen. Pay attention to the size of the spot and check how grows when the laser beam arrives with an angle to the fibre axis. When you still increase the incoming angle, the spot on the screen becomes a ring. You have to adjust the laser angle in order to minimize the spot size. Under this condition measure the spot diameter and determine the F/# (= f/d, where f is the distance between the fibre output end and the screen, and d the diameter of the projected spot on the screen). The spot is not well defined and you will find difficulties to determine the edge to measure the diameter. Indeed, the sharpness of the spot gives information of the FRD: smaller the FRD, sharper is the spot.
As a rule of thumb, fibres with excellent FRD fibres provide beams with aperture smaller to F/15. Bad fibres emit apertures faster than F/4 (F/# < F/4). This evaluation is not precise and should be taken just as a reference.
4.5 Which FRD is acceptable?
As explained in Section 4.3.1, the main causes of the FRD are the micro-bends and micro-roughness between the core and cladding along the fibre. These defects are highly increased when the fibre is passed through the protective jacket and glued inside the connectors. The fibre is not anymore free but in constant contact with the jacket. Some of these points may strongly press the fibre, especially when the fibre is longer than the jacket. In this case the fibre adopts a serpent shape with strong bends. This problem is quite common in telecommunication fibres where the protection cable includes Kevlar bread to avoid breaking the fibre when it is pulled. Indeed, if the fibre is a bit longer, the tension is received by the Kevlar bread and not by the fibre. In addition to these problems the stresses on the fibre created by the polymerization process of the glue may be so strong that this is the main cause of FRD. The adhesives with high hardness coefficients induce very strong stresses on the surface of the fibre. This effect is even higher when air bubbles remains at the moment the glue is applied. The pressure of these cavities on the glue when the polymerization occurs can create strong pinches on the fibre.
Figure 9 shows the FRD in fibres ready for use at the telescope. Left side shows a cabled fibre of 34 m. The core diameter is 60 µm. For an input-output F/3 beam, the FRD is 12% (or a relative efficiency of 88%). For F/6, the efficiency falls to 59%. We can see that the FRD is higher from the values for bare-free fibres (Figure 7).
Figure 9. FRD in cabled fibres for use at the telescope
In practical cases, if you procured a commercial cabled fibre, there are few chances to reduce the FRD already produced by the contact with the protective jacket and glue stresses. You can only evaluate it and hopping that it is acceptable. Of course you have to avoid additional stresses and bends on the fibre not to increase the FRD when the fibre is installed between the spectrograph and telescope! The question here is to know what is acceptable. In professional telescopes a throughput of more than 65 % is usually tolerable. This figure includes everything: internal transmission, Fresnel losses, FRD and lens coupling losses. Best fibres are close to 80 % ! In amateur spectroscopy most the fibres are commercial and they are not optimized to reduce the FRD. Therefore fibres showing a total transmission bigger than 40 % are quite acceptable.
Of course you can design the aperture of the spectrograph collimator to work at higher apertures, but as we will discuss in Section 5.1 the price to pay is a reduction in the resolving power of the spectrograph. If you still want to keep the resolving power, you have to increase the size of the optics: bigger diameter of the collimator, larger diffraction grating and bigger CCD objective.
5 Photometrical scrambling
Notice: this section is for the moment not relevant for most amateur spectroscopic applications. However, it is important in high accuracy measurements of radial velocities.
A limiting factor in the high accuracy measurements of radial velocities is the stability of the photometrical distribution of the light emerging from the fibre and feeding the spectrograph collimator. The point spread function (PSF) on the detector may shift slightly along the dispersion due to minute movements of the star in front of the fibre (guiding errors, for instance). This residual “memory” of the fibre is quite evident when the image of the star on the fibre input end shifts from the centre to the edge. In other words, the fibre keeps a bit of information on the position of the input spot. The photometrical scrambling is the ability of an optical fibre to reduce this remaining position information. In the case of HARPS spectrograph, this spurious shift limits the accuracy in radial velocity to few meters per second.
The residual information of the light distribution at the input fibre end has basically radial symmetry. The skew rays are mainly the generators of this radial symmetry (Section 4). There are two cases showing this effect: the near and far fields. In the astronomical jargon, the light distribution on the surface of the fibre is called “near field” and the angular distribution of the beam is called “far field”. Since the pupil of a fibre is at infinity, the far field is indeed the photometrical distribution of the pupil.
5.1 Scrambling gain
Figure 10 shows the effect of the fibre scrambling on the spurious shift of the point spread function (PSF) on the detector of the spectrograph. The image of the star is projected on the centre of the fibre input end (blue star). This configuration generates a PSF on the CCD as represented by the blue curve. When the star shifts with respect to the centre by a distance d, the PSF might move along the dispersion by a small shift f. The scrambling gain is defined by the ratio between the relative shift of the star and the shift of the PSF.
In this equation D is the fibre diameter and F the full width maximum of the PSF.
The photometrical stability of the PSF on the spectrograph detector is affected by the flux distribution on the output fibre surface (Near Field) end and by the angular illumination variations on the pupil of the spectrograph (Far Field, here, the pupil of the fibre is at infinity). In the hypothetical case where the flux distribution is constant on the surface of the output fibre end (near field) when the star moves in front of the fibre input end, the centre of mass of the PSF will depend only of the flux distribution on the pupil of the spectrograph (far field). Even if the spectrograph optics is diffraction limited, any variation on the pupil illumination will change the shape of the PSF and therefore its centre of mass.
A way to measure the shift of the PSF is to use a lens to image the fibre end on a CCD camera. The photometrical centre of mass or barycentre can be computed and therefore the shift can be estimated. Any variation on the surface of the fibre end will affect the barycentre. We have to point out that the far field variations may affect also the PSF even if the near field is constant. In the case of the study of the far field variations, the output beam from the fibre can be projected directly on the CCD, provided that the distance fibre-CCD is enough large (few centimetres). One way to quantify the variations of the far field would be to record the profile of the far field spot on the CCD and see how evolves when the star shifts in front of the fibre input. The ratio of the profile of the “shifted” spot to the profile of the spot on the “centre” gives a profile of distortion factor.
5.2 Scrambling gain measurements
Laboratory tests show an increase of the scrambling gain with the length of the fibre. It also increases when the fibre is submitted to bends. A fibre with a core of 60 μm diameter, illuminated with a spot of 50 μm, a beam input of F/2.5, a length of 3 m and free of strong bends shows a scrambling gain of G = 120. When this fibre is forced to follow a small bend in such a way that the FRD increases of 20%, the scrambling gain jumps up to 500. On the other side when a 600 μm fibre of 3 m, Injected with a F/3 beam, a spot of 250 μm and free of bends, the gain is G = 150. When submitted to follow a serpent shape with bends of 15 mm radius giving a FRD of 20, the scrambling gain boost up 3000!
Characterization of Optical Fibres by CAOS group is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 Germany License.
Based on a work at spectroscopy.wordpress.com.