A model is developed to evaluate the light collection of a diffuse light source located at the tip of an optical fibre. The model is confirmed experimentally and used to evaluate and compare the light collection efficiency of different fibre-coupled luminescence dosimeter probe designs. The model includes contributions from both meridional and skew rays, and considers the light collection from an optically attenuating scintillator. Hence the model enables the optimisation of different, but useful and new probe materials such as BeO ceramic. Four different dosimeter architectures are considered, including previously investigated probe designs; the butt-coupled and reflective wall, along with two novel designs. The novel designs utilise a combination of the scintillating material and transparent media to increase the light collection. Simulations indicate that the novel probes are more efficient in light collection for applications in which it is necessary to minimise the volume of the scintillating material.
© 2014 Optical Society of America
Collection of the emission of point sources at the tip of an optical fibre is conceptually important since it can be employed in various applications, such as fluorescence-based sensing [1, 2], fluorometers , chemical and biological optical spectroscopy , and radiation detection and dosimetry [5–12]. Thus it is important to identify the parameter regime for optimum collection of light emitted by point sources.
Optical fibre-coupled luminescence dosimetry has been an increasingly investigated topic due to its attractive attributes for dosimetry in radiation oncology [5–11] and in the general detection of ionising radiation . In general, this is where a phosphor probe is attached to an optical fibre. When exposed to ionising radiation, light is emitted from the probe, known as radioluminescence (RL) or scintillation. A portion of this light can then be collected by the optical fibre and guided to a reader . Plastic scintillators have been the most common probe material. Another branch of fibre-coupled luminescence dosimetry has been the use of a light source to stimulate charges stored in some probes due to the exposure to ionising radiation. Upon stimulation of the stored or trapped charges, light is emitted. This process is known as optically stimulated luminescence (OSL). In fibre-coupled luminescence dosimetry a portion of the OSL can be collected by the optical fibre [8, 9]. Among available detectors, Al2O3:C crystals have been the most common probes for OSL. Both RL and OSL have been shown to be able to potentially satisfy the need for a small and sensitive dosimeter in radiation oncology for such applications as the dosimetry of small field sizes  and in-vivo brachytherapy dose verification [7–10].
Optimisation of the light collection, either RL or OSL, by the optical fibre is of great importance in fibre-coupled luminescence dosimetry in order to increase the sensitivity of the dosimeter. A very important phenomenon in fibre-coupled luminescence dosimetry is the “stem effect”, which is the term given for light produced within the optical fibre when it is exposed to ionising radiation. This produces unwanted signal in the reader, hence reducing the signal-to-noise ratio . Therefore, increasing the light collected from the phosphor probe reduces the effect of the stem effect.
The light collection of optically transparent, plastic scintillator coupled fibre optic dosimeters has been modelled for a number of designs using ray optics [14, 15]. Modelling of the light coupled from a cylindrical scintillator into an optical fibre has shown that the addition of reflections from the scintillator wall improves the light collection from a butt-coupled geometry. The use of a reflective coating on the end face of the scintillator effectively doubles the length of the scintillator. The use of optical elements such as lens between the scintillator and optical fibre can also improve the performance, though in some cases the improvement is marginal .
The purpose of this study is to investigate the light collection of different phosphor probe designs, for an optically attenuating phosphor, in order to maximise the light collection for a given probe design. The light collection is modelled by the use of ray tracing. We include skew rays in the model since they make up the majority of rays in an optical fibre. Also by explicitly including the attenuation of phosphor probes in our models, we have investigated the collection efficiency of different probe designs in the presence of optically attenuating scintillating materials such as beryllium oxide (BeO) ceramics . Recently BeO ceramics have been investigated for their use in fibre-coupled luminescence dosimetry because of their attractive properties; near tissue equivalence, exhibiting RL and OSL .
The theory for the light collection modelling was developed by the consideration of multimode optical fibres with comparatively large cores. Hence a ray optics approach is valid and used in the modelling of the light collection from the different probes. Rays in a multimode fibre are either meridional or skew. Meridional rays, depicted in Fig. 1(a) pass through the optical axis after every reflection from the core-cladding boundary, whereas skew rays, depicted in Fig. 1(b) spiral down the fibre, never passing through the optical axis. Meridional rays are defined by a single angle, the longitudinal propagation angle,, which is the angle between the ray and the fibre optical axis. Skew rays are defined by two angles; similarly to meridional rays, the longitudinal propagation angle,, and the azimuthal propagation angle, . This is the angle that the projection of the ray onto the fibre cross section, makes with the tangent to the fibre core at the point of incidence. The total angle of incidence for a skew ray, α, depicted in Fig. 1(c) is given by cos = sinsin.
The rays in a multimode fibre can be categorized into; bound, refracting and tunnelling. Bound rays are able to propagate in the core of the fibre and are dependent on the longitudinal propagation angle,. These are confined within the critical angle,, defined by cos = ncl/nco. The range of longitudinal propagation angles for which the rays are bound is hence given by; 0≤ <.
Refracting rays, unlike bound rays, are lost at the core-cladding boundary. They are defined on their total incidence angle,, with ranges: 0<<. Where, the critical incident angle is defined by sin = ncl/nco.
Tunnelling rays are skew rays which are not bound rays but are still able to propagate significant distances along a fibre. Their longitudinal propagation and total incident angles are confined within the ranges given by≤ ≤π/2 and≤ ≤ π/2. Whereas Tunnelling rays are due to the curvature of the optical fibre, and thus as an example a slab waveguide would not have tunnelling rays .
In the current model, the scintillator is treated as many point sources. Due to the symmetry of the cylindrical scintillators, any source point of the same distance from the optical fibre and the optical axis will contain identical light collection properties. Therefore an entire annulus of points with the same distance from the optical fibre and the optical axis, can simply be modelled by a single point, as depicted in Fig. 2.
The parameter t, R1 and R2 shown in Fig. 2(a), determine the resolution of the number of diffuse source points modelled. The total power of either bound, refracted or tunnelling rays is given by Eq. (1):
Here, Ωb, Ωr, and Ωt are the solid angles produced by the bound, refracting and tunnelling rays, respectively. They are determined by the fraction of each class of rays from the total rays modelled, given by Eq. (3).
Therefore the total power of a slice of the scintillator is calculated by the summation of all annuli of the same distance from the optical fibre.
Figure 3 shows the probe arrangements investigated here. The butt-coupled and reflective wall configurations, shown in Figs. 3(a) and 3(b) respectively, have been previously investigated for transparent plastic scintillators but without considering skew rays . Two novel arrangements are modelled here; named double-cladding and cladding-coupled, shown in Figs. 3(c) and 3(d) respectively. These novel designs utilise a combination of the scintillating material and a transparent media to increase the light collection.
The butt-coupled design shown in Fig. 3(a), is where a cylindrical scintillator is simply coupled directly to the optical fibre. In modelling the light collection from the butt-coupled architecture, only rays which are directly incident on the optical fibre core can be collected. All longitudinal propagation angles within the range; 0≤ ≤ π/2 are modelled, along with the cross sectional angle,, depicted in Fig. 4, with range; 0<≤ π. This will model all forward propagating rays towards the optical fibre. All backward propagating rays are not modelled since they will not be incident on the optical fibre. With these two angles, the Cartesian coordinates of the ray path can now be described by Eqs. (4)–(6), where r is the ray path length.
The azimuthal angle,, can be calculated from the modelled cross sectional angle,, shown in Fig. 4, where b is the distance of the emitting source at point A from the optical axis, and rs is the radius of the scintillator, by using the sine rule:Eqs. (4)–(6), those rays not incident on the core of the fibre can be rejected since they will not contribute to collected light. The rays can then be characterised as following:
- (1) If < then the ray is a bound ray within the optical fibre. is the maximum acceptance angle of the fibre, determined from the complement of the critical angle of the fibre and given by Eq. (9).
- (2) If < then the ray is a refracted ray in the optical fibre. is the maximum total incidence angle, determined from the critical angle and given by Eq. (10).
- (3) Any other ray which is not either bound or refracting is thus a tunnelling ray within the optical fibre.
Any optical attenuation of the ray can then be applied to the intensity collected as follows:
2.2 Reflective wall
In the case of the reflective wall design shown in Fig. 3(b), a reflective interface surrounds the scintillator, produced by a lower refractive index medium, n1. For the modelling of the reflective wall architecture, there are two possibilities for rays to be incident on the core of the optical fibre.
- (1) Rays directly incident on the optical fibre core, which are modelled in the same way as for the butt-coupled design.
- (2) Rays reflected off the wall of the scintillator.
In tracing the rays after reflection after incidence on a reflective interface, as long as rs ≤ rco all rays reflected off the scintillator wall will be incident on the fibre core. For the purpose of increasing the light collection, this is optimal. Since the propagation properties of the ray, i.e. the longitudinal propagation angle and azimuthal angle do not change after multiple reflections, then all rays that satisfy the condition that their propagation angle, , is less than the complement of the critical angle,, of the ns, n1 interface, given by Eq. (12), will be incident of the fibre core.Eq. (13).Eqs. (4)–(6). Once a ray is incident upon the scintillator wall, it will incur a reflection which will rotate the modelled cross-sectional angle, given in Eq. (14).
The double-cladding design shown in Fig. 3(c), consists of a scintillator surrounded by a lower refractive index, transparent layer, n1, which itself is surrounded by a reflective layer, n2. Hence n2 < n1 < ns, where the purpose of this probe design is for when the scintillating material is optically attenuating, an optically transparent layer surrounding the scintillator may significantly increase the light collection.
For modelling the double-cladding design, the following are the possibilities for ray incidence on the fibre core;
- (1) Rays directly incident on the optical fibre core, which are modelled in the same way as for the butt-coupled design.
- (2) Rays reflected off the wall of the scintillator, at the ns, n1 interface, modelled in the same way as for the reflective wall design.
- (3) Rays incident on the fibre core after refraction into n1.
Once rays are refracted in n1 there are many scenarios which can occur on fibre core incidence, i.e.
This results in two possibilities for incidence; either via the scintillator, ns, or via the optically transparent layer, n1. If the scintillator has a higher refractive index then the transparent layer, refraction will result in a smaller propagation angle in the transparent layer,, then the initial propagation angle,, in the scintillator. Hence with some optimisation there could be a significant increase in light collection with an appropriate choice of n1. This is depicted in 2D in Fig. 5, where via n1 a lower propagation angle is incident on the fibre core, due to the refraction into n1. With a closely matched lower refractive index, n1, and fibre core refractive index, then effectively a larger acceptance angle of rays incident from n1 is obtained.
In modelling the double-cladding design, rays directly incident on the fibre core, or via reflections off the n1, ns interface have been previously discussed in the butt-coupled and reflective wall probes. If rays are incident upon the n1, ns interface and the total incidence angle of the ray is less that the critical angle,, given in Eq. (15), then the ray will be refracted into n1.Eqs. (16)–(18), are solved to calculate the properties of the ray after each refraction and reflection. These are based on the fact that the longitudinal propagation angle and azimuthal angle are constant along a particular ray path.Eqs. (19)–(21).Fig. 6, which can be easily calculated using Eq. (22).
If the ray is also incident on the n1, n2 interface before reaching the fibre core then a reflection can also occur, which has been previously discussed in the reflective wall design. Unlike the reflective wall situation, the azimuthal angle that the ray makes with the n2, n1 interface is different to that it makes with the n1, ns interface. This is due to the difference in radii and hence curvature of the two interfaces, which has been depicted in Fig. 6. The azimuthal angle which is made with the n2 interface,, is calculated using the law of cosines, as:
This method is then continually applied to any refractions and reflections until the ray reaches the optical fibre core. Depending on whether the ray is incident on the fibre core from the scintillator or the transparent layer, then the appropriate refractive index must be applied, ns or n1 respectively.
2.4 Cladding coupled
In the case of the cladding coupled design shown in Fig. 3(d), the probe consists of a hollow cylinder scintillator surrounded by a reflective layer, n2, and filled with a lower refractive index, transparent material, n1. The purpose of this probe design is similar to that of the double-cladding, when the scintillating material is optically attenuating, an optically transparent central region may significantly improve the light collection efficiency.
The modelling of the light collection is similar to that of the double-cladding.
2.5 Probe design comparison
An in-house Matlab code was developed to simulate the models discussed. Figure 7 shows the two probe scenarios that have been modelled in order to compare the different probe designs, these are:
- (1) Figure 7(a) shows the scenario where the overall size of the probes are kept constant, corresponding to a radius, r, of 0.4 mm. Therefore all probes are evaluated with the same physical size, though this may correspond to the probes having different scintillator volumes. This is important for applications where the size of the detector needs to be minimised.
- (2) Figure 7(b) shows the scenario where the volumes of the scintillating materials are kept constant, Vs, such that in each case the same volume of scintillating material is investigated. This is important for applications where the sensitive volume of the detector needs to be minimised.
The optical fibre parameters, which have been used in the simulations are given in Table 1. For all probe designs the refractive index of the scintillating material was 1.73, corresponding to that of BeO ceramic, and a unit value is assumed for the power emitted per unit volume per steradian of the scintillator, I0. In the case of the novel architectures, PMMA of the same refractive index as the cladding of the optical fibre, ncl, was used as the transparent medium in the probe, n1. The dimensions used for simulating the different probe designs are given in Table 2.
3. Experimental validation
In order to validate the model discussed above, it was compared to experimentally measured data from commercially available scintillators. These included:
- (1) The butt-coupled design using a ~1.2 mm × 2 mm × 200 mm BC-400 plastic scintillator (Bicron), cut from a thick sheet of BC-400.
- (2) The reflective wall design using a 1.5 mm diameter BCF-10 plastic scintillating fibre (Bicron).
- (3) An optically attenuating butt-coupled design using eight 1 mm diameter × 1 length BeO ceramic cylinders (Thermalox 995, Materion).
These scintillators were placed at the tip of a 20 m long polymethyl methacrylate (PMMA) optical fibre, ESKA Ck-20 with ~0.5 mm core diameter and ~1.0 mm outer jacket diameter (Mitsubishi Rayon Co., LTD, Tokyo, Japan) and the scintillation intensity measured using a photomultiplier tube, Burle 8575 (Burle Technologies, Inc., USA). A light tight heat shrink was used to encapsulate the scintillator and the optical fibre. This jacket was not modelled in the simulations as it was assumed that all light which reaches the jacket is lost.
A superficial x-ray unit (SXR), Gulmay D3150 (Gulmay Medical LTD., UK), was used to expose the scintillators to ionising radiation. A 3 mm thick lead plate was used to shield the scintillators such that length of the scintillator exposed to x-rays could be controlled, as shown in Fig. 8. In the case of the BeO ceramic cylinders, they were simply placed against each other one by one to control the length of scintillator exposed.
The simulated probe parameters are given in Table 3, and optical fibre parameters in Table 1. An power emitted per unit volume per steradian, I0, of 1.25 was modelled for the butt-coupled design to account for the higher light output of BC-400 compared to BCF-10.
4. Results and discussion
4.1 Transparent scintillators
In order to verify the model, we compare the results found for the bound ray light collection of the butt coupled and reflective wall designs with those experimentally measured. The results shown in Fig. 9 are comparing both the modelled and measured normalised to the highest power. The uncertainties graphed for the measured data is for two standard deviations from the mean readings, and a 0.25 mm uncertainty on the placement of the lead plate.
These results agree with that previously reported , where the bound light collection from the butt-coupled increases linearly with scintillator length, until a point, beyond which it increases to an asymptotic limit. The bound light collection from the reflective wall on the other hand continually increases linearly with scintillator length, due to the reflections occurring from the wall.
Both the reflective wall and butt-coupled measured and modelled results have an average relative agreement to each other of within 5% and 12.5%, respectively. With this partial validation, the model is now used to compare the performance of the various investigated probes.
Figures 10(a) and 10(b) depict the modelling results for a cladding-coupled and double-cladding architecture, respectively. Both bound and tunnelling rays’ power within the fibre is shown. Results show that not only is there a significant bound power collected via the transparent medium, n1, but that there is also a significant amount of tunnelling power collected via n1.
Figure 11(a) shows the comparison of the total light collection of the four probes designs when constraining the overall size of the probe. It is clear that the reflective wall achieves the most light collection over the novel probes. This is due to the fact that when simply constraining the overall size of the probe, that there is more emitting scintillator possible for the reflective wall over the novel. In fact, as the volume of scintillator in the novel probes is increased, the light collection increases until reaching the reflective wall situation.
Figure 11(b) shows the total light collection of the four investigated probes, when constraining the volume of the scintillator. Results show that the novel probe designs have an increase in light collection over the butt-coupled design. Though when compared to the reflective wall design there is no significant increase in light collection.
4.2 Optically attenuating scintillators
The attenuating properties of BeO ceramics are now applied, with an optical attenuation coefficient, µ = 2.69x10−3 m−1 . To validate the model for the addition of the optically attenuating scintillator properties, the modelled results for a butt-coupled BeO ceramic design were compared to that experimentally measured, shown in Fig. 12. The results are in good agreement of each other, with an average relative difference of within 1%. The results show that beyond 1 mm of BeO ceramic, that there is virtually no increase in the light collection by the optical fibre, therefore light collection optimisation is crucial.
Figure 13(a) shows the results for when constraining the overall size of the probe. Results now show that in both the reflective wall and butt-coupled cases, that the total light collection increases linearly with BeO length, until a point, beyond which it increases to an asymptotic limit. Hence the significant increase in the light collection when using a reflective wall over the butt-coupled is lost. This is due to the greater the path length of the ray within the BeO, then the greater the optical attenuation encountered.
Results also indicate that both the butt-coupled and reflective wall collect more light over the novel geometries. As before, this is due to the increase in BeO volume capable with the previous geometries over that of the novel, since the novel geometries size needs to include that of the transparent layer. Even though the rays are more attenuated in the butt-coupled and reflective wall, the overall increase in light emission gives these geometries the most light collection. It can also be seen that the cladding-coupled architecture achieves a higher light collection than the double-cladding. This can be explained by the increase in the thickness of the transparent layer n1 for the cladding-coupled design.
Again a comparison of the total light collected while the volume of BeO has been kept constant, such that the efficiency of the probes can be investigated is shown in Fig. 13(b). It can be seen that the novel geometries are more efficient in collecting light from the BeO, which is to be expected since some rays have a less attenuating route to the fibre unlike the butt-coupled and reflective wall. The important factor of the use of n1 is that light continues to be collected by the fibre beyond the thickness of BeO where the previous probes platoe in their light collections. This is simply due to the fact that the transparent layer, n1, is shortening the path length of some rays in the BeO, hence lowering the amount by which they are attenuated.
The importance of these two results comes about with the application of the fibre–coupled luminescence dosimeter. For example, where the overall size of the probe is important such as for the application of in-vivo dosimetry, then the simple butt-coupled or reflective wall designs are the best option. When the overall size is perhaps not of concern but the sensitive volume of the dosimeter, such as in the dosimetry of small fields and high dose gradient regions, then the novel designs can be employed to significantly increase light collection.
A simple model for simulating the light collection from sources located at the tip of a multimode optical fibre has been developed, with the application primarily being fibre-coupled luminescence dosimetry. The model is based on ray optics and includes both meridional and skew rays. It also includes the optical attenuation of the medium, hence enabling the modelling of new scintillator materials, such as beryllium oxide (BeO) ceramics. Using the model, we proposed and evaluated the light collection of four different designs: the butt-coupled, reflective wall, double-cladding and cladding-coupled, for which we considered the combination of transparent and attenuating scintillator (BeO ceramics) materials.
The model was partially verified by the comparison of measured results for the butt-coupled and reflective wall geometries using a transparent plastic scintillator, and also an optically attenuating design using BeO ceramics. The modelled light collection was in good agreement with the measured results. The results for the transparent butt-coupled and reflective wall design also agrees with those previously reported : the reflective wall geometry has a linear increase in light collection with scintillator length, and the coupling power in the butt-coupled geometry increases linearly to a point and then to an asymptotic limit as the scintillator length increases.
It is important to note that this is only a partial validation of the model, as the light collection of the novel investigated designs has not been verified. Similar designs to the novel probe designs are commercially available, such as double cladding scintillating optical fibres. While these have not been developed for light collection from the inner cladding interface, construction on the novel designs discussed here is certainly possible. Construction and experimental light collection measurements would be of interest for a complete verification of the model discussed here.
It was found that the new designs (double-cladding and cladding-coupled) can significantly increase light collection when constraining the volume of the optically attenuating probe. These results are of significance in possible BeO fibre optic dosimetry , especially the fact that the increase in light collection from reflective wall architectures over butt-coupled is minimised due to the optical attenuation of BeO. We therefore conclude that when using non-transparent phosphors in fibre-coupled luminescence dosimetry, the most practical approach for probe architecture is the use of the simple butt-coupled design when the goal is minimising the overall size of the probe, such as that for in-vivo brachytherapy dosimetry. However, in some situations minimising the size of the sensitive volume is the major concern, as required in the dosimetry of small field sizes where dose averaging needs to be minimised. The use of more transparent material, as in the double-cladding and cladding-coupled designs, can substantially increase the light collection.
S. Afshar V. acknowledges the support of T. Monro’s Laureate Fellowship.
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