Light leakage in optical fibers: experimental results, modeling and the consequences for solar concentrators

Pergamon

PII:

Solar Energy Vol. 72, No. 3, pp. 195–204, 2002  2002 Elsevier Science Ltd S 0 0 3 8 – 0 9 2 X ( 0 1 ) 0 0 1 0 0 – 1 All rights reserved. Printed in Great Britain 0038-092X / 02 / $ - see front matter

www.elsevier.com / locate / solener

LIGHT LEAKAGE IN OPTICAL FIBERS: EXPERIMENTAL RESULTS, MODELING AND THE CONSEQUENCES FOR SOLAR CONCENTRATORS DANIEL FEUERMANN*, JEFFREY M. GORDON** ,† and MAHMOUD HULEIHIL* *Ben-Gurion University of the Negev, Jacob Blaustein Institute for Desert Research, Department of Solar Energy and Environmental Physics, Sede Boqer Campus 84990, Beersheva, Israel **The Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion University of the Negev, Beersheva 84105, Israel Received 15 May 2001; revised version accepted 20 October 2001 Communicated by VOLKER WITTWER

Abstract—Optical fibers used to transport sunlight exhibit considerable light leakage within their nominal numerical aperture. Of particular interest in the design and diagnosis of solar fiber-optic concentrators is the dependence of this leakage on: (a) incidence angle, (b) the optical properties of the core and the cladding, and (c) fiber length. We present measurements of fiber angular transmission, along with a theoretical model. The implications for solar fiber-optic concentrators are also assessed.  2002 Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

How imperfect is total internal reflection in optical fibers, i.e. how severe is radiation leakage from the fiber’s core, in particular for multi-mode broad-spectrum fibers? A quantitative answer to this query is essential to the accurate design, modeling and diagnostics of recent innovations in solar energy concentration (Feuermann and Gordon, 1998, 1999) as well as remote lighting (Feuermann et al., 1998). Experimental studies have indicated surprisingly high leakage losses in optical fibers developed for these applications, but have not offered a modeling capability (Liang et al., 1997; Nakamura et al., 2000). This begs several questions: What is the physical basis for the observed light leakage, especially in fibers where attenuation in the core is negligible? How does leakage depend on the optical properties of the core and cladding? Are materials readily available that can reduce these losses to negligible levels? The transmission of light in optical fibers is usually predicated on the assumption of perfect total internal reflection within the fiber’s numerical aperture. Attenuation in the fiber’s core and Fresnel reflections at the proximate and distal ends are accounted for, but leakage at the corecladding interface is invariably viewed as negli†

Address to whom correspondence should be addressed. Tel.: 1972-8-659-6923; fax: 1972-8-659-6921; e-mail: [email protected]

gible. Fiber manufacturers typically provide the core’s attenuation spectrum for normal incidence radiation, but do not offer information on leakage losses. Some attempts to account for leakage (in multimode fibers) have adopted a model where there is a constant absorptive loss for each reflection at the core-cladding interface (Kapany, 1967; Mildner and Chen, 1994). However, model validation was restricted to narrow-spectrum sources and fibers of low numerical aperture. Our measured data (reported below), as well as other angular transmission data (Liang et al., 1997; Nakamura et al., 2000), for a wide-spectrum source and fibers of high numerical aperture, indicate that this type of model does not account for experimental observations. Dugas et al. (1987) demonstrated that substantial light leakage can occur from the core to the cladding in large-diameter multi-mode optical fibers. They identified the source of the problem as the non-zero imaginary component of the refractive index (k) of the cladding, in particular when the incident light undergoes a large number of reflections at the core-cladding interface. Their experiments employed a monochromatic laser source as well as fibers with substantial core attenuation. Irvin and Nakamura observed modest extents of light leakage in multi-mode optical fibers, and surmised that absorption in the cladding material (i.e. a non-zero imaginary component of the cladding’s refractive index) was responsible (Irvin

195

196

D. Feuermann et al.

and Nakamura, 1991). However, no physical modeling was offered to account for the impact of the complex refractive index of the cladding on light leakage or its angular dependence, in particular within the fiber’s numerical aperture. Their studies were further complicated by substantial attenuation in the fiber’s core. Numerical aperture, NA, is a measure of the angular range supported by total internal reflection in the fiber. Light entering the fiber from air at angle u with respect to the fiber’s axis refracts to angle uin in the core (see Fig. 1). From Snell’s law (with the refractive index of air being approximately unity): sin(u ) 5 n 1 sin(uin )

(1a)

]] max NA 5 sin(u max) 5 n 1 sin(u in ) 5œn 12 2 n 22

(1b)

where the superscript max indicates the largest value for which total internal reflection nominally is respected; and n 1 and n 2 are the real part of the refractive indices of the core and cladding, respectively. Fiber optical properties, principally refractive index and attenuation, are functions of wavelength. However, the spectral dependencies are sufficiently weak that the use of a single energyweighted average turned out to be acceptable for the materials and spectral ranges in this study. For

Fig. 1. Optical fiber cross-section. The incidence angle in air u refracts to the angle uin in the core. u max (refracted into angle max u in ) indicates the largest angle ideally supported by total internal reflection. N1 and N2 are the complex refractive indices of the core and cladding, respectively. Because of the non-zero imaginary component of the cladding’s refractive index, some rays leak from the core to the cladding even at incidence angles u , u max . f is the projection of the angle between the ray propagating in the core and the normal to the core-cladding interface.

example, in the experiments reported below, for which all fiber cores were fused silica, and a quartz-halogen lamp served as the light source (an effective blackbody close to 3000 K), the spectrum-averaged refractive index for the fiber core is knl 5 1.45. Were the same fibers used in sunlight (an effective blackbody at 5800 K), the core’s knl would be 1.46. The cladding’s knl invariably changes correspondingly, so that the fiber’s nominal NA turns out to be about the same in this lamp light or sunlight. Fig. 2 illustrates the angular response of an ideal optical fiber: one where the refractive indices of the core and cladding can be treated as real variables, and core attenuation is negligible. Fresnel reflective losses at the entrance and exit apertures have been filtered out. Transmission t is normalized to its value at normal incidence. The onset of leakage occurs at angles beyond that corresponding to the fiber’s NA. Since the NA required of solar concentrators is large and matched to that of the fiber, the shoulder region of Fig. 2 is usually not of practical interest. However, we retain the complete angular range in our analysis toward full validation of the model, as well as a diagnostic capability for ascertaining the actual NA. Radiation leakage primarily stems from absorption in the cladding, i.e. the fact that its refractive index is a complex, and not simply a real, variable. The imaginary part of the cladding’s refractive index may be orders of magnitude smaller than its real component; but the number of reflections incurred can be so large that the net effect is measurable leakage. Our experimental measurements support the assertion that Fresnel’s equations of reflection, complemented by the proper geometric weighting for rays of varying skewness, are adequate to account for the observations, and to predict the optical throughput. We will demonstrate the sensitivity of leakage losses to cladding optical properties, and the degree to which available cladding materials can offer negligible leakage losses. 2. MEASUREMENT OF RADIATION LEAKAGE IN OPTICAL FIBERS

Fig. 3 is a schematic of our experimental rig for measuring the angular transmission of optical fibers. The fiber’s proximate tip was irradiated by an approximate point source (collimated light from a quartz-halogen lamp at far-field). Radiation intensity was measured with a silicon photodiode sensor inside a calibrated integrating box.

Light leakage in optical fibers: experimental results, modeling and the consequences for solar concentrators

197

Fig. 2. Angular response of an ideal optical fiber (no absorption in the core or cladding). Transmission t as a function of incidence angle u is expressed relative to t at normal incidence. t is insensitive to the ratio of fiber length L to core diameter d for L /d greater than around 10 (to within 60.001). (The graph was generated for L /d 5 3000.) Fresnel reflections at the fiber tips are filtered out.

The incidence angle was varied by rotating the proximate end of the fiber about its tip relative to a static light source. The fiber tips were polished, and the aperture discs were checked to be free of defects that could have lessened light injection or emission. The polishing process necessitates stripping a short length of the fiber’s buffer. With hard cladding materials, stripping does not affect the fiber core. But teflon cladding unavoidably pares off and thereby: (a) exposes a short extent of the core; and (b) engenders the risk of scratching the core’s exposed periphery, which in turn can leak light.

The significance of this problem will be addressed in Section 6. Table 1 lists the key physical characteristics of the optical fibers we were able to procure, and for which the angular transmission curves were measured. We used two methods to ascertain that attenuation in the fiber cores was negligible. First, transmission measurements at normal incidence, for different lengths of the same fiber, yielded differences smaller than the experimental error (less than 1% at normal incidence). Second, we measured radiation intensity at normal incidence

Fig. 3. Schematic of our experimental rig for measuring optical fiber angular transmission. Light from a quartz-halogen lamp is collimated with two lenses, and provides narrow-angle far-field irradiation onto the proximate end of an optical fiber. Longer fiber runs comprise bent fibers, but with radii of curvature that do not incur measurable additional leakage losses. An integrating box with a photodiode sensor is used for radiation detection.

198

D. Feuermann et al.

Table 1. Physical characteristics of the optical fibers for which angular transmission was measured. All fibers listed have a core of fused silica with a refractive index n 1 5 1.45 (a spectrum-averaged value) Manufacturer and fiber ID

Polymicro FLHAC

Ceramoptec WF1000 / 1100

3M FT1EMT

Spectran HCN-N1000T-14

Polymicro FLUA1000

Nominal NA

0.33

0.37

0.39

0.44

0.66

Cladding

Hard polymer

Doped silica

TECSE

Hard polymer

Teflon

Buffer

TefzelE

Polyimide

TefzelE

TefzelE

Acrylate

Core diameter, d

1.00 mm

1.00 mm

1.00 mm

1.00 mm

1.00 mm

Cladding diameter

1.035 mm

1.10 mm

1.035 mm

1.035 mm

1.04 mm

Buffer diameter

1.40 mm

1.30 mm

1.40 mm

1.40 mm

1.30 mm

Length L

3000 and 7000 mm

1780 mm

133 and 7000 mm

5100 mm

166, 3000 and 20 000 mm

k2

0.861310 26

3.67310 26

1.70310 26

1.18310 26

2.10310 26

Relative transmission T integrated over the fiber’s NA (excluding Fresnel losses at the entrance and exit apertures)

0.96 (both lengths)

0.80

0.98 (133 mm) 0.94 (7000 mm)

0.91

0.93 (166 mm) 0.82 (3000 mm) 0.78 (20 000 mm)

with and without the fiber, and corrected for Fresnel reflective losses at the two fiber tips, to reach the same conclusion. Specifically, in measurements without the fiber, the collimated lamp output irradiated a spot larger than the aperture of the integrating box. We confirmed flux uniformity within the irradiated spot by moving an optical fiber with a diameter of 1.0 mm over the spot and finding the same measured power at all positions. The flux corresponding to a diameter of 1.0 mm was then calculated by scaling the corresponding areas. The finding of negligible core attenuation for the fiber lengths considered here is consistent with

the attenuation spectra provided by the fiber manufacturers (Fig. 4). Optical path length increases with incidence angle, so attenuation could, in principle, grow non-negligible for angles within the fiber’s NA. Optical path length is proportional to 1 / cos(uin ). Since 1 / cos(uin ) is in the approximate range of 1.0–1.1 within the nominal NA of these optical fibers, our conclusion of negligible attenuation remains valid for incidence angles of practical interest. Fig. 5 presents measurements for fibers intended for high-flux solar concentrators, i.e. large core diameters (up to 1 mm), high NA, and low

Fig. 4. Attenuation coefficient k as a function of wavelength l for the fused silica typically used in the core of the optical fibers studied here (Polymicro, 2000).

Light leakage in optical fibers: experimental results, modeling and the consequences for solar concentrators

199

Fig. 5. Measured angular response of five commercial optical fibers intended for high-flux solar applications. For three of the fibers, more than one length was available (see Table 1). The best-fit model curves are also included.

core attenuation integrated over the solar spectrum. Leakage losses due to failed total internal reflection are evident for incidence angles within the nominal NA. Incidence angles were measured to within 60.58. The experimental uncertainty for transmission varied with incidence angle, with larger uncertainties in the vicinity of the NA, where the sensitivity of transmission to incidence angle is most pronounced. The plots of Fig. 5 show positive angles only, but measurements were performed by scanning back and forth from negative to positive angles (in part to insure repeatability), and averaging the results. Toward establishing the reproducibility of these angular

transmission curves, we also confirmed angular transmission measurements on some of the identical fibers reported in (Nakamura et al., 2000). 3. MODELING RADIATION LEAKAGE

Reflective properties at the core-cladding interface can be accounted for with Fresnel’s equations of reflection, provided: (1) both the real (n 2 ) and imaginary (k 2 ) parts of the cladding’s refractive index N2 are accounted for: N2 5 n 2 2 ik 2 and (2) the geometric contributions of rays of assorted skewness are weighted correctly. Attenuation coefficient k and k are related by (Koltun, 1988)

200

D. Feuermann et al.

Fig. 5. (continued)

k 5 kl /(4p)

(2)

where l 5wavelength. Measurements of negligible attenuation in the core, at normal incidence, attest to its k value being extremely small. The equations for the perpendicular and parallel components of the reflectivity R at the corecladding interface, as well as the average reflectivity R avg , are: ]]]]]]] 2 2 2 (n 2 2 ik 2 ) 2 n 1 sin (f ) 2 n 1 cos(f ) œ R ' 5 ]]]]]]]]]]] ]]]]]]] 2 2 2 œ(n 2 2 ik 2 ) 2 n 1 sin (f ) 1 n 1 cos(f )

H

J

2

(3a)

Ri 5 ]]]]]] 2 2 2 2 n 1œ(n 2 2 ik 2 ) 2 n 1 sin (f ) 2 (n 2 2 ik 2 ) cos(f ) ]]]]]]]]]]]] ]]]]]] n 1œ(n 2 2 ik 2 )2 2 n 12 sin 2 (f ) 1 (n 2 2 ik 2 )2 cos(f )

H

J

2

(3b) R' 1 Ri R avg 5 ]]] 2

(3c)

where f denotes the angle between a ray propagating in the core and the normal to the corecladding interface (Fig. 1). The arithmetic average is taken in Eq. (3c) because the light is essentially unpolarized.

Light leakage in optical fibers: experimental results, modeling and the consequences for solar concentrators

201

Fig. 5. (continued)

Each incidence angle u embodies a cone of rays with varying skewness values, which in turn corresponds to different u values in Eq. (3) and hence different reflectivities at the core-cladding interface. The basic geometric relations are reviewed in Mildner and Chen (1994). Fig. 6 is a cross-section of the fiber in which the polar coordinates are depicted: r is the radial position at which a ray enters the fiber (r being the mag→ nitude of the radial vector r ); and w is the

azimuthal angle of the ray trajectory →direction relative to the plane that includes both r and the fiber’s axis. For meridional rays, f 5 (p / 2) 2 uin but for rays of non-zero skewness, f is given by ]]]]] 4r 2 sin 2 (w ) cos(f ) 5 sin(uin ) 1 2 ]]] (4) d2

œ

where d denotes the core diameter. Assuming a uniform cone of rays at each angle u (and exploiting axial symmetry), we can express the normalized angular transmission for a fiber of length L as

t (u ) ]] t (0) r 5d / 2 w 5p / 2

4F(u ) 5 ]] p

E E

r 50

L tan(u in ) ]]]

]]] œd 224r 2 sin 2(w ) r dr dw R avg

(5)

w 50

where the function F(u ) accounts for the angular dependence of Fresnel reflective losses entering and leaving the fiber (F(u ) is only noticeably less than unity at large angles – in the cases considered here at angles beyond the NA). Eq. (5) accounts for the number of reflections (at the core-cladding interface) and skewness distribution of rays of a given incidence angle. Should the core attenuation not be negligible, then the integrand in Eq. (5) should be multiplied by Exp(2k1 L / hd cos(uin )j), where k1 is the core’s attenuation coefficient. Fig. 6. The angular and radial coordinates noted in the 3D calculations for leakage losses.

3.1. Estimating model parameters Eq. (5) furnishes a model with no adjustable

202

D. Feuermann et al.

parameters to predict fiber transmission (and hence light leakage), provided the attenuation spectrum of the cladding, or at least its spectrumaveraged k value, is known. The manufacturers of the fibers we tested did not provide attenuation spectra or k values for their cladding materials (neither initially nor upon request). Compared to the fused silica core (a typical attenuation spectrum is graphed in Fig. 4), the fiber claddings invariably have higher impurity levels and enhanced attenuation. As the simplest model, with only one adjustable parameter, we adopted the approximation k 2 5 constant. We then applied our model with this single adjustable parameter to see: (a) if good agreement between theory and experiment can exist; and (b) if so, whether that best-fit value is physically tenable. When the precise identity and absorption spectrum of the cladding materials are provided, more rigorous tests of the model, with no adjustable parameters, can be effected. We also investigated whether alternative oneparameter functional forms for k 2 could better correlate the fiber transmission measurements. Functions with k 2 linear in l, as well as proportional to a power of l, were introduced. We also examined a two-band k 2 , i.e. different (constant) values for two spectral regimes. However, the improvement in fitting the measurements was negligible in all instances. Hence we proceeded with the constant-k 2 approximation.

3.2. Consistency check for fiber numerical aperture In some cases, the reported NA turns out to be too low to account for our observations. This apparent inconsistency can be detected by considering the angular transmission of an ideal fiber (k50) at angles beyond the nominal NA. This exercise is illustrated in Fig. 7 where many of the measured points (for u . umax ) for a fiber with a manufacturer-reported NA of 0.39 lie above the curve of the corresponding ideal fiber. Fiber manufacturers typically report nominal NA values with an uncertainty of 60.02. In the example of Fig. 7, by adjusting the NA from 0.39 to 0.41, the experimental measurements are not inconsistent with the upper bound of the ideal fiber. This type of consistency check is best performed for short fiber lengths (minimal leakage). Within the narrow band of NA values cited by the fiber manufacturers, we performed model fits for NA values that were consistent both with the fundamental upper bound of the k 2 5 0 limit, as well as with the reported uncertainty in NA.

4. COMPARISON BETWEEN EXPERIMENT AND THEORY

The accuracy with which our model can correlate the data is illustrated in Fig. 5, for optical

Fig. 7. Consistency check for NA. Relative transmission is plotted against incidence angle for an optical fiber with an aspect ratio L /d 5 133, and a manufacturer-reported NA of 0.3960.02. In addition to our experimental measurements, the ideal fiber (k 5 0) curves are plotted for NA50.39 (solid curve) and NA50.41 (broken curve).

Light leakage in optical fibers: experimental results, modeling and the consequences for solar concentrators

fibers (listed in Table 1) that span a factor of two in (relatively large) NA values. Our measured points are presented along with the one-parameter non-linear regression fits based on Eq. (5) and the approximation of constant k 2 . When more than one length of the same fiber was available, we required that the same k 2 and NA apply to all samples. At wavelengths in the visible to near infrared, k 2 for materials commonly used as fiber cladding is of order 10 26 , which is consistent with our regressed k 2 values (Table 1). However, the length dependence of light leakage for teflon-coated fibers appears to be anomalous: unusually high leakage for short extents. The same peculiarity was noted by (Nakamura et al., 2000), but no explanation was ventured. As noted in Section 2, the problem may stem from a short length of the core’s periphery being exposed and scratched when the buffer is stripped and the teflon cladding is inextricably shaved off with it. Although light leakage from these exposed circumferential elements are visible, we have not yet found a way to resolve this uncertainty quantitatively. Since there would appear to be a substantial essentially length-independent leakage loss in these instances, there is little meaning to the model regression fits. The dearth of good agreement between a consistent model fit and the data in Fig. 5e reflects this. Nonetheless, by nature of the functional dependence of light leakage on fiber length, the regression fit is effectively more highly weighted by longer fibers, and the best-fit k 2 value is of the same order as known absorption coefficients for teflon.

5. CONSEQUENCES FOR SOLAR FIBEROPTIC CONCENTRATORS

Table 1 also includes values of integrated relative transmission T, with a uniform light source the NA of which is matched to that of the fiber: sin(u )5N A

E

t (u ) ]] d(sin 2 (u )) t (0) 0 T 5 ]]]]]]] . (NA)2

(6)

Solar concentrator (as well as remote lighting) designs have been based on the assumption that leakage losses (within the NA) are negligible. In some cases, considerable overestimates of optical throughput have resulted, or a posteriori explanations for unexpectedly low throughput were attrib-

203

uted to attenuation coefficients having been reported as below their actual values. In fact, the missing element in the energy balance was most likely radiation leakage. For the types of optical fibers considered here, some manufacturer catalogs report that NA decreases noticeably as a given fiber grows longer. Declarations of this sort derive from the type of behavior shown in Fig. 5c. Since NA depends only on the real refractive indices of the core and cladding (Eq. (1b)) and cannot depend on fiber length, these manufacturer claims reflect the rapid increase of light leakage with fiber length (Eq. (5)). However, leakage losses can be kept small were cladding materials with adequately low k 2 values used. The quality of these prospective claddings lies intermediate between the high purity of the core materials, and the lower quality claddings currently used in broad-spectrum optical fibers. 6. CONCLUSIONS

Total internal reflection at the core-cladding interface in optical fibers is an imperfect process. Even with the best materials, a small fraction of incident light is absorbed in the cladding. Alternatively, the imaginary component of the refractive index may be orders of magnitude smaller than its real component, but is still non-zero. Although the leakage loss at each intersection with the core-cladding interface may be extremely small, the number of reflections can be large, and the net effect is non-negligible leakage. The problem is especially pronounced for the largeNA broad-spectrum multi-mode fibers needed in solar fiber-optic concentrator applications. When the complex refractive index of the cladding is introduced into Fresnel’s equations for reflection, and the correct weighting for the contribution of skew rays is correctly incorporated, one can account for the full angular transmission curve, i.e. for leakage as a function of angle and fiber characteristics. We have confirmed this assertion for optical fibers that cover a range of NA values and cladding materials. Do similar leakage problems plague telecommunication fiber-optic systems? Although telecommunication fibers have very low NA, their small diameters and extensive lengths result in large aspect ratios (L /d), such that the situation may superficially appear no more favorable than for high-NA fibers. However, the very low NA results in the use of cladding materials that are

204

D. Feuermann et al.

quite close in quality (and all optical properties) to the low-attenuation core materials. Furthermore, telecommunications use monochromatic sources, for which cladding k values over the narrow prescribed spectral window can be kept tolerably small. Their k values are vastly smaller than the averaged k values of cladding materials for broadspectrum fibers, especially compared to high-NA fibers where differences between core and cladding materials are more appreciable. In addition, the principal aim in telecommunications is signal integrity rather than high optical transmission. In solar concentrator systems, high transmission is essential to system viability; so sizable leakage losses are inadmissible. Light leakage in optical fibers is not reported in manufacturer specifications. Leakage can be noticeable for the high-NA broad-spectrum fibers suitable to solar concentrator (and remote lighting) systems. Model predictions enable the designer to estimate the quality of cladding material necessary to insure that leakage losses are kept to a prescribed level for a given fiber length. Alternatively, one could select the fiber’s NA to exceed that of the optical system, and thereby work in the angular regime where leakage losses are acceptably small. Either way, one can resolve the uncertainty surrounding the source of surprisingly large losses in large-NA optical fibers for broad-spectrum sources, and can model their optical throughput. NOMENCLATURE d F k L n N NA r Ri R' R avg T

f w k

fiber core diameter function that accounts for the angular dependence of Fresnel reflections at the fiber’s tips imaginary part of the complex refractive index fiber length real part of the complex refractive index complex refractive index numerical aperture of fiber core radial position at which a ray enters the fiber parallel component of reflectivity perpendicular component of reflectivity arithmetic average of reflectivity relative transmission integrated over the fiber’s nominal numerical aperture angle between a ray propagating in the fiber core and the normal to the core-cladding interface azimuthal angle of a ray trajectory attenuation coefficient

l u uin t

wavelength incidence angle in air, relative to fiber axis refracted angle inside fiber core, relative to fiber axis transmission of the fiber core

Indices 1 fiber core 2 fiber cladding Acknowledgements—This research was supported by grants from the Israel Ministry of National Infrastructures (Jerusalem), and the Rita Altura Foundation (Los Angeles, CA). We are indebted to Michael Altura for his encouragement and sponsorship. We are grateful to Takashi Nakamura of Physical Sciences Inc. (San Francisco, CA) for valuable discussions, and for providing us with three of the optical fibers on which our measurements were performed. We also thank Jonathan Molcho of Ben-Gurion University of the Negev’s Department of Electrical and Computer Engineering for generously granting the use of facilities and space in his laboratory, as well as his valuable recommendations in the selection of measurement equipment.

REFERENCES Dugas J., Sotom M., Martin L. and Cariou J. M. (1987) Accurate characterization of the transmissivity of largediameter multimode optical fibers. Applied Optics 26, 4198–4208. Feuermann D. and Gordon J. M. (1998) Solar surgery: remote fiber-optic irradiation with highly concentrated sunlight in lieu of lasers. Optical Engineering 37, 2760–2767. Feuermann D. and Gordon J. M. (1999) Solar fiber-optic mini-dishes: a new approach to the efficient collection of sunlight. Solar Energy 65, 159–170. Feuermann D., Gordon J. M. and Ries H. (1998) Non-imaging optical designs for maximum power density irradiation. Applied Optics 37, 1835–1844. Irvin B. R. and Nakamura T. (1991) Characterization of optical fibers for solar energy transmission. Second ASMEJSES-JSME International Solar Energy Conference, Reno, Nevada, 17–22 March 1991, Proceedings pp. 359–366. ASME, New York. Kapany N. S. (1967). Fiber Optics, Academic Press, New York. Solar Cells, Koltun M. M. (Ed.), (1988). p. 10, Allerton Press, New York. Liang D., Nunes Y., Fraser-Monteiro L., Fraser-Monteira M. L. and Collares-Pereira M. (1997) 200 W solar energy delivery with optical fiber bundles. SPIE Proceedings 3139, 217– 224. Mildner D. F. R. and Chen H. (1994) The neutron transmission through a cylindrical guide tube. Journal of Applied Crystallography 27, 316–325. Nakamura T., Comaskey B. and Bell M. (2000) Development of optical components for space-based solar plant lighting. S. A.E. 30 th Int. Conf. on Environmental Systems, Toulouse, France, 10–13 July 2000. Technical Paper No. 00ICES-361. Polymicro Technologies LLC (2000) 18019 N. 25th Ave., Phoenix, AZ, USA. Company prospectus: low-OH fiber attenuation chart.