- Email: [email protected]
Rayleigh scattering in sapphire test mass for laser interferometric gravitational-wave detectors
15 August 1999
Optics Communications 167 Ž1999. 7–13 www.elsevier.comrlocateroptcom
Rayleigh scattering in sapphire test mass for laser interferometric gravitational-wave detectors I. Measurement of scattering attenuation coefficient F. Benabid ) , M. Notcutt, L. Ju, D.G. Blair Physics Department, UniÕersity of Western Australia, Nedlands, W.A. 6907, Australia Received 28 September 1998; received in revised form 26 May 1999; accepted 28 May 1999
Abstract We give a theoretical estimate of the lower limit of Rayleigh scattering in sapphire crystals. We report measurements of scattering attenuation coefficient in sapphire at wavelengths of 633 and 1064 nm using an imaging measurement technique. The measured scattering loss in sapphire is ; 13 ppmrcm. Using sapphire as a mirror and beamsplitter substrate in laser interferometric gravitational-wave detectors, Rayleigh scattering losses are similar in magnitude to estimates of mirror losses. However, the scattering level still remains higher than the theoretical limit because of the presence of impurities– vacancies in the sapphire crystal. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 78.20-D; 04.80 Keywords: Rayleigh scattering; Sapphire; Gravitational wave detector
1. Introduction The choice of sapphire as a test mass for the next generation of laser interferometric gravitational wave ŽLIGW. detectors would decrease the internal thermal noise of the test masses by more than an order of magnitude compared with fused silica ones w1x. A study of sapphire’s optical properties is therefore necessary to quantify its optical performance and compare the resulting equivalent gravitational signal from substrate losses and distortions to the other sources of noise. Reports of the measurements of )
Corresponding author. E-mail: [email protected]
sapphire absorption w2x and birefringence w3x have shown promising results. The effect of the scattered light from a test mass substrate would firstly limit the recycling factor of the interferometer by decreasing the amount of the circulating power inside the interferometer w4x. The second effect is similar to light scattered from mirror surfaces w5–7x, recombining with the main beam and in doing so adding an undesirable phase modulation. These effects will be the subjects of a coming paper. In order to have an estimate of the lower limit of scattering loss in sapphire, we describe the parameters in light scattering in sapphire and distinguish the intrinsic Žpure crystal. processes from the extrinsic
0030-4018r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 2 8 7 - 4
F. Benabid et al.r Optics Communications 167 (1999) 7–13
8
ones. We give an estimate of the intrinsic light scattering from crystals in the absence of any imperfection or impurities, which is set by thermodynamic fluctuations in density and temperature. We also give a numerical estimate of this loss in the case of sapphire at wavelengths of 633 and 1064 nm. Moreover, we show the effect of the point defects in a crystal on the enhancement of Rayleigh scattering Žextrinsic processes. using a theoretical model w8x. In Section 3, we describe a method of measurement of light scattering using an imaging technique, which gives good SrN ratio against a background of extraneous light. Using this method, we report scattering measurements on a number of high purity sapphire samples and, for comparison, samples of high purity fused silica and benzene. 2. Theoretical background of Rayleigh scattering from crystals
A hypothetically pure, perfect crystal scatters light only from spatial fluctuations of permittivity D ´ Ž r . w9,10x. In the regime where the wavelength l is greater than the length scale of such fluctuations, the scattered light power is given by the Rayleigh–Born approximation, where the Rayleigh ratio Žfractional scattered power per unit solid angle. at an observation angle u is:
p2 2 l4
Ž 1 q cos 2u . w D´ 2 x n´
Ž 1.
or Ru ,u s R 90,u Ž 1 q cos 2u .
Ž 2.
where R 90 is the Rayleigh ratio at an observation angle of 908; ‘u’ in the subscript stands for unpolarized incident light. D ´ is dielectric fluctuation magnitude inside the associated correlation volume n´ . The attenuation coefficient per unit length a sca and which we will call for brevity the scattering coefficient, is deduced by integrating the Rayleigh ratio over all the angles of a sphere of radius R.
a sca s
16p 3
R 90,u .
a sca s
8p 3 27l4
n8 kT b T Ž p 11 q 2 p 12 .
2 q9b S p 12 Cdep ,
2.1. Density fluctuation in perfect crystals
Ru ,u s
For a polarized incident laser beam, a multiplication factor due to the anisotropy of scatter centers Žthe Cabannes factor. must be included w10x. In a perfect crystal, the only processes that contribute to the dielectric fluctuations are density fluctuations and Raman effects. The density fluctuations include both isobaric and adiabatic density fluctuations. The isobaric density fluctuations give rise to the Rayleigh central line and the adiabatic density fluctuations to acoustic waves and to Brillouin satellite lines. For both sapphire and silica, the Raman scattering components are much smaller compared with thermal scattering. We thus limit our case to density fluctuation scattering. Inserting expressions for D ´ 2 of the processes mentioned above, the total fractional scattered power a sca in isotropic crystals is given as follows w11x:
Ž 3.
2
Ž 4.
where l, n, b T , b S , T, pi j are the wavelength of the incident light, the refractive index of the crystal, its isothermal compressibility, adiabatic compressibility, temperature and its photo-elastic coefficients respectively. The coefficient Cdep is the Cabannes depolarization factor w10x due to the optical anisotropy of the scattering molecules. Although sapphire is anisotropic, Eq. Ž4. is an excellent approximation for a sca as the differences in the relevant matrix elements due to crystal anisotropy are only 5%. The calculation of the scattered light due to density fluctuation in sapphire is approximately 1.6 ppmrcm at 633 nm and 0.21 ppmrcm at 1064 nm. Whereas in practice in glassy materials such as fused silica the density fluctuation limit has been reached w12,13x, few real ionic crystals ever exhibit this low limit case. 2.2. Point defects in real crystals Recent theoretical work w8,14x showed that the presence of point defects in crystals such as impurities or vacancies contribute to Rayleigh scattering. For individual point defects, the permittivity change
F. Benabid et al.r Optics Communications 167 (1999) 7–13
9
depends on the three following processes described by Arora w8x: Ža. the change in electronic polarisability at the defect site; Žb. the elastic strain field around the defect due to ionic size mismatch; Žc. the strain field arising due to effective charge on the defect. Experimentally, Fredericks et al. w15x have demonstrated in the case of KBr crystals the linearity between the number of impurities–vacancies and the elastic scattering in crystals. Taking into account the above results, one can calculate the scattering coefficient due to individual point defects in the following expression:
a sca ,def s f = 10y2 7 cm2 n def Ž cmy3 .
ž
633 nm
l
4
/
,
Ž 5. where f is a factor which depends on the nature of the point defect and can take values in the range 0.2 to 14 at an incident wavelength of 633 nm for typical impurities and vacancies. Here, n def is the density of the impurities–vacancies and l is the incident wavelength expressed in nanometer. Averaging over the typical density numbers of the common impurities and vacancies found in high purity sapphire samples w16,17x, we find that the total impurities–vacancies might be about 100 ppm of the density number of Al 2 O 3 , namely ; 10 21 to 10 22 point defects per cm3. Using these densities in Eq. Ž5. with f s 7, the scattering level at 1064 nm, due to individual point defects is between 1 to 10 ppmrcm. We notice here that this extrinsic Rayleigh scattering is more than one order of magnitude greater than the intrinsic Rayleigh scattering due to permittivity fluctuations. 3. Experimental set-up Light scattering measurements were performed using the setup shown in Fig. 1. The incident collimated laser light passes through the middle of the sample. The transmitted beam is measured with a calibrated power meter. The scattered light passes through a first pupil, which determines the measured length of the scattered light, then a focusing telescope and finally a second pupil. The image of the scattered light is detected by a 16-bit cooled Žy108C. CCD camera placed at the image plane of the telescope. The whole collection system is mounted on a
Fig. 1. Scattering measurement set-up. Ds Pupil; u sangle of observation; V s collection solid angle; T s telescope; PDs power meter.
rotating arm with an angular range of 908. The image is then processed by a personal computer. Extraneous light is eliminated by enclosing the whole of the light gathering optics in a light-tight, matt black tube. The laser source is either an unpolarized He–Ne laser of 4 mW output or a vertically polarized diode pumped Nd:YAG laser of 100 mW output. Both lasers show power fluctuations less than 2% over a period of 6 h and less than 0.5% of fluctuations per 5 min Žthe average time for taking an image.. In this paper we will present the results of light scattering measurements on 3 sapphire samples, 2 fused silica samples, and benzene. The 3 sapphire samples are all high purity single crystals obtained from Crystal Systems ŽSalem, USA.. They are all cylindrical shaped with a c-axis parallel to the cylinder axis. One is polished HEMEX Ultra single crystal, a cylinder of 50 mm diameter and 100 mm length Žsample C.. The two other samples ŽA and B. are CSI pure sapphire samples of 30 mm diameter and 100 mm length. Their c-axis and the laser beam were aligned in parallel. The 2 fused silica samples are Suprasil high purity glass of 50 mm diameter and 100 mm length.
10
F. Benabid et al.r Optics Communications 167 (1999) 7–13
Fig. 2. Typical data treatment. Ža. Scattered light recorderd for different exposure times. Žb. The image is digitized, and the number of counts C Žt . is determined. Žc. After plotting the number of counts as a function of exposure time, the count rate CR is determined from the slope of the line.
Both are optically polished Žsample 1. and the second is in addition chemically polished Žsample 2. Žprovided from the VIRGO project.. Benzene ŽHPLC grade. was contained in a cylindrical shaped glass cell of 30 mm diameter and 100 mm length.
determined by comparing a calibrated measurement of laser power with a known fraction of the laser falling on the CCD Žsee Fig. 3.. The relation between the power and count rate can be written as follows:
3.1. Data treatment
Where CR is the count rate per pixel expressed in countsrsrpixel, CR 0 is the count rate due to dark
CR s qP y CR 0 .
Ž 6.
As illustrated in Fig. 2 for a typical example, to determine the Rayleigh ratio we carry out the three following steps. First, we take images of the scattered light at a certain angle u , for different exposure times. Secondly, the CCD readout of the scattered light is integrated over a nominated area, slightly larger than the image area ŽFig. 2b.. The integrated data C Žt . Žcount number due to the scattered light for the unit solid angle defined by our collection aperture. is plotted versus integration time t . The slope of the graph gives the count rate CR, which is directly proportional to the collected power ŽFig. 2c. The measurement of the count rate is a method of averaging as well as giving the readout noise as the y-axis intercept. Consequently, one can deduce via the slope an accurate measurement of CR and use of the full linear dynamic range of the CCD by choice of the right range of exposure time. 3.2. Calibration of CCD camera The coefficient of proportionality between the laser power per unit area and the count rate is
Fig. 3. CCD camera calibration set-up. The mirror mounted on a translation stage and is either positioned at M to reflect the light X into the CCD camera or M to reflect the light into a calibrated power meter ŽPD.. F1 and F2 are sets of neutral density filters.
F. Benabid et al.r Optics Communications 167 (1999) 7–13
current and background noise, P is the power and q is the coefficient of proportionality. Eq. Ž6. then gives us the power corresponding to the determined count rate. 3.3. Experimental errors The noise in the digitized image is due to the background light, CCD dark current and readout noise. The count rate corresponding to the dark current has been measured by completely covering the CCD array and proceeding as in Section 3.2, to be 10 countsrsrpixel. This is negligible compared with the background light count rate noise which was measured to be about 400 countsrsrpixel for a signal level of 15 000 countsrpixel. This represents a relative error of 1–2% of our typical signal due to light scattering, which is of the order of 10 4 countsrsrpixel. The power meter specifications show a relative error of 2% in the measured power. Finally, with the power fluctuation of 0.5%, the experimental error remains less than 5%.
4. Measurement of the Rayleigh ratio and scattering coefficient Taking into account the geometry of our experiment, the value of the Rayleigh ratio is given by the expression: PLS Ž u . s Ptrans = V
Ž2yt . t 1
ž / n2
Ž1 y k Ž u . 2 .
Ru d.
Ž 7.
In this equation PLS Ž u . is the scattered laser power determined using the method described in Section 3. Ptrans is the power of the transmitted light by the sample. t and k are the transmission and reflection coefficients of the sample respectively.
Ž2yt . t
Ž 1 y k Ž u . 2.
is the reflection correction term due to the multireflection of the transmitted beam inside the sample and to the multireflection of the scattered light. The
11
details of this correction can be found in Ref. w18x. V is the collection solid angle in air, n is the refractive index of the sample, and d the actual path length observed in the sample. Ru the Rayleigh ratio at an observation angle of u . In our experiment the measurements were carried out at 2 different wavelengths; one at 633 nm with an unpolarized He–Ne laser, and a second at 1064 nm with a vertically polarized diode pumped Nd:YAG laser. The attenuation coefficient due to scattering a sca is deduced from Eq. Ž3.. 4.1. Results and discussion We have confirmed the assumption that we are in the Rayleigh regime of light scattering Žsmall scattering centers compared with the wavelength. by measuring the Rayleigh ratio at a range of observation u between 90 and 308. In the case of unpolarized incident light, the angular distribution of scattered light fits the 1 q cos 2 Ž u . curve given in Eq. Ž2. within 5%. For vertically polarized incident light, the scattered light is almost constant Žwithin 5%. with respect to the observation angle in the horizontal plane as expected w19x. Knowing the angular dependence of Ru , it is sufficient to determine the Rayleigh ratio at an observation angle of 908 Ž R 90 . via Eq. Ž7. and the scattering loss factor a sca can be deduced from Eq. Ž3.. Table 1 shows the results of Rayleigh ratio and scattering coefficient for the different samples at 633 and 1064 nm, respectively. The measurement results of Rayleigh scattering in the benzene samples are in the range reported in the literature w10,18x. The results obtained with the fused silica show a scattering loss 3 or 4 times higher than the best reported in the literature w12x. Also, one can notice that the Rayleigh ratio R 90 at 633 and 1064 nm follows the ly4 Rayleigh law in all the samples. Considering the sapphire samples, the results of the scattering loss of the samples A and B are the lowest results reported on the sapphire scattering w20x but are however still ; 2 times higher than our fused silica. Reducing defects such as F centers, known w21x to be present in sapphire grown in a vacuum Žheat
F. Benabid et al.r Optics Communications 167 (1999) 7–13
12 Table 1 Results of Rayleigh ratio measurement. Sample
R 90 at 633 nm Žppmrcm.
R 90 at 1064 nm Žppmrcm.
a sca at 1064 nm Žppmrcm.
Theoretical a sca at 1064 nm Žppmrcm.
Sapphire A B C
6.1 6.4 9.3
0.75 0.8 1.1
12.6 13.4 18.4
0.2 0.2 0.2
F-Silica 1 2
5.4 3.2
0.7 0.4
11.7 6.7
0.4 0.4
Benzene
41.0
5.1
85
67 a
Columns 2 and 3 report the results of the measurement of Rayleigh ratio at wavelengths of 633 and 1064 nm, respectively. The scattering coefficient is reported in column 3 and deduced from Eq. Ž3.. The last column shows the theoretical scattering coefficient. a cf. Ref. w10x.
exchanger method. which results in stoichiometric imbalance of cations and anions, would suggest that sapphire with substantially less scattering loss may be possible. 5. Conclusion A simple and accurate light scattering measurement using an imaging technique has been used successfully. According to thermodynamic fluctuation theory, the fundamental limit of scattering loss in pure sapphire is 0.21 ppmrcm at 1064 nm, half that of fused silica. Our experimental results of Rayleigh scattering showed a scattering loss in sapphire more than 50 times higher than the intrinsic scattering loss predicted by the theory. We attribute this enhancement to nano-scale point defects such as impurities and vacancies. It has been demonstrated w15x that the reduction of these point defects can decrease substantially the scattering loss in a crystal. High purity sapphire samples grown by the heat exchange method ŽHEM. are known to still have non-negligible amounts of vacancies ŽF centers. and ions impurities such as Cr 3q and Ti 3q Žfew ppm.. Their reduction of at least one order of magnitude would reduce substantially the scattering loss in sapphire. We also observed a high scattering loss in fused silica compared with other results reported and the theoretical
limit, making the Rayleigh scattering measurement of every test mass substrate an important and necessary consideration in the design for all laser interferometric gravitational-wave detectors. Considering the performance of LIGW detectors, a scattering loss of 13 ppmrcm reported here, is a promising result indeed. The loss due to Rayleigh scattering in a Fabry–Perot Michelson interferometer will contribute in a round trip loss through the 10 cm thick beamsplitter and the 10 cm thick input mirror substrate of ; 500 ppm. This is comparable to 100 bounces cavity round trip, where the mirror scatteringrabsorption loss are of ; 600 ppm Ž3 ppmrmirror. w22x. In conclusion, these results confirm that sapphire is a viable test mass material for LIGW detectors given the measurements of quality factor and birefringence reported in Refs. w1x and w3x, respectively. References w1x V.B. Braginskii, et al., Systems with Small Dissipation, The University of Chicago Press, Chicago, IL, 1985. w2x D.G. Blair, F. Cleva, C.N. Man, Opt. Mater. 8 Ž1997. 233–236. w3x F. Benabid, M. Notcutt, L. Ju, D.G. Blair, Phys. Lett. A 237 Ž1998. 337–342. w4x W. Winkler, K. Danzmann, A. Rudiger, R. Schilling, Phys. Rev. A 44 Ž11. Ž1991. 7022–7036. w5x K.S. Thorne, Caltech Report, 1989. w6x J.Y. Vinet, B. Meers, C.N. Man, A. Brillet, Phys. Rev. D 38 Ž2. Ž1988. 433–447.
F. Benabid et al.r Optics Communications 167 (1999) 7–13 w7x J.Y. Vinet, V. Brisson, S. Braccini, Phys. Rev. D 54 Ž2. Ž1996. 1276–1286. w8x A.K. Arora, R. Kesavamoorth, D. Sahoo, J. Phys. C 15 Ž1982. 4591–4598. w9x B. Chu, Laser Light Scattering, Academic Press, New York, 1974. w10x I.L. Fabelinskii, Molecular Scattering of Light, Plenum, New York, 1968. w11x J. Tauc, Highly transparent glasses, in: S.S Mitra, B. Bendow ŽEds.., Highly Transparent Solids, Plenum, New York, 1975. w12x S. Sakaguchi, S.-I. Todoroki, J. Am. Ceram. Soc. 79 Ž11. Ž1996. 2821–2824. w13x D.A. Pinnow, Appl. Phys. Lett. 22 Ž10. Ž1973. 527–529. w14x D.L. Mills, J. Appl. Phys. 51 Ž11. Ž1980. 5864–5867. w15x W.J. Fredericks, P.R. Collins, D.F. Edwards, Phys. Rev. B 35 Ž6. Ž1987. 2999–3002.
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w16x M.E. Innocenzi, R.T. Swimm, M. Bass, R.H. French, A.B. Villaverde, M.R. Kokta, J. Appl. Phys. 67 Ž12. Ž1990. 7542– 7546. w17x B.D. Evans, J. Appl. Phys 70 Ž7. Ž1991. 3995–3997. w18x E.R. Pike, W.R. Pomeroy, J.M. Vaughan, J. Chem. Phys. 62 Ž8. Ž1974. 3188–3192. w19x H.C. Hulst, Light Scattering by Small Particles, Wiley, New York, 1957. w20x M. Burka, Notes on sapphire, CNRS, Groupe de Recherches sur les ondes de gravitation, Private Communication. w21x B.D. Evans, G.J. Pogatshnik, Y. Chen, Nucl. Instrum. Methods Phys. Res. B 91 Ž1994. 258–262. w22x J.E. Logan, N.A. Robertson, J. Hough, Opt. Commun. 107 Ž1994. 342–346.
Optics Communications 167 Ž1999. 7–13 www.elsevier.comrlocateroptcom
Rayleigh scattering in sapphire test mass for laser interferometric gravitational-wave detectors I. Measurement of scattering attenuation coefficient F. Benabid ) , M. Notcutt, L. Ju, D.G. Blair Physics Department, UniÕersity of Western Australia, Nedlands, W.A. 6907, Australia Received 28 September 1998; received in revised form 26 May 1999; accepted 28 May 1999
Abstract We give a theoretical estimate of the lower limit of Rayleigh scattering in sapphire crystals. We report measurements of scattering attenuation coefficient in sapphire at wavelengths of 633 and 1064 nm using an imaging measurement technique. The measured scattering loss in sapphire is ; 13 ppmrcm. Using sapphire as a mirror and beamsplitter substrate in laser interferometric gravitational-wave detectors, Rayleigh scattering losses are similar in magnitude to estimates of mirror losses. However, the scattering level still remains higher than the theoretical limit because of the presence of impurities– vacancies in the sapphire crystal. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 78.20-D; 04.80 Keywords: Rayleigh scattering; Sapphire; Gravitational wave detector
1. Introduction The choice of sapphire as a test mass for the next generation of laser interferometric gravitational wave ŽLIGW. detectors would decrease the internal thermal noise of the test masses by more than an order of magnitude compared with fused silica ones w1x. A study of sapphire’s optical properties is therefore necessary to quantify its optical performance and compare the resulting equivalent gravitational signal from substrate losses and distortions to the other sources of noise. Reports of the measurements of )
Corresponding author. E-mail: [email protected]
sapphire absorption w2x and birefringence w3x have shown promising results. The effect of the scattered light from a test mass substrate would firstly limit the recycling factor of the interferometer by decreasing the amount of the circulating power inside the interferometer w4x. The second effect is similar to light scattered from mirror surfaces w5–7x, recombining with the main beam and in doing so adding an undesirable phase modulation. These effects will be the subjects of a coming paper. In order to have an estimate of the lower limit of scattering loss in sapphire, we describe the parameters in light scattering in sapphire and distinguish the intrinsic Žpure crystal. processes from the extrinsic
0030-4018r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 2 8 7 - 4
F. Benabid et al.r Optics Communications 167 (1999) 7–13
8
ones. We give an estimate of the intrinsic light scattering from crystals in the absence of any imperfection or impurities, which is set by thermodynamic fluctuations in density and temperature. We also give a numerical estimate of this loss in the case of sapphire at wavelengths of 633 and 1064 nm. Moreover, we show the effect of the point defects in a crystal on the enhancement of Rayleigh scattering Žextrinsic processes. using a theoretical model w8x. In Section 3, we describe a method of measurement of light scattering using an imaging technique, which gives good SrN ratio against a background of extraneous light. Using this method, we report scattering measurements on a number of high purity sapphire samples and, for comparison, samples of high purity fused silica and benzene. 2. Theoretical background of Rayleigh scattering from crystals
A hypothetically pure, perfect crystal scatters light only from spatial fluctuations of permittivity D ´ Ž r . w9,10x. In the regime where the wavelength l is greater than the length scale of such fluctuations, the scattered light power is given by the Rayleigh–Born approximation, where the Rayleigh ratio Žfractional scattered power per unit solid angle. at an observation angle u is:
p2 2 l4
Ž 1 q cos 2u . w D´ 2 x n´
Ž 1.
or Ru ,u s R 90,u Ž 1 q cos 2u .
Ž 2.
where R 90 is the Rayleigh ratio at an observation angle of 908; ‘u’ in the subscript stands for unpolarized incident light. D ´ is dielectric fluctuation magnitude inside the associated correlation volume n´ . The attenuation coefficient per unit length a sca and which we will call for brevity the scattering coefficient, is deduced by integrating the Rayleigh ratio over all the angles of a sphere of radius R.
a sca s
16p 3
R 90,u .
a sca s
8p 3 27l4
n8 kT b T Ž p 11 q 2 p 12 .
2 q9b S p 12 Cdep ,
2.1. Density fluctuation in perfect crystals
Ru ,u s
For a polarized incident laser beam, a multiplication factor due to the anisotropy of scatter centers Žthe Cabannes factor. must be included w10x. In a perfect crystal, the only processes that contribute to the dielectric fluctuations are density fluctuations and Raman effects. The density fluctuations include both isobaric and adiabatic density fluctuations. The isobaric density fluctuations give rise to the Rayleigh central line and the adiabatic density fluctuations to acoustic waves and to Brillouin satellite lines. For both sapphire and silica, the Raman scattering components are much smaller compared with thermal scattering. We thus limit our case to density fluctuation scattering. Inserting expressions for D ´ 2 of the processes mentioned above, the total fractional scattered power a sca in isotropic crystals is given as follows w11x:
Ž 3.
2
Ž 4.
where l, n, b T , b S , T, pi j are the wavelength of the incident light, the refractive index of the crystal, its isothermal compressibility, adiabatic compressibility, temperature and its photo-elastic coefficients respectively. The coefficient Cdep is the Cabannes depolarization factor w10x due to the optical anisotropy of the scattering molecules. Although sapphire is anisotropic, Eq. Ž4. is an excellent approximation for a sca as the differences in the relevant matrix elements due to crystal anisotropy are only 5%. The calculation of the scattered light due to density fluctuation in sapphire is approximately 1.6 ppmrcm at 633 nm and 0.21 ppmrcm at 1064 nm. Whereas in practice in glassy materials such as fused silica the density fluctuation limit has been reached w12,13x, few real ionic crystals ever exhibit this low limit case. 2.2. Point defects in real crystals Recent theoretical work w8,14x showed that the presence of point defects in crystals such as impurities or vacancies contribute to Rayleigh scattering. For individual point defects, the permittivity change
F. Benabid et al.r Optics Communications 167 (1999) 7–13
9
depends on the three following processes described by Arora w8x: Ža. the change in electronic polarisability at the defect site; Žb. the elastic strain field around the defect due to ionic size mismatch; Žc. the strain field arising due to effective charge on the defect. Experimentally, Fredericks et al. w15x have demonstrated in the case of KBr crystals the linearity between the number of impurities–vacancies and the elastic scattering in crystals. Taking into account the above results, one can calculate the scattering coefficient due to individual point defects in the following expression:
a sca ,def s f = 10y2 7 cm2 n def Ž cmy3 .
ž
633 nm
l
4
/
,
Ž 5. where f is a factor which depends on the nature of the point defect and can take values in the range 0.2 to 14 at an incident wavelength of 633 nm for typical impurities and vacancies. Here, n def is the density of the impurities–vacancies and l is the incident wavelength expressed in nanometer. Averaging over the typical density numbers of the common impurities and vacancies found in high purity sapphire samples w16,17x, we find that the total impurities–vacancies might be about 100 ppm of the density number of Al 2 O 3 , namely ; 10 21 to 10 22 point defects per cm3. Using these densities in Eq. Ž5. with f s 7, the scattering level at 1064 nm, due to individual point defects is between 1 to 10 ppmrcm. We notice here that this extrinsic Rayleigh scattering is more than one order of magnitude greater than the intrinsic Rayleigh scattering due to permittivity fluctuations. 3. Experimental set-up Light scattering measurements were performed using the setup shown in Fig. 1. The incident collimated laser light passes through the middle of the sample. The transmitted beam is measured with a calibrated power meter. The scattered light passes through a first pupil, which determines the measured length of the scattered light, then a focusing telescope and finally a second pupil. The image of the scattered light is detected by a 16-bit cooled Žy108C. CCD camera placed at the image plane of the telescope. The whole collection system is mounted on a
Fig. 1. Scattering measurement set-up. Ds Pupil; u sangle of observation; V s collection solid angle; T s telescope; PDs power meter.
rotating arm with an angular range of 908. The image is then processed by a personal computer. Extraneous light is eliminated by enclosing the whole of the light gathering optics in a light-tight, matt black tube. The laser source is either an unpolarized He–Ne laser of 4 mW output or a vertically polarized diode pumped Nd:YAG laser of 100 mW output. Both lasers show power fluctuations less than 2% over a period of 6 h and less than 0.5% of fluctuations per 5 min Žthe average time for taking an image.. In this paper we will present the results of light scattering measurements on 3 sapphire samples, 2 fused silica samples, and benzene. The 3 sapphire samples are all high purity single crystals obtained from Crystal Systems ŽSalem, USA.. They are all cylindrical shaped with a c-axis parallel to the cylinder axis. One is polished HEMEX Ultra single crystal, a cylinder of 50 mm diameter and 100 mm length Žsample C.. The two other samples ŽA and B. are CSI pure sapphire samples of 30 mm diameter and 100 mm length. Their c-axis and the laser beam were aligned in parallel. The 2 fused silica samples are Suprasil high purity glass of 50 mm diameter and 100 mm length.
10
F. Benabid et al.r Optics Communications 167 (1999) 7–13
Fig. 2. Typical data treatment. Ža. Scattered light recorderd for different exposure times. Žb. The image is digitized, and the number of counts C Žt . is determined. Žc. After plotting the number of counts as a function of exposure time, the count rate CR is determined from the slope of the line.
Both are optically polished Žsample 1. and the second is in addition chemically polished Žsample 2. Žprovided from the VIRGO project.. Benzene ŽHPLC grade. was contained in a cylindrical shaped glass cell of 30 mm diameter and 100 mm length.
determined by comparing a calibrated measurement of laser power with a known fraction of the laser falling on the CCD Žsee Fig. 3.. The relation between the power and count rate can be written as follows:
3.1. Data treatment
Where CR is the count rate per pixel expressed in countsrsrpixel, CR 0 is the count rate due to dark
CR s qP y CR 0 .
Ž 6.
As illustrated in Fig. 2 for a typical example, to determine the Rayleigh ratio we carry out the three following steps. First, we take images of the scattered light at a certain angle u , for different exposure times. Secondly, the CCD readout of the scattered light is integrated over a nominated area, slightly larger than the image area ŽFig. 2b.. The integrated data C Žt . Žcount number due to the scattered light for the unit solid angle defined by our collection aperture. is plotted versus integration time t . The slope of the graph gives the count rate CR, which is directly proportional to the collected power ŽFig. 2c. The measurement of the count rate is a method of averaging as well as giving the readout noise as the y-axis intercept. Consequently, one can deduce via the slope an accurate measurement of CR and use of the full linear dynamic range of the CCD by choice of the right range of exposure time. 3.2. Calibration of CCD camera The coefficient of proportionality between the laser power per unit area and the count rate is
Fig. 3. CCD camera calibration set-up. The mirror mounted on a translation stage and is either positioned at M to reflect the light X into the CCD camera or M to reflect the light into a calibrated power meter ŽPD.. F1 and F2 are sets of neutral density filters.
F. Benabid et al.r Optics Communications 167 (1999) 7–13
current and background noise, P is the power and q is the coefficient of proportionality. Eq. Ž6. then gives us the power corresponding to the determined count rate. 3.3. Experimental errors The noise in the digitized image is due to the background light, CCD dark current and readout noise. The count rate corresponding to the dark current has been measured by completely covering the CCD array and proceeding as in Section 3.2, to be 10 countsrsrpixel. This is negligible compared with the background light count rate noise which was measured to be about 400 countsrsrpixel for a signal level of 15 000 countsrpixel. This represents a relative error of 1–2% of our typical signal due to light scattering, which is of the order of 10 4 countsrsrpixel. The power meter specifications show a relative error of 2% in the measured power. Finally, with the power fluctuation of 0.5%, the experimental error remains less than 5%.
4. Measurement of the Rayleigh ratio and scattering coefficient Taking into account the geometry of our experiment, the value of the Rayleigh ratio is given by the expression: PLS Ž u . s Ptrans = V
Ž2yt . t 1
ž / n2
Ž1 y k Ž u . 2 .
Ru d.
Ž 7.
In this equation PLS Ž u . is the scattered laser power determined using the method described in Section 3. Ptrans is the power of the transmitted light by the sample. t and k are the transmission and reflection coefficients of the sample respectively.
Ž2yt . t
Ž 1 y k Ž u . 2.
is the reflection correction term due to the multireflection of the transmitted beam inside the sample and to the multireflection of the scattered light. The
11
details of this correction can be found in Ref. w18x. V is the collection solid angle in air, n is the refractive index of the sample, and d the actual path length observed in the sample. Ru the Rayleigh ratio at an observation angle of u . In our experiment the measurements were carried out at 2 different wavelengths; one at 633 nm with an unpolarized He–Ne laser, and a second at 1064 nm with a vertically polarized diode pumped Nd:YAG laser. The attenuation coefficient due to scattering a sca is deduced from Eq. Ž3.. 4.1. Results and discussion We have confirmed the assumption that we are in the Rayleigh regime of light scattering Žsmall scattering centers compared with the wavelength. by measuring the Rayleigh ratio at a range of observation u between 90 and 308. In the case of unpolarized incident light, the angular distribution of scattered light fits the 1 q cos 2 Ž u . curve given in Eq. Ž2. within 5%. For vertically polarized incident light, the scattered light is almost constant Žwithin 5%. with respect to the observation angle in the horizontal plane as expected w19x. Knowing the angular dependence of Ru , it is sufficient to determine the Rayleigh ratio at an observation angle of 908 Ž R 90 . via Eq. Ž7. and the scattering loss factor a sca can be deduced from Eq. Ž3.. Table 1 shows the results of Rayleigh ratio and scattering coefficient for the different samples at 633 and 1064 nm, respectively. The measurement results of Rayleigh scattering in the benzene samples are in the range reported in the literature w10,18x. The results obtained with the fused silica show a scattering loss 3 or 4 times higher than the best reported in the literature w12x. Also, one can notice that the Rayleigh ratio R 90 at 633 and 1064 nm follows the ly4 Rayleigh law in all the samples. Considering the sapphire samples, the results of the scattering loss of the samples A and B are the lowest results reported on the sapphire scattering w20x but are however still ; 2 times higher than our fused silica. Reducing defects such as F centers, known w21x to be present in sapphire grown in a vacuum Žheat
F. Benabid et al.r Optics Communications 167 (1999) 7–13
12 Table 1 Results of Rayleigh ratio measurement. Sample
R 90 at 633 nm Žppmrcm.
R 90 at 1064 nm Žppmrcm.
a sca at 1064 nm Žppmrcm.
Theoretical a sca at 1064 nm Žppmrcm.
Sapphire A B C
6.1 6.4 9.3
0.75 0.8 1.1
12.6 13.4 18.4
0.2 0.2 0.2
F-Silica 1 2
5.4 3.2
0.7 0.4
11.7 6.7
0.4 0.4
Benzene
41.0
5.1
85
67 a
Columns 2 and 3 report the results of the measurement of Rayleigh ratio at wavelengths of 633 and 1064 nm, respectively. The scattering coefficient is reported in column 3 and deduced from Eq. Ž3.. The last column shows the theoretical scattering coefficient. a cf. Ref. w10x.
exchanger method. which results in stoichiometric imbalance of cations and anions, would suggest that sapphire with substantially less scattering loss may be possible. 5. Conclusion A simple and accurate light scattering measurement using an imaging technique has been used successfully. According to thermodynamic fluctuation theory, the fundamental limit of scattering loss in pure sapphire is 0.21 ppmrcm at 1064 nm, half that of fused silica. Our experimental results of Rayleigh scattering showed a scattering loss in sapphire more than 50 times higher than the intrinsic scattering loss predicted by the theory. We attribute this enhancement to nano-scale point defects such as impurities and vacancies. It has been demonstrated w15x that the reduction of these point defects can decrease substantially the scattering loss in a crystal. High purity sapphire samples grown by the heat exchange method ŽHEM. are known to still have non-negligible amounts of vacancies ŽF centers. and ions impurities such as Cr 3q and Ti 3q Žfew ppm.. Their reduction of at least one order of magnitude would reduce substantially the scattering loss in sapphire. We also observed a high scattering loss in fused silica compared with other results reported and the theoretical
limit, making the Rayleigh scattering measurement of every test mass substrate an important and necessary consideration in the design for all laser interferometric gravitational-wave detectors. Considering the performance of LIGW detectors, a scattering loss of 13 ppmrcm reported here, is a promising result indeed. The loss due to Rayleigh scattering in a Fabry–Perot Michelson interferometer will contribute in a round trip loss through the 10 cm thick beamsplitter and the 10 cm thick input mirror substrate of ; 500 ppm. This is comparable to 100 bounces cavity round trip, where the mirror scatteringrabsorption loss are of ; 600 ppm Ž3 ppmrmirror. w22x. In conclusion, these results confirm that sapphire is a viable test mass material for LIGW detectors given the measurements of quality factor and birefringence reported in Refs. w1x and w3x, respectively. References w1x V.B. Braginskii, et al., Systems with Small Dissipation, The University of Chicago Press, Chicago, IL, 1985. w2x D.G. Blair, F. Cleva, C.N. Man, Opt. Mater. 8 Ž1997. 233–236. w3x F. Benabid, M. Notcutt, L. Ju, D.G. Blair, Phys. Lett. A 237 Ž1998. 337–342. w4x W. Winkler, K. Danzmann, A. Rudiger, R. Schilling, Phys. Rev. A 44 Ž11. Ž1991. 7022–7036. w5x K.S. Thorne, Caltech Report, 1989. w6x J.Y. Vinet, B. Meers, C.N. Man, A. Brillet, Phys. Rev. D 38 Ž2. Ž1988. 433–447.
F. Benabid et al.r Optics Communications 167 (1999) 7–13 w7x J.Y. Vinet, V. Brisson, S. Braccini, Phys. Rev. D 54 Ž2. Ž1996. 1276–1286. w8x A.K. Arora, R. Kesavamoorth, D. Sahoo, J. Phys. C 15 Ž1982. 4591–4598. w9x B. Chu, Laser Light Scattering, Academic Press, New York, 1974. w10x I.L. Fabelinskii, Molecular Scattering of Light, Plenum, New York, 1968. w11x J. Tauc, Highly transparent glasses, in: S.S Mitra, B. Bendow ŽEds.., Highly Transparent Solids, Plenum, New York, 1975. w12x S. Sakaguchi, S.-I. Todoroki, J. Am. Ceram. Soc. 79 Ž11. Ž1996. 2821–2824. w13x D.A. Pinnow, Appl. Phys. Lett. 22 Ž10. Ž1973. 527–529. w14x D.L. Mills, J. Appl. Phys. 51 Ž11. Ž1980. 5864–5867. w15x W.J. Fredericks, P.R. Collins, D.F. Edwards, Phys. Rev. B 35 Ž6. Ž1987. 2999–3002.
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w16x M.E. Innocenzi, R.T. Swimm, M. Bass, R.H. French, A.B. Villaverde, M.R. Kokta, J. Appl. Phys. 67 Ž12. Ž1990. 7542– 7546. w17x B.D. Evans, J. Appl. Phys 70 Ž7. Ž1991. 3995–3997. w18x E.R. Pike, W.R. Pomeroy, J.M. Vaughan, J. Chem. Phys. 62 Ž8. Ž1974. 3188–3192. w19x H.C. Hulst, Light Scattering by Small Particles, Wiley, New York, 1957. w20x M. Burka, Notes on sapphire, CNRS, Groupe de Recherches sur les ondes de gravitation, Private Communication. w21x B.D. Evans, G.J. Pogatshnik, Y. Chen, Nucl. Instrum. Methods Phys. Res. B 91 Ž1994. 258–262. w22x J.E. Logan, N.A. Robertson, J. Hough, Opt. Commun. 107 Ž1994. 342–346.