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Effect of atmospheric turbulence on propagation of ultraviolet radiation
Optics & Laser Technology 32 (2000) 39±48
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Eect of atmospheric turbulence on propagation of ultraviolet radiation Daniel L. Hutt a,*, 1, David H. Tofsted b a
Defence Research Establishment Valcartier, 2459 Pie XI Blvd. North, Val-Belair, Quebec, G3J 1X5, Canada US Army Research Lab, Battle®eld Environment Division, AMSRL-IS-EW, W.S.M.R., NM 88002-5501, USA
b
Received 13 October 1999; accepted 3 February 2000
Abstract We investigate the eect of atmospheric optical turbulence on ultraviolet (UV) radiation with a wavelength of 253.7 nm. The normalized irradiance variance (scintillation index) was measured using a UV scintillometer with a path length of 185 m. The dependence of the UV scintillations on the atmospheric turbulence structure parameter and inner scale was determined through simultaneous measurements of these quantities made with a visible laser scintillometer. The dependence of the UV scintillation index and its probability density function on receiver aperture size was also measured. It was found that the scintillation predicted by currently available models which take into account the eects of inner scale, saturation and aperture averaging was in good agreement with the measurements made under various conditions in weak turbulence. Crown Copyright 7 2000 Published by Elsevier Science Ltd. All rights reserved. Keywords: Ultraviolet radiation; Scintillation; Optical turbulence; Inner scale
1. Introduction There has been increasing interest in exploiting the ``solarblind ultraviolet'' (SBUV) waveband from 240 to 280 nm for military surveillance and target detection. Solar radiation in this waveband is absorbed by stratospheric ozone resulting in a complete absence of solar background radiation at ground level. Systems operating in the SBUV have the advantage of high signal-to-noise ratio and low scene clutter, but the eects of atmospheric optical turbulence can degrade performance. These eects include irradiance ¯uctuations (scintillation), beam wander and random distortion and displacement of images which reduce the system * Corresponding author. Tel.: +1-902-426-3100, ext. 218; fax: +1902-426-9654. E-mail address: [email protected] (D.L. Hutt). 1 Current address: Defence Research Establishment Atlantic, Ocena Acoustics Group, 9 Grove Street, Dartmouth, Nova Scotia, B2Y 3Z7, Canada
angular resolution. The eects of atmospherical optical turbulence can be a limiting factor for UV systems operating near the ground where turbulence is greatest. To date, most work on the eects of optical turbulence has been done for visible or near-infrared wavelengths. According to the familiar Rytov solution to the wave equation [1], the log-amplitude variance scales as a wavelength to the ÿ7/6 power which implies that the eects of scintillation due to atmospheric turbulence are two to three times greater in the SBUV than in the visible. The same wavelength dependence also implies that the log-amplitude variance in the SBUV would become saturated at levels of turbulence approximately half those required to cause saturation of visible light. Thus, according to the Rytov solution, UV radiation should be much more susceptible to turbulence eects than visible light. To investigate turbulence eects in the UV we measured irradiance scintillations at a wavelength of l=253.7 nm over a path length of L = 185 m. Simul-
0030-3992/00/$ - see front matter Crown Copyright 7 2000 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 0 - 3 9 9 2 ( 0 0 ) 0 0 0 1 4 - 1
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D.L. Hutt, D.H. Tofsted / Optics & Laser Technology 32 (2000) 39±48
taneous measurements of the atmospheric turbulence structure parameter C n2 and inner scale l0 were made with a laser scintillometer operating at wavelength of 670 nm. The eects of receiver aperture size on the measured scintillation and on the probability density of scintillations were also measured by acquiring data using a wide range of receiver aperture sizes. It was p found that for the Fresnel zone scale lL used, the measured UV scintillations were considerably less than those predicted by the Rytov theory. The observed reduction is in accordance with the known combined eects of inner scale, saturation, and aperture averaging at both transmitter and receiver. 2. UV scintillation measurements The data used in this study were acquired during September 1997 at the Defence Research Establishment Valcartier (DREV), located 20 km northwest of Quebec City. The coordinates of DREV are approximately 46.9N, 71.5W. The site is a level, grass covered ®eld with sandy soil. It is about 300 m across, surrounded by spruce trees on the West and North sides and by low buildings in the East. The height of the surrounding trees and buildings is approximately 5± 7 m. The UV source and receiver were housed in small instrument sheds separated by a distance of 185 m. The propagation path was 1.8 m above the ground and oriented approximately North±South with prevailing winds coming from the West. Due to the ¯at terrain, turbulence parameters should be approximately constant over the test line of sight, though there may have been some fetch in¯uences due to the buildings and trees bordering the area. According to the 100-to-1 rule for fetch eects [2], the site half width of approximately 150 m produces a fully established internal boundary layer of 1.5 m. A deeper, but less well established internal boundary layer of 3± 5 m is also established. When compared to the path height of 1.8 m, these rules of thumb indicate that our measurement conditions were nearly free of fetch in¯uences. Near the surface, the outer scale of the turbulence spectrum is given approximately by L0 1 1.68kvh, using results from Lewellen [3] and Paulson [4]. Here, kv 1 0.4 is von Karman's constant and h is the height above the surface, assuming h<
The dangerous ozone-producing radiation at 185 nm is absorbed by an outer quartz jacket. The lamp has a 25 mm long bulb with a power output of 36 mW/cm2 at a distance of 20 cm. The quartz jacket also acts as an insulator to make the lamp less sensitive to ambient temperature changes. The lamp requires a high frequency, 900 V power supply available from BHK. The power supply output oscillates at 50 kHz with the result that the light output of the lamp has a 50% ripple at 50 kHz. The size of the emitting surface of the lamp was masked to 3.2 mm radius. This is the smallest size that could be used while maintaining an adequate signal at the receiver. The receiver used for the UV scintillation measurements was a UV-sensitive photomultiplier tube with a multi-element absorption ®lter that limited the spectral response to the solar blind waveband from 240 to 280 nm. The ®lter greatly reduced the solar background signal. The output of the sensor was a pulse train representing received photons. The receiver aperture was controlled using aperture plates with circular holes with radii between 0.8 and 12.7 mm. The unrestricted receiver aperture had a radius of 13 mm. The receiver output was recorded by a data logging computer with a sampling period of 1 ms. Photons were counted for 90% of the sampling period thus averaging over approximately 45 cycles of the 50 kHz ripple of the UV source. As a result, the source intensity was for all intents and purposes constant and the 50 kHz ripple did not interfere with measurements of the turbulence induced intensity ¯uctuations. The measured intensity ¯uctuations were expressed in terms of the normalized variance sI2 or scintillation index, de®ned as sI2 hI 2 i ÿ hIi 2 =hIi 2 ,
1
where I is the instantaneous measured intensity, and hIi is the time-averaged intensity. 3. Turbulence measurements The eect of atmospheric turbulence on light propagation may be characterized by the refractive index structure parameter C n2 and the inner scale l0. For homogeneous and isotropic turbulence C n2 is de®ned as [1] C n2 hn ~r1 ÿ n ~r2 2 i= j r~1 ÿ r~2 j 2=3 ,
2
where n is the index of refraction at points rÄ1 and rÄ2. C n2 has the dimensions of mÿ2/3 and varies from approximately 10ÿ17±10ÿ12 mÿ2/3 in the atmospheric boundary layer. l0 characterizes the boundary between the inertial subrange of the turbulence spectrum and the dissipation region. In the surface layer the value of
D.L. Hutt, D.H. Tofsted / Optics & Laser Technology 32 (2000) 39±48
l0 is of the order of millimeters. For the de®nition of inner scale we use the distance at which the large and small asymptotic forms of the temperature structure functions are equal [6]. C n2 and l0 were measured with an SLS-20 scintillometer manufactured by Scintec GmhB, TuÈbingen, Germany (www.scintec.com). The SLS-20 uses the independently measured scintillation of two displaced laser beams to simultaneously derive C n2 and l0. The source is a 1 mW laser diode with a wavelength of 670 nm and divergence of approximately 5 mrad. The beam passes through a birefringent calcite crystal which splits the beam into two displaced but parallel components with crossed polarizations. The beams are displaced by 2.7 mm. At the receiver the two overlapping beams are resolved by a polarizing beam splitter, allowing the scintillation of each beam to be measured independently by two 2.5 mm diameter detectors. The transmitted laser beams are modulated at 20 kHz to allow AC coupling of the detector outputs which eliminates background ¯uctuations. The SLS-20 is described in detail in Ref. [7]. The calculations to derive C n2 and l0 are carried out in real time by the instrument's laptop computer. The SLS-20 provided path-averaged values of C n2 and l0 (weighted toward the center of the path) every 60 s. Each recorded value is the average of six measurements based on a 10 s subperiod. The SLS-20 source and receiver were located in the same sheds as the UV source and receivers so that both scintillometer paths were parallel but displaced laterally by 15±20 cm. Due to the close proximity of the scintillometer paths, Taylor's hypothesis of frozen turbulence between the two is reasonable. Thus the measurements of C n2 and l0 were carried out over essentially the same path as the UV scintillation measurements and with comparable averaging times. The SLS-20 provides C n2 at a wavelength of 670 nm while the wavelength of interest is near 254 nm. C n2 varies slightly from the infrared to UV wavebands with a 1/l 2 dependence. The wavelength dependence of C n2 can be calculated from the spectral refractivity of air which is de®ned as N = 106 (n ÿ 1). Using EdleÂn's [8] expression for N with a temperature of 208C and atmospheric pressure of 1013 mb, N (670 nm)=271.6 and N (254 nm)=294.0. When the values of n = 106 N + 1 are used in Eq. (2) it is found that there is negligible dierence between C n2 at 670 and at 254 nm. Thus, the C n2 values provided by the SLS-20 at 670 nm are taken to be applicable at the wavelength of interest, 253.7 nm. 4. Scintillation and optical turbulence For weak optical turbulence and a point source, the
41
Rytov solution to the wave equation for spherical wave propagation describes scintillation in terms of the variance in log-amplitude (w ) of the received wave. Assuming l0=0, L0=1 and constant turbulence along the propagation path [1], sw20 0:124k7=6 L11=6 C n2 ,
3
where sw20 is the log-amplitude variance over path length L and k is the wavenumber 2p/l. The subscript in sw20 refers to a zero inner scale statistic. Based on the assumption that ¯uctuations of log-amplitude are normally distributed, Rytov theory predicts the normalized variance of the received irradiance to be sI2 exp 4sw20 ÿ 1:
4
The weak turbulence condition is met if the actual logamplitude variance sw2 satis®es the following inequality [9]: sw2 < 0:3:
5
Rytov theory with zero inner scale would approximate sw2 using sw20 : Therefore, according to Eq. (4), the weak turbulence condition is met for sI2 < 1: Most of the measurements reported here were made under weak turbulence conditions. The strong scintillation regime sI2 > 1 includes the saturation region where sI2 decreases as sw20 increases. Saturation is the result of decorrelation of wavefront perturbations. Further defocusing eects as sw20 increases, merely produce a more random signal whose limit is unity, i.e. sI2 41 as sw20 4 1: The Scintec scintillometer requires nonsaturation conditions for measurement of C n2 : According to Eq. (3), for the scintillometer wavelength of 670 nm and pathlength of 185 m path, the turbulence can be considered weak for C n2 up to 1 10ÿ12 mÿ2/3. Such high values of C n2 were not observed during the measurements. Therefore, the measurements of C n2 provided by the Scintec system are assumed to be unaected by saturation and are considered to be accurate. Rytov theory is approximately correct for the analysis of our experimental data. This theory predicts a linear correlation between measured scintillation variance and C n2 : An example of this behavior is shown in Fig. 1 where independent measurements of UV scintillation sI2 derived using Eq. (1)] and C n2 data acquired over a seven hour period are well correlated as expected from Eqs. (3) and (4). However, the scintillation values fall below theoretical calculations using Rytov theory with zero inner scale.
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D.L. Hutt, D.H. Tofsted / Optics & Laser Technology 32 (2000) 39±48
Fig. 1. Time plot of turbulence structure parameter C n2 and UV scintillation index sI2 : The measurements were made on 22 September 1997 under partly cloudy conditions. Source and receiver apertures were both 3.2 mm radius.
5. Eect of inner scale and saturation on scintillation Measured UV scintillation is plotted against C n2 in Fig. 2 along with the theoretical curve for sI2 given by Eqs. (3) and (4). As expected, the measured scintillation is much less than that predicted by Eqs. (3) and (4) because the zero inner scale form for sw20 is not appropriate. A nonzero inner scale reduces the value of unsaturated scintillation because turbules smaller than l0 are not available to perturb the propagating wavefront. These smaller turbules would be the most eective ones at causing scintillation, and their absence, due to ®nite l0, hinders the onset of saturation. An additional eect that reduces the measured
scintillation in Fig. 2 is aperture averaging at both the source and receiver. To better understand the measurements shown in Fig. 2, a propagation model that includes accurate inner scale eects is needed. Hill [10] proposed a spectrum to describe the inertial subrange to dissipation range transition for atmospheric refractive index ¯uctuations. Hill and Cliord [6] then calculated its eect on the unsaturated log-irradiance variance. Numerous authors later considered the eects of inner scale on saturation [9,11]. The theoretical curves of Churnside and Cliord [11], which describe the eects of inner scale and saturation on sI2 , were then parameterized by Tofsted [12] using analytic expressions. To employ these corrections, sw20 is ®rst derived from Eq. (3) and used to evaluate sw2 sw20 1 ÿ exp ÿj 2 =H 0:48 exp ÿj=14:43 1, 6 p where j is the normalized Fresnel scale lL=l0 and H is given by H j 11:1 ÿ 4:7 expÿ j ÿ 1:7 2 =7:632:
Fig. 2. Measured UV scintillation as a function of C n2 and that predicted by Eqs. (3) and (4). Source and receiver apertures were both 3.2 mm radius.
7
The normalized Fresnel scale represents the turbule size, relative to the inner scale l0, which is most eective in causing scintillations. Eqs. (6) and (7) are parameterizations which represent the eects of two competing features of the Hill turbulence spectrum: viscous dissipation of ¯uctuation energy at scale sizes less than l0 (the Kolmogorov microscale) and an increase in turbule density above that predicted by the Kolmogorov spectrum around p the size of l0 (the so-called spectral bump). For lL < 7l0 the available turbules cannot eectively focus the incident wavefront, resulting in reduced scintillation. The factor 1 ÿ exp ÿj 2 =H in Eq. (6) represents this
D.L. Hutt, D.H. Tofsted / Optics & Laser Technology 32 (2000) 39±48
suppression of scintillations for small Fresnel zone scales. The second factor in Eq. (6) acts in combination with the ®rst to simulate the eects of the spectrum peak p which produces a response maximum around lL 7l0 :
43
Using sw2 from Eq. (6), sI2 is derived from a parameterization based on recent observations by Consortini et al. [13], sI2 P ÿ 1X 11 ÿ G0 AZ=P :
8
The ®rst term on the right of Eq. (8) represents the outer envelope of the saturation curve and the second term represents the response prior to saturation. P(j ) is given by P j 2:91 93j ÿ5=2 :
9
Eq. (9) is valid over the range 2.4 < j < 8, based on the range of inner scales 3 mm < l0 < 10 mm available from the Consortini data set. The X function in the ®rst term of Eq. (8) is de®ned as X Z, j C jG1 Z=A 1 ÿ C jG0 Z=A,
10
where Z 4sw2 =A and A = 62.23 is a constant. The G functions represent the rate at which the curve envelope collapses toward its limiting value of unity as Z 4 1. The G functions are de®ned by G0 x exp ÿx and G1 x 1 ÿ exp ÿx=x: C(j ) is a function which blends the eect of the G functions and is given by C(j )=[1+exp(4ÿj )]ÿ1. At small sw2 , G0 1 G1 1 1 producing X 1 1. Thus, to single term accuracy at small Z, Eq. (8) becomes sI2 11 ÿ exp ÿAZ=P P11 ÿ 1 AZ=P P AZ 4sw2 ,
Fig. 3. Comparison of sI2 calculated using parameterization of Eq. (8) with measurements of Consortini et al. [13] and to the numerical experiments of Hill and Frehlich [14] and Flatte et al. [15]. (a) 4 mm < l0 < 6 mm, (b) 6 mm< l0 < 8 mm, (c) 8 mm < l0 < 10 mm.
11
having the same limiting form as the Rytov model. For large sw2 , X 1 0, and sI2 4 1, giving the correct asymptotic behavior. sI2 calculated with the above empirical model is compared to measurements from Consortini et al. [13] and to the numerical experiments of Hill and Frehlich [14] and Flatte et al. [15], in Fig. 3. To facilitate comparison with Refs. [13±15], sI2 is plotted as a function of b0 where b02 0:496k7=6 L11=6 C n2 is the point receiver spherical wave irradiance variance with zero inner scale. Data from Consortini's 4±5 and 5±6 mm cases are combined in Fig. 3(a); from the 6±7 and 7±8 mm cases in Fig. 3(b); and from the 8±10 mm case in Fig. 3(c). Data from Consortini's l0 > 10 mm case were not considered as the upper bound of the inner scale p was not reported. For the Consortini data, Rf L=k was reported to be 9.65 mm, corresponding to p Fresnel zone scale of lL 24:2 mm. The curves from Refs. [14,15] are for l0/Rf =0.4 and 0.6 (Fig. 3(a)), 0.6 and 0.8 (Fig. 3(b)), and 0.8 and 1.0 (Fig. 3(c)). These same values of l0/Rf were also used for the parametric curves from Eq. (8). In all plots of Fig. 3, the parametric curves [Eq. (8)] and both sets of numerical propagation results predict
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D.L. Hutt, D.H. Tofsted / Optics & Laser Technology 32 (2000) 39±48
similar values for b0 < 0.5. For b0 1 2, the data appear to be better characterized by the parametric curves than the numerical propagation results. For example, in Fig. 3(b) the parametric curves predict a somewhat slower rise than the numerical simulations which tend to capture the maximum extent of the data ¯uctuations. Plots 3(a) and (b) show that the parametric curves are similar to the numerical propagation results of Flatte et al. beyond b0=4. Diculties in interpretation occur in regions where data are sparse such as in plot 3(b) beyond about b0=8 and in plot 3(c) beyond b0=4. Within the data shown in Fig. 3 are points which appear well below all the mathematical curves. The measurement techniques employed by Consortini et al. attempted to ensure uniform turbulence conditions over the optical path, so nonuniformity should not be
the cause of the scatter. The parametric method attempts to characterize some of this below-the-line scatter. This appears particularly eective in plot 3(c), where the parametric curves are seen to roughly pass through the centroid of a cluster of data points near b0=3. We may conclude that while none of the models adequately explain all the variations in the data, the parametric curves provide a rapid means of estimating scintillation eects over a reasonable range of b0 and j. Ultraviolet scintillation data is plotted as a function of sw0 in Fig. 4(a) and (b) with the parametric curve in Eq. (8). The parameters applicablep to our experiment are L = 185 m and l=254 nm or lL 6:9 mm. The plots show data for two dierent conditions where l0 was between approximately 4 and 6 mm and between 9 and 13 mm. The maximum measured value of sI2 in the ®gures is 1.1, so the UV scintillation measurements are mostly within the weak turbulence regime. As with the data of Consortini et al. [13], good agreement is observed between the predictions of Eq. (8) and the measured scintillation for the range of l0 shown. Although the UV scintillation values shown in Fig. 4 are much smaller than those of Fig. 3, the fact that Eq. (8) gives good agreement with these independent data sets, measured under quite dierent conditions, is an indication of its general validity.
6. Eect of aperture averaging on scintillation
Fig. 4. UV scintillation compared to predictions of Eq. (8). The measurement range was 185 m at a wavelength of 253.7 nm. (a) 4 mm < l0 < 6 mm, (b) 9 mm < l0 < 13 mm. Source and receiver apertures were both 3.2 mm radius.
Turbulence induced irradiance scintillations exist in both temporal and spatial domains giving rise to a rapidly varying pattern of bright and dark patches or ``speckles'' at the receiver plane. The characteristic size of the speckles is of the order of the Fresnel scale p lL: The total power transmitted through a ®nite aperture source and receiver is the integral of the spatially varying intensity across both apertures. This spatial averaging has the eect of reducing measured scintillation. In our experiments, the source aperture had a relatively large radius of 3.2 mm to reduce photon shot noise eects. Consequently the measurements contained both source and receiver aperture eects. These eects can be characterized in terms of an aperture averaging factor a for weak turbulence [16,17]. Integrations over the source and receiver for spherical waves involve the form, p 16 1 xJ0 2ux cos x ÿ1 ÿ x 1 ÿ x 2 dx K u p 0 2 2J1 u 12 u where u is a dimensionless function of the spatial fre-
D.L. Hutt, D.H. Tofsted / Optics & Laser Technology 32 (2000) 39±48
quency variable k, the dimensionless path variable u, and either the transmitter or receiver aperture radius, at or ar, respectively. For parameterization purposes we normalizep our variables by the inner scale; k~ k=l0 , j lL=l0 , Nt at =l0 and Nr ar =l0 : This allows the transmitter and receiver aperture arguments ~ ÿ u and ur Nr ku, ~ respectto be written as ut Nt k 1 ively. The aperture averaging factor is then computed by dividing the evaluated scintillation variance over ®nite apertures, sln2 I Nt , Nr , j 48:596s 2 1 1 ~ n k ~ dkC du sinj 2 k~ 2 U uK ut K ur , 8=3 5=3 k~ 0 0 j
13
45
tures with radii ranging from 0.8 to 12.7 mm were used. The source aperture radius was 3.2 mm for each 2 measurement. The measured scintillation values sIm 2 were normalized by sI determined from Eq. (8) using measured values of C n2 and l0. Assuming that Eq. (8) 2 =sI2 repgives the correct value of sI2 , the ratios sIm resent estimates of the aperture averaging factor a. In 2 =sI2 are compared to aperture Fig. 5 the measured sIm averaging factors calculated with Eq. (14). Points along the unity line of Fig. 5 would represent perfect agreement between the computed results and the 2 =sI2 : Error bars in the graph re¯ect net measured sIm 2 standard deviation due to C n2 , l0 and sIm variations during the measurement periods. A least squares linear ®t to the data of Fig. 5 has a slope of 1.24 (r = 0.97) which constitutes reasonably good agreement.
by a similar calculation for point statistics. Thus, the aperture averaging factor is given by a
sln2 I Nt , Nr , j : sln2 I 0, 0, j
14
In Eq. (13), U u u 1 ÿ u= 4p and s 2 0:496k7=6 L11=6 C n2 is the point receiver spherical wave irradiance variance with zero inner scale. The spectrum of refractive index ¯uctuations, given by Fn ~ has already been expanded and 0:033C n2 k ÿ11=3 Cn k, ~ represents the combined with other variables. Cn k bump structure of Fn in the spectral dissipation range [10,18,19]. To determine the eects of aperture averaging during our measurements, nine dierent receiver aper-
2 Fig. 5. Measured ratios sIm =sI2 as a function of computed aperture averaging factor a from Eq. (14). Error bars are the net standard de2 viation due to C n2 , l0 and sIm variations during the measurement 2 period. sIm was measured using nine dierent receiver apertures with radii from 0.8 to 12.7 mm. The source aperture was 3.2 mm radius for all measurements.
Fig. 6. UV scintillation calculated from Eqs. (8) and (15) using measured l0 and C n2 vs measured UV scintillation. The measurements were made with source and receiver aperture radii of 3.2 mm.
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D.L. Hutt, D.H. Tofsted / Optics & Laser Technology 32 (2000) 39±48
In Fig. 5, the two data points corresponding to the smallest receiver apertures (a30.62, a30.69) are obviously further from the unity line than the other points. Here, the possibility exists that photon noise has compromised the measurements. There was also a 2 and C n2 dierence in integration times between the sIm measurements which may also account for some of the discrepancies between the calculated and measured 2 measurement integration times varied from results. sIm 10 to 40 s while the C n2 values were all based on 60 s averages. 2 =sI2 to Tofsted's We also compared our measured sIm [12] weak turbulence parameterization of Churnside's aperture averaging equations based on the Tatarskii spectrum: act f1 Nr2 =4=9 j 2 = 2pg ÿ1 :
15
2 =sI2 data A least squares linear ®t to the measured sIm plotted against act from Eq. (15) had a slope of 0.86 (r = 0.98). Thus, Eq. (15) appears to perform slightly better than Eq. (14), which is based on a more complete theory. This unexpected result may be explained by non-stationary conditions [18,19] which Hill and Ochs [20] claim to be particularly sensitive in the dissipation region where the weighting functions peaked for our measurement conditions. The UV scintillation measurements shown in Fig. 4 were used, along with the associated C n2 and l0 measurements, to calculate sI2 with Eq. (8), and the aperture averaging correction of Eq. (15). Both transmitter and receiver radii were 3.2 mm for these measurements. The resulting ``predicted'' sI2 is plotted 2 in Fig. 6(a) as a function of measured scintillation sIm and (b). A least squares linear ®t to the combined data of Fig. 6(a) and (b) has a slope of 1.05 (r = 0.94) indicating that the predicted and measured scintillation are in good agreement. This is another indication that Eqs. (8) and (15) are a good model of inner scale eects and aperture averaging at 254 nm even though these expressions have previously only been compared to 633 nm data.
7. Probability density function for intensity It is well known that the probability distribution of intensity ¯uctuations p(I ) caused by weak turbulence is well represented by a log-normal function. Churnside and Hill [21] showed that p(I ) should be log-normal even in strong turbulence if a suciently large receiver aperture is used. Since our measurements were made under weak turbulence conditions, p(I ) should be log-normal with the distribution function given by
! ln I 2sw2 2 1 p exp ÿ p I : 8sw2 2Isw 2p
16
Measured p(I ) are shown in ®g. 7(a)±(d) for four dierent aperture sizes ranging from 1.6 mm to 12.7 mm radius. Each p(I ) is based on 30 s of time series data measured at 1 kHz. For each measurement period, values of C n2 and l0 measured with the Scintec scintillometer were used to calculate sw2 with Eq. (6). The resultant values of sw2 were used in Eq. (16) to calculate the expected intensity probability function. The data in Fig. 7 was collected with a source aperture of 3.2 mm and C n2 in the range of 1 10ÿ13 to 5 10ÿ13 mÿ2/3. The scatter of the measured p(I ) points in the data for the 1.6 mm aperture (Fig. 7(a)) is due to the fact that the signal is lower for a smaller aperture which results in a lower signal to noise ratio. Fig. 7 shows excellent agreement between the measured p(I ) and those predicted by Eqs. (6) and (16). Since the p(I ) curves were not ``®tted'' to the data but were generated using sw2 based on measured values of C n2 and l0, the good agreement supports the validity of Eqs. (6) and (16) at 254 nm. 8. Conclusions We investigated the eect of atmospheric optical turbulence on solarblind UV radiation through measurements of intensity scintillation at 253.7 nm and simultaneous measurements of C n2 and l0. The zero inner scale Rytov solution for log-amplitude variance predicts that radiation in the SBUV band would experience two to three times as much scintillation as visible light for a given level of turbulence and that saturation eects would start at levels of turbulence approximately half those required to cause saturation of visible light. However, we showed that the combined eects of l0 and aperture averaging greatly reduced the measured scintillation for our measurement conditions. The eect of l0 on scintillation was modeled using expressions of Tofsted [12] which are parameterizations of theoretical expressions of Hill and Cliord [6]. Although Tofsted's expressions have only been compared with measurements at wavelengths of 488 nm [13] and 633 nm [11], there was good agreement between measured UV scintillation and predicted values when the eects of aperture averaging were taken into account. Aperture averaging eects were modeled in the weak turbulence regime using two dierent approaches: evaluating the combined eect of source and receiver averaging with the bump spectrum [6] and using Tofsted's parameterization of Churnside's receiver aperture averaging analysis [16] with a Tatarskii spectrum. The two-
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47
Fig. 7. (a)±(d) Probability density functions of UV (254 nm) scintillations measured with dierent receiver radii, ar. The curves are p(I ) from Eq. (16) using sw2 derived from Eq. (6) with measured l0 and C n2 : The source aperture radius at was 3.2 mm for all measurements.
ended solution showed good agreement with the measurements except for the two smallest receiver apertures used, while the single-ended parameterization showed slightly better overall agreement with the data. The poorer agreement between the combined model and measurements with the smallest apertures may be due to intermittency eects which alter the form of the turbulence spectrum. The eect of receiver aperture averaging on the probability density of scintillations was also measured using a wide range of receiver aperture sizes. The measured distributions were compared to log-normal curves based on log-amplitude variances corrected for the eect of l0 using Tofsted's expression. Consistent
results were obtained for measured and predicted probability densities, thus establishing that the expressions given are valid in the UV and that they provide a convenient means of modeling UV turbulence eects. Acknowledgements The authors would like to thank Chris Cully for writing the software to acquire the UV signals, Mike Duy for processing the raw UV scintillation data, and MicheÁle Cardinal for assistance in setting up the experiment.
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References [1] Beland R. Propagation through atmospheric optical turbulence. In: Smith FG, editor. The infrared and electro-optical systems handbook. Bellingham: SPIE Optical Engineering Press, 1993. [2] Oke TR. Boundary layer climates. London: Methuen and Co Ltd, 1978. [3] Lewellen WS. Use of invariant modeling. In: Frost W, Moulden TH, editors. Handbook of turbulence. New York: Plenum, 1977 (Chapter 9). [4] Paulson CA. The mathematical representation of wind speed and temperature pro®les in the unstable atmospheric surface layer. Boundary Layer Meteorol 1970;9:857±61. [5] Monin AS, Obukhov AM. Dimensionless characteristics of turbulence in the surface layer. Dokl Akad Nauk SSSR 1953;93:223±6. [6] Hill RJ, Cliord SF. Modi®ed spectrum of atmospheric temperature ¯uctuations and its application to optical propagation. J Opt Soc Am 1978;68:892±9. [7] Thiermann V. A displaced-beam scintillometer for line-averaged measurements of surface layer turbulence. In: Proceedings of the 10th Symposium on Turbulence and Diusion, Portland, Oregon. Amer. Meteorol. Soc. Boston, 29 September±2 October, 1992. p. 244±7. [8] EdleÂn B. The dispersion of standard air. J Opt Soc Am 1953;43:339±44. [9] Cliord SF, Ochs GR, Lawrence RS. Saturation of optical scintillation by strong turbulence. J Opt Soc Am 1974;64:148±54. [10] Hill RJ. Models of the scalar spectrum for turbulent advection. J Fluid Mech 1978;88:541±62. [11] Churnside JH, Cliord SF. Log-normal Rician probability density function of optical scintillations in the turbulent atmosphere. J Opt Soc Am A 1987;4:1923±30.
[12] Tofsted DH. Empirical modeling of laser propagation eects in intermediate turbulence. In: Proceedings of SPIE 1487 Propagation Engineering: Fourth in a Series, Orlando, FL, 1991. p. 372±81. [13] Consortini A, Cochetti F, Churnside JH, Hill RJ. Inner-scale eect on irradiance variance measured for weak-to-strong atmospheric scintillation. J Opt Soc Am A 1993;10:2354±62. [14] Hill RJ, Frehlich RG. Onset of strong scintillation with application to remote sensing of turbulence inner scale. Appl Opt 1996;35:986±97. [15] Flatte SM, Wang G, Martin J. Irradiance variance of optical waves through atmospheric turbulence by numerical simulation and comparison with experiment. J Opt Soc Am A 1993;10:2363±70. [16] Churnside JH. Aperture-averaging factor for optical propagation through the turbulent atmosphere. NOAA Tech. Memorandum, ERL WPL-169, Wave propagation laboratory, Boulder, CO, 1990. [17] Ochs GR, Cliord SF, Wang T. Laser wind sensing: the eects of saturation of scintillation. Appl Opt 1976;15:403±8. [18] Frehlich R. Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer. J Atmos Sci 1992;49:1494±509. [19] Sisterman EA, Andrews LC. A new look at bump spectra models for temperature and refractive index ¯uctuations. In: Proceedings of SPIE 1487 Propagation Engineering: Fourth in a Series, Orlando, FL, 1991. p. 345±55. [20] Hill RJ, Ochs GR. Fine calibration of large-aperture optical scintillometers and an optical estimate of inner scale of turbulence. Appl Opt 1978;17:3608±12. [21] Churnside JH, Hill RJ. Probability density of irradiance scintillations for strong path-integrated refractive turbulence. J Opt Soc Am A 1987;4:727±33.
www.elsevier.com/locate/optlastec
Eect of atmospheric turbulence on propagation of ultraviolet radiation Daniel L. Hutt a,*, 1, David H. Tofsted b a
Defence Research Establishment Valcartier, 2459 Pie XI Blvd. North, Val-Belair, Quebec, G3J 1X5, Canada US Army Research Lab, Battle®eld Environment Division, AMSRL-IS-EW, W.S.M.R., NM 88002-5501, USA
b
Received 13 October 1999; accepted 3 February 2000
Abstract We investigate the eect of atmospheric optical turbulence on ultraviolet (UV) radiation with a wavelength of 253.7 nm. The normalized irradiance variance (scintillation index) was measured using a UV scintillometer with a path length of 185 m. The dependence of the UV scintillations on the atmospheric turbulence structure parameter and inner scale was determined through simultaneous measurements of these quantities made with a visible laser scintillometer. The dependence of the UV scintillation index and its probability density function on receiver aperture size was also measured. It was found that the scintillation predicted by currently available models which take into account the eects of inner scale, saturation and aperture averaging was in good agreement with the measurements made under various conditions in weak turbulence. Crown Copyright 7 2000 Published by Elsevier Science Ltd. All rights reserved. Keywords: Ultraviolet radiation; Scintillation; Optical turbulence; Inner scale
1. Introduction There has been increasing interest in exploiting the ``solarblind ultraviolet'' (SBUV) waveband from 240 to 280 nm for military surveillance and target detection. Solar radiation in this waveband is absorbed by stratospheric ozone resulting in a complete absence of solar background radiation at ground level. Systems operating in the SBUV have the advantage of high signal-to-noise ratio and low scene clutter, but the eects of atmospheric optical turbulence can degrade performance. These eects include irradiance ¯uctuations (scintillation), beam wander and random distortion and displacement of images which reduce the system * Corresponding author. Tel.: +1-902-426-3100, ext. 218; fax: +1902-426-9654. E-mail address: [email protected] (D.L. Hutt). 1 Current address: Defence Research Establishment Atlantic, Ocena Acoustics Group, 9 Grove Street, Dartmouth, Nova Scotia, B2Y 3Z7, Canada
angular resolution. The eects of atmospherical optical turbulence can be a limiting factor for UV systems operating near the ground where turbulence is greatest. To date, most work on the eects of optical turbulence has been done for visible or near-infrared wavelengths. According to the familiar Rytov solution to the wave equation [1], the log-amplitude variance scales as a wavelength to the ÿ7/6 power which implies that the eects of scintillation due to atmospheric turbulence are two to three times greater in the SBUV than in the visible. The same wavelength dependence also implies that the log-amplitude variance in the SBUV would become saturated at levels of turbulence approximately half those required to cause saturation of visible light. Thus, according to the Rytov solution, UV radiation should be much more susceptible to turbulence eects than visible light. To investigate turbulence eects in the UV we measured irradiance scintillations at a wavelength of l=253.7 nm over a path length of L = 185 m. Simul-
0030-3992/00/$ - see front matter Crown Copyright 7 2000 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 0 - 3 9 9 2 ( 0 0 ) 0 0 0 1 4 - 1
40
D.L. Hutt, D.H. Tofsted / Optics & Laser Technology 32 (2000) 39±48
taneous measurements of the atmospheric turbulence structure parameter C n2 and inner scale l0 were made with a laser scintillometer operating at wavelength of 670 nm. The eects of receiver aperture size on the measured scintillation and on the probability density of scintillations were also measured by acquiring data using a wide range of receiver aperture sizes. It was p found that for the Fresnel zone scale lL used, the measured UV scintillations were considerably less than those predicted by the Rytov theory. The observed reduction is in accordance with the known combined eects of inner scale, saturation, and aperture averaging at both transmitter and receiver. 2. UV scintillation measurements The data used in this study were acquired during September 1997 at the Defence Research Establishment Valcartier (DREV), located 20 km northwest of Quebec City. The coordinates of DREV are approximately 46.9N, 71.5W. The site is a level, grass covered ®eld with sandy soil. It is about 300 m across, surrounded by spruce trees on the West and North sides and by low buildings in the East. The height of the surrounding trees and buildings is approximately 5± 7 m. The UV source and receiver were housed in small instrument sheds separated by a distance of 185 m. The propagation path was 1.8 m above the ground and oriented approximately North±South with prevailing winds coming from the West. Due to the ¯at terrain, turbulence parameters should be approximately constant over the test line of sight, though there may have been some fetch in¯uences due to the buildings and trees bordering the area. According to the 100-to-1 rule for fetch eects [2], the site half width of approximately 150 m produces a fully established internal boundary layer of 1.5 m. A deeper, but less well established internal boundary layer of 3± 5 m is also established. When compared to the path height of 1.8 m, these rules of thumb indicate that our measurement conditions were nearly free of fetch in¯uences. Near the surface, the outer scale of the turbulence spectrum is given approximately by L0 1 1.68kvh, using results from Lewellen [3] and Paulson [4]. Here, kv 1 0.4 is von Karman's constant and h is the height above the surface, assuming h<
The dangerous ozone-producing radiation at 185 nm is absorbed by an outer quartz jacket. The lamp has a 25 mm long bulb with a power output of 36 mW/cm2 at a distance of 20 cm. The quartz jacket also acts as an insulator to make the lamp less sensitive to ambient temperature changes. The lamp requires a high frequency, 900 V power supply available from BHK. The power supply output oscillates at 50 kHz with the result that the light output of the lamp has a 50% ripple at 50 kHz. The size of the emitting surface of the lamp was masked to 3.2 mm radius. This is the smallest size that could be used while maintaining an adequate signal at the receiver. The receiver used for the UV scintillation measurements was a UV-sensitive photomultiplier tube with a multi-element absorption ®lter that limited the spectral response to the solar blind waveband from 240 to 280 nm. The ®lter greatly reduced the solar background signal. The output of the sensor was a pulse train representing received photons. The receiver aperture was controlled using aperture plates with circular holes with radii between 0.8 and 12.7 mm. The unrestricted receiver aperture had a radius of 13 mm. The receiver output was recorded by a data logging computer with a sampling period of 1 ms. Photons were counted for 90% of the sampling period thus averaging over approximately 45 cycles of the 50 kHz ripple of the UV source. As a result, the source intensity was for all intents and purposes constant and the 50 kHz ripple did not interfere with measurements of the turbulence induced intensity ¯uctuations. The measured intensity ¯uctuations were expressed in terms of the normalized variance sI2 or scintillation index, de®ned as sI2 hI 2 i ÿ hIi 2 =hIi 2 ,
1
where I is the instantaneous measured intensity, and hIi is the time-averaged intensity. 3. Turbulence measurements The eect of atmospheric turbulence on light propagation may be characterized by the refractive index structure parameter C n2 and the inner scale l0. For homogeneous and isotropic turbulence C n2 is de®ned as [1] C n2 hn ~r1 ÿ n ~r2 2 i= j r~1 ÿ r~2 j 2=3 ,
2
where n is the index of refraction at points rÄ1 and rÄ2. C n2 has the dimensions of mÿ2/3 and varies from approximately 10ÿ17±10ÿ12 mÿ2/3 in the atmospheric boundary layer. l0 characterizes the boundary between the inertial subrange of the turbulence spectrum and the dissipation region. In the surface layer the value of
D.L. Hutt, D.H. Tofsted / Optics & Laser Technology 32 (2000) 39±48
l0 is of the order of millimeters. For the de®nition of inner scale we use the distance at which the large and small asymptotic forms of the temperature structure functions are equal [6]. C n2 and l0 were measured with an SLS-20 scintillometer manufactured by Scintec GmhB, TuÈbingen, Germany (www.scintec.com). The SLS-20 uses the independently measured scintillation of two displaced laser beams to simultaneously derive C n2 and l0. The source is a 1 mW laser diode with a wavelength of 670 nm and divergence of approximately 5 mrad. The beam passes through a birefringent calcite crystal which splits the beam into two displaced but parallel components with crossed polarizations. The beams are displaced by 2.7 mm. At the receiver the two overlapping beams are resolved by a polarizing beam splitter, allowing the scintillation of each beam to be measured independently by two 2.5 mm diameter detectors. The transmitted laser beams are modulated at 20 kHz to allow AC coupling of the detector outputs which eliminates background ¯uctuations. The SLS-20 is described in detail in Ref. [7]. The calculations to derive C n2 and l0 are carried out in real time by the instrument's laptop computer. The SLS-20 provided path-averaged values of C n2 and l0 (weighted toward the center of the path) every 60 s. Each recorded value is the average of six measurements based on a 10 s subperiod. The SLS-20 source and receiver were located in the same sheds as the UV source and receivers so that both scintillometer paths were parallel but displaced laterally by 15±20 cm. Due to the close proximity of the scintillometer paths, Taylor's hypothesis of frozen turbulence between the two is reasonable. Thus the measurements of C n2 and l0 were carried out over essentially the same path as the UV scintillation measurements and with comparable averaging times. The SLS-20 provides C n2 at a wavelength of 670 nm while the wavelength of interest is near 254 nm. C n2 varies slightly from the infrared to UV wavebands with a 1/l 2 dependence. The wavelength dependence of C n2 can be calculated from the spectral refractivity of air which is de®ned as N = 106 (n ÿ 1). Using EdleÂn's [8] expression for N with a temperature of 208C and atmospheric pressure of 1013 mb, N (670 nm)=271.6 and N (254 nm)=294.0. When the values of n = 106 N + 1 are used in Eq. (2) it is found that there is negligible dierence between C n2 at 670 and at 254 nm. Thus, the C n2 values provided by the SLS-20 at 670 nm are taken to be applicable at the wavelength of interest, 253.7 nm. 4. Scintillation and optical turbulence For weak optical turbulence and a point source, the
41
Rytov solution to the wave equation for spherical wave propagation describes scintillation in terms of the variance in log-amplitude (w ) of the received wave. Assuming l0=0, L0=1 and constant turbulence along the propagation path [1], sw20 0:124k7=6 L11=6 C n2 ,
3
where sw20 is the log-amplitude variance over path length L and k is the wavenumber 2p/l. The subscript in sw20 refers to a zero inner scale statistic. Based on the assumption that ¯uctuations of log-amplitude are normally distributed, Rytov theory predicts the normalized variance of the received irradiance to be sI2 exp 4sw20 ÿ 1:
4
The weak turbulence condition is met if the actual logamplitude variance sw2 satis®es the following inequality [9]: sw2 < 0:3:
5
Rytov theory with zero inner scale would approximate sw2 using sw20 : Therefore, according to Eq. (4), the weak turbulence condition is met for sI2 < 1: Most of the measurements reported here were made under weak turbulence conditions. The strong scintillation regime sI2 > 1 includes the saturation region where sI2 decreases as sw20 increases. Saturation is the result of decorrelation of wavefront perturbations. Further defocusing eects as sw20 increases, merely produce a more random signal whose limit is unity, i.e. sI2 41 as sw20 4 1: The Scintec scintillometer requires nonsaturation conditions for measurement of C n2 : According to Eq. (3), for the scintillometer wavelength of 670 nm and pathlength of 185 m path, the turbulence can be considered weak for C n2 up to 1 10ÿ12 mÿ2/3. Such high values of C n2 were not observed during the measurements. Therefore, the measurements of C n2 provided by the Scintec system are assumed to be unaected by saturation and are considered to be accurate. Rytov theory is approximately correct for the analysis of our experimental data. This theory predicts a linear correlation between measured scintillation variance and C n2 : An example of this behavior is shown in Fig. 1 where independent measurements of UV scintillation sI2 derived using Eq. (1)] and C n2 data acquired over a seven hour period are well correlated as expected from Eqs. (3) and (4). However, the scintillation values fall below theoretical calculations using Rytov theory with zero inner scale.
42
D.L. Hutt, D.H. Tofsted / Optics & Laser Technology 32 (2000) 39±48
Fig. 1. Time plot of turbulence structure parameter C n2 and UV scintillation index sI2 : The measurements were made on 22 September 1997 under partly cloudy conditions. Source and receiver apertures were both 3.2 mm radius.
5. Eect of inner scale and saturation on scintillation Measured UV scintillation is plotted against C n2 in Fig. 2 along with the theoretical curve for sI2 given by Eqs. (3) and (4). As expected, the measured scintillation is much less than that predicted by Eqs. (3) and (4) because the zero inner scale form for sw20 is not appropriate. A nonzero inner scale reduces the value of unsaturated scintillation because turbules smaller than l0 are not available to perturb the propagating wavefront. These smaller turbules would be the most eective ones at causing scintillation, and their absence, due to ®nite l0, hinders the onset of saturation. An additional eect that reduces the measured
scintillation in Fig. 2 is aperture averaging at both the source and receiver. To better understand the measurements shown in Fig. 2, a propagation model that includes accurate inner scale eects is needed. Hill [10] proposed a spectrum to describe the inertial subrange to dissipation range transition for atmospheric refractive index ¯uctuations. Hill and Cliord [6] then calculated its eect on the unsaturated log-irradiance variance. Numerous authors later considered the eects of inner scale on saturation [9,11]. The theoretical curves of Churnside and Cliord [11], which describe the eects of inner scale and saturation on sI2 , were then parameterized by Tofsted [12] using analytic expressions. To employ these corrections, sw20 is ®rst derived from Eq. (3) and used to evaluate sw2 sw20 1 ÿ exp ÿj 2 =H 0:48 exp ÿj=14:43 1, 6 p where j is the normalized Fresnel scale lL=l0 and H is given by H j 11:1 ÿ 4:7 expÿ j ÿ 1:7 2 =7:632:
Fig. 2. Measured UV scintillation as a function of C n2 and that predicted by Eqs. (3) and (4). Source and receiver apertures were both 3.2 mm radius.
7
The normalized Fresnel scale represents the turbule size, relative to the inner scale l0, which is most eective in causing scintillations. Eqs. (6) and (7) are parameterizations which represent the eects of two competing features of the Hill turbulence spectrum: viscous dissipation of ¯uctuation energy at scale sizes less than l0 (the Kolmogorov microscale) and an increase in turbule density above that predicted by the Kolmogorov spectrum around p the size of l0 (the so-called spectral bump). For lL < 7l0 the available turbules cannot eectively focus the incident wavefront, resulting in reduced scintillation. The factor 1 ÿ exp ÿj 2 =H in Eq. (6) represents this
D.L. Hutt, D.H. Tofsted / Optics & Laser Technology 32 (2000) 39±48
suppression of scintillations for small Fresnel zone scales. The second factor in Eq. (6) acts in combination with the ®rst to simulate the eects of the spectrum peak p which produces a response maximum around lL 7l0 :
43
Using sw2 from Eq. (6), sI2 is derived from a parameterization based on recent observations by Consortini et al. [13], sI2 P ÿ 1X 11 ÿ G0 AZ=P :
8
The ®rst term on the right of Eq. (8) represents the outer envelope of the saturation curve and the second term represents the response prior to saturation. P(j ) is given by P j 2:91 93j ÿ5=2 :
9
Eq. (9) is valid over the range 2.4 < j < 8, based on the range of inner scales 3 mm < l0 < 10 mm available from the Consortini data set. The X function in the ®rst term of Eq. (8) is de®ned as X Z, j C jG1 Z=A 1 ÿ C jG0 Z=A,
10
where Z 4sw2 =A and A = 62.23 is a constant. The G functions represent the rate at which the curve envelope collapses toward its limiting value of unity as Z 4 1. The G functions are de®ned by G0 x exp ÿx and G1 x 1 ÿ exp ÿx=x: C(j ) is a function which blends the eect of the G functions and is given by C(j )=[1+exp(4ÿj )]ÿ1. At small sw2 , G0 1 G1 1 1 producing X 1 1. Thus, to single term accuracy at small Z, Eq. (8) becomes sI2 11 ÿ exp ÿAZ=P P11 ÿ 1 AZ=P P AZ 4sw2 ,
Fig. 3. Comparison of sI2 calculated using parameterization of Eq. (8) with measurements of Consortini et al. [13] and to the numerical experiments of Hill and Frehlich [14] and Flatte et al. [15]. (a) 4 mm < l0 < 6 mm, (b) 6 mm< l0 < 8 mm, (c) 8 mm < l0 < 10 mm.
11
having the same limiting form as the Rytov model. For large sw2 , X 1 0, and sI2 4 1, giving the correct asymptotic behavior. sI2 calculated with the above empirical model is compared to measurements from Consortini et al. [13] and to the numerical experiments of Hill and Frehlich [14] and Flatte et al. [15], in Fig. 3. To facilitate comparison with Refs. [13±15], sI2 is plotted as a function of b0 where b02 0:496k7=6 L11=6 C n2 is the point receiver spherical wave irradiance variance with zero inner scale. Data from Consortini's 4±5 and 5±6 mm cases are combined in Fig. 3(a); from the 6±7 and 7±8 mm cases in Fig. 3(b); and from the 8±10 mm case in Fig. 3(c). Data from Consortini's l0 > 10 mm case were not considered as the upper bound of the inner scale p was not reported. For the Consortini data, Rf L=k was reported to be 9.65 mm, corresponding to p Fresnel zone scale of lL 24:2 mm. The curves from Refs. [14,15] are for l0/Rf =0.4 and 0.6 (Fig. 3(a)), 0.6 and 0.8 (Fig. 3(b)), and 0.8 and 1.0 (Fig. 3(c)). These same values of l0/Rf were also used for the parametric curves from Eq. (8). In all plots of Fig. 3, the parametric curves [Eq. (8)] and both sets of numerical propagation results predict
44
D.L. Hutt, D.H. Tofsted / Optics & Laser Technology 32 (2000) 39±48
similar values for b0 < 0.5. For b0 1 2, the data appear to be better characterized by the parametric curves than the numerical propagation results. For example, in Fig. 3(b) the parametric curves predict a somewhat slower rise than the numerical simulations which tend to capture the maximum extent of the data ¯uctuations. Plots 3(a) and (b) show that the parametric curves are similar to the numerical propagation results of Flatte et al. beyond b0=4. Diculties in interpretation occur in regions where data are sparse such as in plot 3(b) beyond about b0=8 and in plot 3(c) beyond b0=4. Within the data shown in Fig. 3 are points which appear well below all the mathematical curves. The measurement techniques employed by Consortini et al. attempted to ensure uniform turbulence conditions over the optical path, so nonuniformity should not be
the cause of the scatter. The parametric method attempts to characterize some of this below-the-line scatter. This appears particularly eective in plot 3(c), where the parametric curves are seen to roughly pass through the centroid of a cluster of data points near b0=3. We may conclude that while none of the models adequately explain all the variations in the data, the parametric curves provide a rapid means of estimating scintillation eects over a reasonable range of b0 and j. Ultraviolet scintillation data is plotted as a function of sw0 in Fig. 4(a) and (b) with the parametric curve in Eq. (8). The parameters applicablep to our experiment are L = 185 m and l=254 nm or lL 6:9 mm. The plots show data for two dierent conditions where l0 was between approximately 4 and 6 mm and between 9 and 13 mm. The maximum measured value of sI2 in the ®gures is 1.1, so the UV scintillation measurements are mostly within the weak turbulence regime. As with the data of Consortini et al. [13], good agreement is observed between the predictions of Eq. (8) and the measured scintillation for the range of l0 shown. Although the UV scintillation values shown in Fig. 4 are much smaller than those of Fig. 3, the fact that Eq. (8) gives good agreement with these independent data sets, measured under quite dierent conditions, is an indication of its general validity.
6. Eect of aperture averaging on scintillation
Fig. 4. UV scintillation compared to predictions of Eq. (8). The measurement range was 185 m at a wavelength of 253.7 nm. (a) 4 mm < l0 < 6 mm, (b) 9 mm < l0 < 13 mm. Source and receiver apertures were both 3.2 mm radius.
Turbulence induced irradiance scintillations exist in both temporal and spatial domains giving rise to a rapidly varying pattern of bright and dark patches or ``speckles'' at the receiver plane. The characteristic size of the speckles is of the order of the Fresnel scale p lL: The total power transmitted through a ®nite aperture source and receiver is the integral of the spatially varying intensity across both apertures. This spatial averaging has the eect of reducing measured scintillation. In our experiments, the source aperture had a relatively large radius of 3.2 mm to reduce photon shot noise eects. Consequently the measurements contained both source and receiver aperture eects. These eects can be characterized in terms of an aperture averaging factor a for weak turbulence [16,17]. Integrations over the source and receiver for spherical waves involve the form, p 16 1 xJ0 2ux cos x ÿ1 ÿ x 1 ÿ x 2 dx K u p 0 2 2J1 u 12 u where u is a dimensionless function of the spatial fre-
D.L. Hutt, D.H. Tofsted / Optics & Laser Technology 32 (2000) 39±48
quency variable k, the dimensionless path variable u, and either the transmitter or receiver aperture radius, at or ar, respectively. For parameterization purposes we normalizep our variables by the inner scale; k~ k=l0 , j lL=l0 , Nt at =l0 and Nr ar =l0 : This allows the transmitter and receiver aperture arguments ~ ÿ u and ur Nr ku, ~ respectto be written as ut Nt k 1 ively. The aperture averaging factor is then computed by dividing the evaluated scintillation variance over ®nite apertures, sln2 I Nt , Nr , j 48:596s 2 1 1 ~ n k ~ dkC du sinj 2 k~ 2 U uK ut K ur , 8=3 5=3 k~ 0 0 j
13
45
tures with radii ranging from 0.8 to 12.7 mm were used. The source aperture radius was 3.2 mm for each 2 measurement. The measured scintillation values sIm 2 were normalized by sI determined from Eq. (8) using measured values of C n2 and l0. Assuming that Eq. (8) 2 =sI2 repgives the correct value of sI2 , the ratios sIm resent estimates of the aperture averaging factor a. In 2 =sI2 are compared to aperture Fig. 5 the measured sIm averaging factors calculated with Eq. (14). Points along the unity line of Fig. 5 would represent perfect agreement between the computed results and the 2 =sI2 : Error bars in the graph re¯ect net measured sIm 2 standard deviation due to C n2 , l0 and sIm variations during the measurement periods. A least squares linear ®t to the data of Fig. 5 has a slope of 1.24 (r = 0.97) which constitutes reasonably good agreement.
by a similar calculation for point statistics. Thus, the aperture averaging factor is given by a
sln2 I Nt , Nr , j : sln2 I 0, 0, j
14
In Eq. (13), U u u 1 ÿ u= 4p and s 2 0:496k7=6 L11=6 C n2 is the point receiver spherical wave irradiance variance with zero inner scale. The spectrum of refractive index ¯uctuations, given by Fn ~ has already been expanded and 0:033C n2 k ÿ11=3 Cn k, ~ represents the combined with other variables. Cn k bump structure of Fn in the spectral dissipation range [10,18,19]. To determine the eects of aperture averaging during our measurements, nine dierent receiver aper-
2 Fig. 5. Measured ratios sIm =sI2 as a function of computed aperture averaging factor a from Eq. (14). Error bars are the net standard de2 viation due to C n2 , l0 and sIm variations during the measurement 2 period. sIm was measured using nine dierent receiver apertures with radii from 0.8 to 12.7 mm. The source aperture was 3.2 mm radius for all measurements.
Fig. 6. UV scintillation calculated from Eqs. (8) and (15) using measured l0 and C n2 vs measured UV scintillation. The measurements were made with source and receiver aperture radii of 3.2 mm.
46
D.L. Hutt, D.H. Tofsted / Optics & Laser Technology 32 (2000) 39±48
In Fig. 5, the two data points corresponding to the smallest receiver apertures (a30.62, a30.69) are obviously further from the unity line than the other points. Here, the possibility exists that photon noise has compromised the measurements. There was also a 2 and C n2 dierence in integration times between the sIm measurements which may also account for some of the discrepancies between the calculated and measured 2 measurement integration times varied from results. sIm 10 to 40 s while the C n2 values were all based on 60 s averages. 2 =sI2 to Tofsted's We also compared our measured sIm [12] weak turbulence parameterization of Churnside's aperture averaging equations based on the Tatarskii spectrum: act f1 Nr2 =4=9 j 2 = 2pg ÿ1 :
15
2 =sI2 data A least squares linear ®t to the measured sIm plotted against act from Eq. (15) had a slope of 0.86 (r = 0.98). Thus, Eq. (15) appears to perform slightly better than Eq. (14), which is based on a more complete theory. This unexpected result may be explained by non-stationary conditions [18,19] which Hill and Ochs [20] claim to be particularly sensitive in the dissipation region where the weighting functions peaked for our measurement conditions. The UV scintillation measurements shown in Fig. 4 were used, along with the associated C n2 and l0 measurements, to calculate sI2 with Eq. (8), and the aperture averaging correction of Eq. (15). Both transmitter and receiver radii were 3.2 mm for these measurements. The resulting ``predicted'' sI2 is plotted 2 in Fig. 6(a) as a function of measured scintillation sIm and (b). A least squares linear ®t to the combined data of Fig. 6(a) and (b) has a slope of 1.05 (r = 0.94) indicating that the predicted and measured scintillation are in good agreement. This is another indication that Eqs. (8) and (15) are a good model of inner scale eects and aperture averaging at 254 nm even though these expressions have previously only been compared to 633 nm data.
7. Probability density function for intensity It is well known that the probability distribution of intensity ¯uctuations p(I ) caused by weak turbulence is well represented by a log-normal function. Churnside and Hill [21] showed that p(I ) should be log-normal even in strong turbulence if a suciently large receiver aperture is used. Since our measurements were made under weak turbulence conditions, p(I ) should be log-normal with the distribution function given by
! ln I 2sw2 2 1 p exp ÿ p I : 8sw2 2Isw 2p
16
Measured p(I ) are shown in ®g. 7(a)±(d) for four dierent aperture sizes ranging from 1.6 mm to 12.7 mm radius. Each p(I ) is based on 30 s of time series data measured at 1 kHz. For each measurement period, values of C n2 and l0 measured with the Scintec scintillometer were used to calculate sw2 with Eq. (6). The resultant values of sw2 were used in Eq. (16) to calculate the expected intensity probability function. The data in Fig. 7 was collected with a source aperture of 3.2 mm and C n2 in the range of 1 10ÿ13 to 5 10ÿ13 mÿ2/3. The scatter of the measured p(I ) points in the data for the 1.6 mm aperture (Fig. 7(a)) is due to the fact that the signal is lower for a smaller aperture which results in a lower signal to noise ratio. Fig. 7 shows excellent agreement between the measured p(I ) and those predicted by Eqs. (6) and (16). Since the p(I ) curves were not ``®tted'' to the data but were generated using sw2 based on measured values of C n2 and l0, the good agreement supports the validity of Eqs. (6) and (16) at 254 nm. 8. Conclusions We investigated the eect of atmospheric optical turbulence on solarblind UV radiation through measurements of intensity scintillation at 253.7 nm and simultaneous measurements of C n2 and l0. The zero inner scale Rytov solution for log-amplitude variance predicts that radiation in the SBUV band would experience two to three times as much scintillation as visible light for a given level of turbulence and that saturation eects would start at levels of turbulence approximately half those required to cause saturation of visible light. However, we showed that the combined eects of l0 and aperture averaging greatly reduced the measured scintillation for our measurement conditions. The eect of l0 on scintillation was modeled using expressions of Tofsted [12] which are parameterizations of theoretical expressions of Hill and Cliord [6]. Although Tofsted's expressions have only been compared with measurements at wavelengths of 488 nm [13] and 633 nm [11], there was good agreement between measured UV scintillation and predicted values when the eects of aperture averaging were taken into account. Aperture averaging eects were modeled in the weak turbulence regime using two dierent approaches: evaluating the combined eect of source and receiver averaging with the bump spectrum [6] and using Tofsted's parameterization of Churnside's receiver aperture averaging analysis [16] with a Tatarskii spectrum. The two-
D.L. Hutt, D.H. Tofsted / Optics & Laser Technology 32 (2000) 39±48
47
Fig. 7. (a)±(d) Probability density functions of UV (254 nm) scintillations measured with dierent receiver radii, ar. The curves are p(I ) from Eq. (16) using sw2 derived from Eq. (6) with measured l0 and C n2 : The source aperture radius at was 3.2 mm for all measurements.
ended solution showed good agreement with the measurements except for the two smallest receiver apertures used, while the single-ended parameterization showed slightly better overall agreement with the data. The poorer agreement between the combined model and measurements with the smallest apertures may be due to intermittency eects which alter the form of the turbulence spectrum. The eect of receiver aperture averaging on the probability density of scintillations was also measured using a wide range of receiver aperture sizes. The measured distributions were compared to log-normal curves based on log-amplitude variances corrected for the eect of l0 using Tofsted's expression. Consistent
results were obtained for measured and predicted probability densities, thus establishing that the expressions given are valid in the UV and that they provide a convenient means of modeling UV turbulence eects. Acknowledgements The authors would like to thank Chris Cully for writing the software to acquire the UV signals, Mike Duy for processing the raw UV scintillation data, and MicheÁle Cardinal for assistance in setting up the experiment.
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