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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 19 — Sep. 13, 2010
  • pp: 20350–20359

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Infrared lidar overlap function: an experimental determination

Juan Luis Guerrero-Rascado, Maria João Costa, Daniele Bortoli, Ana Maria Silva, Hassan Lyamani, and Lucas Alados-Arboledas  »View Author Affiliations


Optics Express, Vol. 18, Issue 19, pp. 20350-20359 (2010)
http://dx.doi.org/10.1364/OE.18.020350


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Abstract

The most recent works demonstrate that the lidar overlap function, which describes the overlap between the laser beam and the receiver field of view, can be determined experimentally for the 355 and 532 nm channels using Raman signals. Nevertheless, the Raman channels cannot be used to determine the lidar overlap for the infrared channel (1064 nm) because of their low intensity. In addition, many Raman lidar systems only provide inelastic signals with reasonable signal-to-noise ratio at nighttime. In view of this fact, this work presents a modification of that method, based on the comparison of attenuated backscatter profiles derived from lidar and ceilometer, to retrieve the overlap function for the lidar infrared channel. Similarly to the Raman overlap method, the approach presented here allows to derive the overlap correction without an explicit knowledge of all system parameters. The application of the proposed methodology will improve the potential of Raman lidars to investigate the aerosol microphysical properties in the planetary boundary layer, extending the information of 1064 nm backscatter profiles to the ground and allowing the retrieval of microphysical properties practically close to the surface.

© 2010 OSA

1. Introduction

During the last decades, the lidar systems have been recognized as very useful to aerosol and cloud research, both in the troposphere and the stratosphere, mainly due to their multi-spectral capabilities and high vertical spatial and temporal resolution. However, lidar-based studies related to the dynamics of the planetary boundary layer or air pollution are often not accurately performed, due to an incomplete knowledge of the lidar response at all ranges. To interpret the near field lidar observations properly, lidar data must be corrected with the so-called overlap function, geometrical form factor or crossover function. The well-known lidar equation assumes that the volume of space containing the transmitted pulse is completely imaged onto the detector. Nevertheless, as a result of the typical separation distance between the biaxial lidar transmitter and receiver (order of centimeters) and the narrow beam width, the irradiated volume at ranges below a certain altitude (altitude of full overlap) are detected incompletely, and the signal at that near range is dominated by this effect. Even the coaxial systems can be affected by incomplete overlap, when the optical axes of the transmitter and receiver are misaligned. Thus, the overlap function provides a measure of the amount of backscattered power coming from a distance z, which is collected by the receiver telescope and imaged onto the detector.

Since the 70s, literature has reported different methods to determine the overlap function that can be grouped into two different categories: theoretical and experimental approaches. The theoretical computations are based on the specifications and configuration of optical elements as the laser beam cross section, the beam direction, beam divergence and the telescope optics. The inconvenient of such approaches is that they require a good knowledge of these specifications to derive an overlap function with enough accuracy [1

T. Halldórsson and J. Langerholc, “Geometrical form factors for the lidar function,” Appl. Opt. 17(2), 240–244 (1978). [CrossRef] [PubMed]

–5

K. Stelmaszczyk, M. Dell’Aglio, S. Chudzyński, T. Stacewicz, and L. Wöste, “Analytical function for lidar geometrical compression form-factor calculations,” Appl. Opt. 44(7), 1323–1331 (2005). [CrossRef] [PubMed]

]. In addition, the parallelism between the laser beam and the telescope optical axis is necessary, which is quite difficult to ensure, especially for mobile lidars. Consequently, an experimental determination of the overlap function is required to derive an accurate correction for real lidar data. The most relevant studies are reviewed here. The first experimental works were limited to particular atmospheric conditions. Thereby, Sasano et al. [6

Y. Sasano, H. Shimizu, N. Takeuchi, and M. Okuda, “Geometrical form factor in the laser radar equation: an experimental determination,” Appl. Opt. 18(23), 3908–3910 (1979). [CrossRef] [PubMed]

] presented a simple experimental procedure assuming an atmospheric transmission equal to unity and an homogeneous aerosol distribution. Following this paper, it is possible determine the overlap function during very clean conditions, but in general this is not reliable, especially at urban locations where pollutants are present, or at places close to deserts affected frequently by mineral dust outbreaks. Tomine et al. [7

K. Tomine, C. Hirayama, K. Michimoto, and N. Takeuchi, “Experimental determination of the crossover function in the laser radar equation for days with a light mist,” Appl. Opt. 28(12), 2194–2195 (1989). [CrossRef] [PubMed]

] suggested an experimental method to derive the overlap function under conditions of light fog and mist, assuming distributions statistically homogeneous for these atmospheric constituents. Both approaches assume a homogeneous distribution of a certain atmospheric component (aerosol, fog or mist), which often does not hold in the low troposphere close to the surface. Dho et al. [8

S. W. Dho, Y. J. Park, and H. J. Kong, “Application of geometrical form factor in differential absorption lidar measurement,” Opt. Rew. 4(4), 521–526 (1997). [CrossRef]

] also suggested an experimental determination using the slope method in a homogeneous atmosphere, which is not applicable in general conditions as well. A useful development in inhomogeneous atmospheres was reported in [9

S. W. Dho, Y. J. Park, and H. J. Kong, “Experimental determination of a geometric form factor in a lidar equation for an inhomogeneous atmosphere,” Appl. Opt. 24(24), 6009–6010 (1997). [CrossRef]

], based on a polynomial regression in the range of full overlap. This approach assumes that the coefficients obtained in this region can also describe the overlap features at lower altitudes (extrapolation in the incomplete overlap ranges).

The most recent study about experimental determination of the lidar overlap function was reported by [10

U. Wandinger and A. Ansmann, “Experimental determination of the lidar overlap profile with Raman lidar,” Appl. Opt. 41(3), 511–514 (2002). [CrossRef] [PubMed]

]. From our point of view, this is the most powerful and applicable method because it works under homogeneous and inhomogeneous conditions without any critical assumption, and it is a widely used approach in EARLINET (European Aerosol Research LIdar NETwork) [11

J. Bösenberg, A. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hagard, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: a European aerosol research lidar network”. In Laser Remote Sensing of the Atmosphere, A. Dabas, C. Loth, and J. Pelon, eds., selected papers of the 20th International Laser Radar Conference (Edition Ecole Polytechnique, Palaiseau, France, 2001), pp. 155–158.

]. The technique is based on the measurements, using a Raman lidar system, of a pure molecular Raman signal together with the corresponding elastic signal. The procedure assumes that the overlap functions for the elastic and Raman channels are identical, which is the basic assumption required to obtain aerosol backscatter coefficient profiles using the Raman method [12

A. Ansmann, U. Wandinger, M. Riebesell, C. Weitkamp, and W. Michaelis, “Independent measurement of extinction and backscatter profiles in cirrus clouds by using a combined Raman elastic-backscatter lidar,” Appl. Opt. 31(33), 7113–7131 (1992). [CrossRef] [PubMed]

]. Following this method, it is possible to retrieve an overlap function at 355 and 532 nm, if the corresponding Raman shifted signals are measured. However, the performance at 1064 nm is still limited because of the low intensity for the Raman shifted signal in the infrared range. The present paper presents a modification of the method proposed by [10

U. Wandinger and A. Ansmann, “Experimental determination of the lidar overlap profile with Raman lidar,” Appl. Opt. 41(3), 511–514 (2002). [CrossRef] [PubMed]

], from now on denoted as Raman overlap method, to retrieve the overlap function for the lidar infrared channel, based on the combination of lidar and ceilometer measurements of attenuated backscatter coefficient profile. The proposed method complements the Raman overlap method for Raman lidars and it is also applicable to elastics lidar systems. In addition, this work complements the previous works devoted to improve the quality of 1064 nm-lidar profiles (e. g. see [13

F. Navas-Guzmán, J. L. Guerrero-Rascado, and L. Alados-Arboledas, “Calibration of 1064nm-backscatter profiles with a multiwavelength Raman lidar,” Rom. J. Physiol. (to be published).

, 14

F. Navas-Guzmán, J. L. Guerrero-Rascado, J. A. Bravo-Aranda, and L. Alados-Arboledas, “On the use cirrus clouds for elastic lidar calibration”, Ópt. Pur, Apl. submitted.

]).

The paper is organized as follows. Section 2 presents the instruments used to characterize the infrared overlap function. Section 3 describes the method that allows for retrieving the overlap function through the combination of elastic lidar and ceilometer data. The results obtained during the CAPEX field campaign are presented in Section 4. Finally, Section 5 provides a summary and the main conclusions of this work.

2. Instrumentation

2.1 Raman lidar LR321D400

The Raman lidar model LR321D400 (Raymetrics S.A., Greece) is a robust system configured in a monostatic biaxial alignment, pointing vertically to the zenith. It is based on a pulsed Nd:YAG laser with fundamental emission at 1064 nm, and additional emissions at 532 and 355 nm by using second and third harmonic generators. Output energies are 110, 65 and 60 mJ at 1064, 532 and 355 nm, respectively and pulses of 7-9 ns can be fired with a pulse repetition frequency (PRF) of 1, 2, 5 and 10 Hz (typically a PRF of 10 Hz is used). The receiving system consists of a 400mm-diameter Cassegrainian telescope with a full field-of-view of 2 mrad, and a wavelength separation unit with dichroic mirrors, interferential filters and a polarization cube, that discriminates six channels corresponding to elastic wavelengths (1064, 532 parallel-polarized, 532 perpendicular-polarized, and 355 nm), and to nitrogen and water vapor Raman-shifted wavelengths (387 and 408 nm). Due to the low intensity of Raman signals, they are only used for night-time retrievals when the sky background is low and steady enough. For the purpose of this work, only the signals at 1064 nm are considered. The lidar backscattered signals are registered with a vertical resolution of 7.5 m and temporal resolution of 1-minute. To increase the signal-to-noise ratio, single profiles were averaged over 30 minutes following the EARLINET protocols.

2.2 Ceilometer CL31

The VAISALA Ceilometer CL31 is based on an eye-safe laser InGaAs diode at 910 nm, sending pulses out along the zenital direction. The reflection of light (backscattered signal) caused by any atmospheric component is analyzed. This system provides the cloud base height, up to three simultaneous layers, and the total attenuated backscatter coefficient, including the contribution of molecules, aerosols and clouds. The CL31Vaisala ceilometer is described in deep by [15

C. Münkel, N. Eresmaa, J. Räsänen, and A. Karppinen, “Retrieval of mixing height and dust concentration with Lidar ceilometer,” Boundary-Layer Meteorol. 124(1), 117–128 (2007). [CrossRef]

]. The main difference from the previous models is the way the common lens is used for transmitting and receiving light. The centre of the lens is used for collimating the outgoing laser beam, whereas the outer part of the lens is used for focusing the backscattered light onto the receiver. The separation between transmitting and receiving areas is provided by an inclined mirror with a hole in the centre. In this way, a very good performance is achieved, even at the lowest heights. The full field-of-view for this system is 1.66 mrad. The ceilometer has a measurement range from 0 to 7.5 km, with a maximum resolution of 5 m and programmable measurement cycle (from 2 to 120 s). In order to improve the signal-to-noise ratio the ceilometer was run for this study at the vertical resolution of 20 m with a sampling rate of 30 s. Then single profiles were smoothed with a height-dependent sliding window and also averaged over 30 minutes to improve the signal-to-noise ratio and to ensure the same sampling interval than the lidar data. The 30-minute ceilometer and lidar averages correspond to the same time intervals.

3. Methodology

The fundamental idea that motivates our approach is coincident with that proposed by [10

U. Wandinger and A. Ansmann, “Experimental determination of the lidar overlap profile with Raman lidar,” Appl. Opt. 41(3), 511–514 (2002). [CrossRef] [PubMed]

], that is, the deviation between the elastic lidar profile and other profiles unaffected by incomplete overlap allows for retrieving the lidar overlap function. Whereas the Raman overlap method assumes that the aerosol backscatter profile derived from the lidar is unaffected by overlap artifacts and is considered as a reference profile, our approach assumes that the ceilometer profile is the reference. Since the CL31 ceilometer uses a one lens system with overlapping transmitting and receiving optics, the beam overlap occurs at shorter distances in comparison with the lidar, providing nearly full overlap of the transmitter and the receiver field-of-view at distances below 30 m [16

C. Münkel, and R. Roininen, “Investigation of boundary layer structures with ceilometer using a novel robust algorithm”, Proc. 90th American Meteorological Society Annual Meeting: 15th Symposium on Meteorological Observation and Instrumentation (2010), 5.3.

]. The following paragraphs describe in detail the proposed procedure.

The elastic lidar equation in its simplest form, valid for quasi-monochromatic emission of the laser, instantaneous scattering and negligible multiple scattering, can be written as (for simplicity the wavelength dependence is omitted):
P(z)= C⋅O(z) z2⋅β(z)⋅ T2(z)
(1)
where P(z) is the backscattered laser power from the altitude z, C is an instrumental constant that includes all range-independent instrumental parameters (as the detector’s efficiency, receiving telescope area and laser pulse width), O(z) is the overlap function, β(z) is the volume backscatter coefficient and T(z) is the atmospheric transmission between the lidar and the altitude z. In general, the last two parameters take into account the contribution of molecules, aerosols and clouds. For the purpose of this paper, it is convenient to re-write the elastic lidar equation in terms of the so-called attenuated backscatter as follows:
P(z)= C⋅O(z) z2⋅ β att(z)
(2)
where βatt(z) is the attenuated backscatter coefficient defined as the product of the volume backscatter coefficient and the two-way optical transmission [17

C. A. Hostetler, Z. Liu, J. Reagan, M. Vaughan, D. Winker, M. Osborn, W. H. Hunt, K. A. Powell, and C. Trepte, “CALIOP Algorithm Theoretical Basis Document”, PC-SCI-201, NASA Langley Res. Cent., Hampton, Va. http://www-calipso.larc.nasa.gov/resources/project_documentation.php (2006).

]. In the last years, this has become a well-known parameter for the lidar community, because it is provided as one of the level-1 products for the Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) satellite mission. The data used in the present study includes ceilometer and lidar measurements. The CL31 ceilometer measurements provide the total attenuated backscatter profiles at 910 nm, practically unaffected by incomplete overlap effect as explained above. The lidar measurements provide the aerosol backscatter coefficient at 1064 nm, obtained using the Klett-Fernald-Sasano method [18

F. G. Fernald, B. M. Herman, and J. A. Reagan, “Determination of Aerosol Height Distribution by Lidar,” J. Appl. Meteorol. 11(3), 482–489 (1972). [CrossRef]

–23

Y. Sasano, E. V. Browell, and S. Ismail, “Error caused by using a constant extinction/backscattering ratio in the lidar solution,” Appl. Opt. 24(22), 3929–3932 (1985). [CrossRef] [PubMed]

], which are affected by incomplete overlap. The lidar profiles are subsequently converted into attenuated backscatter profiles using the method proposed by [24

L. Mona, A. Amodeo, G. D’Amico, and G. Pappalardo, “First comparisons between CNR-IMAA multiwavelength Raman lidar measurements and CALIPSO measurements,” Proc. SPIE 6750, 6750–35 (2007).

], in order to be combined with the ceilometer attenuated backscatter profiles. Two distinct processes are proposed for the determination of the overlap function (iteratively or directly) and both procedures are described next.

3.1 Iterative method

The ceilometer and lidar systems provide data in the infrared range of the spectrum at 910 and 1064 nm, respectively. In order to be able to compare these profiles, the ceilometer attenuated backscatter profiles must be scaled to the lidar wavelength. The 910 nm-ceilometer attenuated backscatter profiles are converted in 1064 nm-ceilometer attenuated backscatter profiles through a height-independent conversion factor computed as the ratio of the mean value of 1064 nm-lidar and 910 nm-ceilometer profiles in a window of the region with full overlap (typically somewhere between 1 and 2 km above the instruments). Above 2.5 to 3 km, the ceilometer signal-to-noise ratio presents values that limit the value of detection [25

K. M. Markowicz, P. J. Flatau, A. E. Kardas, J. Remiszewska, K. Stelmaszczyk, and L. Woeste, “Ceilometer Retrieval of the Boundary Layer Vertical Aerosol Extinction Structure,” J. Atmos. Ocean. Technol. 25(6), 928–943 (2008). [CrossRef]

].

The basic idea underlying the iterative method is that the elastic lidar signal, after correction for the distance and overlap function (see Eq. (2), is proportional to the ceilometer attenuated backscatter profile; whereas the elastic lidar signal, corrected only by the distance effect, is proportional to the lidar attenuated backscatter profile that is affected by the incomplete overlap:

β att,ceilom,unaffected(z)∝P(z)⋅ z2⋅ O −1(z)
(3)
β att,lidar,affected(z)∝P(z)⋅ z2
(4)

Following the procedure described in [10

U. Wandinger and A. Ansmann, “Experimental determination of the lidar overlap profile with Raman lidar,” Appl. Opt. 41(3), 511–514 (2002). [CrossRef] [PubMed]

], we will use the relative difference, ΔO(z), between the ceilometer and lidar attenuated backscatter profiles to reduce iteratively the effect on the elastic lidar signal:

β att,ceilom(z)− β att,lidar(z) β att,ceilom(z)∝ P(z)⋅ z2⋅ O −1(z)−P(z)⋅ z2 P(z)⋅ z2⋅ O −1(z)=1−O(z)≡ΔO(z)
(5)

In step i=1, the uncorrected elastic lidar signal is used to retrieve the aerosol backscatter profile that afterwards is converted in a lidar attenuated profile. By applying Eq. (5), the relative difference can be computed and used to correct the elastic lidar signal that will be the input for the next iteration as follows:

Δ O (i)= β att,ceilom,unaffected(z)− β (i) att,lidar,affected(z) β att,ceilom,unaffected(z)
(6)
P (i+1)(z)= P (i)(z)⋅ [ 1+Δ O (i)(z)]
(7)

The iterative application of this procedure allows for correcting the lidar attenuated profile for the overlap effect. Our results indicate that 20 to 25 iterations are enough to remove the overlap effect completely. After the final step, the overlap function can be computed as:

O(z)= P (1)(z)/ P (end)(z)
(8)

3.2 Direct method

The direct method provides a faster determination of the overlap function. By rewriting Eq. (2), the overlap function can be computed as:

O(z)= P(z)⋅ z2 C⋅ β att(z)
(9)

Again, we consider the ceilometer attenuated backscatter as the reference profile, since it is unaffected by incomplete overlap already from above 30 m height, i.e., βatt(z) = βatt,ceilom(z). In this approach, it is not necessary to explicitly scale the ceilometer profile to the lidar wavelength, since the scaling factor can be included in the unknown constant C. The value for this unknown constant is selected such that the maximum values of O(z) must tend to 1. These maximum values are found in the far range for a well-adjusted lidar when the Raman overlap method is applied [10

U. Wandinger and A. Ansmann, “Experimental determination of the lidar overlap profile with Raman lidar,” Appl. Opt. 41(3), 511–514 (2002). [CrossRef] [PubMed]

]. Nevertheless, using ceilometer profiles as reference, the maximum values are found just above the full overlap altitude, typically in the region between 1 and 2 km above the instruments.

4. Experiment

Data used in this paper were measured at the Observatory of the Évora Geophysics Centre (Portugal, 38.57°N, 7.90°W, 293 m asl) during the CAPEX field campaign. CAPEX (Clouds and Aerosols over Portugal Experiment) was an European project, partly funded by EC under the 6th Framework Program within the EUFAR Initiative, coordinated by Évora Geophysics Centre (CGE), to investigate aerosol particles, radiation, cloud properties, precipitation and radioactivity over Portugal using both airborne and ground based instrumentation. It took place from 30th May until 18th June 2006 over central and southern Portugal, and consisted of three individual projects: AEROPOR (AERosols Over Portugal: Optical and Radiation Measurements), CLAPREC (Clouds, Aerosols, Precipitation) and VPRACOP (Vertical Profiles of Radioactive Aerosol Constituents Over Portugal). The lidar measurements used here were obtained within AEROPOR project, which counted with the involvement of two research teams, one from the University of Évora (CGE) and the other from the University of Granada. The ceilometer CL31 is working in the CGE since the beginning of May 2006, whereas the lidar was moved from Granada to the CGE observatory in Évora, for the duration of the campaign.

Figure 1a presents an example of the averaged infrared attenuated backscatter profiles obtained from lidar and ceilometer, performed on 7th June 2006 from 09:30 to 10:00 UTC. The comparison of the two profiles shows the consequence of ignoring (lidar profile) or including (ceilometer profile) the overlap effect. The ceilometer profile indicates the planetary boundary layer with a top height of about 1.0 km asl. The troposphere above this altitude also includes two additional aerosol loaded layers, with North Africa as source region (backtrajectories not shown here). One layer is confined between 1.0 and 2.0 km, and it is well recognized by both instruments (Fig. 1a). The second layer is only detected by the lidar above 2.5 km. In fact, as mentioned before, above these heights the ceilometer signal-to-noise ratio is too low to allow for the detection of atmospheric constituents other than clouds, therefore its performance for aerosol vertical distribution research is limited to the first kilometers above the surface. Nevertheless, at the lower levels the ceilometer is routinely used not only for cloud base height determination (up to 7.5 km height), but also to determine vertical visibility, boundary layer monitoring, visibility mixing height or dust concentration [15

C. Münkel, N. Eresmaa, J. Räsänen, and A. Karppinen, “Retrieval of mixing height and dust concentration with Lidar ceilometer,” Boundary-Layer Meteorol. 124(1), 117–128 (2007). [CrossRef]

, 16

C. Münkel, and R. Roininen, “Investigation of boundary layer structures with ceilometer using a novel robust algorithm”, Proc. 90th American Meteorological Society Annual Meeting: 15th Symposium on Meteorological Observation and Instrumentation (2010), 5.3.

, 25

K. M. Markowicz, P. J. Flatau, A. E. Kardas, J. Remiszewska, K. Stelmaszczyk, and L. Woeste, “Ceilometer Retrieval of the Boundary Layer Vertical Aerosol Extinction Structure,” J. Atmos. Ocean. Technol. 25(6), 928–943 (2008). [CrossRef]

]. In order to manage high quality ceilometer profiles for comparison with lidar, several criteria have been taken into account: (i) single ceilometer profiles were averaged over 30 minutes to improve the signal-to-noise ratio, (ii) only segments of profiles below 2.0 km were considered for comparison, and (iii) middle and high aerosol load conditions were considered. The last criterion follows the work reported by [26

T. Elias, A. M. Silva, N. Belo, S. Pereira, P. Formenti, G. Helas, and F. Wagner, “Aerosol extinction in a remote continental region of the Iberian Peninsula during summer,” J. Geophys. Res. 111(D14), D14204 (2006), doi:. [CrossRef]

] who established the conditions for the distinction of aerosol events during summer, in Évora, in terms of the aerosol optical depth (AOD) and its spectral dependence in the atmospheric column, using a multiwavelength Sun/sky photometer (CE318-2, CIMEL Electronique, France). Consistently, the procedure to derive the infrared overlap function was applied during daytime only under non-clean conditions, when the AOD at 440 and 870 nm was larger than 0.12 and 0.04, respectively. Figure 1b illustrates the iterative method used to correct the elastic lidar signal for the measurement of 7th June 2006 from 09:30 to 10:00 UTC. The consequence of the overlap correction in the determination of the attenuated backscatter coefficient lidar profiles is illustrated for each step. A lidar ratio value of 30 sr is assumed in the retrieval of the lidar attenuated backscatter profiles. The lidar solution matches quite well (maximum differences of 0.3%) the ceilometer solution in Fig. 1b, when the elastic lidar signal is overlap corrected after 25 iterations. As it can be seen, the lidar measurements are useless for overlap effect at altitudes below 1.0 km if the overlap effect is ignored. Similar improvements are obtained using the direct method.

Fig. 1 a) Attenuated backscatter profiles derived from lidar (black, overlap effect is ignored) and from ceilometer (grey) on 7th June 2006. For the lidar procedure a lidar ratio of 30 sr was assumed. b) Attenuated backscatter profiles derived from lidar for step 1 (black, overlap effect is ignored), next steps (dashed-grey, overlap is corrected iteratively), and from ceilometer (grey) on 7th June 2006. 25 iterations were required.

The transmission of a laser beam through a dense media is increased by the multiple scattering effects, because photons scattered out of the sampling volume of the lidar or ceilometer after a single scattering event may be redirected to the receiver field-of-view in the following scattering events. The evaluation of this effect is complicated and depends on different factors as the laser penetration depth, altitude, field-of-view of the receiver, size distribution and shape of scatterers [27

C. M. R. Platt, J. C. Scott, and A. C. Dilley, “Remote sounding of high clouds. Part VI: Optical properties of mid-latitudemand tropical cirrus,” J. Atmos. Sci. 44(4), 729–747 (1987). [CrossRef]

–30

H. Chepfer, J. Pelon, G. Brogniez, C. Flamant, V. Trouillet, and P. H. Flamant, “Impact of cirrus cloud ice crystal shape and size on multiple scattering effect: application to spaceborne and airborne backscatter lidar measurements during the LITE mission and E LITE campaign,” Geophys. Res. Lett. 26(14), 2203–2206 (2000). [CrossRef]

]. Following some works as Chen et al. [31

W. N. Chen, C. W. Chiang, and J. B. Nee, “Lidar ratio and depolarization ratio for cirrus clouds,” Appl. Opt. 41(30), 6470–6476 (2002). [CrossRef] [PubMed]

], the multiple scattering effect should be considered for optical depths larger than 1. Taking into account the technical specifications for the instruments used here and the aerosol loads prevailing during the experiment (aerosol optical depths lower than 1), the multiple scattering effects in our data may be considered negligible.

The aforementioned iterative and direct methods have been applied to the lidar measurements that were performed during CAPEX. From the whole period, 31 profiles averaged over 30 min, including measurements in the morning and in the afternoon, fulfilled the requirements of high quality for such kind of retrieval. These profiles were measured with synoptic conditions favoring air masses coming to Évora from Europe, Northern Africa and Mediterranean basin. To calculate the contribution of the aerosol transmission to the attenuated backscatter profile obtained by lidar, the backscatter coefficient profile must be converted into an aerosol extinction coefficient profile assuming a lidar ratio value. Using the Raman overlap method, the lidar measurements performed under clear conditions (particle transmission >0.9) are preferable for the determination of the overlap profile [10

U. Wandinger and A. Ansmann, “Experimental determination of the lidar overlap profile with Raman lidar,” Appl. Opt. 41(3), 511–514 (2002). [CrossRef] [PubMed]

]. Such conditions are not applicable to the infrared overlap method and, thus, an uncertainty is expected. To minimize a potential impact of the aerosol lidar ratio, as well as of the aerosol layering, the infrared overlap functions are averaged. Figure 2 shows the experimental lidar overlap function for the 1064 nm channel computed as the mean value of the individual overlap functions derived from 30-min averaged profiles during CAPEX, applying the direct and iterative methods. The error bars (one standard deviation) show the variability of the infrared overlap retrieval as a result of the variability of the aerosol concentration.

Fig. 2 Experimental infrared mean overlap function obtained, for CAPEX (full line), using the direct and iterative methods. Error bars denote one standard deviation.

For the system LR321D400, different parameterizations were used and fitted to the experimental overlap function computed during CAPEX data. Table 1 shows the coefficients for the different parameterizations investigated and Fig. 3 (upper and middle panels) illustrates the curves plotted together with the experimental data for comparison. Our experimental data fits with high accuracy all the proposed models (R2>0.99), although model 2 achieves the best result for the direct and iterative method (R2≈0.999). The larger discrepancies are confined between 0.8 and 0.9 km and are found using the iterative method. This might be caused by a different recognition of the aerosol layering using ceilometer or lidar, and additionally this effect could be amplified if the assumed lidar ratio is very far from the actual value. The absolute differences of the modeled overlap functions are low with mean value of 0.02 and a maximum of 0.06 (Fig. 3 bottom panel).

Table 1  Different parameterizations tested for the mean experimental infrared overlap function computed during CAPEX using the direct and iterative methods.
Model Direct Iterative
Model 1 O(z)= A1− A2 1+ ( z/ z0)p+ A2 Model 2 O(z)=a⋅exp { −exp [ −k⋅(z− zc)]} Model 3 O(z)= a 1+exp [ −k⋅(z− zc)] R2: 0.99891 A1: −0.0199 ± 0.0011 A2: 1.0003 ± 0.0001 z0: 0.6297 ± 0.0004 p: 7.60 ± 0.03 R2: 0.99919 a: 1.00028 ± 0.00009 zc: 0.58673 ± 0.00024 k: 8.545 ± 0.021 R2: 0.99754 a: 0.99984 ± 0.00015 zc: 0.6416 ± 0.0004 k: 12.30 ± 0.05R2: 0.9975 A1: −0.0254 ± 0.0019 A2: 1.00055 ± 0.00016 z0: 0.6389 ± 0.0007 p: 6.30 ± 0.03 R2: 0.99809 a: 1.00036 ± 0.00014 zc: 0.5901 ± 0.0004 k: 7.006 ± 0.024 R2: 0.99533 a: 0.99986 ± 0.00021 zc: 0.6569 ± 0.0006 k: 9.92 ± 0.06
Fig. 3 Different models tested for the mean experimental infrared overlap function computed during CAPEX using the direct and iterative methods. The comparison for the best fitting (model 2) is also shown.

5. Summary and conclusions

A method to retrieve the overlap function for the lidar infrared channel is proposed here. It is based on the modification of the Raman overlap method, widely applied amongst the EARLINET community. The method presented allows for retrieving the lidar overlap function at 1064 nm through the combination of lidar and ceilometer attenuated backscatter data. Therefore, it complements the Raman overlap method for Raman lidars and it is also applicable to elastics lidar systems. Due to the requirements of the retrieval, especially those referred to have atmospheric conditions with non-negligible aerosol loads, some uncertainty could be present in the individual infrared overlap functions retrieved over 30 min. To minimize the potential impact of aerosols, caused by a deviation of the assumed lidar ratio with respect to the actual value and/or of the aerosol layering recognition from the two different instruments, the individual infrared overlap functions are averaged and parameterized successfully. Due to its simple and faster formulation together with the low absolute differences found, the use of the direct method is more advisable than the iterative one.

The study of the exchange processes of anthropogenic particles between the sources and the lowermost atmospheric layers is not possible without an appropriate overlap correction of the lidar signals. This is especially relevant for the aerosol microphysical properties, where the so called 3 + 2 lidars (systems providing backscatter coefficients at 355, 532 and 1064 nm, and extinction coefficients at 355 and 532 nm) are the minimum requirement for the retrieval of vertical profiles of microphysical properties [32

I. Veselovskii, A. Kolgotin, V. Griaznov, D. Müller, U. Wandinger, and D. N. Whiteman, “Inversion with regularization for the retrieval of tropospheric aerosol parameters from multiwavelength lidar sounding,” Appl. Opt. 41(18), 3685–3699 (2002). [CrossRef] [PubMed]

]. The application of the proposed methodology will improve the potential of Raman lidars to investigate the aerosol microphysical properties in the planetary boundary layer, extending the information of 1064-nm backscatter profiles to the ground and allowing the retrieval of microphysical properties practically close to the surface.

Acknowledgements

This work has been supported by Spanish Ministry of Science and Innovation under the Acciones Complementarias CGL2008-01330-E/CLI and CGL2009-08031-E/CLI; project CGL2007-66477-C02-01, CSD2007-00067 and CGL2004-05984-C07-03 of the Spanish Ministry of Education; project P08-RNM-3568 and P06-RNM-01503 of the Autonomous Government of Andalusia; and by the EARLINET-ASOS project (EU Coordination Action, contract nº 025991 (RICA)). This work is also supported by FCT through projects PTDC/CTE-ATM/65307/2006 and PTDC/CTE-ATM/102142/2008. Dr. Guerrero-Rascado was funded by FCT under grant SFRH/BPD/63090/2009. The authors wish to acknowledge Frank Wagner, working at the University of Évora, for comments regarding this paper.

References and links

1.

T. Halldórsson and J. Langerholc, “Geometrical form factors for the lidar function,” Appl. Opt. 17(2), 240–244 (1978). [CrossRef] [PubMed]

2.

K. Sassen and G. C. Dodd, “Lidar crossover function and misalignment effects,” Appl. Opt. 21(17), 3162–3165 (1982). [CrossRef] [PubMed]

3.

G. M. Ancellet, M. J. Kavaya, R. T. Menzies, and A. M. Brothers, “Lidar telescope overlap function and effects of misalignment for unstable resonator transmitter and coherent receiver,” Appl. Opt. 25(17), 2886–2890 (1986). [CrossRef] [PubMed]

4.

H. Kuze, H. Kinjo, Y. Sakurada, and N. Takeuchi, “Field-of-view dependence of lidar signals by use of Newtonian and Cassegrainian telescopes,” Appl. Opt. 37(15), 3128–3132 (1998). [CrossRef]

5.

K. Stelmaszczyk, M. Dell’Aglio, S. Chudzyński, T. Stacewicz, and L. Wöste, “Analytical function for lidar geometrical compression form-factor calculations,” Appl. Opt. 44(7), 1323–1331 (2005). [CrossRef] [PubMed]

6.

Y. Sasano, H. Shimizu, N. Takeuchi, and M. Okuda, “Geometrical form factor in the laser radar equation: an experimental determination,” Appl. Opt. 18(23), 3908–3910 (1979). [CrossRef] [PubMed]

7.

K. Tomine, C. Hirayama, K. Michimoto, and N. Takeuchi, “Experimental determination of the crossover function in the laser radar equation for days with a light mist,” Appl. Opt. 28(12), 2194–2195 (1989). [CrossRef] [PubMed]

8.

S. W. Dho, Y. J. Park, and H. J. Kong, “Application of geometrical form factor in differential absorption lidar measurement,” Opt. Rew. 4(4), 521–526 (1997). [CrossRef]

9.

S. W. Dho, Y. J. Park, and H. J. Kong, “Experimental determination of a geometric form factor in a lidar equation for an inhomogeneous atmosphere,” Appl. Opt. 24(24), 6009–6010 (1997). [CrossRef]

10.

U. Wandinger and A. Ansmann, “Experimental determination of the lidar overlap profile with Raman lidar,” Appl. Opt. 41(3), 511–514 (2002). [CrossRef] [PubMed]

11.

J. Bösenberg, A. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hagard, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: a European aerosol research lidar network”. In Laser Remote Sensing of the Atmosphere, A. Dabas, C. Loth, and J. Pelon, eds., selected papers of the 20th International Laser Radar Conference (Edition Ecole Polytechnique, Palaiseau, France, 2001), pp. 155–158.

12.

A. Ansmann, U. Wandinger, M. Riebesell, C. Weitkamp, and W. Michaelis, “Independent measurement of extinction and backscatter profiles in cirrus clouds by using a combined Raman elastic-backscatter lidar,” Appl. Opt. 31(33), 7113–7131 (1992). [CrossRef] [PubMed]

13.

F. Navas-Guzmán, J. L. Guerrero-Rascado, and L. Alados-Arboledas, “Calibration of 1064nm-backscatter profiles with a multiwavelength Raman lidar,” Rom. J. Physiol. (to be published).

14.

F. Navas-Guzmán, J. L. Guerrero-Rascado, J. A. Bravo-Aranda, and L. Alados-Arboledas, “On the use cirrus clouds for elastic lidar calibration”, Ópt. Pur, Apl. submitted.

15.

C. Münkel, N. Eresmaa, J. Räsänen, and A. Karppinen, “Retrieval of mixing height and dust concentration with Lidar ceilometer,” Boundary-Layer Meteorol. 124(1), 117–128 (2007). [CrossRef]

16.

C. Münkel, and R. Roininen, “Investigation of boundary layer structures with ceilometer using a novel robust algorithm”, Proc. 90th American Meteorological Society Annual Meeting: 15th Symposium on Meteorological Observation and Instrumentation (2010), 5.3.

17.

C. A. Hostetler, Z. Liu, J. Reagan, M. Vaughan, D. Winker, M. Osborn, W. H. Hunt, K. A. Powell, and C. Trepte, “CALIOP Algorithm Theoretical Basis Document”, PC-SCI-201, NASA Langley Res. Cent., Hampton, Va. http://www-calipso.larc.nasa.gov/resources/project_documentation.php (2006).

18.

F. G. Fernald, B. M. Herman, and J. A. Reagan, “Determination of Aerosol Height Distribution by Lidar,” J. Appl. Meteorol. 11(3), 482–489 (1972). [CrossRef]

19.

F. G. Fernald, “Analysis of atmospheric lidar observations: some comments,” Appl. Opt. 23(5), 652–653 (1984). [CrossRef] [PubMed]

20.

J. D. Klett, “Stable analytical inversion solution for processing lidar returns,” Appl. Opt. 20(2), 211–220 (1981). [CrossRef] [PubMed]

21.

J. D. Klett, “Lidar inversion with variable backscatter/extinction ratios,” Appl. Opt. 24(11), 1638–1643 (1985). [CrossRef] [PubMed]

22.

Y. Sasano and H. Nakane, “Significance of the extinction/backscatter ratio and the boundary value term in the solution for the two-component lidar equation,” Appl. Opt. 23(1), 11–13 (1984). [CrossRef]

23.

Y. Sasano, E. V. Browell, and S. Ismail, “Error caused by using a constant extinction/backscattering ratio in the lidar solution,” Appl. Opt. 24(22), 3929–3932 (1985). [CrossRef] [PubMed]

24.

L. Mona, A. Amodeo, G. D’Amico, and G. Pappalardo, “First comparisons between CNR-IMAA multiwavelength Raman lidar measurements and CALIPSO measurements,” Proc. SPIE 6750, 6750–35 (2007).

25.

K. M. Markowicz, P. J. Flatau, A. E. Kardas, J. Remiszewska, K. Stelmaszczyk, and L. Woeste, “Ceilometer Retrieval of the Boundary Layer Vertical Aerosol Extinction Structure,” J. Atmos. Ocean. Technol. 25(6), 928–943 (2008). [CrossRef]

26.

T. Elias, A. M. Silva, N. Belo, S. Pereira, P. Formenti, G. Helas, and F. Wagner, “Aerosol extinction in a remote continental region of the Iberian Peninsula during summer,” J. Geophys. Res. 111(D14), D14204 (2006), doi:. [CrossRef]

27.

C. M. R. Platt, J. C. Scott, and A. C. Dilley, “Remote sounding of high clouds. Part VI: Optical properties of mid-latitudemand tropical cirrus,” J. Atmos. Sci. 44(4), 729–747 (1987). [CrossRef]

28.

K. Sassen and B. Y. Cho, “Subvisual-thin cirrus lidar dataset for satellite verification and climatological research,” J. Appl. Meteorol. 31(11), 1275–1285 (1992). [CrossRef]

29.

E. W. Eloranta, “Practical model for the calculation of multiply scattered lidar returns,” Appl. Opt. 37(12), 2464–2472 (1998). [CrossRef]

30.

H. Chepfer, J. Pelon, G. Brogniez, C. Flamant, V. Trouillet, and P. H. Flamant, “Impact of cirrus cloud ice crystal shape and size on multiple scattering effect: application to spaceborne and airborne backscatter lidar measurements during the LITE mission and E LITE campaign,” Geophys. Res. Lett. 26(14), 2203–2206 (2000). [CrossRef]

31.

W. N. Chen, C. W. Chiang, and J. B. Nee, “Lidar ratio and depolarization ratio for cirrus clouds,” Appl. Opt. 41(30), 6470–6476 (2002). [CrossRef] [PubMed]

32.

I. Veselovskii, A. Kolgotin, V. Griaznov, D. Müller, U. Wandinger, and D. N. Whiteman, “Inversion with regularization for the retrieval of tropospheric aerosol parameters from multiwavelength lidar sounding,” Appl. Opt. 41(18), 3685–3699 (2002). [CrossRef] [PubMed]

OCIS Codes
(010.3640) Atmospheric and oceanic optics : Lidar
(010.0280) Atmospheric and oceanic optics : Remote sensing and sensors

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: June 28, 2010
Revised Manuscript: August 13, 2010
Manuscript Accepted: August 13, 2010
Published: September 9, 2010

Citation
Juan Luis Guerrero-Rascado, Maria João Costa, Daniele Bortoli, Ana Maria Silva, Hassan Lyamani, and Lucas Alados-Arboledas, "Infrared lidar overlap function: an experimental determination," Opt. Express 18, 20350-20359 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-20350


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References

  1. T. Halldórsson and J. Langerholc, “Geometrical form factors for the lidar function,” Appl. Opt. 17(2), 240–244 (1978). [CrossRef] [PubMed]
  2. K. Sassen and G. C. Dodd, “Lidar crossover function and misalignment effects,” Appl. Opt. 21(17), 3162–3165 (1982). [CrossRef] [PubMed]
  3. G. M. Ancellet, M. J. Kavaya, R. T. Menzies, and A. M. Brothers, “Lidar telescope overlap function and effects of misalignment for unstable resonator transmitter and coherent receiver,” Appl. Opt. 25(17), 2886–2890 (1986). [CrossRef] [PubMed]
  4. H. Kuze, H. Kinjo, Y. Sakurada, and N. Takeuchi, “Field-of-view dependence of lidar signals by use of Newtonian and Cassegrainian telescopes,” Appl. Opt. 37(15), 3128–3132 (1998). [CrossRef]
  5. K. Stelmaszczyk, M. Dell’Aglio, S. Chudzyński, T. Stacewicz, and L. Wöste, “Analytical function for lidar geometrical compression form-factor calculations,” Appl. Opt. 44(7), 1323–1331 (2005). [CrossRef] [PubMed]
  6. Y. Sasano, H. Shimizu, N. Takeuchi, and M. Okuda, “Geometrical form factor in the laser radar equation: an experimental determination,” Appl. Opt. 18(23), 3908–3910 (1979). [CrossRef] [PubMed]
  7. K. Tomine, C. Hirayama, K. Michimoto, and N. Takeuchi, “Experimental determination of the crossover function in the laser radar equation for days with a light mist,” Appl. Opt. 28(12), 2194–2195 (1989). [CrossRef] [PubMed]
  8. S. W. Dho, Y. J. Park, and H. J. Kong, “Application of geometrical form factor in differential absorption lidar measurement,” Opt. Rew. 4(4), 521–526 (1997). [CrossRef]
  9. S. W. Dho, Y. J. Park, and H. J. Kong, “Experimental determination of a geometric form factor in a lidar equation for an inhomogeneous atmosphere,” Appl. Opt. 24(24), 6009–6010 (1997). [CrossRef]
  10. U. Wandinger and A. Ansmann, “Experimental determination of the lidar overlap profile with Raman lidar,” Appl. Opt. 41(3), 511–514 (2002). [CrossRef] [PubMed]
  11. J. Bösenberg, A. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hagard, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: a European aerosol research lidar network”. In Laser Remote Sensing of the Atmosphere, A. Dabas, C. Loth, and J. Pelon, eds., selected papers of the 20th International Laser Radar Conference (Edition Ecole Polytechnique, Palaiseau, France, 2001), pp. 155–158.
  12. A. Ansmann, U. Wandinger, M. Riebesell, C. Weitkamp, and W. Michaelis, “Independent measurement of extinction and backscatter profiles in cirrus clouds by using a combined Raman elastic-backscatter lidar,” Appl. Opt. 31(33), 7113–7131 (1992). [CrossRef] [PubMed]
  13. F. Navas-Guzmán, J. L. Guerrero-Rascado, and L. Alados-Arboledas, “Calibration of 1064nm-backscatter profiles with a multiwavelength Raman lidar,” Rom. J. Physiol. (to be published).
  14. F. Navas-Guzmán, J. L. Guerrero-Rascado, J. A. Bravo-Aranda, and L. Alados-Arboledas, “On the use cirrus clouds for elastic lidar calibration”, Ópt. Pur, Apl. submitted.
  15. C. Münkel, N. Eresmaa, J. Räsänen, and A. Karppinen, “Retrieval of mixing height and dust concentration with Lidar ceilometer,” Boundary-Layer Meteorol. 124(1), 117–128 (2007). [CrossRef]
  16. C. Münkel, and R. Roininen, “Investigation of boundary layer structures with ceilometer using a novel robust algorithm”, Proc. 90th American Meteorological Society Annual Meeting: 15th Symposium on Meteorological Observation and Instrumentation (2010), 5.3.
  17. C. A. Hostetler, Z. Liu, J. Reagan, M. Vaughan, D. Winker, M. Osborn, W. H. Hunt, K. A. Powell, and C. Trepte, “CALIOP Algorithm Theoretical Basis Document”, PC-SCI-201, NASA Langley Res. Cent., Hampton, Va. http://www-calipso.larc.nasa.gov/resources/project_documentation.php (2006).
  18. F. G. Fernald, B. M. Herman, and J. A. Reagan, “Determination of Aerosol Height Distribution by Lidar,” J. Appl. Meteorol. 11(3), 482–489 (1972). [CrossRef]
  19. F. G. Fernald, “Analysis of atmospheric lidar observations: some comments,” Appl. Opt. 23(5), 652–653 (1984). [CrossRef] [PubMed]
  20. J. D. Klett, “Stable analytical inversion solution for processing lidar returns,” Appl. Opt. 20(2), 211–220 (1981). [CrossRef] [PubMed]
  21. J. D. Klett, “Lidar inversion with variable backscatter/extinction ratios,” Appl. Opt. 24(11), 1638–1643 (1985). [CrossRef] [PubMed]
  22. Y. Sasano and H. Nakane, “Significance of the extinction/backscatter ratio and the boundary value term in the solution for the two-component lidar equation,” Appl. Opt. 23(1), 11–13 (1984). [CrossRef]
  23. Y. Sasano, E. V. Browell, and S. Ismail, “Error caused by using a constant extinction/backscattering ratio in the lidar solution,” Appl. Opt. 24(22), 3929–3932 (1985). [CrossRef] [PubMed]
  24. L. Mona, A. Amodeo, G. D’Amico, and G. Pappalardo, “First comparisons between CNR-IMAA multiwavelength Raman lidar measurements and CALIPSO measurements,” Proc. SPIE 6750, 6750–35 (2007).
  25. K. M. Markowicz, P. J. Flatau, A. E. Kardas, J. Remiszewska, K. Stelmaszczyk, and L. Woeste, “Ceilometer Retrieval of the Boundary Layer Vertical Aerosol Extinction Structure,” J. Atmos. Ocean. Technol. 25(6), 928–943 (2008). [CrossRef]
  26. T. Elias, A. M. Silva, N. Belo, S. Pereira, P. Formenti, G. Helas, and F. Wagner, “Aerosol extinction in a remote continental region of the Iberian Peninsula during summer,” J. Geophys. Res. 111(D14), D14204 (2006), doi:. [CrossRef]
  27. C. M. R. Platt, J. C. Scott, and A. C. Dilley, “Remote sounding of high clouds. Part VI: Optical properties of mid-latitudemand tropical cirrus,” J. Atmos. Sci. 44(4), 729–747 (1987). [CrossRef]
  28. K. Sassen and B. Y. Cho, “Subvisual-thin cirrus lidar dataset for satellite verification and climatological research,” J. Appl. Meteorol. 31(11), 1275–1285 (1992). [CrossRef]
  29. E. W. Eloranta, “Practical model for the calculation of multiply scattered lidar returns,” Appl. Opt. 37(12), 2464–2472 (1998). [CrossRef]
  30. H. Chepfer, J. Pelon, G. Brogniez, C. Flamant, V. Trouillet, and P. H. Flamant, “Impact of cirrus cloud ice crystal shape and size on multiple scattering effect: application to spaceborne and airborne backscatter lidar measurements during the LITE mission and E LITE campaign,” Geophys. Res. Lett. 26(14), 2203–2206 (2000). [CrossRef]
  31. W. N. Chen, C. W. Chiang, and J. B. Nee, “Lidar ratio and depolarization ratio for cirrus clouds,” Appl. Opt. 41(30), 6470–6476 (2002). [CrossRef] [PubMed]
  32. I. Veselovskii, A. Kolgotin, V. Griaznov, D. Müller, U. Wandinger, and D. N. Whiteman, “Inversion with regularization for the retrieval of tropospheric aerosol parameters from multiwavelength lidar sounding,” Appl. Opt. 41(18), 3685–3699 (2002). [CrossRef] [PubMed]

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