Lidars are efficient tools for retrieving the aerosol optical and microphysical properties in the planetary boundary layer (PBL) and free troposphere. More precisely, the lidar techniques that are widely used for aerosol research are capable of providing range-resolved information for (a) the aerosol backscatter coefficient βαer using the backscatter lidar technique (e.g., Fernald et al., 1972; Klett, 1981), (b) the aerosol extinction coefficient (ααer) using the Raman lidar technique (Ansmann et al., 1990, 1992) and (c) the volume and particle linear depolarization ratios using the depolarization lidar technique (e.g., Sassen, 2005; Freudenthaler et al., 2009). Many studies have demonstrated that the provision of the aforementioned aerosol optical properties for multiple wavelengths facilitates the retrieval of aerosol microphysical properties through inversion techniques (Müller et al., 1999; Veselovskii et al., 2002, 2010). The accuracy of the optical properties used as inputs for the inversions is critical. Uncertainties involved for the retrieval of the aerosol optical properties with lidar techniques are linked both to systematic and statistical sources of error. Statistical errors are due to noise associated to the photon nature of the signal itself and to noise added in the detection process (Ansmann et al., 1992; Bösenberg, 1997; Iarlori et al., 2015). On the other hand, systematic errors may arise both from assumptions or uncertain values that enter the lidar data analysis and from the system setup and geometry. The first category may include uncertainties introduced by the estimation of temperature and pressure profiles along with the wavelength dependence parameter required in the Raman technique (Ansmann et al., 1990; Whiteman, 1999) and the assumption of lidar ratio, reference height and backscattering ratio, required in the backscatter technique. The second broad category of systematic errors may include uncertainties introduced, for example, by interferences caused by the laser source in the analog receiving channels, the range-dependent overlap factor, the calibration of the system, etc. Over the last decades, a lot of work towards estimating and minimizing the errors in aerosol lidar retrievals has been done in the framework of the European Aerosol Research Lidar Network (EARLINET; Pappalardo et al., 2014). For example, and in order to optimize the optical performance and control the quality of aerosol measurements, a number of quality assurance (QA) tests have been adopted and applied in EARLINET lidar systems (Freudenthaler, 2008). Moreover, increased effort has been made by the European lidar community to develop and apply accurate depolarization calibration techniques (Freudenthaler, 2016) and quantify and correct the influence of systematic error introduced by imperfections of lidar optical elements on the depolarization related retrievals (Mattis et al., 2009; Bravo-Aranda et al., 2016; Belegante et al., 2016). These studies are based on the description of the state of polarization of light and lidar optical elements by means of the Müller–Stokes formulation. The present study tackles basic lidar design trade-offs, and more advanced topics such as depolarization measurements are out of its scope.

The lidar equation in its simplest form includes the overlap function (O(z)) and the overall optical efficiency of the system. The overlap function is range-dependent and thus related to the lidar system geometry, since it describes the fraction of the light scattered within the receiver field of view, taking values from 0 to 1 (Wandinger, 2005). More precisely, at the height range where the overlap function reaches the value of 1, each point of the telescope aperture collects the scattered light entirely and with the same efficiency (Fig. 1a). This height range is determined by the intersection point between the outer edge of the laser beam divergence (LBD) and the lateral surface of receiver field of view (RFOV) cone, with the apex on the point of the telescope aperture farthest away from the laser (Fig. 1a). This range is known as the distance of full overlap (DFO) and is usually found from 500 to 1500 m for EARLINET aerosol lidar systems. The overlap function depends on the range up to the DFO, adding a significant drawback to the retrieval of aerosol optical properties from lidar systems, since it becomes difficult to obtain useful and accurate information regarding the aerosol present below that height range. More precisely, Wandinger and Ansmann (2002) demonstrated that when not applying overlap correction in lidar signals, the retrieved aerosol extinction coefficient may take even non-physical negative values for heights up to the DFO. However, they proposed that the effect of the incomplete overlap can be corrected, and trustworthy retrievals may finally be obtained in the case of a system operating both elastic and Raman channels under the assumption that they are affected by the same overlap function.

Thus, in order to optimize the performance of a lidar at lower altitudes and effectively retrieve optical properties of the aerosol entrapped below the PBL height, it is of great importance that the receiving telescope is able to detecting the emitted laser pulse, already at short ranges from the lidar system. Therefore, a low full overlap height is needed. The wide-angle RFOV is not an optimal solution for minimizing the DFO, since in that case (a) the signal will be contaminated with more sky background light, and (b) multiple scattering effects have to be taken into account, especially for case studies of optically thick targets (i.e., water and ice clouds; e.g., Eloranta, 1998; Wandinger, 1998). Nevertheless, there are systems employing two telescopes, one for the short ranges and the other for the far ones, with the short-range one having a wider RFOV angle (e.g., Engelmann et al., 2016). However, such setups are more complicated and their signals demand special treatment during the retrieval processing schemes.

Case studies, but also long-term lidar observations performed during the last decade at various EARLINET stations, revealed that the DFO have to be much lower than 600 m in order to detect the boundary layer at European latitudes, especially during wintertime (e.g., Matthias and Bösenberg, 2002; Matthias et al., 2004; Amiridis et al., 2007; Baars et al., 2008). Measurements of the aerosols within the PBL are in particular required during daytime, when the convection is stronger. However, daytime lidar operation suffers from the increased sky radiance contaminating the lidar signal, which needs suppression. In order to suppress the bright daytime sky radiance and enhance the signal to noise ratio (SNR) of the lidar signal, small bandwidth (BW) interference filters (IFFs) are widely used in EARLINET. Such IFFs have recently become commercially available with small BW narrower than 0.2 nm (FWHM) at the visible spectrum and high transmission values (greater than 90 %) at peak (Alluxa, CA, http://www.alluxa.com). Their high transmission and narrow BW characteristics have been recently used for rotational Raman measurements at visible (Veselovskii et al., 2015) and infrared spectrum (Haarig et al., 2016). A significant drawback of these filters is that their narrow bandwidth can cause low acceptance angle (AIFFmax), which in turn limits the possible DFO. Alternative methods for efficiently suppressing the background are based on the shaping of the receivers field of view diaphragm (FOVD) along with their geometry and their relative position on the optical axis, as has been proposed by Abramochkin and Tikhomirov (1999) and Freudenthaler (2003).

There are several studies in the literature related to the determination of the overlap function of lidar systems analytically (e.g., Halldórsson and Langerholc, 1978; Jenness et al., 1997; Chourdakis et al., 2002; Stelmaszczyk et al., 2005; Comeron et al., 2011) or experimentally (e.g., Sasano et al., 1979; Tomine et al., 1989; Dho et al., 1997; Guerrero-Rascado et al., 2010; Wandinger and Ansmann, 2002). For the theoretical approaches, a good understanding of the actual light distribution in the laser beam cross section, and the characteristics of the receiving unit are needed to obtain an overlap profile with sufficient accuracy. Stelmaszczyk et al. (2005) also proposed an analytical formula to decrease the DFO based only on the laser telescope geometry and specifically the introduction of a small inclination between the transmitter and receiver central axis. However, all the aforementioned theoretical studies provide information regarding the overlap function based explicitly on the laser telescope setup.

This study focuses on the extension of the paraxial approximation down to the detector, revealing all the possible constraints of a lidar setup, since DFO depends on the overall optical path of the detected backscattered radiation. The distance of full overlap, as presented in this work, depends on the entire geometry considering all the parts of the lidar as one optical system. The analysis in this study highlights the need to take into consideration the acceptance angle of the interference filter when designing an optimized lidar system and the possible limitations that this imposes. The corresponding geometrical formulation is presented in Sect. 2, describing the basic characteristics (focal lengths, distances and diameters) of all the optical components, compromised with the EARLINET QA standards. The derived formulation also includes the characteristics of an eyepiece lens, which has to be used in a lidar setup in order to form an image of the entrance pupil on the surface of the photodetector so as to spread the collected light uniformly over this image. The results of ray-tracing simulations with respect to lidar design and alignment according to geometrical formulation are presented in Sect. 3.

The entire set of equations has been integrated in a Microsoft Excel worksheet through Visual Basic for Applications (VBA) code and distributed with supplementary documentation to the members of EARLINET network (http://www.meteo.physik.uni-muenchen.de/~stlidar/earlinet_asos/raytracing/Basic_design/basic_lidar_design.html), but can also be widely available upon request from the author (panko@noa.gr). The aforementioned worksheet cannot substitute advanced optical design software. However, it may be complementary to the preliminary design of a system, can be used as a quick checkup tool for an existing lidar system or even used as a learning tool to familiarize a system with an optical lidar setup.

Lidar systems are using large telescopes to collect the weak light, backscattered from the atmosphere. This portion of light has to be further transmitted and projected to small detectors without any range-dependent losses. For example, with a telescope diameter of 300 mm and a detector diameter of 5 mm, an overall magnification of the optical system of 0.0166 is necessary. On the other hand, the angular magnification is increased by 60, which means that 1.25 mrad field of view of the receiver setup (i.e., determined by telescope with focal length 600 mm and a field stop with diameter 1.5 mm), is magnified to about 75 mrad (∼ 4.3∘).

For single but more particularly for multiwavelength systems, the wavelength separation unit is not capable of accepting such a divergent received light beam, since (a) it would soon be too wide for 1 or 2 inch optical elements, and (b) the transmission of interference filters is very sensitive to the incidence angle. Therefore, the magnification of the receiver optics is split in two parts, i.e., the telescope with a collimation lens and another objective with an eyepiece, with a low divergent light path (parallel received light beam) in between (Fig. 1b). The inclination of the parallel beam is determined by the laser beam divergence, the laser telescope axis distance, the tilt of the laser beam with respect to the telescope axis, determining the field of incidence angles into the telescope, and the magnification of the telescope together with the collimating lens.

The limitations for this inclination of the received rays are the following: the field of view of the telescope should be as small as possible in order to reduce the background light collected from the sky; the laser beam diameter and its divergence must both be small enough to fit through the 1 inch optics for all necessary beam splitters; the inclination must be less than the maximum acceptance angles of the interference filters.

For evaluating the formulation presented in this study, ray-tracing simulations with ZEMAX software (www.zemax.com) have been performed. Considering that, unlike ZEMAX, various aberration effects are not taken into account with thin lens approximation, in this section it is investigated how close the calculations are in reality when derived in comparison with real ray-tracing simulations.

The geometrical properties of the simulated lidar system used as input parameters in paraxial approximation lead to a DFO=257 m. More precisely, a laser with initial parameters DL= 8 mm and LBD=0.8 mrad was considered and expanded by an ideal laser beam expansion unit (EX=×4). The expansion unit finally results in an emitted laser beam with a diameter of 32 mm and divergence LBD=0.2 mrad. Moreover, the laser beam is considered to be tilted towards the telescope central axis with an angle of Atilt= 0.4 mrad. The laser light scattered in the atmosphere is collected by an ideal telescope with DT = 300 mm and FT = 600 mm, guided through a circular field stop (DFS= 1.5 mm) to the collimator. The distance Z1 has to be as high as possible to mount all the needed optical elements (i.e., beam splitters), always keeping Amax lower than AIFFmax.. Considering a receiving system with RFOV= 1.25 mrad, effective Dobj.=23.5 mm and a reasonably low Amax value (Amax=1.15∘), a distance of Z1= 160 mm, is estimated through Eq. (8). Moreover, for these values Eq. (2) implies that the focal length of the collimator should be Fcol=37.36 mm.

The 3-D ray-tracing simulations have been initiated at 532 nm using the aforementioned values and assuming an optimized Newtonian telescope. The curvature radii of the primary mirror was set to 1200 mm and its radius to 150 mm. The surface type was set to a sphere and the conic constant -1. At 500 mm above the primary mirror, a secondary diagonal mirror was set with a radius of 25 mm. The backscattering from the laser beam was simulated by a disk of source rays, which was placed at distances of 257 m and 10 000 m from the telescope. Assuming that the telescope and the laser beam are at the same level (Fig. 1b), then the initial distance between their central axis was considered to be DTL= 180 mm. However, as the laser beam propagates in the atmosphere with a tilting angle of 0.4 mard, the distance between the laser beam and telescope central axis decreases with a constant rate of Atilt. For each distance, the size of the laser disk was calculated from the LBD. Regarding the optical components mounted after the field stop, we used lenses available from the ZEMAX database, with parameters (i.e., effective focal length and diameter) similar to the ones revealed from the paraxial approximation calculations. More precisely, x2 Newport (model SPX019) lenses were used as collimator (effective Fcol= 37.72 mm and Dcol= 25.40 mm), x2 Melles Griot (models 01LUP033 and 01LUP031) were used as an objective (effective Fobj= 74.69 and Dobj= 25.00 mm) and x2 Melles Griot (model 01LUP009) lenses were used as an eyepiece lens (effective Feye= 15.31 and Deye= 12.50 mm; Table 1). In ZEMAX we set the distance between the collimator and the IFF at exactly 160 mm (Z1), the eyepiece at 85.64 mm after the IFF (Z2), while the distance between the eyepiece and PMT was at 13 mm. The value of distance Z3 was not kept constant in the ZEMAX simulation as it was estimated from the paraxial calculations. Instead, this distance was slightly increased up to some millimeters in order to sufficiently image the telescope aperture on the PMT (DPMT=5mm), without any truncation of the rays. Please note here that, in the case of real simulations with ZEMAX, the lenses cease to be ideal thin lenses. The distances, namely Z1, Z2 and Z3, are measured from the points on the principal axis of the last surface of each lens up to the first surface of the next optical component. For example, the measured distance Z1 refers to the distance between the last surface of the collimator up to the first surface of the IFF filter, located on the principal axis. In Fig. 4 the spot diagrams of rays from far (green spots; 10 000 m) and near range (blue spots; 257 m) are demonstrated. The five field points selected so as to make an object height with radius equal to LBDxZ+DL2, where Z is the atmospheric distance from the lidar. The central field point is located at the center of the disk, with coordinates xC,yC=0,DTL-Z×Atilt, with respect to the Cartesian system shown in Fig. 1a, while the remaining four are selected to be on the perimeter of this disk with coordinates xR,yR=-Z×LBD,yC, xLyL=Z×LBD,yC,xU,yU=0,yC+Z×LBD,xD,yD=(0,yC-Z×LBD). The spot diagrams in Fig. 4a are in cosine space, demonstrating the angle with which each field point of far- and near-range rays falls on the first surface of the IFF filter. The maximum field incident angle on the IFF was found to be equal to 0.0196 mrad. The full field spot diagram demonstrated in Fig. 4b refers to the surface of the PMT. As can be seen in Fig. 4b a homogeneous distribution of far- and near-range rays on PMT surface have been achieved, covering the same area. The spot diameter was found to be 4.6 mm, within the 5 mm diameter of effective detector aperture, revealing an overall sufficient imaging of far- and near-range rays on the detector.

The relative differences between the calculated parameters from paraxial approximation and the simulations with ZEMAX are demonstrated in Table 1, and the slight discrepancies are attributed to the following reasons: (a) the slightly different parameters of lenses used from ZEMAX database compared to those used as input (Fcol) or estimated (FobjFeye) with the paraxial approximation formulation and (b) the inability of the paraxial approximation to take into account the sag of each lens surface to better model the refraction of off-axis rays in contrast to the ZEMAX simulation. However, the effect of the telescope's defocus seems to be taken into account by paraxial optics. Rays arriving at the telescope from near points intersect behind the focal plane. Where they will finally intersect depends on how near the origin point is. Here the near points are considered to be from 257 m (blue lines), which they intersect at 1.4 mm behind the focal plane of the telescope. As the points are taken farther away, the intersection point will tend to approach the focal plane, eventually being virtually on the focal plane for long enough distances (i.e., the points from 10 000 m considered here with green lines). From the paraxial formula relating object and image positions, a simple calculation shows that, for a thin lens of 600 mm focal length, the image of a point at 10 000 m from the lens will be at 600.04 mm from the lens plane, while the image of a point at 257 m will be at 601.40 mm, the difference being 1.36 mm, very close to the 1.40 mm revealed by ZEMAX simulations (Fig. 5). Thus, the telescope processes the rays according to the paraxial optics laws (or at least to a good degree of approximation), so the considered rays satisfy the paraxial conditions.

Based on thin lens approximation formulas, a set of equations is derived, describing the optical design of a typical EARLINET lidar system. The limitations of a lidar optical setup are revealed through geometric optics, from the emitted laser beam to the projection of the entrance pupil on the photomultiplier. The main lidar issue studied here concerns the distance of full overlap and how this depends on the entire geometry which describes the optical path in the detection unit of a lidar system, not only on the laser telescope geometry. The usage of IFF with small bandwidth for background suppression is limited by their small acceptance angle, especially if the alignment uncertainties of the mechanical setup of the lidar optics are taken into account. The evaluation of the paraxial approximation formulation has been performed with ZEMAX ray-tracing simulations, showing an overall good performance with a relative difference (between ZEMAX and paraxial approximation) of the order of 5 % (see Table 1) and a negligible impact on the system performance. These differences are mainly attributed to the inability of thin lens formulations to better model the refraction of the off-axis rays. The described formulation cannot substitute an advanced optical design software, since 3-D ray-tracing simulations of realistic lidar systems are required to reveal the necessity of using the highest-quality optical parts mounted with the highest possible accuracy.