The mathematical formulation for the optical setup of a typical EARLINET lidar system is given here. The equations describing a lidar system from the emitted laser beam to the projection of the telescope aperture on the final receiving unit (i.e., photomultiplier or photodiode) are presented, based on paraxial approximation and geometric optics approach. The receiving optical setup includes a telescope, a collimating lens, an interference filter and the ensemble objective eyepiece. The set of the derived equations interconnects major parameters of the optical components (e.g., focal lengths, diameters, angles of incidence), revealing their association with the distance of full overlap of the system. These equations may used complementarily with an optical design software, for the preliminary design of a system or can be used as a quick check up tool of an existing lidar system. The evaluation of the formulation on a real system is performed with ray-tracing simulations, revealing an overall good performance with relative differences of the order of 5 % mainly attributed to the limitations of the thin lens approximation.

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

The lidar equation in its simplest form includes the 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

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,

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
(

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 (

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.

The setup of a biaxial lidar system is schematically given in Fig. 1. The laser telescope geometry is demonstrated in the upper part (Fig. 1a) while the optical setup of the lidar receiving unit behind the telescope, is presented in the lower part (Fig. 1b). The abbreviations used in this study in order to describe the lidar parameters are summarized in the Appendix as a list of abbreviations.

The modeling of a transmitted laser beam in the atmosphere has been
approximated by a truncated cone of an ideally circularly shaped beam with
initial diameter DL and divergence LBD (half angle).
The DL and LBD values provided by the manufacturers
usually correspond to the 86.5 % (2

For simplicity, all the optical components of the system (telescope and lenses) are presented as thin lenses in Fig. 1b. In addition to the paraxial approximation assumption, namely that rays are not too distant from the system axis and their angles with respect to that axis are small, it also implies that aberration effects are not considered.

Regardless of the origin point of the incoming rays, any rays incident at the
telescope with angles higher than

In principle, the system presented in Fig. 1b spreads uniformly over the image of the entrance aperture formed on the photodetector surface the light coming from the illuminated parts of the atmosphere within the system field of view. The effective diameter of the photomultiplier is maximum 8 mm for Hamamatsu PMTs R7400 series (Hamamatsu Photonics, 2006). However, the useful diameter of the PMT is about 5 mm, including mounting and adjustment tolerances.

For ranges above the DFO, the laser beam stays entirely inside the telescope's full field of view. For those ranges, any ray coming from a point in the illuminated volume and reaching the telescope aperture will pass through the field stop diaphragm.

The parameter of RFOV is chosen as the coupling link between the laser telescope part and the detection optics, which are located after the telescope focus. This choice can be explained, since on one side the given telescope and laser geometrical characteristics determine the DFO, and on the other side towards the PMT, all rays entering the field stop have to be collected by the PMT.

As also demonstrated from Stelmaszczyk et al. (2005) from Fig. 1a we have the following:

In the case that a laser beam expander with an expansion factor of

The dominator of Eq. (1) must be positive (i.e.,

where

With

The ratio

Consequently,

Additional constraints are the limited diameters of the lenses, filters and
beam splitters in combination with the diameter and the focal lengths of the
telescope and the collimator (i.e.,

All these parameters must be balanced for optimum lidar performance and for
a specific scientific objective. The following system of equations (Eq. 8)
is derived with paraxial approximation (Fig. 1b). The first one is given by
definition (Fig. 1b), the second one gives the plane where the image of the
entrance pupil by the collimating lens is formed, and the third one gives
the minimum diameter that the objective lens must have in order to let all
the rays within the field of view pass:

Panels

Furthermore, in the

The rays collected by the IFF and the objective lens (L3 in Fig. 1b) are
guided through the eyepiece lens (L4 in Fig. 1b), creating the final image
of the entrance pupil at a distance

From the similarity of the triangles shown in Fig. 2b we find that

Considering as object the image of the aperture by L2, which is at distance

PMTs and APDs suffer from a non-uniform spatial response of their effective
surface, which may cause artifacts to lidar signals during its transduction
into electrical signal. Simeonov et al. (1999) revealed that
the normalized spatial uniformity on the active area of the detector varies
from 0.2 up to almost three times the average value, defined for the central
part of the detector. In order to avoid lidar signal deviations due to the
spatial inhomogeneity PMT sensitivity, the detector must be placed at an
image of the telescopes aperture. At this place (distance

For boundary layer measurements a low

Tilting the laser by an angle

More clearly, for the former scenario

and with

The variability of the maximum angle of incident rays on the IFF
(

The IFF allows for acceptable transmission of the backscattered rays with
incident angles lower than

In Fig. 3 the variation of the maximum angle of incident rays on the IFF
(

For evaluating the formulation presented in this study, ray-tracing
simulations with ZEMAX software (

The geometrical properties of the simulated lidar system used as input
parameters in paraxial approximation lead to a

The optical parameters (distances and focal lengths) estimated with paraxial approximation and simulated by ZEMAX, along with their relative differences.

Spot diagrams of far- (green; 10 000 m) and near- (blue; 257 m) range
rays on the

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

ZEMAX simulations regarding the focal plane of the telescope where
the field stop
(

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 (

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.

The Excel worksheet that integrates the entire set of equations as described in this paper is available upon request from the author (panko@noa.gr).

The authors declare that they have no conflict of interest.

This article is part of the special issue “EARLINET, the European Aerosol Research Lidar Network”. It is not associated with a conference.

I would like to thank Volker Freudenthaler for the helpful discussions, ideas and useful comments.

The publication was supported by the European Union Seventh Framework Programme (FP7-REGPOT-2012-2013-1) in the framework of the project BEYOND, under grant agreement no. 316210 (BEYOND-Building Capacity for a Center of Excellence for EO-based monitoring of Natural Disasters). Edited by: Albert Ansmann Reviewed by: two anonymous referees