Analytical function for lidar geometrical compression form-factor calculations

Analytical function for lidar geometricalcompression form-factor calculations

Kamil Stelmaszczyk, Marcella Dell’Aglio, Stanislaw Chudzynski, Tadeusz Stacewicz,and Ludger Wöste

A simple model of image formation in a Newtonian telescope was used for calculating an analyticalformula, that describes the geometric compression form factors of coaxial and biaxial lidars. Calculationswere successfully validated by comparison with real measurements, confirming the accuracy of ourapproach. The need for different alignment of coaxial and biaxial systems to increase the overlap betweenthe lidar emitter and receiver is also discussed. © 2005 Optical Society of America

OCIS codes: 010.3640, 280.0280, 110.6770.

1. Introduction

For many years, lidar systems have been successfullyused in remote sensing of the atmosphere. They havebeen applied in local- and global-scale studies of at-mospheric aerosols1 and climate-relevant gases suchas ozone2 or vapor3 as well as for analyzing the prop-erties of clouds4 and for dynamic processes within theplanetary boundary layer.5

Regardless of type, the effective range of a lidarmeasurement is limited by strong atmospheric ex-tinction (depending on the wavelength, at �0.3 to�0.5 dB�km in clear atmosphere) and by the geomet-rical 1�R2 factor. As a consequence, the best accuracyand precision in signal inversion are obtained close tothe device (typically not more than a few kilometersaway), where the signal-to-noise ratio is relativelyhigh.6 Unfortunately, in this region recorded data areusually affected by geometrical compression.

Geometrical compression (GC) is an effect of re-duced detector response to the return signal causedby a lack of perfect coincidence between the tele-scope’s full field of view (defined below) and the laser

beam. Also, other factors, such as the presence ofobstacles inside the telescope and insufficient size ofa detector can contribute to GC. In this paper wediscuss only the first of the aforementioned effects,which leads to large systematic errors in short-ranged aerosol backscattering profiles.7 These errorsare also of great importance for differential-absorption lidars, for which additional misalignmentbetween on and off beams can occur.8

The influence of GC on the lidar signal is describedin terms of GC form factor, ��R� [see Eq. (1) below].To determine the dependence of GC on distance,analytical,9–14 ray-tracing,15,16 and experimental7,17–19

methods have been proposed. As was reported byWandinger and Ansmann,19 the last-named method isan ideal choice during field campaigns, during whichinformation on laser or telescope parameters, as wellas on their mutual orientation, is not easily available.An experimental approach, however, cannot be usedfor modeling the lidar signal. It is also useless duringthe design of transmitting, or receiving optics. For suchtasks, analytical or ray-tracing methods are superior.

Although many theoretical calculations have beenreported, the first (and to the authors’ knowledge theonly) analytical function that describes the GC formfactor was proposed by Kuze et al.14 In that paper,only biaxial lidars (in which emitter and receiver axesdo not overlap) with parallel laser and telescope axeswere considered. Unfortunately, the assumption ofparallel laser and telescope axes cannot easily beverified during measurements, when a small inclina-tion between transmitter and receiver can exist. Aswas admitted by those authors,14 the postulated for-mula had no physical basis and was chosen some-

K. Stelmaszczyk ([email protected]) andL. Wöste are with the Fachbereich Physik der Freien UniversitätBerlin, Institut Für Experimentalphysik, Arnimallee 14, 14195Berlin, Germany. M. Dell’Aglio is with the Dipartimento Chimica,Università degli Studi di Bari, Via Orabona, 4-70125 Bari, Ital-y. St. Chudzynski and T. Stacewicz are with the Instytut FizykiDoswiadczalnej Uniwersytetu Warszawskiego, ul Hoza 69, 00-681Warsaw, Poland.

Received 19 January 2004; revised manuscript received 13 Sep-tember 2004; accepted 26 October 2004.

0003-6935/05/071323-09$15.00/0© 2005 Optical Society of America

1 March 2005 � Vol. 44, No. 7 � APPLIED OPTICS 1323

what arbitrarily; it also included a fitting parameterthat could not be resolved before signal registration.

In this paper we propose a new analytical formulathat describes the GC form factor. To be distin-guished from that described by Kuze et al., it usesmore easily readable parameters, which can be de-duced from simple analysis of lidar signals. We alsopresent a comparison of measured and calculated GCcurves for three different lidars, one located at theFreie Universität Berlin, one at Warsaw University,and one at the Centro Laser of Bari, Italy.

2. Definitions

Attenuation of the laser pulse energy in the atmo-sphere is described by lidar equation. It is usuallyexpressed in terms of average power P received by thephotodetector from a distance R. Given single, elasticscattering and wavelength �L, a lidar equation can bewritten in the form20

P(R, �L) � PL

c�L

2 �(�L)�(R, �L)A

R2 �(R)

� exp��2 �0

R

�(r, �L)dr�, (1)

where PL represents the initial laser pulse (mean)power, �L denotes the pulse duration, c is the speed oflight, ��L� describes the spectral efficiency of the re-ceiver for �L, ��R, �L� is the volume backscatteringcoefficient, A�R2 represents the acceptance angle ofthe receiver optics (A is the area of the telescope’sprimary mirror), and ��R, �L� is the atmospheric ex-tinction (scattering plus absorption) coefficient. TheGC form factor is denoted �(R). It can be defined asthe ratio of the energy transferred to the photodetec-tor, Edet to the energy reaching the telescope primarymirror, Escat (Ref. 16):

�(R) �Edet(R)Escat(R). (2)

The geometrical compression form factor is oftenused for defining the telescope’s effective area, Aeff.

Aeff (R) � �(R)A. (3)

Small values of �(R) correspond to a situation inwhich a major part of the light is focused outside thedetector’s sensitive area. For longer distances, signalcompression becomes smaller, resulting in highervalues of the GC form factor. At the point at whichbackscattered radiation is completely registered,��R� � 1. We refer to this distance as the full-overlapdistance, Rmin (see Fig. 1). For R Rmin the beamstays entirely inside the telescope’s full field of view,and each point of the telescope mirror is used with thesame light-collecting efficiency, a characteristic thatholds true, provided that there are small inclinationangles � between emitter and receiver.

As will be shown below, Eq. (3) can be consideredan alternative definition of the GC form factor.

3. Full-Overlap Distance

To determine the full-overlap distance we considerthe biaxial lidar arrangement presented schemati-cally in Fig. 2. For figure clarity, only the primarymirror of diameter T and aperture s of the receivertelescope are depicted. The secondary telescope mir-ror, which in reality gives rise to the shadow in thefocal plane, is omitted from this schematic. A simpli-fied configuration was chosen because our calcula-tions are valid only for an ideal reflector with nocentral obstruction. Because the secondary mirror ina Newtonian-type telescope has no focusing (defocus-ing) properties and its role is diminished to shiftingthe focus outside the telescope’s primary mirror area(beyond the telescope tube), the setup from Fig. 2 isequivalent to that of a real Newtonian telescope withno central obstruction.

A laser beam with an ideal circular shape is sentinto the atmosphere at an angle � with respect to thetelescope axis. We assume that this inclination issmall (single milliradians) and that for R � 0 thedistance between the beam and the telescope axis isequal to d0. Obviously, d0 � 0 corresponds to an on-axis configuration, so our formulas can be generalizedfor coaxial lidars. The initial beam diameter andbeam divergence are g0 and �, respectively. We as-sume that � is smaller than the telescope’s field ofview (FOV), � s�f, where s refers to the apertureand f to the focal length. As indicated in Fig. 2, wetreat only the case in which the aperture is placed inthe focal plane and is coaxial with the telescope axis,a configuration that is standard for most lidars.

Let us choose first an arbitrary point X on theprimary mirror of diameter T. The incoming light atthis point is registered by a photodetector situateddirectly behind the aperture but only when it is emit-ted inside a cone with opening angle �, whose vertexoverlaps chosen point X. This quality follows from the

Fig. 1. Lidar signal and the corresponding calculated GC formfactor. The uncompressed signal proportional to 1�R2 is also plot-ted.

1324 APPLIED OPTICS � Vol. 44, No. 7 � 1 March 2005

telescope’s FOV definition. The space formed by theintersection of all cones that have opening angles �and vertices situated on the mirror is also a cone, andit is shown shaded in Fig. 2. Obviously, the openingangle of the shaded cone is also �. Light scatteredinside the shaded cone that reaches the primary mir-ror will be completely focused on the detector’s sen-sitive area. For this reason we can call this cone thetelescope’s full field of view (full FOV). For light back-scattered inside the FOV but outside the shaded cone,only a fraction of the primary mirror will be used. Itsresidual area will focus light outside aperture s. Forthis case, Aeff � A as soon as the beam’s cross sectionhas completely entered the telescope’s full FOV. Rminis then determined by the intersection point betweenthe outer edge of a laser beam and the lateral surfaceof the full FOV.

Using Fig. 2, we can therefore calculate that

Rmin � p sin��

2 � � �

2�, (4)

where

p � (T�2 d0 g)cos(�2)

sin(� �2 � ��2). (5)

Because laser’s divergence and the receiver’s FOV,as well as the inclination angle, are small (typically asingle milliradian or less) we can combine Eqs. (4)and (5) in a compact form:

Rmin �2d0 g0 T2� � �

. (6)

In some publications13 the difference between atelescope’s FOV and its full FOV is not clearly stated.Such confusion leads to underestimations of full-overlap distance, as the erroneous criterion of enter-ing the beam inside the FOV rather than inside thefull FOV was used for calculations.

If rmin denotes the distance in which the laser beamis fully contained within telescope’s FOV (not its fullFOV), the difference between Rmin and rmin can beestimated by means of the formula

Rmin � rmin �2T

2� � �. (7)

Equation (7) is independent of both d0 and g0. Thelargest differences between the two distances occurfor a laser beam with � � 0 and a divergence approx-imately equal to the telescope’s FOV �� , which isa typical lidar alignment, ensuring the best overlap

Fig. 2. Entry of the laser beam into the telescope’s FOV (at rmin) and into the telescope’s full-FOV (at Rmin). Starting from Rmin, the signalis free from GC [��R� � 1].


and a high signal-to-noise ratio. In such a case,Rmin � rmin can easily exceed several or several tens ofkilometers. Figure 3 illustrates the difference betweenthe two distances calculated for three values of �.

4. Geometrical Compression Form Factor: AnalyticalFormula

Equation (6) allows for a quick and accurate calcula-tion of the full-overlap distance, but it is not sufficient

to provide a complete description of the GC form fac-tor. To determine the variability of �(R), one can ap-ply a simple model of the image formation. In thispaper we consider only a Newtonian telescope withno central obstruction.

Figure 4 shows a circular object of diameter G po-sitioned a distance R close to a lidar detection unit(unlike astronomical telescopes, lidar receivers mustbe capable of collecting light from the region neigh-boring the device). In such a case the image of G isformed not in the focal plane but slightly behind it.Using the fundamental concave mirror equation andthe equation for image magnification, we can calcu-late the diameter of this image, B:

B � GbR, b �

fRR � f . (8)

where b denotes the distance to the image plane.Because imaged object G is a radiating circular

laser beam, it gives rise to a blurred circular area ofillumination in the focal plane. The energy distribu-tion across this area is constant in the middle andtends to zero at the edges.11,12 The ratio between theenergy accumulated in the nonuniformly illuminatedarea to an overall energy transmitted to the focalplane depends on laser and telescope parameters aswell as on the distance to the object. However, forshort distances this ratio is usually not greater than0.1.11,12 Therefore we can assume that the imageformed in the focal plane is a circle with uniformly

Fig. 3. Systematic error Rmin � rmin resulting from estimating thefull-overlap distance by the criterion of entry of the laser beam intothe FOV and not into the full FOV. Calculations were performedfor a 150�mm-diameter telescope.

Fig. 4. Image formation in the Newtonian-type telescope without central obstruction. For short distances, image B of divergent laserbeam G is formed behind the focal plane. The image observed in the focal plane forms a blurred area of illumination, e, which is usuallybigger than the telescope’s aperture s and shifted from optical axis �.


distributed energy. Diameter e of this circle is ex-pressed by

e �Bf T(b � f)

b . (9)

Equations (8) and (9) can be transformed into

e �f (G T)

R . (10)

The laser beam’s diameter increases with distanceaccording to the formula

G(R) � g0 �R, (11)

where g0 and � were previously introduced as theinitial laser beam diameter and divergence, respec-tively. To derive Eq. (11) we also assumed that laserinclination � is small. Substituting the expression forG [Eq. (11)] into Eq. (10), we finally find that

e(R) � f�� g0 T

R �. (12)

The smallest area of illumination, e� � f�, is ob-tained for light backscattered from infinity (R → �). Iftelescope aperture s is smaller than e�, some part oflight will always be blocked at the telescope aperture.To make sure that the parallel beam is focused with-out losses, relation s e� must be fulfilled. This in-equality is consistent with the well-known principlethat for lidar applications the telescope’s FOV cannotbe smaller than the laser beam’s divergence. Equa-

tion (12) can therefore be used for estimating theneeded detector size.

If object G is somewhat shifted from the telescope’saxis, its image will undergo similar displacement.This effect must be taken into account when one iscalculating a geometrical form factor. For small � wecan write that this displacement, ��R�, is equal to

�(R) � fd0 � �R

R . (13)

Assuming uniform energy distribution across thearea of illumination, the energy transferred to thephotodetector, Edet, is proportional to the overlap be-tween the aperture and the image. Additionally, theimage area itself is proportional to Escat, i.e., the totalenergy reaching the primary mirror (with an accu-racy of losses that are to reflection and transmission).Using Eq. (2), we can write

�(R) �Edet(R)Escat(R) �

O(R)��e(R)�22. (14)

In Eq. (14), O�R� corresponds to the overlap area (Fig.5). The denominator of the fraction is directly ob-tained from Eq. (12). The numerator is derived formthe circular sector area. An analytical function todescribe the GC form factor can thus be derived:

�(R) �

�1(R) � sin��1(R)�s2 �2(R) � sin��2(R)�e2(R)

2�e2(R).

(15)

Angles �1 and �2 are defined by

Fig. 5. Illumination e of the focal plane and its position with respect to telescope aperture s. (a) The laser beam out of the telescope’s FOV(the detector does not register any light); (b) the beam partially inside the telescope’s FOV (the recorded lidar signal is affected bygeometrical compression); (c) illumination area inside the telescope aperture (the laser beam is completely contained in the telescope’s fullFOV).


�1(R) ��2 arccos s2 4�2(R) � e2(R)4�(R)s ��, (16)

�2(R) ��2 arccos e2(R) 4�2(R) � s2

4�(R)e(R) ��. (17)

Equations (16) and (17) are valid when |s�e(R)|�2� �(R) � [s e(R)]�2. This condition does not describeall relative positions between aperture s and image e.In particular, it does not include the situations of fulloverlap, in which image e is formed inside aperture s,or zero overlap, in which image e is formed beyond theaperture. To extend our analytical formula for anysituation, we must modify Eq. (15):

Equation (18) describes the effect of GC for any arbi-trary distance from the lidar.

The method described here can be used for thecalculation of full-overlap-distance Rmin. In this re-gard we note that, for R Rmin, light reflected fromthe primary mirror passes freely through the aper-ture because image e is formed inside the aperturearea [Fig. 5(c)]. This observation leads to the inequal-ity s 2��Rmin� e�Rmin�, which can be solved bymeans of Eqs. (12) and (13):

R Rmin �2d0 g0 T2� � �

. (19)

The result in inequality (19) is consistent with thecalculations presented in Section 3. It confirms com-plete correspondence between the analysis of the GCeffect by means of the spatial position of the laserbeam with respect to the telescope’s full FOV and theposition of the beam’s cross-section image observed inthe focal plane.

5. Coaxial and Biaxial Lidars

The form of inequality (19) indicates that telescopeswith small primary mirrors are much more suitablefor measurements at short distances than are tele-scopes with large mirror diameters. Moreover, thelaser beam should be sent as close as possible to theoptical axis, which means that coaxial rather thanbiaxial arrangements are preferred for sounding theatmosphere close to the lidar. However, to restrainthe GC effect these two lidar types require somewhatdifferent alignments.

For biaxial lidars, the laser beam should be slopedslightly toward the telescope, especially when the op-tical transmitter is situated relatively far from thereceiver. The importance of such alignment becomesevident when one compares the curves � � 0 mradand � � 0.375 mrad in Fig. 6(a). A small inclinationof the beam suppresses GC to the first 1200 m for� � 0.375 mrad and to 3000 m for � � 0.

The inclination angle cannot exceed half of the dif-ference between the telescope’s FOV and the laserdivergence �� 2. For the example given inFig. 6(a) the maximal inclination is equal to �� 0.375 mrad. Further increase of this parameterwill result in the exit of the beam from the telescope’sfull FOV. For extreme inclinations, the lidar signal is

�(R) ��0 �(R) s e(R)�2

�1(R) � sin��1(R)�s2 �2(R) � sin��2(R)�e2(R)

2�e2(R)|s � e(R)|�2 � �(R) � s e(R)�2

s2�e2(R) �(R) � [e(R) � s]�2, e(R) � s1 �(R) � [s � e(R)]�2, e(R) � s

. (18)

Fig. 6. GC form factor calculated for (a) biaxial and (b) coaxiallidar systems. Calculations performed for the telescope with a T� 400 mm diameter primary mirror and a � 1.25 mrad FOV forfour inclination angles, � � 0, 0.375, 0.8, and 2 mrad. Laser beaminitial shift, d0 � 900 mm; the beam divergence, ��2� 0.25 mrad.


always affected by geometrical compression. Such asituation can be observed in Fig. 6(a) for � � 2 mrad.

The laser beams of coaxial lidar systems should bealigned collinearly with the telescope axis. Any otherconfiguration shifts Rmin away from the receiver [Fig.6(b)]. Assuming the absence of a secondary mirror,the effect of obstruction of light at short distancesdoes not occur, permitting transmission of high-intensity light directly to the photodetector. This ide-alized situation illustrates the fact that coaxial lidarscan in general be much more affected by the problemof detector saturation that appears soon after thelaser is shot. Coaxial lidar can also encounter theproblem of direct light reflection at the exit of trans-mitting optics. To overcome both problems, a shorttube is usually mounted around the secondary mir-ror. This tube blocks high-intensity light scatteredclose to the receiver. Another solution is to apply agated photomultiplier.

For a real coaxial system, additional reduction ofthe detector’s irradiance at short distances will ap-pear. It originates from the secondary mirror’sshadow that forms in the focal plane. Its presencebecomes critical when the initial beam diameter ismuch smaller than that of a secondary mirror.12 Insuch a case the relevance of Eq. (18) can be reduced.

6. Comparison of Analysis with Measurements

To validate the usefulness of proposed model, wecompared the results of analytical calculations withthe measurements. The comparison encompassedthree different lidar systems: biaxial lidars at theFreie Universität Berlin21 (FUB) and the Centro La-ser of Bari22 (CLB) as well as the coaxial lidar at theInstitute of Experimental Physics, Warsaw (IEPW).23

The first of these systems was designed for the inves-tigation of tropospheric aerosols with a three-wavelength (355-, 532-, and 1064�nm) Nd:YAG laser.For the comparison we chose a signal of 355 nm. Thesecond lidar was a differential absorption systembased on a Ti:sapphire laser with nonlinear mixingbetween Ti:sapphire and Nd:YAG lasers, allowing forwavelengths to extend to the UV–visible region. Inthis case a recorded signal of 446 nm was employed.The last-named setup was a differential-absorptionlidar based on a tunable Ti:sapphire laser withsecond-harmonic and third-harmonic crystals. Forthe comparison ozone, �-off wavelength 286 nm was

used. Relevant parameters of each device are com-piled in Table 1.

The result of the comparison is shown in Fig. 7.Each continuous curve represents theoretical valuesthat were calculated by means of Eq. (18) and param-eters from Table 1. Experimental values of the GCform factor are plotted as open circles. They werereconstructed by analysis of the recorded lidar sig-nals. For the reconstruction we applied the methodproposed by Sasano et al.7

For each theoretical curve, laser inclination � wasdetermined by Eq. (6) and corresponded to the mea-sured full-overlap distance. Because all parametersof the receiver, as well as the distance between thelaser and the telescope together with initial beamdiameter, are well known or may be determined withhigh precision, the estimation of Rmin and laser diver-gence � becomes critical for evaluating the GC.

Consequently, the absolute error of � can be es-timated by means of the differential equationd� � |��2d0 g0 T��2�Rmin�2|dRmin 1�2d�.

Provided that the estimation of Rmin is better than101 m, the first summand is negligible for systemswith |��2d0 g0 T��2�Rmin�2| � 10�6 m (the errorof � usually does not exceed 10�3 rad). In such a casethe accuracy of the laser’s inclination is determinedby the accuracy of the beam divergence. If the accu-racy of Rmin is of the order of 101–102 m, both sum-mands contribute equally to the error of �. The lidarsat the IEPW and the CLB can be classified as systemsof this type.

For the FUB lidar, |��2d0 g0 T��2�Rmin�2|� 10�5 m, larger than that of the other two lidars.This value results from a relatively short 189 � 15 mfull-overlap distance, as opposed to 380 � 50 and946 � 100 m in the IEPW and CLB lidars, respec-tively. Nevertheless, for this device the higher valueof the ratio |��2d0 g0 T��2�Rmin�2| is compen-sated for by a small, only 15 m, inaccuracy of Rmin. Asa consequence, the first summands of the differentiald� become comparable for all systems.

For the comparison presented in Fig. 7 the accu-racy of the beam divergence for each lidar could notbe resolved. The only data available were the laserdivergence specified by each laser manufacturer (seeTable 1). For the IEPW lidar, original data were ad-

Table 1. Specifications of the Lidar Systems Used for the Validation Test

Location of Lidar System

Telescope Lasera

T �mm� f �mm� � (mrad, adjusted) g0 �cm� d0 �cm�

Freie Universität, Berlin, Germany 150 900 3.3 0.1 16Institute of Experimental Physics, Warsaw, Poland 400 1200 1.7 3 0Centro Laser, Bari, Italy 600 1600 1.3 0.8 28b

a� is �1 mrad in all-cases.bNote that, owing to the mechanical construction of the CLB lidar, the laser beam is sent from the primary mirror (d0 � T�2). This

solution ensures compactness for the system.


ditionally corrected by the beam expansion factor.Assuming that the error of the beam divergence is notlarger than 10% of the nominal laser divergence, wecan estimate the theoretical accuracy of the laser’sinclination �. Within 16% for the IEPW lidar, 11% forthe FUB lidar, and 15% for the CLB lidar. Thesevalues, however, strongly depend on d�. In any casethe precise estimation of the laser divergence caneasily reduce the error that arises from the uncer-tainty in the laser’s inclination, and it can thereforegreatly improve the accuracy of our method.

Analysis of Fig. 7 indicates excellent correspon-dence between theoretical and experimental GCcurves for the coaxial IEPW lidar. For this device thelargest difference occurs during the first 150 m, dur-ing which calculations do not take the shadow of thesecondary mirror into account. We have already men-tioned that this source of error is particularly impor-tant if the beam diameter at the output of thetransmitting optics is much smaller then that of thesecondary mirror. Such, however, is not the case forthe IEPW lidar, for which an additional beam ex-pander matches the beam’s cross section to the sec-ondary mirror. As a result, the estimation of theshadow effect, which can be made by use of simplegeometrical optics, leads to an error of 1–2% for thedistances from 10 to 100 m. Such a small error cannotfully explain the discrepancy that is apparent in thefigure. Additional reduction of the recorded signalmust be attributed to mechanical blocking of incom-ing light inside the telescope; as mentioned above,this prevents photomultiplier saturation at short dis-tances. In such a case Eq. (18), which is valid only forideal receivers, cannot truly reproduce the experi-mental data. For distances longer than 150 m theshape of the measured GC function was excellentlyreplicated, proving that our model can be successfullyemployed for simulations of the GC.

The result of the comparison for biaxial lidars(FUB and CLB) is also satisfactory [Figs. 7(b) and7(c)]. For these lidars, disagreement between themeasured and the calculated GCs is not more than10% and can be explained by an assumed uniformenergy distribution within the laser beam. Such anassumption leads to a sharp, steplike change in theGC form factor in the region in which the beam entersthe telescope’s FOV (and then completely the fullFOV) and cannot be observed for the experimentallymeasured GC. This result suggests that both lidarsare characterized by a Gaussian profile of the laserbeam. Such a conclusion is in good agreement withthe theoretical prediction of Sassen and Dodd.13

Comparing the curves in Figs. 7(b) and 7(c) alsomakes evident that the FUB lidar, which has a tele-scope with a small 15-cm primary mirror and a large3.3-mrad FOV, is much better for the measurementsat short distances than is the CLB lidar. We must not,however, forget that the CLB lidar uses 60 cm with a2.5-times smaller telescope FOV. A large primarymirror ensures high brightness of the telescope,which makes it more suitable for measurements atlonger distances. Moreover, for this telescope the ini-tial distance between the laser beam and the tele-scope axis is 28 cm (for the FUB lidar it was only12 cm), which according to Eq. (6) leads to a furtherincrease in the full-overlap distance.

7. Conclusions

We have demonstrated that a simple model of imageformation in the focal plane of a Newtonian telescopewithout a central obstruction is appropriate for cal-culating an analytical function to describe a lidargeometrical form factor. Our function is valid for both

Fig. 7. Calculated (continuous curves) and measured (open cir-cles) GC form factors for (a) the coaxial IEPW lidar, (b) the biaxialFUB lidar, and (c) the biaxial CLB lidar. The spike at 100 m for theIEPW lidar is due to the photomultiplier afterpulse. Small (notmore than 10%) differences between curves for biaxial lidars can beattributed to an assumed uniform energy distribution within thelaser beam.


coaxial and biaxial lidar arrangements and uses vari-ables to which one can gain access by simple analysisof the lidar signal. In comparison with formulas re-ported by other authors, it does not contain any freeor fitting parameters and can be applied in a straight-forward manner to designing optical receivers or tosimulating lidar signals. A comparison of calculationswith experimentally measured GC form-factor curvesdemonstrated satisfactory agreement in each of thethree lidar setups.

Based on our model, we have examined the prob-lem of lidar receiver arrangement. We found that, toincrease the effective measurement range at shortdistances, the two lidar types must be aligned differ-ently. For the coaxial lidars, the telescope and laserbeam axes should be parallel, whereas for biaxiallidars these two axes should be slightly inclined, al-lowing for better overlap between receiver and emit-ter.

We have also shown that the classical definition ofa telescope’s field of view leads to large systematicerrors when one is calculating the full-overlap dis-tance [i.e., the distance from which ��R� � 1]. Toovercome this problem we proposed to replace theclassical telescope’s FOV with a new quantity, whichwe call the telescope’s full FOV. To the best of ourknowledge, the definition of a telescope’s full FOV hasthus been introduced for the first time.

Kamil Stelmaszczyk acknowledges the support ofthe Alexander von Humboldt Foundation. MarcellaDell’Aglio thanks Centro Laser of Bari for access tothe lidar system facilities.

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Analytical function for lidar geometrical compression form-factor calculations