optimal bandwidth for topographical differential absorption lidar detection

Optimal bandwidth for topographicaldifferential absorption lidar detection

Avishai Ben-David

A detected laser signal backscattered from a tilted target is modeled with a laser-pulse shape as aresponse of a high-pass filter to an exponential input that describes the gain buildup within the lasercavity before a laser pulse is emitted and a single-pole low-pass RC filter for the electronicamplifier. The model is used to maximize the signal-to-noise ratio of the detected peak signal with aproper choice of the integration time constant t as a function of the laser-pulse shape and the tilt angle ofthe backscattering target.Key words: Lidar, differential absorption lidar 1DIAL2, signal-to-noise ratio, detection. r 1996

Optical Society of America

1. Introduction

In differential absorption lidar 1DIAL2 detection,when the return signal is backscattered from atopographic target, the peak value of the detectedsignal is used in the DIAL analysis and the signal-to-noise ratio 1SNRp2 is determined by the ratio of thepeak value of the detected backscattering signal 1V02to all noise sources. The backscattering signal levelis usually too low to be measured, and an amplifier isneeded to increase the return-signal level to a mea-surable quantity 1millivolts2. The main contribu-tions of noise in CO2 incoherent detection are fromnon-signal-noise currents1 such as a detector darkcurrent, an amplifier-noise current, and a back-ground-noise current. All these noise sources de-pend on the square root of the effective electronicbandwidth used for the detection. As the band-width f increases, V0 increases and so does the noiselevel. The rate of increase of the noise level and thedetected peak signal determine the maximum pos-sible SNR as a function of the bandwidth f. Theoptimal choice of bandwidth for which a maximumSNR is obtained depends on the signal temporalpulse shape and the temporal broadening of thebackscattered signal that results when the target isnot perpendicular to the incidence. For an airbornescanning lidar the angle between the incidence and

The author is with Science and Technology Corporation, 500Edgewood Road, Suite 205, Edgewood, Maryland 21040.Received 23 May 1995; revised manuscript received 20 Septem-

ber 1995.0003-6935@96@091531-06$06.00@0r 1996 Optical Society of America

the target’s normal may vary as a function of thescanning and the flight geometry. Therefore aproper choice of bandwidth and the use of a variablebandwidth may increase the SNRp of the lidarmeasurements. In this paper, the optimal band-width is given as a function of the laser-pulse shapeand the target’s tilt angle to the laser incident beam.The effective bandwidth of two low-pass filters isgiven by2 f 5 f1 [email protected] f12 1 f2221@24, where f is the23-dB high-frequency cutoff of the two components,one of which can be the detector and the other theelectronic amplifier. The lower-frequency cutoff isusually set to a few hundred hertz to avoid DCfluctuations and 1@f noise contribution. Thus thebandwidth of the amplifier is approximately itshigh-frequency cutoff f.Different low-pass filters 1e.g., Butterworth, Che-

byschev, Bessel2 can be designed3 such that thedifferences among them are the slope steepness ofthe frequency roll-off response beyond the 23-dbhigh-frequency cutoff, the flatness of the frequencyresponse within the passband bandwidth, and thetime delay for the different frequency componentscomprising the input signal. When the roll-off re-sponse beyond the 23-db-frequency cutoff is smooth,the time delay is small; as the filter response be-comes steeper for frequencies beyond the bandwidth1i.e., approaching an ideal filter2, the time delayincreases; the output signal is shifted in the timedomain, together with the distortion of the pulseshape that is due to the filter’s spectral attenuationof the different signal-frequency components. Inthis study, a simple single-pole RC low-pass filter,composed of one resistor and one capacitor, is used,

20 March 1996 @ Vol. 35, No. 9 @ APPLIED OPTICS 1531

for which a simple analytical expression can beobtained and, as is shown, provides a reasonableapproximation to the more complicated seventh-order Bessel amplifiers used in the lidar system.The roll-off slope beyond the 23-db cutoff frequencyis less steep for a simple RC filter 1i.e., a first-orderButterworth2 than for the seventh-order Bessel, andthus the output signal from the RC filter will belarger because high-frequency components of thesignal beyond the 23-db amplifier cutoff frequencywill be less attenuated. In addition, the time delayincreases as the order of the filter increases.

2. Laser-Pulse Shape

A gain-switch or a Q-switch laser can be viewed as aprocess for which a capacitor stores energy while apopulation inversion of excited atoms at a higherenergy level is built up in the active medium 1e.g., agas mixture He:CO2:N2 for a CO2 laser or a semicon-ductive material such as neodymium-doped yttriumaluminum garnet in a Nd:YAG laser2. When theexcited atoms return to the ground level, opticalradiation is emitted as a laser pulse. The storedenergy in the population inversion can be releasedspontaneously 1gain switch2 or by an electro-opticQ-switch device. This process can be simulated asthe response of a high-pass filter to an exponentialinput 1 2 exp12t@t12 for which the laser pulse VL1t2 ~exp12t@t22 2 exp12t@t12. The time constant t2 5 RCof the resistor–capacitor high-pass network is re-lated to the fall time of the laser pulse, and the timeconstant t1 is related to the rise time of the pulse.These time constants are a few nanoseconds in aNd:YAG laser and a few tens of nanoseconds in a CO2laser. In a CO2 gas laser, the pulse shape is charac-terized by an additional time constant t3 that resultsin general from the N2 gas and produces a long tail 1afew microseconds2, although the long pulse tail al-ways exists and can be seen in a log scale even ifthere is no nitrogen.4 Thus the falling-edgeexp12t@t22 laser-pulse shape is divided into two com-ponents, a exp12t@t22 and 11 2 a2exp12t@t32 and thelaser-pulse shape is given by

VL1t2 5 a exp12t@t22 1 11 2 a2exp12t@t32 2 exp12t@t12,

112

where t1 , t2 , t3. The peak value of VL1t2 is at t 5tL, where tL 5 t1t21t2 2 t1221ln1t2@at12; is exact for a 51; and for a . 0.8 is within a few percent of the peak.Empirical analytical forms for a CO2 laser-pulse

shape are given by Zhao et al.,4 in which the pulseshape is represented with six free parameters andequations for a short and a long time interval, and byGurdev et al.,5 who used a continuous analyticalform containing three free parameters. Both formscannot be integrated analytically, and thus it isadvantageous to use Eq. 112, which has a simpleanalytical form, can be easily integrated, and quali-tatively relates to the lasing process. An example ofthe 10R20 CO2 laser line measured with a photocon-

1532 APPLIED OPTICS @ Vol. 35, No. 9 @ 20 March 1996

ductive HgCdTe detector 1time constant of 7 ns2 and aseventh-order Bessel 44-MHz bandwidth amplifieris given in Fig. 11a2 with a simulated laser pulse VL1t21a 5 0.91, t1 5 35 ns, t2 5 60 ns, and t3 5 800 ns2,and a simulated short pulse 1a 5 1, t1 5 3 ns, andt2 5 4 ns2 typical of a Nd:YAG laser is shown in Fig.11b2. The bandwidth of the CO2 laser pulse is ap-proximately 10 MHz.

3. Detected Pulse Shape

All photons transmitted within a time interval Dt 52U L tan1c2@C, where L is the distance to the target,C is the speed of light, U is the incidence full-widthbeam divergence 1i.e., UL is the diameter of theilluminated spot size on a perpendicular target at adistance L2, and c is the angle between the targetnormal and the incidence light, will arrive at thedetector at the same time. This time interval Dt is2DL@C, where DL is the projected length of theilluminated spot size on the incidence direction.The broadened pulse shape Vi1t2, which is incidentupon the detectorwhen backscattered fromaLamber-tian tilted target with reflectivity r 5 r0 cos c, wherer0 is the Lambertian reflectivity for a perpendicularincidence, is given by

Vi1t2 5r

Dt e0

t

VL1t82dt8 t , Dt,

Vi1t2 5r

Dt et2Dt

t

VL1t82dt8 t $ Dt. 122

Fig. 1. 1a2 CO2 laser pulse 110R202 measured with a 44-MHzamplifier and a photoconductive HgCdTe detector 1time constantt 5 7 ns2 and a laser pulse VL simulated with Eq. 112 1a 5 0.91,t1 5 35 ns, t2 5 60 ns, t3 5 800 ns2. The pulse bandwidth isapproximately 10 MHz 1t 5 16 ns2. 1b2 Simulated short pulse1a 5 1, t1 5 3 ns, t2 5 4 ns2 typical of a Nd:YAG laser. The pulsebandwidth is approximately 160 MHz 1t 5 1 ns2.

As the target angle c approaches zero, the tempo-ral broadening Dt approaches zero and Vi1t2 backscat-tered from the perpendicular target converges to theincident laser-pulse shape VL1t2. The pulse shapeV01t2 detected by a receiver with a time constant t 5RC for which the rise time tr 5 2.2 t 5 0.35@f and f 51@2pt is the 23-db high-frequency cutoff backscat-tered by a Lambertian target at an incident angle c,is given by

V01t2 5 t21 e0

t

rVi1t82exp321t 2 t82@t4dt8. 132

The incident input pulse shape Vi1t2 and the outputdetected pulse shape V01t2 can be easily integratedanalytically with the laser pulse VL1t2 given by Eq. 112and are not given explicitly. The change in thedetected signal magnitude that is due to the varyingtarget distance within the illuminated spot thataffects the field of view 1~1@L22 and the atmospherictransmission are ignored, and only the detectedpulse shape V01t2 is considered.The peak value of the incident pulse Vi1t2 on the

detector in Eq. 122 can be computed within 10% forVi1t 5 ti2, where ti 5 31tL2 1 Dt221@2 1 tL 1 Dt4@2. Thepeak value of the detected pulse V01t2 in Eq. 132 isgiven within 10% by V01t 5 to2, where to 531ti2 1 t221@2 1 ti 1 t4@2 for t , 2 t2 if a 5 1 3i.e., there isno long N2 tail in the laser pulse VL1t2 in Eq. 1124 andfor t , t3 if a fi 1. When the integration time t islong, the peak value of V01t2 is given within 10% byV01t 5 to2, where to 5 t2t1t 2 t2221ln1t@t22 for t . 2t2and a 5 1, and at to 5 t3t1t 2 t3221ln1t@t32 for a fi 1and t . t3. When the temporal broadening Dt fromthe tilted target is very large, Dt : t2 for a 5 1 andDt : t3 for a fi 1, the input pulse Vi1t2 can beapproximated with a square pulse of width Dt, andthe peak value of the detected pulse is at t 5 Dt,V01Dt2 5 53t3 2 t1 2 a1t3 2 t224Dt21631 2 exp12Dt@t24, inwhich the first term on the right-hand side is thetotal energy in the transmitted laser pulse dividedby the temporal broadening Dt 1i.e., the average peaksignal2 and the second term is the effect of the RCintegration time constant of the amplifier.The transmitted laser pulse and the received laser

pulse 1measured with a cooled photovoltaic HgCdTedetector of 5–8-MHz bandwidth2 from a hard targetat an angle near perpendicular to the incidence 1i.e.,c = 02 were measured with a seventh-order Besselamplifier of 5-, 3-, and 1-MHz bandwidths. Thepeak signal was reproduced within 20% with Eqs. 122and 132. The temporal pulse shape was reproducedwell with the 5- and 3-MHz amplifier bandwidths,with the exception of a very long time delay 1hun-dreds of nanoseconds2 for the peak signal that wasobservedwith the 1-MHz seventh-order Bessel ampli-fier bandwidth and was not reproduced in the simu-lation.

4. Signal-to-Noise Ratio

The SNR of the detected backscattered signal V01t2 isdefined as the ratio of the peak of the mean signal

current 1proportional to the incident optical power inwatts2 to the overall standard deviation of the fluctua-tions in the non-signal-noise current 1proportional tothe noise-equivalent optical power in watts2. WhenPoisson statistics are used, where the variance of thefluctuations is equal to the sampled average, allnon-signal-related noise sources such as dark-current noise, background-noise current, and ampli-fier-noise current contribute a noise current that isproportional to 1@Œt 1i.e., proportional to Œf 2. Thusthe SNR is proportional to V01t2Œt, for which themagnitude of V01t2 depends on t and on the temporalbroadening Dt that is due to the tilted target. Aspeckle noise that is important in CO2 DIAL detec-tion, is inversely proportional to the square root ofthe number of coherent speckle cells in the receiveraperture6 and is proportional to the signalmagnitude.Thus the SNR that is due to speckle is a constantthat depends on parameters such as wavelength, thebeam-divergence, receivers, aperture, and distanceto the target, but does not depend on t.The maximum value of the detected signal V01t2

increases and approaches the maximum value of theinput signal Vi1t2 as the integration time constant tdecreases. However, as the maximum value of thedetected signal increases, so does the noise increase,because a higher bandwidth is required for thedetection. As t increases, the noise decreases 1alower bandwidth is required2, and the maximumvalue ofV01t2 also decreases. Themaximum value ofthe detected peak signal SNRp ~ V01t2Œt is approxi-mately 1i.e., within 15%2 at t 5 t1 1 t2 1 ti and can beeasily computed from the maximum value of V01t2 inEq. 132. For a square pulse of width Dt, it can beshown that the peak output signal from a low-passfilter 1for a unit input signal2 is at t 5 Dt and of avalue 1 2 exp12Dt@t2. A matched filter with abandwidth f 5 1@2Dt and thus t 5 0.3182Dt resultsin a peak output signal of 95.7%. However, themaximum value of SNRp of the peak output signal isfound for t 5 0.7959Dt, for which the peak outputsignal is only 71.5%, resulting in the improvement ofSNRp by a factor of 1.182.SNRp 5 V01t2Œt, assuming a noise value of 1 at t 5

1 ns, and the detected peak signal V0 for a laser pulseof a unit peak value backscattered from a Lamber-tian target, with reflectivity r 5 1, perpendicular tothe incident laser pulse 1i.e., at c 5 02 are shown inFigs. 21a2 and 21b2, which correspond to the laserpulses shown in Figs. 11a2 and 11b2, respectively.Figures 21a2 and 21b2 show that although the peakdetected signal decreaseswith increase of the integra-tion time t, the SNRp reaches a maximum value foran optimal value of t. The improvement of SNRp forthe detected peak signal with an optimal choice ofthe integration time constant t is shown in Fig. 3 fora laser pulse of a unit peak value backscattered froma Lambertian target with reflectivity r 5 1, at adistance L 5 10 km and at an angle c. For the CO2laser pulse shown in Fig. 11a2, the beam divergence is2.8 mrad, and for the short pulse it is 0.5 mrad,


which is typical of a Nd:YAG laser pulse 3Fig. 11b24.Figure 3 shows the temporal broadening Dt from thetilted target and the optimal time constant t tomaximize the SNRp for backscattering from thetilted target, as well as the improvement of the SNRpand the reduction of the peak value of the detectedsignal V0 when the integration time constant t isused over the SNRp and the peak detected signalwhen integration time t 5 16 ns 1i.e., a 10-MHzbandwidth, which is approximately the bandwidth ofthe CO2 laser pulse2 is used in Fig. 31a2, and t 5 1 ns1i.e., a 160-MHz bandwidth, which is approximatelythe bandwidth of the Nd:YAG laser pulse2 is used inFig. 31b2. For the Nd:YAG simulated laser pulse, V0is 0.955 for t 5 1 ns and c 5 0 3Fig. 21b24 and for theCO2 laser pulse, the peak detected signalV0 is 0.9475for t 5 16 ns and c 5 0 3Fig. 21a24. Therefore theimprovement of peak SNR with the optimal choice ofintegration time is SNRp 1t 5 148.2 ns2@SNRp 1t 5 16ns2 5 1.58 for the CO2 pulse and SNRp 1t 5 10.45ns2@SNRp 1t 5 1 ns2 5 1.52 for the Nd:YAG pulse.

Fig. 2. SNRp 5 V01t2Œt, assuming a noise value of 1 at t 5 1 ns,and the detected peak signal V0, for a laser pulse of a unit peakvalue backscattered from a Lambertian target, with reflectivityr 5 1, perpendicular to the incident laser pulse 1i.e., atc 5 02. 1a2Sim-ulated CO2 laser pulse shown in Fig. 11a2. 1b2 Short laser pulsetypical of a Nd:YAG laser pulse as shown in Fig. 11b2.


The effect of the change in the field of view 1~1@L22within the illuminated spot on the target is less than5% for c , 80° and a beam divergence U , 3 mrad.The figures show that as the target’s tilt angle cincreases the temporal broadening Dt increases, andthus a lower bandwidth 1i.e., a higher t2 can be usedto maximize the detected SNRp while the detectedpeak V0 is reduced. In a scanning lidar, in whichthe target’s tilt angle changes, using a variablebandwidth 1i.e., an optimum t2 may improve theSNRp of the detected peak signal V0 compared withthe SNRp for a perpendicular target 1c 5 02 for whicha higher bandwidth 1i.e., a lower t2 is required in thedetection of the maximum signal V0.As c increases, V01t2, V0 1t 5 1 ns2, SNRp1t2, and

SNRp 1t 5 1 ns2 all decreasemonotonically, andV01t2 ,V0 1t 5 1 ns2 and SNRp 1t2 . SNRp 1t 5 1 ns2 alwaysoccur. However, as c increases, the temporal broad-ening Dt also increases, and the backscatteringsignal from the tilted target approaches the shape ofa square pulse for which the detected peak signal is1laser-pulse energy2 Dt2131 2 exp12Dt@t24 1see Section

Fig. 3. Optimal integration time t for maximizing the SNRp of adetected peak signal, improvement of SNRp, reduction of thedetected peak value when the integration time t increased from areference integration time t 3t 5 16 ns in 1a2 and t 5 1 ns in 1b24 tothe optimal integration time t, as well as target temporal broaden-ing Dt as a function of the target tilt angle c. 1a2 Simulated CO2

laser pulse from Fig. 11a2 with a unit peak value with a full-widthbeam divergence U 5 2.8 mrad and a target distance of L 5 10km. For t 5 16 ns and c 5 0 the peak detected signal V0 1t 5 16ns2 5 95% of the peak incident laser pulse VL. 1b2 Short laserpulse typical of a Nd:YAG laser pulse as shown in Fig. 11b2 with aunit peak value with a full-width beam divergence u 5 0.5 mradand a target distance of L 5 10 km. For t 5 1 ns and c 5 0 thepeak detected signal V0 1t 5 1 ns2 5 95% of the peak incident laserpulse VL.

32. Therefore, for a small t, the detected peak signalincreases inversely with Dt, whereas for a large t, thedecrease of the detected peak signal magnitude isslower because of the integration effect of the RClow-pass filter, and a deflection point in the plotV01t2@V0 1t 5 1 ns2 of Fig. 3 is observed.For a square-law detector, the detected current 1or

voltage2 V01t2 is proportional to the incident opticalpower 1in watts2 in the laser pulse. The SNRE of thetotal detected energy 1in joules2 is given by the ratioof the signal area 1in joules2 to the noise area 1i.e., thenoise-equivalent energy in joules2:

SNRE 5

e0

T

V01t2dt

VnT, 142

where Vn is the mean noise level in the detection ofV01t2. When T equals the largest of 35t2, 5t, 5Dt4 fora 5 1 and the largest of 35t3, 5t, 5Dt4 for a fi 1,

e0

`

VL1t2dt 5 e0

T V01t2

rdt 5 t3 2 t1 2 a1t3 2 t22, 152

within 5% 1the effects of distance to the target andatmospheric transmission are ignored2. The meannoise level 1usually defined in terms of a root meansquare of the fluctuations2 is proportional to 1@Œt,and thus the SNRE for the total detected energy isSNRE ~ 3t3 2 t1 2 a1t3 2 t224T21rŒt. The SNRp ofthe detected peak power is SNRp ~ V01t02 Œt, and theratio SNRE@SNRp is given by

SNRE

SNEp5

e0

T

V01t2dt

TV01t025

r e0

`

VL1t2dt

TV01t02

5 rt3 2 t1 2 a1t3 2 t22

TV01t02. 162

Using the mean value theorem, 11@T2 e0T V01t2dt #

V01t02, one can easily see that 1SNRE@SNRp2 # 1.This upper limit of the ratio applies to any fraction ofthe detected energy within the time interval 3a, b43i.e., the integration limit 3a, b4 is substituted for 30, T4and 1b 2 a2 is substituted for T in Eq. 1624. Themaximum detected signal V01t02 # VL1tL2, and there-fore the ratio SNRE@SNRp is bound by

r1laser-pulse energy2

T 1laser-pulse peak power2#SNRE

SNRp# 1, 172

where the peak power can be computed with Eq. 132and the pulse energy is given by Eq. 152. It should benoted, however, that although SNRP . SNRE, it maybe advantageous to process the integrated backscat-

tering energy and not the backscattering peak powerin DIAL analysis when the target backscatteringfluctuates because of wind or surface nonuniformity.

5. Summary

For a lidar system such as a CO2 lidar, for which themain source of noise is from non-signal-noise cur-rents, the SNRp of the detected peak signal isproportional to V01t02Œt, where V01t02 is the peak valueof the detected signal, which depends on the effectiveintegration time constant t of the system, the tempo-ral laser-pulse shape, and the temporal broadeningthat is due to the tilt angle of the backscatteringtarget. As t increases, the noise decreases, but sodoes the peak value of the detected pulse V0, andtherefore a proper choice of the integration timeconstant t of the lidar system will improve the SNRpof the detected peak signal. For an airborne scan-ning lidar the angle between the incidence and thetarget’s normal may vary as a function of the scan-ning and the flight geometry. Therefore a properchoice of bandwidth 1i.e., t2 and the use of a variablebandwidth may increase the SNRp of the lidarmeasurements.The detected laser signal backscattered from a

tilted target is modeled with a laser-pulse shape as aresponse of a high-pass filter to an exponentialinput that describes the gain buildup in the lasercavity before a laser pulse is emitted and a single-pole low-pass RC filter for the electronic amplifier.The model is used to maximize the SNR of thedetected peak signal with a proper choice of theintegration time constant t 5 t1 1 t2 1 ti, where ti isthe location of the peak signal backscattered fromthe tilted target, ti 5 31tL2 1 Dt2211@22 1 tL 1 Dt4@2, tL 5t1t21t2 2 t1221ln1t2@at12 is the location of the peaktransmitted laser pulse, Dt 5 2uL tan1c2@C is thetemporal broadening that is due to the tilted targetat an angle c at a distance L, and 1t1, t2, t3, a2characterizes the laser-pulse shape and its full-width beam divergence U. These results do notinclude the effects of atmospheric transmission andthe variation of the field of view, ~1@L2, within theilluminated spot on the target and are good approxi-mations for c , 80° and a beam divergence U , 3mrad.

This work was supported by the U.S. Army Edge-wood Research, Development & Engineering Center,Aberdeen Proving Ground, Maryland, under grantnumber DAAAM01-94-C-0079. The author thanksSteven Gotoff, Jeff Ahl, Lou Klaras, David Cohoon,and especially Norman Green for useful commentsand discussions.

References1. R. H. Kingston, Detection of Optical and Infrared Radiation

1Springer-Verlag, NewYork, 19782.2. J. Millman and H. Taub, Pulse, Digital and Switching Wave-

forms 1McGraw-Hill, NewYork, 19652.


3. A. Zverev, Handbook of Filter Synthesis 1Wiley, New York,19672.

4. Y. Zhao, T. K. Lea, and R. M. Schotland, ‘‘Correction functionfor the lidar equation and some techniques for incoherent CO2

lidar data reduction,’’Appl. Opt. 27, 2730–2740 119882.


5. L. L. Gurdev, T. N. Dreischuh, and D. V. Stoyanov, ‘‘Deconvolu-tion technique for improving the resolution of long-pulselidars,’’ J. Opt. Soc. Am.A 10, 2296–2306 119932.

6. J. W. Goodman, Statistical Optics 1Wiley, New York, 19852,Chap. 9.

optimal bandwidth for topographical differential absorption lidar detection

Documents