PROCEEDINGS OF THE IRE

Range Equation for Passive-Infrared Devices*LEWIS LARMOREt

pHE solution to the problem of determining themaximum range at which an infrared device can

detect a target requires a detailed analysis of thespectral radiance properties of the target, the absorp-tion in the intervening space, the characteristics of theradiation-gathering system, and the response of thedetector. In this section, the general form of the passive-range equation is developed, while a detailed discussionof the parameters is included elsewhere. For example,target emission characteristics, atmospheric attenu-ation, and detector parameters are given in Section 3.Radiation collection systems and background charac-teristics are discussed in Section 4.

Radiation emitted by a target may have the spectralcharacteristics of a blackbody modified by the emis-sivity of the material; it may have those of a gaseous

emission spectrum; or it may have a combination of thetwo. In any case, the spectral radiant emittance can

be represented by

Wx=Ex(T)e(X) watts cm-2*micronr1 (1)

where Ex(T) is the spectral radiant emittance at wave-

lenigth X of a blackbody at temperature T, aiid e(X) is

the emissivity of the material at wavelength X. In mostsituations, however, the target cannot be characterizedby a single temperature and values of Wx must be ob-tained either from actual measurements or from an

assumed temperature distribution over the target. Inaddition, the values of E((X) may change with tempera-ture also, but this variation is not sufficiently large towarrant a detailed investigation.At a distance R, which is assumed large compared

with the source dimension, the spectral irradiance at theaperture of the infrared-detecting system is

HX =WxA T(X) watts cm-2 micron-'

irR(2)

where A is the projection of the target area and r(X) isthe spectral transmittance of the intervening medium(usually air) between the target and receiving aperture.Note that the value of the spectral tranismittance,r(X), is a function of the range.

In order to determine the amount of radiant power

which actually falls on the detector, one must know thedetails of the detection system. However, these detailscan be represented by our making the assumption of a

spectral transmittance for the system and multiplying

* Original manuscript received by the IRE, June 26, 1959.t Lockheed Aircraft Corp., Burbank, Calif.

it by the area of the aperture. Thus, the radiant power

at wavelength X falling on the detector is

sx = Hxaro(X) watts (3)

where a is the aperture area and ro(X) is the spectraltransmittance of the detecting system.The response of most detectors is wavelength depend-

ent, and they are unable to utilize all of the power whichis incident on them. In order to allow for these response

characteristics, we usually introduce the relative spectralresponse of the detector :(X) and apply it to (3). Hence,the effective radiation at wavelength X falling on thedetector becomes the signal to the detecting system fromthe target and is defined by

Sx = Hxaro(X) 2(X) watts. (4)

The task now remains of putting together all of theserelations and performing an integration over all wave-

lengths. Thus, the total signal provided to the detectionsystem is

r X WAT(X)aTo(X) 2;())dXS = -- watts

Jo wR' (5)

where all terms have been previously defined.In order to have an operating system which will de-

tect a target, we must have two additional pieces ofinformation: 1) the signal-to-noise ratio which can betolerated, and 2) the value of the noise introduced intothe system either by the background radiation, theelectronics, or the detector itself. When the noise prob-lem is introduced into (5), we have

S= p =

1V

Aa rt2 WXT(X)TO(X) 2(X)dX.

irR N(6)

As a rule of thumb, S/N values of unity can be handledfor a tracking system while S/N values of from three tofive are typical for scainning systems.Only in special cases can (6) be solved for the range

since the atmospheric transmittance is also a function ofthis parameter. Thus, graphical or numerical methodsmust be used in general. However, if only narrow win-dows of the atmosphere are used, or if, as in outer space,

there is no attenuation, can we approximate a solution.In these cases the transmittance of the interveningmedium can be removed from under the integral signand a constant value assigned to it, and the maximumdetection range becomes

Paper 3.4.1

1959 1489

PROCEEDINGS OF THE IRE

/TAa "OR T WxTo(X) 2(X)dX (7)

where p is the required signal-to-noise ratio. TJhe noise,N, which limits the performance of the system is frequently one of the most difficult parameters to evaluate.Operation under daytime conditions usually leads to theconclusion that the background radiation furnishes thelimiting noise, while night-time operations are limitedby detector noise. A further discussion of these factors

Paper 3.4.2

is included elsewhere in this text.QAlthough the wavelength inteival for integration in

(6) anld (7) is indicated to be O toco, inm piractice, somie ofthe quantities under the integral are zero except over a

fairly narrow wavelenigth region Sinice the integrationmust be donie either numierically or graphically, it isimportant to exami-ine the values beforehanid in orderto limit the amiount of work as imiuch as possible.

I See R. H. Genioud, "Infrared search-syste ii range perforniance."'Paper 4.4.5, this issue, p. 1581L

Range Equation for Active Devices *KENNETH V. KN-IGHTt

A CTIVE inifrared anid optical raniging can be ac-complished with the use of pulsed radiation in amanner anialogous to that of radar in pulsed RF

techniques. A very-high-intensity lamp at the focus of acollimating milrror is caused to flash, and the eniergyreflected from the target is collected in another miirrorand focused upon a photodetector. The elapsed timlebetweeni the emi.ission of the pulsed light anid the detec-tionl of its reflection from the target is, as in radar, ameasure of the range to the target. This time cani bemeasured on the face of ani oscilloscope whose sweepis triggered with the firing of the flash lamp, or it canbe measured and delivered as an analog voltage bymore sophisticated automatic electroniic techniqujes.Gating and integration techniques can be used to enhanice performanice wheini signal-to-noise ratios are low.

Greater ratige performance can presently be obtainedusing emission of flash lamps in the visual optical regionand using photodetectors senisitive in this region. How-ever, if it is desirable to operate totally within the in-frared spectrum, flash lamps equipped with in-fraredfilters can be employed together with suitably fastinfrared photodetectors.

Farrand Optical Company, Inc., New York, N. Y,has been a leader in. developing these techniques andhas applied them to a wide variety of classified militaryequipmenits.

Although pulsed-light (visible or infrared) rangiing ispresently limited in comparison with pulsed radar,

*Original mantiscript received by the IRE, June 26, 1959.t Ramo-Wooldridge Corp., a division of Thompson Ramio-

Wooldridge Inc., Los Anigeles, Calif.

there are niarly attractive features to reconeiiiiend itsconsideration- where lonig-range performance is niot re-iquired. Several of these are noted. the transmittei fieldof view canl be very small and the receiver field of viewcan be very accurately defined and made as small as oniepleases; consequently, the troublesome ground clutterof radar is n-iot experieniced in operati;ng grounidtoground. Side and rear radiationi lobes are not present.Low power, size, and weight requirements fIurther comi-mend the techniques. High-range accuracy and rangeresolution are attainable with less difficulty than inRF technology. Because of high directivity anid operation outside the RF spectrum, pullsed-light techiniquesare less susceptible to detection and jamnmin-ig thani areRF techniques. Raiigefinders employinig thLe techniiqueare rugged and require little optical adjiustlment Asso-ciated electronic circuits are usually relatively silple.The basic range equatioiis are also simnple.We recall from-n elementary physics that illumination

uponi a scenie varies inversely with the square of the dis-tanice, R, from the illuminatineg source. If we place acollimating mirror behind a source in such a way thatthe target "sees' the iiirror full of light, we have thefam-aiiliar relationship

BsDT21T = (1

(igniorinig atmospheric attenuation) whereIl irradiance of the target,Bs-radiance of the source, an-idDT-=diameter of the transmitting rriirmor

1490 o'eptfember

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