NARC_ΔT Definition

“NARC_ΔT” is the abbreviation adopted here for “narcissus noise equivalent temperature difference”, the  increment of temperature change in a camera scene which produces a  detector signal equal to the electrical  noise in the video signal processing electronics. It is a key figure of merit in infrared imaging systems.

The origin of the  narcissus phenomenon in infrared cameras and  the means to analyze it  are presented in the link to the parent site for this web page. It is shown there that the objectionable “cold spot” in the center of the displayed image is due to ghost image reflections off the lens surfaces of the objective lens.

The relative  distribution of the narcissus irradiance on the detector is established by comprehensive tracing of rays which originate at the detector, travel through the lens to the reflecting surface of the lens  and then are transmitted back through the lens to the detector. There is yet the  need to express this narcissus irradiance distribution on the detector in terms of NARC_ΔT.  This is the subject addressed in the following matter..


The model illustrated  below can be referred to in the following discussion on  NARC_ΔT. The camera scene is a radiating blackbody source comprising elements of different temperature or emissivity. In this analysis, for simplicity,  only temperature variations are addressed.

The lens comprises three infrared transmitting lens elements, a  housing and lens mounts. The lens housing and cells are assumed to be diffusely reflecting  “gray body” radiators, i.e,,their. emissivity is less than unity as it is for a pure blackbody. Their temperature is assumed to be 300 degrees Kelvin, a commonly accepted value for objects near room temperature..  

The lenses are typically fabricated from germanium, silicon, zinc selenide or other materials with good transmission in the infrared spectrum, commonly the 3-5 micron and 8-12 micron wavebands. Aniti-reflection coatings designed for the waveband of interest are applied to all lens surfaces.

The lens images  the scene onto a detector located within the vacuum environment of a dewar. The detector is an array of photo-sensitive cells ,the array being  typically 12 x 9 mm in size. The pixel cells  within the array are spaced at a pitch of five to ten microns. The detector  is cooled to a temperature near 77 degrees Kelvin, the temperature of a cold source applied to the rear surface of the detector. The dewar also houses a spectral “cold” filter with aperture serving as the system aperture “cold” stop in this case. A window seals off the dewar, maintaining the vacuum .and cold interior .allowing photon detection sensitivity by the detector..

Camera Signal and Background Signal

The detector receives radiation emitted from both the scene and a background. The background consists of  radiation emitted from the interior walls of the lens housing which is  reflected from  lens surfaces. back to the detector.

The basis for the following analysis of NARC_ΔT is the Planck function for  blackbody radiation emittance. Radiant emittance W is a function of temperature T  and wavelength λ according to Planck’s  law:

where                  Wλ = spectral radiant emittance, watts / cm/ μ

λ = wavelength, μ

T = absolute temperature, °K

C1 = 2πhc2 =  first radiation constant = 3.7415 x10watts / cm2 / μ

C2 = ch/k =second radiation constant = 1.43879 x 10μ °K

h = Planck’s constant = 6.6256 x 10–34 watt sec2

k = Boltzmann’s constant = 1.38054 x 10-23 watt sec °K-1

c = velocity of light = 2.997925 x 1010 cm/sec

The incremental change of radiant emittance Wλ for an incremental change  in temperature T is :

This incremental change in Wλ  translates directly to a corresponding change in scene radiance Nsc, detector irradiance Hsc and detector electrical output signal via the detector responsivity characteristic

There is also detector irradiance which arises from the lens housing  reflections from the lens surfaces  The radiation is denoted  by red  arrows in the illustration above.

The  background irradiance is not uniformly distributed across the detector because the detector sees more than the reflections of the lens housing off the surfaces of the lens. It also sees an image of way of reflections off the lens surfaces – the narcissus phenomenon. As shown in the related website page on narcissus in infrared cameras the irradiance on the detector  due to the narcissus phenomenon is non-uniform.

The difference between the minimum and maximum background  irradiance can be regarded as a noise signal analogous to the electrical noise in electronic circuitry When this background noise level equals the signal level of the scene,  a signal-to-noise ratio of unity exists. This is the  definition adopted here for NARC_ΔT.

The “noise”  or background signal is the irradiance on the detector generated by the lens housing at  300 deg K.  proportional to the radiant emittance of the housing. which is the integral of the Planck function at 300 deg, K over all wavelengths:

When the background irradiance equals the scene irradiance:


The resultant mathematical expression is complex but  can be readily evaluated using suitable mathematical software. See the link for this and more.

For a 300 °K background, the radiation emitted into a hemisphere is calculated to be 0.003 watts / cm2 of surface area  per micron of waveband at 10μ wavelength. The radiant contrast, the denominator in the above expression calculates to be 5.0 x 10-4. The ratio of the numerator to the denominator, i.e. NARC_DT calculates to be 62.5 °K. It the background radiation is attenuated by the reflectance r of the lens surface which is generating the ghost image irradiance on the detector, NARC_ΔT becomes 0.188 °K.

Now considering the entire waveband of sensitivity of the detector, the mathematical expression for ΔT becomes:


Note that the upper and lower limits of integration for the integrals differ. The limits for the integral in the numerator, the background emittance are  λ1BG and λ2BG,, while those for the denominator, the scene signal emittance,  are  λ1 and λ2. With the waveband of integration for the background set as 9.0-11.0 μ while that for the signal is left unreduced at 8.0-14.0 μ, NARC_ΔT calculates to be 0.07 °K. Thus, a relative  reduction in  background reduces NARC_ΔT.

The above example demonstrates how NARC_DT can be improved by reduction of the radiation on the detector from the background as compared to the signal. It is, of course, not possible to achieve bandpass filtering which affects only the background radiation, as in this hypothetical example. However, filtering out of more background radiation than signal radiation can be effected through the double passage of the rays  through lens surfaces and filters which may be in the optical path..

The expressiion for ΔT for a system which comprises multiple spectrally filtering elements, some of which are double-pass for the background is::


where subscript λ indicates that the parameters are a function of wavelength:

Wλ = background radiant of lens housing at temperature T

dWλ/dT = scene radiant contrast

Sλ = detector responsivity

rλ = reflectance of the lens surface creating the ghost image          of the detector

ελ = emissivity of the lens housing

Π(tλj) = product of the spectral transmittance of N lens  surfaces, j = 1 to N

Π(tλk) = product of the transmittance of lens  surfaces, j = 1 to k , where k  is the number of lens surfaces between the detector and  the ghost generating lens surface

tλfil =   filter transmittance

Background radiation is attenuated by the factors which appear in the numerator but not the denominator, namely, reflectance of the lens surface generating the ghost image on the detector, rλ, and the double pass of radiation through a filter and lens surfaces between the detector and the surface generating the ghost image.

Evaluation of  the above expression for NARC_ΔT can

be numerically determined using a spreadsheet or other means since the weighting functions within the integrands are non-analytical in form.

The NARC_ΔT spreadsheet is a working model not a defining calculation for any specific system The various columns define spectral elements of the integrands of the numerator and denominator of the equation for NARC_ΔT, as defined above plus others which the spreadsheet user may draw on for particular applications. For example, a column for  the spectral responsivity of the detector for a 3-5μ system may be added to the worksheet in addition to the spectral responsivity for an 8-12μ system shown. When glass absorption is relevant an appropriate column can be added to include it in the calculation.Where complete spectral data is not available the user has the option of entering approximations out of convenience as in the case of the present spread sheet.