**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..

**NARC_ΔT Mode**l

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 W_{λ} = spectral radiant emittance, watts / cm^{2 }/ μ

λ = wavelength, μ

T = absolute temperature, °K

C_{1} = 2πhc^{2} = first radiation constant = 3.7415 x10^{4 }watts / cm^{2} / μ

C_{2} = ch/k =second radiation constant = 1.43879 x 10^{4 }μ °K

h = Planck’s constant = 6.6256 x 10^{–34} watt sec^{2}

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

c = velocity of light = 2.997925 x 10^{10} cm/sec

The incremental change of radiant emittance W_{λ} for an incremental change in temperature T is :

This incremental change in W_{λ} translates directly to a corresponding change in scene radiance N_{sc}, detector irradiance H_{sc} 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 itself.by 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. It.is 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:

and

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 λ_{1BG} and λ_{2BG},, while those for the denominator, the scene signal emittance, are λ_{1} and λ_{2}. 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:

W_{λ} = background radiant of lens housing at temperature T

dW_{λ}/dT = scene radiant contrast

S_{λ} = detector responsivity

r_{λ} = reflectance of the lens surface creating the ghost image of the detector

ε_{λ} = emissivity of the lens housing

Π(t_{λj}) = product of the spectral transmittance of N lens surfaces, j = 1 to N

Π(t_{λk}) = 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

t_{λfil} = 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, r_{λ}, 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.

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