Infrared (IR) homing is a passive system in which heat generated by the target is detected and homed on. Typically, it is used in the anti-aircraft role to track the heat of jet engines. It has also been used in the anti-vehicle role with some success. This means the guidance is also referred to as “heat seeking”. The IR heat seeker missiles have exploited techniques to acquire and intercept airborne targets, by passively detecting their IR energy. Developments in IR detection and tracking have led to the increasing effectiveness of IR guided missiles (fire and forget), which are now portable such as MANPADS (Man Portable Air Defense Systems) and easily available. IR heat seeker missiles have been responsible for the majority of aircraft losses since their introduction into service during the 1960s. Some statistics suggest that heat-seeking missiles have been responsible for more than 80% of all combat aircraft losses over the last 40 years , see Fig. 1 for example. There are three main types (generations)of IR heat seeker missiles according to the type of reticle (optical modulator)  and the detector(in missile head): spin-scan, conical-scan, and imaging. The latter is the most advance one because It cannot be easily fooled by counter-measure techniques.
Fig. 1 Damage caused by a missile. The lucky part was that the missile hit the wing.
The IR heat seekers detect the difference of IR energy between the target and back ground, i.e. contrast. Thus, IR stealth technology aims to, ideally, make this difference zero by reducing the target IR signature level (IRSL). Heat seeker missile uses atmosphere windows for detecting and tracking. Those windows are specified bands of wavelength in electromagnetic spectrum that characterized by low IR attenuation. In addition, those windows are interesting to the seeker designers because they include the IR radiation of hot turbojet aircrafts. In an effort to redress the balance between the aircraft and the missile, researches into systems that could be utilized to defeat seekers has been widespread. The development work has followed two main paths: defensive systems such as flares; and active systems such as jammers. Those countermeasures are designed to decreases the lock-on range, R, by increasing the attenuation of IR target. In this paper, we investigates the lock-on range, R, of IR heat seeker missile in details. A model is presented for lock-on range in term of target, which is considered as point source, atmosphere and detector parameters. We solve the R equation in term of Lambert W function. The simulation results show that lock-on range, R, can be decreased by attenuating IR energy of the target or decreasing the target temperature. In addition, the lock on range can be decreased also by decreasing the emissivity of target surface. This can be achieved using special applied coatings/paintings .
IR Radiation and Signature
The spectral radiant emittance (SRE) of blackbody can be written as function of wavelength, λ, and temperature, T (in Kelvin (oK)), according to Planck's low as 
is the spectral radiant emittance (Wcm−2μ−1)
c1 is first radiation constant, c1=3.7415×104Wcm−2μ4 C2 is the first radiation constant, c7=1.43879×104μ°K.
The SRE of blackbody for different values of temperatures is shown in Fig. 2. More general representation of SRE of blackbody is shown in Fig. 3. It can be seen that SRE increases as temperature increases and the maximum SRE peaks lie over the range of short wavelengths. Principally, the range of temperatures (500oK-9000K) is very important to heat seeker designer because it includes the IR energy hot metal tailpipes of turbojet aircraft . The peaks of SRE corresponding to above temperature range lie between wavelengths 3μm−5μm that most of heat seekers are designed to track and lock-on the target. Integrating the above equation with respect to the wavelength from zero to infinity gives us an expression for the radiant emittance, i.e. the flux radiated into hemisphere above blackbody 1 cm2 in area, which can be written as
where S is the radiant emittance (Wcm−2
is Stephan-Boltzman constant, σ=5.667×10−12W cm−2oK−4
Fig. 2 Spectral radiant emittance of black body for different values of temperatures.
Fig. 3 (a) Spectral radiant emittance of black body vs. temperature and wavelength (b) the corresponding contour plot to show different optimum value curves of spectral radiant emittance.
The above law is known as Stephan-Boltzman law .
Stephan-Boltzman law provides a standard comparison; it describes an ideal radiator, i.e. blackbody which can be used to compare the radiation of any other sources. A factor can, be added, so it can be applied to sources that are non blackbodies. This factor is called emissivity, ε
Therefore, SRE of any source can be written as
is the emissivity of the source at specified temperature and wavelength. Fig. 4
shows the radiant emittance as a function of temperature, T, and for different values of emissivity, ε
, where the curve of ε=1
has the highest radiant emittance (black body). It can be shown that radiant emittance decreases as ε
Fig. 4 Radiant emittance for different values of emissivities.
General representation of Stephan-Boltzman law is shown in Fig. 5 where it can be concluded that the target having high ε emits high radiant emittance which can be locked on by IR heat seeker.
Researches in the process of developing an 'electro chromic polymer'. These thin sheets cover the aircraft's white skin and sense the hue, color and brightness of the surrounding sky and ground.
IR energy is generally lost to the atmosphere mainly by absorption and scattering. Absorption is the energy loss due vibration and rotation of molecules . Scattering is the energy loss due to redirection away from the detector. In general, the transmittance of atmosphere can be represented as 
where R is the range between the IR source and the detector and α
is extinction (or attenuation) coefficient which can be written as 
where a and γ
are the absorption (due to water vapour) and scattering (due to fogs and clouds) coefficients respectively. The atmospheric windows are characterized by high τa
, i.e. low α
(attenuation). The presence of high humidity, clouds, fog or smoke can dramatically influence IR absorption.
Fig. 4a (a) General representation of radiant emittance and (b) the corresponding contour plot which show the curves of different optimum values of radiant emittance.
IR signature level (IRSL) of the target (aircraft)
For military systems, generally the targets are aircraft, missiles, ships, ground vehicles, tanks etc. The sources of radiation in military targets are mainly the hot engine parts, the exhaust plume and the high emissivity metal skin. The engine and exhaust plume produce a large amount of heat and makes the exhaust section as a main source of thermal radiation. The typical sources of radiation in jet aircraft, as shown in Fig. 6, are the hot-metal tailpipe, the exhaust gas plume, metallic skin and the aerodynamic heating which increases with the speed of aircraft.
Fig. 6 Sources of IR radiation in an aircraft.
The hot metal tailpipe and the stream of hot exhaust gases known as the plume. Exhaust gas temperature (EGT) is one of the most important criteria of engine performance. The temperature of the plume at the metal tailpipe is given as
is the EGT.
The tailpipe behaves typically as a graybody with total emissivity of about 0.9, with temperature equals to the EGT and an area equals to that of exhaust nozzle. The higher EGT of the aircraft is in takeoff case where the engine thrust is maximum. This high radiation of tailpipe can be easily detected by even less sensitive heat seeker missile. In this paper, we will assume that the thermal radiation is emanating from a single point, i.e. aircraft plume that radiates into hemisphere. According to that, the radiance of the target can be written as
is the radiance of the target (W cm−2
), sr is the sold angle.St
is the radiance emittance of the target (Wcm−2
is the emissivity of the target Tt
is the temperature of the target which we will considered in °C. Note that the relationship between the radiance, N, and the radiant emittance, S, is N=S/π 
In addition the radiant emittance of target background should be taken into account where the heat seeker detection depends on the difference of radiant emittance, i.e. contrast, between the attenuated IR radiant level of the target and its background. The radiance of target background can be modeled as a graybody with specified value of emissivity
is the radiance of background (W cm−2
is the radiance emittance of background (Wcm−2
is the emissivity of the background Tb
is the temperature of background (oC).
Heat seeker Missile
The tactical missile system used against aircraft includes three main sections: the guidance and control, the warhead and the propulsion sections as shown in Fig. 7. The guidance and control system is located in the front part of the missile and consists of the seeker, the guidance control unit and the rudders. Seeker head receives the IR radiation emitted from a heated source, typically the engine of aircraft, and converts this energy into an electric signal. The signal is processed in the guidance control unit that calculates control signals used for directing the missile via rudders and tailfins.
The block diagram of seeker head is shown in Fig. 8. It is comprised of the following major components: (1) IR dome for protection from the aerodynamic forces and weather, (2) optical system (mirrors) to focus the IR target energy onto detector, (3) reticule (or optical modulator) to provide directional information for track (4) detector, to convert the IR energy to electrical signal Also optical filter may be put in front of the detector to pass only specified narrow wavelength band for locking-on and reject background noise.
Fig. 8 Block diagram of seeker head.
The lock on range of heat seeker missile can be written as 
where the definition of above parameters with their values used in calculation are At
is the target area, (3660 cm2
(aircraft nozzle)) Do
is the diameter of optics, (3.8 cm)NA is the numerical aperture of the optics (0.25) τo
is the transmittance of the optics (0.81)D∗
is the detectivity of detector, (5×1010
instance field of view, (1.9×10−5 sr)
Δf is the electrical bandwidth, (200 Hz)SNR is the signal-to noise ratio,(3)
Recalling that the transmittance of atmosphere. τa, is function of the range, R, therefore it appears difficult to solve the above equation for R. Previously the authors have used either numerical method such as Newton Raphson to solve this nonlinear equation. In this paper we present exact and full analytical solution. Substituting equation (5), i.e. τa=e−αR, in equation (10) and re-arrange the terms, on obtains
where we consider the lock-on range, R, is a function of temperature, T, corresponding to the temperature of the contrast, Nt(T)−Nb(T)
, seen by the heat seeker. The solution of equation (11a)
can be written as
is the Lambert's W function.
The lock-on range, R, is calculated as a function of temperature of target and for different values of extinction coefficient of the atmosphere as shown in Fig. 9. The target is assumed as graybody with ε=0.9, i.e. aircraft tailpipe. In addition the background is assumed as a graybody with T=10oCand ε=0.98. According to equation (4) the value of background radiance, Nt, is 0.011 W cm−2sr−1, As shown in this figure, lock-on range, R, increases as the temperature, T, of target increases due to increasing of radiance N, therefore, hot parts of target for instance, the nozzle and tailpipe of jet aircraft, can be detected by the seeker from long distance. In addition, the R decreases as atmosphere extinction coefficient increases.
Fig. 9 Lock-on range of heat seeker against the temperature of the target (tailpipe of jet air craft) for different values of atmosphere extinction coefficient.
In addition, as shown in Fig. 10, the highest lock-on range is obtained at aircraft takeoff case because the temperature of aircraft tailpipe is high where exhaust gases temperature (EGT) reaches 635°C. While it is 515°C and 485°C in continuous and cruise cases respectively. According to above values of temperature, the lock-on range will be 37.167Km, 32.766Km and 31.6 Km respectively where the maximum lock-on range is obtained in aircraft takeoff case. From other hand, lock-on range decreases as extinction coefficient increases. General calculation of lock-on range versus temperature and extinction coefficient is shown in Fig. 11.
Fig. 10 Lock-on range of heat seeker missile against extinction coefficient for different aircraft flying situations, i.e. different tailpipe temperature.
Fig. 11 (a) Lock-on range of IR heat seeker missile against extinction coefficient and aircraft tailpipe temperature. (b) The corresponding contour plot show different curves for different optimum values of lock-on range of heat seeker.
The decreasing of IR signature level of aircraft will decrease the lock-on range of heat seeker missile and provide longer time to the aircraft to manoeuvre, i.e. increase the probability of survivability.
One of the methods used in aircraft protection is the reduction of jet engine IR energy via cooling the exhaust gases plume. This can be done by injection water to the exhaust stream. This leads to decrease the transmittance parameter, τa due to increasing of attenuation coefficient, and then decrease IR signature level. This can be schematically represented by assuming that the exhaust gases plume will pass the cooling medium before the atmosphere one as shown in Fig. 12. Thus, the IR signature received by the seeker head will be attenuated by e−αR×e−αcR=e−(α+αc)R where αcis the attenuation coefficient corresponding to the cooling medium (or any other attenuated medium such as smoke obscurants).
Fig. 12 Schematic representation of aircraft jet engine Exhaust gases plume. In addition to absorption by atmosphere, the IR flux energy can be reduced by cooling the hot exhaust gases plume.