Journal of Engineering Physics and Thermophysics, Vol. 86, No. 4, July, 2013

ANALYSIS OF THE INFLUENCE OF THERMAL CONDITIONS ON THE OPERATION OF OPTOELECTRONIC DEVICES

E. D. Ushakovskaya UDC 535.8;681.7.01

The optoelectronic device has been considered from the viewpoint of system analysis. It has been shown that thermal conditions infl uence the transfer function of this system and, as a consequence, the desired signal. Moreover, the self-radiation of the device′s elements, which also leads to a change in the output signal of the device, depends on the thermal conditions.

Keywords: information and measurement optoelectronic devices, optoelectronic tracking and detecting devices, thermal noise, photon noise, temperature noise, measurement error, detection probability, image quality, thermal conditions, transfer function, desired signals, self-radiation.

Introduction. It is known that the functioning of an optoelectronic device (OED) depends on its thermal conditions, to whose investigation many works were devoted. However, the designer is interested not in the temperature distribution in the device itself, but in the change in the operational characteristics of the device. Until the present time few investigations have been devoted to the infl uence of thermal conditions on the functions of individual elements of the OED, for example, on the thermodeformations and thermooptical aberrations of the optical system (OS) [1–3], on the noise of certain radiation detectors [4, 5], and on the sensitivity and time constant of radiation detectors [6–8]. However, these works lack a complex solution of the problem considering the infl uence of thermal conditions on the OED from the viewpoint of system analysis and permitting to take into account all factors infl uenced by the thermal conditions, as well as to analyze the infl uence of these factors on the fi nite function of the device depending on its purpose.

In the present work, the optoelectronic device has been considered from the viewpoint of the theory of systems. We have analyzed the infl uence of thermal conditions on the transfer functions of individual elements of the OED and of the device as a whole and, as a consequence, on the desired signal of the device. Moreover, we have considered the self-radiation of the device elements and obtained expressions for the change in the output signal of the device. Various kinds of noises have been considered. It has been shown that a change in the transfer function of the device, the self-radiation of the device′s elements, and noises caused by the thermal conditions infl uence the fi nite function of the device: the solution of problems of measurement, detection, tracking, and imaging.

Infl uence of Thermal Conditions on the Transfer Function of the OED and the Desired Signal. From the viewpoint of the system theory the OED represents a combination of individual elements functionally connected with one another so as to fulfi ll a specifi c purpose (e.g., detection, tracking, measurement, imaging).

Frequency methods have found wide use for analyzing complex systems, whose parameters within certain limits do not depend on the signal and time. These methods rely on the frequency representation of the parameters and characteristics of signals, as well as of the dynamic properties of systems, the change-over to which from coordinate and time representations is realized by the Fourier transform. The operation of individual elements of the system, for which the optical, mechanical, and electronic units, the radiation detector, etc. can be considered, is characterized by weighting and transfer functions.

Let us consider individual assemblies and units of the OED and analyze the infl uence of thermal conditions on them and the transfer functions.

Optical systems. The input and output signals for them are the luminance distribution B(x, y) in the object space and the illuminance distribution E(x, y) in the image space. The radiant fl ux transmitted through the optical system is distorted as a consequence of the phenomena of diffraction and aberrations. The distortion is characterized by the scattering function h(x, y, x1, y1) representing the illuminance at the point (x1, y1) when a radiant fl ux equal to unity is directed to the point (x, y).

Scientifi c-Engineering Company "Teplofi zpribor,"4 5th Predportovyi proezd, St. Petersburg, 196240, Russia; email: ushakovskaya@inbox.ru. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 86, No. 4, pp. 894–904, July–August, 2013. Original article submitted October 29, 2012.

952 0062-0125/13/8604-0952 ©2013 Springer Science+Business Media New York

953

The scattering function is the weighting function of the optical system. Its corresponding complex transfer function WOS(ωx, ωy) is equal to [9]*

OS( , ) ( , ) exp ( ( )) ,x y x yW h x y j x y dxdy

∞ ∞

−∞ −∞

ω ω = − ω + ω∫ ∫ (1)

and the frequency spectra of luminance B(ωx, ωy) and illuminance E(ωx, ωy) distributions are related to one another by the relation [9]

2

0 OS( , ) sin ( , ) ( , ) .x y x y x yE u B W′ω ω = πτ ω ω ω ω (2)

A violation of the thermal conditions of the optical system leads to a change in the temperature level of its elements and to the appearance in them of temperature gradients. This in turn causes:

1) a change in the sizes of optical elements and in the distances between them;2) the appearance of thermal stresses and thermal deformations of optical elements;3) a change in the refractive indices (under the infl uence of the temperature and thermal stresses).

In the simplest cases, the approximate values of the sizes of the radii of curvature of the lens or the mirror rt and the distance between elements dt, as well as of the refractive index nt are calculated by the formulas [9]

0 0 0 0 0 0[1 ( )] , [1 ( )] , ( ) .t t t t t tr r T T d d T T n n T T= + α − = + α − = + β − (3)

Note that formulas (3) are highly approximate; they do not take into account the thermal stresses in the elements of the optical system. For more exact calculations, it is necessary to analyze the thermal deformations and the thermal stresses with the use of numerical methods [10–12].

A consequence of the change in the sizes, the surface shape, and the refractive index are changes in the focal length of the system, the position and size of the image (thermooptical aberrations according to the position and magnifi cation), and other aberrations. The character of aberrations and their size depend on the kind of the temperature element and the optical fi eld distribution. Defocussing of the optical system and aberrations lead to a change in the scattering function h(x, y) in the analysis plane.

The change in the transfer functions ΔWOS(ωx, ωy) of the lens is equal to

OS 0( , ) [ ( , ) ( , )] exp ( ( )) .x y x yW h x y h x y j x y dxdy

∞ ∞

−∞ −∞

Δ ω ω = − − ω + ω∫ ∫ (4)

In the simplest cases (e.g., for the defocusing of an ideal optical system under the action of the thermal conditions), using the formulas for h(x, y) and h0(x, y) presented in the literature [9, 13], one can obtain expressions for the change in the transfer function of the OS. In more complicated cases, numerical calculations are needed. For the input data the equations of the optical surfaces, the distances between elements, and the refractive indices under normal thermal conditions and upon their change are given.

The radiation detector transforms the radiant fl ux P(τ) or the illuminance E(τ) to an electric signal. The radiation incident on the detector causes the appearance in the detector circuit of an EMF or a change in one of the parameters, for example, voltage, resistance, or capacity. The transformation of the radiant fl ux to an output signal is estimated by means of the sensitivity SP(E):

, .P ES Q S Q EP= ∂ ∂ = ∂ ∂ (5)

As was shown in [13], in the simplest case the radiation detector can be regarded as an aperiodic link with a transfer function Wtr(ω):

( )tr det(1 .) )( P E jW S + ωτω = (6)

_________________________*It is assumed thereby that the optical system satisfi es the isoplanarity condition, i.e., h(x, y, x1, y1) = h(x – x1, y – y1).

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The change in the transfer function of the radiation detector ΔWtr(ω) is due to the infl uence of the thermal conditions on the sensitivity of the detector and its time constant that increase, as a rule, with decreasing temperature. For instance, for metal and extrinsic-conduction semiconductor resistive-fi lm bolometers the sensitivity is inversely proportional to the tem-perature, and for intrinsic-conduction semiconductor bolometers it is inversely proportional to the square of temperature [14].

To determine the change in the transfer function ΔWtr(ω), let us differentiate expression (6) with respect to the pe-rimeter T and multiply by ΔT:

2

tr det ( ) det det( ) [(1 ) ] (1 ) .EPW j S T j T j Tτ τ τΔ ω = + ω ∂ ∂ − ω∂ ∂ + ω Δ (7)

The thermal conditions infl uence also the transfer functions of other OED elements: rasters, diaphragms, scanners, and other devices, but this infl uence has not been investigated by now.

A change in the transfer functions of OED assemblies and units leads to a change in the transfer function of the whole device. Let us consider this change for an OED representing an open-loop system consisting of elements connected in series. The transfer function W(ω of such a device is defi ned as a product of the transfer functions of elements:

( ) ( ) .

n

ii

W Wω = ω∏

(8)

Differentiating (8) and going over to fi nite differences, we obtain the expression for the change ΔW(ω) in the transfer function of the device as a consequence of the change ΔWk(ω) in the transfer functions of its elements:

0( ) ( ) ( ) .

n n

k ik i k

W W W=

Δ ω = Δ ω ω∑ ∏ (9)

A change in the transfer function of the device causes a change in the desired (deterministic or random) signal.As is known, the spectra of deterministic signals at the output Y(ω) and input G(ω) of the device are related by the

relation [15]

( ) ( ) ( )ω = ω ωY W G . (10)

Then, in view of (9), the change in the spectrum of the deterministic desired signal ΔY(ω) caused by the change in the transfer function of the device will be equal to

0 0

1( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) .

n n

k ik i k

Y W G W G W G G W W= +

Δ ω = ω ω − ω ω = Δ ω ω = ω Δ ω ω∑ ∏

(11)

Applying the inverse Fourier transform to (11), we obtain the expression for the change Δytr in the deterministic signal caused by the change in the transfer function:

tr 1 (2 ) ( ) exp ( ) 1 (2 ) ( ) ( ) exp ( ) .y Y j q d W G j q d

∞ ∞

−∞ −∞

Δ = π Δ ω ω ω = π Δ ω ω ω ω∫ ∫ (12)

For random signals, their spectra at the output Sy(ω) and at the input Sin(ω) of the device are related by the relation [15]

2

in( ) ( ) ( ) .yS W Sω = ω ω (13)

The change in the spectrum of the random signal ΔSy(ω) will be equal to

2 2

0 in( ) { ( ) ( ) } ( ) .yS W W SΔ ω = ω − ω ω (14)

Let us write the transfer functions in algebraic form

1 2 0 01 02 1 2( ) ( ) ( ) , ( ) ( ) ( ) , ( ) ( ) ( ) .W W jW W W jW W W j Wω = ω + ω ω = ω + ω Δ ω = Δ ω + Δ ωThen

955

2 2 2 21 2 10 20 in

2 2 2 210 1 20 2 10 20 in

01 1 02 2 in

( ) {[ ( ) ( )] [ ( ) ( )]} ( )

{[ ( ) ( )] [ ( ) ( )] [ ( ) ( )]} ( )

2[ ( ) ( ) ( ) ( )] ( ) .

yS W W W W S

W W W W W W S

W W W W S

Δ ω = ω + ω − ω + ω ω

= ω + Δ ω + ω + Δ ω − ω + ω ω

≈ ω Δ ω + ω Δ ω ω

.

(15)

The dispersion Δσtr2 of the change in the random signal due to the change in the transfer function is calculated by

the formula [15]

2tr 1 (2 ) ( ) .yS d

∞

−∞

Δσ = π Δ ω ω∫ (16)

In view of (15) and (16) the expression for Δσtr2 will take on the form

2tr 01 1 02 2 in1 [ ( ) ( ) ( ) ( )] ( ) .W W W W S d

∞

−∞

Δσ = π ω Δ ω + ω Δ ω ω ω∫ (17)

Besides the desired signal, in most cases background radiation, which can be considered as a random variable with spectral density Sb(ω), hits the device input. If the optical system transmits partly or completely the background radiation, then the change ΔWb(ω) in the transfer function of the device in the spectral range of the background leads also to a change in the background radiation ΔSby(ω) at the device output. The value of ΔSby(ω) is calculated by formula (15), where Sb(ω) is substituted for Sin(ω), and for W01(ω), ΔW1(ω),W02(ω), and ΔW2(ω) their corresponding values for the transfer function of the W01b(ω), ΔW1b(ω),W02b(ω), and ΔW2b(ω) are substituted.

The dispersion σb2 of the background radiation is calculated by the formula

2b 01b 1b 02b 2b b1 [ ( ) ( ) ( ) ( )] ( ) .W W W W S d

∞

−∞

σ = π ω Δ ω + ω Δ ω ω ω∫ (18)

Thus, under the infl uence of the thermal conditions the transfer function of the device changes and, as a consequence of this, the desired (deterministic or random) signal and the background radiation at the device output change.

Infl uence of Thermal Conditions on the Self-Radiation in the OED. The optoelectronic device converts optical radiation to other kinds of energy. Besides the object under observation, the OED elements, whose radiation hits also the detector and transforms to an output signal, are also radiation sources. The infl uence of the OED self-radiation is determined by the temperatures of the device elements and the spectral range in which the OED operates. As a rule, for the ultraviolet and visible regions of the spectrum the self-radiation of the device elements can be neglected. This radiation becomes substantial in the long-wave region of the spectrum, especially for the cases where the temperatures of the object under observation differs slightly from the device temperature.

Let us consider the change in the illuminance and the heat fl ux in the detector plane depending on the self-radiation of the OED elements. Let a detector of area Sdet be surrounded by bodies with absolute temperatures Ti and radiating capacities εi(l), i = 1, 2, …, N. The irradiance Edeti and the fl ux Pdeti created by the self-radiation of the ith body (i = 1, …, N) on the detector are, according to [16], equal to

1

det det det det det0 0 0

( ) , ( ) ( ) .i i i i i i i i i i iS

E dH R d dH R dP H R d∞ ∞ ∞

λ λ λ= ε λ λ = ε λ λ = ε λ λ∫ ∫ ∫ ∫ (19)

The area of mutual irradiance Hdeti of the detector and the ith body and its element dHdeti are calculated by the formulas [16]

det det det det det det, .i i i idH dS H S= ϕ = ϕ

956

In the presence of mirror-refl ection elements in the system, it is expedient to use instead of the irradiance coeffi cient ϕdeti the resolving angular coeffi cient Φdeti showing what portion of the energy emitted by the ith body hits the detector with account for the multiple refl ections [16]. The method for calculating the resolving angular coeffi cients in optical systems is described in [17].

The total illuminance EdetΣ and the total fl ux of self-radiation ΨdetΣ from all bodies surrounding the detector will be equal to

det det det det

0 0

( ) , ( ) .N N

i i i i i ii i

E dH R d HP R d∞ ∞

Σ λ Σ λ= ε λ λ = ε λ λ∑ ∑∫ ∫ (20)

The spectral concentration of luminous exitance Rλi entering into (20) is determined in accordance with the Planck formula [13]

5 1

1 2[exp ( ) 1] .i iR C C T− −λ = λ λ − (21)

Let us determine the changes in the illuminance of the detector and the self-radiant fl ux from all bodies surrounding it when the temperatures of these bodies change by a value of ΔTi, i = 1, 2, …, N. Differentiating expression (21) with respect to the temperature Ti and multiplying by ΔTi, upon substitution into (20) we get

6 2 2det det 1 2 2

0

2 4 2 2 4 2det 2 1 det det 2 1

0 0

( ( ) ((exp ( ) 1) ))

( ( ) ) , ( ( ) ) .

N

i i i i ii

N N

i i i i i i i i i ii i

E dH c c T c T T d

dH c c PR T T d H c c R T T d

∞−

Σ

∞ ∞− −

λ Σ λ

Δ = ε λ λ Δ λ − λ

= ε λ λ Δ λ = ε λ λ Δ λΔ

∑ ∫

∑ ∑∫ ∫

(22)

In [18, 19], expressions for the illuminance distribution for more complicated cases with account for the nonuniform temperature fi elds of optical elements were obtained. The above expressions are awkward and inconvenient for practical use. At the same time, as the estimate has shown, at temperature differences in OED elements of 10 K at the level of 300 K the error of calculations by the approximate formulas (20) and (22) does not exceed 5% and at a temperature difference of 1 K it does not exceed 0.5%. Taking this into account, as well as the uncertainty of the input information εi(λ), we see that the application of the approximate formulas (20) and (22) is justifi ed in the majority of cases.

The additional illuminance ΔEdetΣ or fl ux ΔPdetΣ on the detector change the output signal y by a value of ΔyE(P) equal to

( ) tr ( )1 (2 ) ( ) ( ) exp ( ) .E P PEW G jy q d

∞

−∞

= π ω Δ ω ωΔ ω∫ (23)

The spectral concentrations ΔGE(P)(ω) of the additional illuminance or fl ux that are due to the self-radiation of optical elements are calculated by means of the direct Fourier transformation of expressions (22):

det det( ) exp ( ) , ( ) exp ( ) .PEG E j q dq G P j q dq

∞ ∞

Σ Σ−∞ −∞

Δ ω = Δ − ω Δ ω = Δ − ω∫ ∫ (24)

Summarizing, we can say that the result of the action of thermal conditions is a change in the transfer function of the device, which in turn leads to a change in the desired signal of deterministic or random background radiation at the device output. Moreover, the thermal conditions determine the self-radiation of the device elements hitting the detector and transforming to an output signal.

Infl uence of Thermal Conditions on the OED Noise. Many processes proceeding in the OED: radiation, radiative and convective heat transfer, electric charge transfer, etc. are random in nature. These processes are characterized by certain quantities fl uctuating randomly about their mean. These fl uctuations just represent noise. Practically all elements of the optoelectronic device, as well as the radiator itself and the signal propagation medium are noise sources.

Noise is a random process and is characterized by the power spectral density S(ω). To describe noises, equivalent circuits with EMF or current sources are used. Depending on the kind of circuit, the power spectral density of EMF SE(ω) or current SI(ω) is meant.

957

Let us consider certain kinds of noises, whose parameters are determined to a particular degree by the thermal conditions, namely thermal, photon, and temperature noises.

Thermal (Nyquist or Johnson) noise is a variable electromotive force that arises from thermal chaotic motion at the leads of any resistor. The sources of thermal noises in OEDs are the radiation detector and the active (resistors) and reactive (capacitors and inductors) elements of electronic circuits. For frequencies below 10 Hz, the thermal noise can be assumed to be white, and the EMF spectral density E

TS and the current spectral density ITS can be calculated by the Nyquist formulas [13]

4 , 4 .E IT TS KTR S KT R= = (25)

The photon noise, called also the radiation noise, is associated with fl uctuations of the radiant fl ux incident on the detector and is explained by the corpuscular properties of the radiation. The photon noise calculation is different for thermal radiation detectors reacting to the mean value of the fl ux and photon detectors whose signal is determined by the quantity of charge carriers released by radiation quanta and either leaving the substance (photoemission) or staying in it, thus increasing the conductivity (photoresistors) or creating an EMF (photovoltaic detectors).

In [4, 5, 13], formulas for the photon noise of a thermal detector located in a medium with a constant temperature were obtained. In [13], the expression for the photon noise of the radiation detector in the case where the ambient temperature Tamb is a function of coordinates is given. This expression contains the integral of the function of the ambient temperature over the spatial coordinates, which impedes considerably the process of calculations. It is interesting to obtain simpler dependences for the photon noise power in the case where the detector is surrounded by N bodies with constant temperatures.* The author of [15] has shown that the expression for the power spectral density of photon noise Sph of the thermal detector has the form

5 5ph 0 det det det det

216 .

N

i i ii

S K S T T=

⎛ ⎞⎜ ⎟= πσ ε + ϕ ε⎜ ⎟⎝ ⎠

∑ (26)

Temperature noise is observed only in thermal detectors. Its cause is the detector temperature fl uctuation resulting from the statistical nature of the heat exchange with the environment. Heat exchange occurs mainly by radiation and heat conduction. If the heat exchange due to the heat conduction is insignifi cant compared to the radiation, then the temperature noises are identifi ed with photon noises and can be calculated by formula (26).

In the presence of heat conduction the expression for the fl uctuation dispersion has the form [13]

2 2 2hc b hc4 .TP P KT RΔ = Δ + Δω (27)

The fl uctuation dispersion of the fl ux and the power spectral density of the radiation fl uctuation for white noise in the frequency band Δw are related by the relation [13]

2

( ) 2 .S Pω = Δ π Δω (28)

Substituting expressions (27) and (26) into (28), we obtain a formula for the power spectral density of temperature noises Shc(ω):

5 5 2hc ph det det 0 det det det det det

2( ) 16 16 8 .

n

T i i i Ti

S S KT T R K S T T K T R=

⎛ ⎞⎜ ⎟ω = + Δ π = πσ ε + ϕ ε + π⎜ ⎟⎝ ⎠

∑ (29)

The noise spectrum Sn(w) at the output from the system is composed of all noises nn(ω) arising from the infl uence of the thermal conditions in the OED that have passed from the point where they arose to the device output. In accordance with (13), Sn(ω) is equal to

n2

n ( ) ( ) ( ) .n

k kk

S W Sω = ω ω∑ (30)

________________________________*As noted above, the given condition is satisfi ed, as a rule, in the OED. The temperature differences in the regions do not exceed fractions of a degree.

958

In view of expression (8), for the transfer function the noise spectrum at the output from the system will be equal to

n2

n ( ) ( ) ( ) .knn

ik kk i

S W Sω = ω ω∑∏ (31)

The dispersion σn2 of noises at the output from the device will be equal to [15]

2n n1 (2 ) ( ) .S d

∞

−∞

σ = π ω ω∫ (32)

Infl uence of Thermal Conditions on the Solution of Tracking and Measurement Problems. In tracking and measurement OEDs, the desired signal and the background radiation, as well as the signals due to the self-radiation of OED elements and noises, after the detector and the amplifi er, hit the device, where the desired signal is developed and the desired signal is processed with the aim of obtaining information on the value of the measured parameter (in measurement OEDs) or mismatch (in tracking OEDs). The signal processing algorithm is determined by the purpose of the device, its operational conditions, and the character of the desired signal.

One of the basic requirements imposed upon measurement and tracking OEDs is the guarantee of the given accuracy of measurement of object or process parameters or the accuracy of tracking objects and monitoring physical or technological processes. Above, we obtained expression (12) for the error Δytr of the deterministic desired signal, expression (17) for the variance error σtr

2 of the random input signal Sin(w), and expression (18) for the background radiation dispersion σb2due to the

change ΔW(ω) in the transfer function of the OED. The error due to the self-radiation of the optical system elements ΔyE(P) can be calculated by formula (23). The error due to the noises in the device can be calculated by formulas (25), (29), (31), and (32). Errors Δytr and ΔyE(P) are systematic, and if the temperatures of the OED elements are constant or vary by a known law, then the values of Δytr and ΔyE(P) can be estimated and excluded afterward. However, often only the range of variation of temperatures of OED elements is known. In this case, one can determine only the boundaries of the nonexcluded systematic error (NSE) Θtr for the nondeterministic signal error caused by the change in the transfer function and Θself for the error caused by the self-radiation of the OED elements. The NSE boundaries can be estimated approximately by calculating the change in the transfer function and the self-radiation at limiting values of the change in the range of variation of temperatures. The boundaries can be estimated more exactly as a result of the statistical modeling, whose algorithm is given in [17]. It is assumed thereby that the NSE caused by the change in the transfer function of the device and the self-radiation of OED elements obeys the laws of random errors with an equally probable distribution.

Provided that the error components are independent, the confi dence limit ΘS for the total error that is due to the infl uence of the thermal conditions is determined by the formula [20]

2 2 2 2 2 1/ 2n tr b tr self

* 2 2 1/ 2 2 2 2 1/ 2 2 2 2 1/ 2 2 2 1/ 2tr self n tr b n tr b tr self

[ 1 3 ( )] ,

( ( )( ) ( )( ) ) (( ) 1 3 ( ) ) .

R

R K t

ΣΘ = σ + σ + σ + Θ + Θ

= α Θ + Θ + α σ + σ + σ σ + σ + σ + Θ + Θ

(33)

Here the coeffi cient K*(α) is equal to 1.1 at α = 0.95.Tracking systems are characterized by the presence of feedback. The spectra of signals at the input and output of such

systems are related by the relation [21]

2

OS OS in( ) ( ) (1 ( )) ( ) , ( ) ( ) 1 ( )) ( ) .Y W K W G Sy W K W Sω = ω + ω ω ω = ω + ω ω `(34)

In this case, expressions for errors are derived analogously. For the deterministic signal error due to the change in the transfer function

tr OS 0 OS 0

2OS

1 (2 ) [ ( ) (1 ( )) ( ) (1 ( ))] ( ) exp ( )

1 (2 ) [ ( ) (1 ( )) ] ( ) exp ( ) .

y W K W W K W G j q d

W K W G j q d

∞

−∞

∞

−∞

Δ = π ω + ω − ω + ω ω ω ω

≈ π Δ ω + ω ω ω ω

∫

∫

(35)

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The error variance at a random input signal is

2 22tr OS 0 OS 0 in1 (2 ) [ ( ) (1 ( )) ( ) (1 ( )) ] ( )W K W W K W S d

∞

−∞

σ = π ω + ω − ω + ω ω ω∫ . (36)

For the noise error

22n OS1 (2 ) [ ( ) (1 ( )) ] ( )

n

k kt

W K W S d∞

−∞

σ = π ω + ω ω ω∑∫ . (37)

The background radiation dispersion

2 22b OS 0 OS 0 b1 (2 ) [ ( ) (1 ( )) ( ) (1 ( )) ] ( )W K W W K W S d

∞

−∞

σ = π ω + ω − ω + ω ω ω∫ . (38)

The error due to the self-radiation of OED elements is determined by the formula

( ) OS ( )1 (2 ) [ ( ) (1 ( ))] ( ) exp ( ) .E P E Py W K W G j q d

∞

−∞

Δ = π ω + ω Δ ω ω ω∫ (39)

The confi dence interval ΘΣ for the total error is determined, as in the previous case, by formula (23).Infl uence of Thermal Conditions on the Solution of the Detection Problem. The theoretical basis for the solution

of the detection problem is the present-day theory of statistical solutions [22, 23], since the presence of noise, the background radiation, and the radiation of the OED elements permits detecting an object (target) only with a certain probability.*

Figure 1 shows the typical block diagram of the OED designed for detecting a distant source [13]. In this device, analysis of the fl ux consisting of the desired signal, the background radiation, the radiation of the OED elements, as well as noises is carried out by a threshold unit that generates a signal about the presence of a target only when the signal exceeds a certain threshold value. The threshold unit can compare thereby, according to a particular task, the fl uxes, the maximum values of voltages, the so-called probability ratio, etc.

For stationary signals, the condition for detection is the excess of the fl ux PS incident on the detector over a certain threshold value of Pdet:

detP pPΣ = . (40)

Fig. 1. Typical scheme of the optoelectronic device for detecting a distant source .

___________________________*It should be noted that the signal processing can be different. The detection algorithms can be changed or improved. The aim of the present work is to show, with the example of one of the algorithms, the relation of the OED temperature fi eld to the detection problem solution. The same applies to devices of other purposes.

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The total radiant fl ux PΣis composed of the desired signal power Pds, the radiant fl ux from the background Pb, and the self-radiation Pself of the OED elements:

ds b self .P P P PΣ = + + (41)

For pulsed signals of frequency Δfself and duration Δτself, the condition for detection is the r-fold excess of the

maximum voltage Umax over the root-mean-square voltage 2nU of noise at the output from the amplifi er [13]:

2nmax nU U U= ρ = ρ . (42)

In this case ρ and p are related by the relation [13]

self self( )p f= ρ Δτ Δ . (43)

The voltage Umax is composed of the voltage Uds caused by the desired signal, and the voltages Ub and Uself caused, respectively, by the background radiation and the radiation of the OED elements:

max ds b selfU U U U= + + . (44)

The relations between signals (the power or the maximum value of voltage) and their threshold values determine the probability of making a correct decision about the presence or absence of a target.

A change in the transfer function of the device due to the infl uence of the thermal conditions and, as a consequence, in the desired and the background signals, as well as due to the self-radiation and noises, leads to a change in the relation between the signals (PΣ and Umax) and their threshold values. This in turn changes the probability of making a correct decision about the presence or absence of a detection object.

The relation between signals and the detection probability depends on the detection algorithm used in a particular OED. The choice of the algorithm is determined by the features of the detection object and the background radiation, the design of the device, etc. In [17], the author considered some of the detection algorithms and obtained expressions for the change in the probability of correct detection and false alarm under the infl uence of thermal conditions.

Infl uence of Thermal Conditions on the Image Quality. Let us consider information OEDs designed for obtaining images of the object under investigation. The operation of optoelectronic devices of this class is characterized by the degree of conformity between the photometric and geometric parameters of the image and the imaged object [9]. The degree of conformity between the image and object parameters can be estimated by different characteristics.

The most promising methods for estimating the image quality are those based on the analysis of the scattering function and the optical transfer function (OTF) [9, 13, 24, 25]. In [9], it was shown that the image quality of the optical system and the optoelectronic device is described fairly well by the frequency-contrast characteristic (FCC) and the phase-frequency characteristic (PhFC). These characteristics determine the resolution of the optical system, the wavefront aberrations, the root-mean-square displacement of the image, and other parameters characterizing the image quality [9, 13, 24, 25].

Mathematically, the FCC is expressed as follows [9]:

FCC F F′= , (45)

where

max min max min max min max min( ) ( ) , ( ) ( ) .F E E E E F B B B B′ = − + = − +

The phase-frequency characteristic defi nes the phase difference between the output ϕy(ω) and input ϕg(ω) signals [9]:

PhFC ( ) ( )y g= ϕ ω − ϕ ω . (46)

The frequency-contrast characteristic is the transfer function modulus, and the FCC is its argument [9]:

2 2 1/2

1 2 2 1FCC ( ) [ ( ) ( )] , PhFC arctan ( ( ) ( )) .W W W W W= ω = ω + ω = ω ω (47)

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Many image quality characteristics either are based on the OTF or can be calculated with its help. For instance, in [24, 25] the distortion function Fd(ω) of the FCC equal to

d r id r id( ) FCC FCC ( ) ( ) .F W Wω = = ω ω (48)

is used for quality assessment.In [24], the expressions for the FCC of an ideal lens are given.* The author of the given paper considered also the

other characteristics of the image quality: the noise equivalent Tn, the relative edge response e, the equivalent transmission band Ne, and the Strehl number D, and showed that these parameters can be calculated on the basis of the OTF:

0 0

0 0

222

n id0 0 0 0

2 22

id0 0 0 0 0

1 (4 ) ( ) , ( ) ( ) ,

1 (2 ) ( ) , ( , ) ( ) .e

T W d d W d W d

N W d D W d d W d d

π ∞ ω ω

∞ π ω π ω

= π ω ω ω ϕ ε = ω ω ω ω

= π ω ω = ω ϕ ω ω ϕ ω ω ω ϕ

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫ ∫

(49)

Here integration is carried out with respect to the circular spatial frequency w and the angular coordinate of the object. As was noted in [13, 24, 25], the OTF and the criteria based on it give a clear idea about the quality of the image formed by the optical system. The above criteria can be used not only for the OS, but also for the optoelectronic device on the whole. However, for careful assessment of the image quality, it is necessary to introduce thereby an additional criterion characterizing the presence of noise in the device. Such a criterion is the signal-to-noise ratio (SNR) [10, 11].

Thus, the image quality measurement is connected with the determination of the resolution threshold, i.e., the maximum spatial frequency of the test object, whose lines can be resolved. The resolution corresponds to the case where the signal-to-noise ratio at the spatial frequency exceeds a certain threshold value. For the simplest detection (development of a signal against the background of noises) it is essential that the NSR exceeds 3.

In the previous sections, it was shown that thermal conditions infl uence the transfer functions and noises of the OED elements and, consequently, the transfer function of the whole device, as well as the signal-to-noise ratio, which in turn infl uences the image quality.

The degree of infl uence can be characterized by the change in the FCC and the parameters Tn, ε, Ne, D, SNR, having calculated them under normal thermal conditions and after a change in them.

Thus, it may be stated that the thermal conditions changing the transfer functions of the OED elements and of the whole device, change the desired signal of the device. Moreover, they infl uence the background radiation, the noises, and the self-radiation of OED elements, which in turn has an effect on the performance of the functions of the device connected with its purpose and on the parameters characterizing the quality of the OED operation (detection probability, measurement and tracking errors, image quality).

NOTATION

Bmax and Bmin, maximum and minimum luminances of test object elements; C1 and C2, Planck constants; Emax and Emin, maximum and minimum luminances of the resulting image; h0(x, y), scattering function of the lens at normal temperature; KOS, feedback coeffi cient; K, Boltzmann constant; nk, number of elements from the point of application of the kth noise to the system output; n, number of elements in the system; p, depreciation factor; Q, parameter characterizing the

________________________________*The lens is thought to be ideal if it satisfi es the following three conditions: 1) all rays emerging from the point O(x, y) and passing through the optical system should converge at the point O1(x1, y1); 2) each element of the plane normal to the optical axis and containing the point O(x, y) should be imaged by an element of the plane normal to the axis and containing the point O1(x1, y1); 3) the image height should be proportional to the height of the object, and the proportionality coeffi cient should be constant whatever the location of the point O(x, y) in the object plane.

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output signal of the detector; Rλi, spectral concentration of radiant exitance of a blackbody with temperature Ti; R, value of internal resistance; Rt, conductive thermal resistance between the detector and the bodies surrounding it; r0, d0, and n0, rated values of the radius, the distance between elements, and the refractive index at temperature T, usually T0 = 20oC; Sk(ω), power spectral density of the kth noise; Sdet, detector area; t(a), Student distribution quantile; Tdet and Ti, temperature of the detector and the ith body; u′, rear aperture angle; |Wk (ω)| , transfer function modulus from the point of application of the kth noise to the system output; FCCr and FCCid, FCC of the real and ideal systems; |W(ω)|, transfer function modulus of the device; Wtr(ω), transfer function of the part of the OED path from the radiation detector to the device output; Wr(ω) and Wid(ω),transfer functions of the real and ideal lenses; Wi0(ω) (i = k, …, n), rated value of the transfer function of the ith element of the system; Wi(ω),transfer function of the ith assembly or unit; Wik(ω),transfer function of the kth noise for the ith assembly or unit of the OED; αt, linear expansion coeffi cient of the material; βt, temperature coeffi cient of the refractive index; σ0, Stefan–Boltzmann constant; 2

phPΔ , photon noise dispersion; εdet, emissive factor of the detector; εi, emissive factor of the ith body; τ0, spectral transmission coeffi cient of the optics; τdet, time constant of the detector; ϕdeti, coeffi cient of illumination of the detector by the ith body; ω0, cutoff spatial transmission frequency of the optical system; w, circular frequency. Subscripts: in, input; d, distortion; id, ideal; tr, transfer; det, detector; r, real; self, self; hc, heat conducting; b, background; ph, photon; n, noise; t, temperature.

REFERENCES

1. Yu. P. Shramko, Infl uence of thermal conditions of the illuminator on the wave front deformation, Opt.-Mekh. Prom., No. 3, 14–17 (1972)

2. G. F. Pishchik, Stresses and Strain in the Parts of Optical Devices [in Russian], Mashinostroenie, Leningrad (1968).3. K. V. Mazer, Space Optics [in Russian], Mashinostroenie, Moscow (1980).4. A. Van der Ziel, Noise in Measurements [Russian translation], Mir, Moscow (1979).5. A. Van der Ziel, Noise: Sources, Characterization, Measurement [Russian translation], Sovetskoe Radio, Moscow (1973).6. M. N. Markov, Infrared Radiation Detectors [in Russian], Nauka, Moscow (1968).7. L. S. Kremenchugskii, Ferroelectric Radiation Detectors [in Russian], Naukova Dumka, Kiev (1971).8. G. G. Ishanin, É. D. Pankov, and V. S. Radiikin, Radiation Sources and Detectors [in Russian], Mashinostroenie,

Moscow (1982).9. A. S. Dubovik (Ed.), Applied Optics [in Russian], Nedra, Moscow (1982).

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Moscow (1979).13. M. M. Miroshnikov, Theoretical Principles of Optoelectronic Devices [in Russian], Mashinostroenie, Moscow (1977).14. L. P. Lazarev, Infrared and Light Guidances and Homers [in Russian], Mashinostroenie, Moscow (1977).15. A. Ango, Mathematics for Radio Engineers [Russian translation], Nauka, Moscow (1964).16. E. M. Sparrow and R. D. Cess, Radiation Heat Transfer [Russian translation], Énergiya, Moscow (1971).17. E. D. Ushakovskaya, Methods for Calculating the Thermal Regime of Optoelectronic Devices, Candidate's Dissertation

(in Engineering), LITMO, Leningrad (1984).18. B. I. Golub', I. I. Pakhomov, and A. M. Khorokhorov, Self-Radiation of the Elements of Optical Systems of Optoelec-

tronic Devices [in Russian], Mashinostroenie, Moscow (1978).19. I. I. Pakhomov and A. M. Khorokhorov, Determination of the radiant fl ux in the plane of image analysis from the self-

radiation of lenses, in: Optoelectronic Devices, Sb. Tr. MVTU, No. 174, Moscow (1974), pp. 105–110.20. L. S. Zazhigaev, A. A. Kin'yan, and Yu. I. Romanikov, Methods of Planning and Processing the Results of Physical

Experiment [in Russian], Atomizdat, Moscow (1978).21. V. A. Bessekerskii and E. A. Popov, The Theory of Automatic Regulation Systems [in Russian], Nauka, Moscow (1975).22. G. Van Trees, Detection, Estimation, and Modulation Theory [Russian translation], Sovetskoe Radio, Moscow (1972).23. M. S. Shestov, Development of Optical Signals against the Background of Random Noises [in Russian], Sovetskoe

Radio, Moscow (1967).24. W. Wetherell, The Use of Image Quality Criteria in Designing [Russian translation], Mir, Moscow (1983), pp. 178–332.25. W. B. Wetherell, Criterion of the Image Quality of Large Space Telescopes. Space Optics [Russian translation],

Mashinostroenie, Moscow (1980), pp. 43–72.