MICROWAVE TECHNOLOGY SERIES 5
INFRAREDTHERMOGRAPHY
G. GAUSSORGUES
mS P R I N G E R - S C I E N C E + B U S I N E S S MEDIA, B.V.
English language edition 1994
© 1994 Springer Science+Business Media Dordrecht Originally published by Chapman & Hall in 1994 Softcover reprint o f the hardcover 1st edition 1994
Original French language edition - La Thermographie Infrarouge - Principes, Technologies, Applications (3rd edition, revised) - © 1989 Technique e t Documentation - Lavoisier
ISBN 978-94-010-4306-9 ISBN 978-94-011 -0711 -2 (eBook)DOI 10.1007/978-94-011 -0711 -2
A part from any fair dealing for the purposes of research or private study, or criticism or review, as perm itted under the UK Copyright Designs and Patents Act, 1988, this publication may not be reproduced, stored, or transm itted, in any form or by any means, without the prior permission in writing of the publishers, or in the case of reprographic reproduction only in accordance with the term s of the licences issued by the Copyright Licensing Agency in the UK, or in accordance with the term s of licences issued by the appropriate Reproduction Rights Organization outside the UK. Enquiries concerning reproduction outside the term s stated here should be sent to the publishers a t the London address printed on this page.
The publisher m akes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions th a t may be made.
A catalogue record for this book is available from the British Library
C ontents
Colour and black and white plates showing thermograms and images recorded in false colours appear at the end of the book
Foreword xiii
Historical Background xv
1 Revision of Radiometry 11.1 The radiometric chain 11.2 Radiant flux 21.3 Geometrical spreading of a beam 21.4 Radiance 31.5 Irradiance 41.6 Radiant exitance 51.7 Radiant intensity of a source in a given direction 51.8 Quantity of radiation and exposure 51.9 Bouguer’s law 61.10 Radiation scattering 61.11 Note on units 7
2 Origins of Infrared Radiation 8
3 Thermal Emission by Matter 1 13.1 Black-body radiation 11
3.1.1 Planck’s law 123.1.2 Wien’s law 133.1.3 Stefan-Boltzmann law 153.1.4 Exitance of black-body in a given spectral band 163.1.5 Evaluation of exitance of a body by the method of
reduced coordinates 173.1.6 Thermal derivation of Planck’s law 223.1.7 Thermal contrast 23
3.2 Different types of radiator 243.3 Problems with the emissivity of a material 253.4 Thermodynamic equilibrium 26
vi Infrared Thermography
3.5 Problems with the reflectance of a material 263.6 Example of an application 30
3.6.1 Calculation of Te and et 323.6.2 Calculation of £0 and T0 34
3.7 Emissivity of materials 363.7.1 Spectral emissivity 363.7.2 Emissivity of dielectrics - the effect of temperature 373.7.3 Emissivity of metals - the effect of temperature 393.7.4 The effect of the angle of incidence on emissivity 403.7.5 Measurement of emissivity 423.7.6 The effect of emissivity in thermography 433.7.7 Emissivity of a rough surface 433.7.8 The emissivity of dihedrons and trihedrons 45
3.8 Emission from the interior of a medium 463.9 Other sources of infrared radiation 50
3.9.1 The Nernst filament (Nemst glower) 503.9.2 The globar 513.9.3 Electroluminescent junctions 513.9.4 Sources employing stimulated emission (lasers) 52
4 Transmission by the Atmosphere 614.1 Self-absorption by gases 624.2 Scattering by particles 674.3 Atmospheric turbulence 68
4.3.1 Diffraction by inhomogeneities 704.3.2 The structure function 714.3.3 Measurements of turbulence 74
4.4 Methods for calculating atmospheric transmission 754.4.1 The ‘line-by-line’ method 764.4.2 The band model method 764.4.3 Empirical methods employing band models 774.4.4 The multiparametric model 78
4.5 A practical method for calculating atmospheric transmission 784.5.1 Molecular absorption 784.5.2 Scattering by particles 814.5.3 Example of application 94
5 Optical Materials for the Infrared 1035.1 Propagation of an electromagnetic wave in matter 1035.2 Optical properties of a medium 109
5.2.1 Refraction 1105.2.2 Dispersion 1105.2.3 Absorption, transmission and reflection 111
5.3 Physical properties of optical materials 1145.3.1 Hardness 115
Contents vii
5.3.2 Thermal properties 1155.3.3 Cost of materials 116
5.4 Types of material 1175.4.1 Glasses 1175.4.2 Crystals 1185.4.3 Plastics 1185.4.4 Metals 118
5.5 Properties of some optical materials 1195.5.1 Glasses 1195.5.2 Crystals 125
6 Optical Image Formation 1356.1 Geometrical optics 1356.2 Aberrations of optical systems 136
6.2.1 Chromatic aberrations 1366.2.2 Geometrical aberrations 139
6.3. Calculation of geometrical aberrations 1596.3.1 Path of a marginal ray - imaging by an objective 1606.3.2 The path of a principal - stop imaging 1626.3.3 Paraxial rays - the Gaussian approximation 1646.3.4 The third-order approximation 1676.3.5 Spherical aberration 1676.3.6 The case of aplanatic optics - the Abbe’s sine condition 1686.3.7 Calculation of coma 1696.3.8 Astigmatism and field curvature 1716.3.9 Distortion 172
6.4 Diffraction 1736.4.1 Diffraction by an aperture 1736.4.2 Image formation - linear filter theory 1786.4.3 The optical transfer function 1816.4.4 Optics for the infrared 1886.4.5 Reflecting telescopes 1886.4.6 Catadioptric telescopes 1966.4.7 Evaluation of image-spot abberration for different simple
optical systems 1966.4.8 Refractive optics 2006.4.9 Simple germanium lens for A = 10 ^m 201
7 Scanning and Imaging 2137.1 Radiometers 2137.2 Radiometers for spatial analysis 2147.3 Thermography 2207.4 Scanning methods 220
7.4.1 Line scanners 2217.4.2 Image scanning 231
viii Infrared Thermography
7.5 Imaging____________ 2327.6 Imaging with multi-element detectors 234
7.6.1 Two-dimensional scanning with a single detector 2347.6.2 Scanning by a parallel array of n elements 2357.6.3 Scanning using an array of p elements in series 2397.6.4 Serial-parallel scanning with a two-dimensional array 240
7.7 Electronic imaging 2407.7.1 The pyroelectric image tube 2427.7.2 Pyroelectric arrays 2437.7.3 Solid state arrays 243
8 Spectral Filtering 2448.1 Spectral transmittance of materials 2448.2 The properties of thin layers 2468.3 Antireflective thin films 249
8.3.1 Antireflective coating using a single layer 2498.3.2 Two-layer antireflective coating 2518.3.3 Multilayer antireflective coating 2518.3.4 Examples of surface treatments for improving the
transmission of materials 2528.4 Filters 254
8.4.1 Different types of filter 2558.4.2 Filter fabrication technologies 256
9 Radiation Detectors 2619.1 Generalities 2619.2 Characteristics of detectors 262
9.2.1 Current-voltage characteristic 2629.2.2 Shape of signal 264
9.3 Noise 2649.3.1 The spectral distribution and technological causes of noise 2649.3.2 Signal-to-noise ratio ____________ 2659.3.3 The noise equivalent power (NEP) 2679.3.4 Detectivity 2679.3.5 Detectivity limit of a perfect detector 268
9.4 Detector sensitivity 2689.4.1 Local variation of sensitivity 2689.4.2 Spectral sensitivity 2689.4.3 Global sensitivity 2699.4.4 Sensitivity as a function of frequency 270
9.5 Thermal detectors 2719.5.1 Fluctuations 2719.5.2 General principle of operation 2719.5.3 Signal-to-noise ratio 273
Contents ix
9.5.4 Detectivity of heat detectors 2739.6 Different types of thermal detector 274
9.6.1 Bolometers 2749.6.2 Pyroelectric detectors 2759.6.3 Thermopiles 2769.6.5 Pneumatic detectors 276
9.7 Quantum detectors 2769.7.1 Fluctuations 2779.7.2 Detectivity of quantum detectors 278
9.8 Different types of quantum detector 2809.8.1 Photoemissive detectors 2809.8.2. Summary of solid state physics 2829.8.3. Photoconductive detectors 2849.8.4 Photovoltaic detectors 287
9.9 Applications of detectors 2899.9.1 Spectral sensitivity range 2899.9.2 Sensitivity 2919.9.3 Noise and detectivity 2929.9.4 Frequency response of detectors 2939.9.5 Detector bias arrangements 2949.9.6 Effect of detector field angle 2959.9.7 Passivation of detectors 295
9.10 Multielement detectors 2969.11 Detectors used in thermography 2979.12 Charge coupled devices 298
9.12.1 Three-phase CCD 2989.12.2 Two-phase CCD 2999.12.3 Transfer efficiency 3009.12.4 Reading of a detector array with a CCD 3019.12.5 Imaging with a CCD matrix 3019.12.6 Charge injection devices (CIDs) 3029.12.7 Spectral response and characteristics of CCD and
CID imaging devices 3049.13 Infrared charge coupled devices (IRCCD) 304
9.13.1 HgTeCd detectors 3059.13.2 Indium antimonide 3069.13.3 Silicon-platinum Schottky diode 3069.13.4 Performance of IRCCDs 307
9.14 Sprite detectors 3089.15 Detector cooling 311
9.15.1 Cooling by liquified gas 3119.15.2 Cooling by Joule-Thomson expansion 3119.15.3 Cooling by cryogenic cycles 3139.15.4 Thermoelectric cooling 316
x Infrared Thermography
10 Signal Processing 31910.1 The analogue signal 31910.2 Processing of analogue signals 32310.3 Processing of digital signals 32310.4 Example of application 324
10.4.1 Analogue acquisition 32510.4.2 Digitisation of the signal 32510.4.3 Visualisation 32910.4.4 Architecture of image reconstruction 33110.4.5 Image processing 33310.4.6 Temperature calibration of images 33410.4.7 Description of program 337
11 Characterisation of infrared systems 34011.1 Generalities 340
11.1.1 Noise equivalent irradiance (NEI) 34111.1.2 Thermal resolution 34111.1.3 Spatial resolution 34211.1.4 Spectral response 34211.1.5 The signal - temperature relation 34211.1.6 Temporal stability and drift 343
11.2 Characteristics of infrared detectors 34311.2.1 Sensitivity 34311.2.2 Time constant 34411.2.3 Noise equivalent power (NEP) 34411.2.4 Noise equivalent irradiance (NEI) 34511.2.5 Detectivity 345
11.3 Calculation of the characteristics of infrared systems 34711.3.1 Calculation of noise equivalent irradiance (NEI) 34711.3.2 Calculation of noise equivalent temperature difference
(NETD) 35111.4 Measurement of the characteristics of an infrared system 353
11.4.1 Measurement of NEI 35411.4.2 Measurement of NETD 35711.4.3 Measurement of MRTD 35711.4.4 Measurement of MDTD 35911.4.5 Measurement of relative spectral response 35911.4.6 Measurement of spatial resolution - the modulation
transfer function 36011.4.7 Determination of the signal-temperature relation 36311.4.8 Measurement of drift 368
11.5 Example: Characterisation of a system 36911.5.1 Evaluation of NEI 37011.5.2 Evaluation of NETD 37211.5.3 Measurement of NEI 37411.5.4 Measurement of NETD 373
where T < 1,7" = 1 and T > 1 for absorbing, non-absorbing and emitting media, respectively. For an extended beam, we have
J f T L s ^ Gf_R S_R________
~ Fs ~ f f Ls d2GS R
The radiance is conserved on refraction
n 2d2G = const
where n is the refractive index of the medium in which the geometrical spread d2G is evaluated.
This invariance implies
4 Infrared Thermography
1.5 IR R A D IA N C E
This is defined as the local value of the ratio of the flux dFR received by the detector and the area dR of the detector, i.e., the power received per unit area (W m -2 ). Symbolically,
E (x tY) = n)cos0rt<JQn (1.3)
where the integral is evaluated over the half-space, E(x ,Y) is in W m "2 and X, Y are the spatial coordinates relative to the points at which the irradiance is evaluated. The quantities £, rj define the direction of the beam.
Fig. 1.3 Irradiance
O rigins o f In fra re d R a d ia tio n
* f 2£-<*> *J0
M atter continuously emits and absorbs electromagnetic radiation. The process of emission involves molecular excitations in the material, which generate radiative transitions in its constituent particles.
The fundamental laws of classical electromagnetic theory show tha t the electric field E due to a uniformly moving charge q is little different from the static field
47r£0r 2
where the distance r a t which observations are made is large compared with the displacement of the charge and €o is the permittivity of empty space.
On the other hand, if the electric charge is accelerated, the Maxwell
12 Infrared Thermography
Radiation
F ig . 3 .1 A black body
such an object is also an emitter of radiation of all wavelengths. It transfers energy to the surroundings until a state of thermodynamic equilibrium isreached.
F ig . 3 .2 R ad ia to r surface elem ent
3.1.1 P lanck’s law
The emission of radiation by a black body is described by Planck’s law which employs the concepts of statistical thermodynamics and takes the form
dR( X}T) 2tt hc2X~5dX e x p ( h c / X k T ) - l }
where d R( X)T) /dX is the spectral exitance, i.e., the power emitted per unit area per unit wavelength, h = 6.6256 x 10-34 J s (or W s2) is Planck’s constant, k = 1.38054 x 10“ 23J K_1 is Boltzmann’s constant, c = 2.998 x 108m s_1 is the speed of light and T is the absolute temperature of the black body in degrees Kelvin (K). The temperature conversions are:
5degrees Celsius = (°F — 32) x -
y
degrees Kelvin =°C + 273.16
9degrees Fahrenheit = °C x - + 32
5
which assumes tha t the skin is a black body, which is valid in the infrared. We note, however, tha t this loss of energy is compensated by the absorption of radiation from the surroundings, in particular, clothes.
3.1.4 E xitance o f a black body in a given spectral band
The exitance of a black body between Xa and A* is obtained by integrating Planck’s law over the spectral interval:
Ak_ f Xb dR(X, T)R x - ~ A . dX
The following table gives the values of exitance of a black body, calculated for the spectral band AA = A& — Aa at temperatures normally encountered in thermography.
16 Infrared Thermography
A, p m / x. -ST"* < W cm' 7 'A. At T-280°K 7*290°K T- 300 °K Tz 310 ° K T z 750°K Tz1000°K
3 5 2,78•104 4,11 .10“4 .45,97. 10 8,48 . 10’ 4 5,84 . 10-1
3 5,5 5,44 .10*4 7,87.10*4 1,11 .toJ 1,54 . 10“3 7,09 . 10*1
3.5 5 2,88 . 10'4 3,97. icr4 5,75-10*4 8,13 . 10*4 4,42. 10'1
3.5 5,5 5,38 . 10"4 7.73. 10‘4 1,09 .10° 1,50 • 10"3 5,68 10”1 2,3 8
4 5 2,38. 10*4 3,49. 10’4 5,01 .10* 4 7,02. 10”4 2,8£ to"1
4 5,5 5,08 . 10"4 7,25.10'4 1.02.10*3 1,39- 10° 4,15 10* ’
8 10 4,20. 10*3 5,12 • 10-3 6,15 .10*3 7, 32.10’3 1,74 10’ ’
8 12 8,59 . 10'3 1,03-10’2 1,22 . 10*J 1,43. 10‘ 2 2,74 10_18 14 1,26 . 10’2 1,48- 10*2 1j74 .10*2 2,01. 10'2 3,34 10"1 6,05
10 12 4,39. 10’3 5,17 . 10*3 6,02. 10*3 6,95 . 10" 3 9,99 10”2
10 14 8,35. 10° 9,72.10*3 1,12 • 102 1,28 • I©* 2 1,60 10"’
1 2 14 3,96. 10° 4,55. 10*3 5,19- '0* 3 5,86 .10’3 6(04 IQ"2
We note that the quantities given above in energy units can in certain cases be expressed in terms of the number of photons. This refers above all to those cases where emission occurs between discrete energy levels and the emitted quantum is the photon (in W )
The spectral energy exitance given by Planck’s law can be expressed in terms of the number of photons emitted per second per square centimetre
20 Infrared Thermography
so thatx - 5
y = 142.3------------------- (3.13)exp 4.96/x — 1
The function y = f ( x ) is tabulated above (the numerical values are taken from L ’introduction a la photojnetrie by Charles Fabry).
The exitance of a black body between two wavelengths Aa and A&, i.e., the energy exitance, is given by
•A
* Jx. dAa
In terms of the reduced coordinates
dR{\T) dR(Xm,T)
dX y dX
dX = Am dx
The above integral can then be written in the form
nAt r> dR(Xm,T) dR(Xm,T) r~ J , m y — Tx— Xmdx ~ ~ d x — Am J , m yd x
whereAa j Abx a = — and x b = —
We saw above that Stefan’s law leads to the following expression for total exitance of a black body:
* = [ d«%P-dx = „T«
•oo
or, in terms of the the reduced coordinates,
D dR(Xm , T ) x r ,=----dX---- J
we now definef f 6 y dx
Z{x) = (3.14)J o y d x
This normalised function represents the area under the plot of the spectral exitance of a black body in terms of the reduced coordinates between x a and xi, . The values of the function Z(x) are also listed in Fabry’s tables.
We can thus calculate the energy emitted per unit surface area of the black body between, for example, Xa = 7.3 ^m and A& = 8.6 f im a t T = 1 273K (i.e., 1000°C).
24 Infrared Thermography
6(A) i
Spectralemissivity
7i
I
■ "
1i i / 1
/ 1 1 1/ 1 9 X
/ V \ A/
/
/✓\\\
Black body
Grey body
Selective body Mirror
Wavelength Fig. 3.11 Spectral emissivity
3.2 D IF F E R E N T T Y P E S O F R A D IA T O R
Normal objects are not generally black bodies and the above laws do not apply to them unless certain corrections are made.
Non-black bodies absorb only a fraction A of incident radiation; they reflect a fraction 7Z and transm it a fraction T . These different factors are selective, i.e., they depend on the wavelength.
Consider an object of this kind, exposed to a given amount of incident radiation. When the system is in a state of thermodynamic equilibrium, the energy released into the ambient medium as radiation plus energy reflected and transm itted, must equal the energy introduced into the system by absorption.
It is thus necessary to introduce the spectral emissivity s(A) whose role is to balance the absorptance .4(A), where
Fig. 3.10 Radiant energy balance
^ciden t
A ( A) = e(A)
£(A) + 7v(A) -f r(A) = 1 (3.19)
Some special case are listed below:
opaque body : T(A) = 0 and e(A) -j- 7£(A) = 1
shiny body : V, (A) large and e(A) almost zero
blackbody : e (A) = 1 and 71 (A) = 0 r (A) = 0
grey: body e(A) = constant and 7£(A) = constant
28 Infrared Thermography
Surroundings: tem perature Te, emissivity e t O bject
being examined: tem perature To reflectance 7 le emissivity £0
M irror' = 1
Infraredimager
R i , Z?2 “ images of object and stray radiation
R [ , R'2 - images ofsurroundings in the mirror
Fig. 3.14 Characterisation of an object in a perturbed environment
from the background (noise). T he apparatus is illustrated schematically in Fig. 3.14.
T he infrared image of the object is generated by the exitances of two sources, namely, self-emission and background reflection, in the spectral band of the detector under consideration:
r tj = G [ (A) ee (A) d R ^ ’Te )dX + [ e0 (A) — 1[̂ o)- dXJ A A » a A J A A i dX
(3.20)
R 2 = Q / K ' ( A ) ( A ) d R { * ' Te)dX + [ e0 (A) d- ^ T° U x Jb a3 <*a yAAa dX
where Q is a geometrical param eter th a t depends on the distance between the two objects and on their surfaces.
T he infrared image of the neighboring body viewed in a m irror is generated by the exitance:
d R ( X , T t )R[ = f 6e (X) 'm-” V ' y e / dX in AX\
J AAi »A
(3.21)
d R ( X , T e)R'2 = f e .(A ) " " ' " ’^ ' dA in AA2
J AAa “ A
R\ = 0.289 W cm "2 (3.23)
32 Infrared Thermography
A similar calculation for the other temperatures would give
EMITTANCE OF EMITTANCE OFMEASURED OBJECTS, BACKGROUND OBJECTS,
W cm‘2 W cm"^R l= 0.05802 R’l = 0.289
A \2 R2 = 0.05057 R’2 = 0.251~ ̂~ v • * m m
This calculation can be made with the help of Fabry’s table in terms of the reduced coordinates, but the precision of the result is degraded when the small spectral band widths employed are taken into account. In spite of this, we can carry out the calculation to illustrate the process of evaluation of, for example, R \ .
We have seen that for T[ = 1059 K and Am = 2.737 /im, the reduced coordinates are
Ala = 3.5 fim x ia = Ala/Am = 1.279 Z ia = 42.50 x 10“ 2
Au, = 3.7 /im x ib = Alft/Am = 1.352 Z \t = 46.58 x 10“ 2
The values of Z \a and Z n are extrapolated from the Fabry tables. The exitance in the band AAi is
R[ = <tT4 (Z n - Z ta) = 5.87 x 10-12 x (1059)4 x 4.078 x 10-2
andR[ = 0.291 W cm "2 (3.24)
This value is quite close to that found by the direct method. We can now calculate the parameters relative to a neighbouring object, considered parasitic.
3.6.1 C alculation o f Te and ee
The estimates of T[ and T £, i.e., of R l and R!2, are derived from the following set of equations:
R , f £/ _ * ( ^ l dX J a Xi
K = l . " f c Z - k *Aa
(3.25)
36 Infrared Thermography
T 200 K 300 K 400 K 500 K
C -0.2890 -0.2890 -0.2884 -0.2849
D -0.2892 -0.2891 -0.2882 -0.2829
C-D 2.00x10"* 1.00x1 (T4 -2.00x1 cr4 -20.00x10-4
(c - d). 104
Fig. 3.18 Plot of C — D Temperature, K
£o - 0.8 (3.33)
The results are summarised below:
T0 = 350K » 80°C e0 = 0.80Te = 1 070K w 800°C ee = 0.96
These values may be compared with the raw thermographic data which, given an estimated apparent temperature of the object under measurement of the order of 740 K, yield roughly 460°C instead of the 80°C effective temperature.
This precise and definite example, illustrates the importance of the errors tha t can be introduced in the absence of any correcting factors.
3.7 E M IS S IV IT Y O F M A T E R IA L S
3.7.1 S p e c tra l em issiv ity
The spectral emissivity e(A) of a surface is defined as the ratio of the spectral exitance d R ( \ ,T ) / d \ and the exitance of a black body dRcn( \ )T ) / d \
and increases as its electrical conductivity y decreases. An increase in the temperature corresponds to a reduction in electrical conductivity due to the thermal motion of the molecular lattice, which produces an increase in the emissivity.
The spectral emissivity £a(T) and the total emissivity e(T) are given by
40 Infrared Thermography
e x (T) = 0 .365\/p[l + a ( T - 2 9 3 ) ] i - 0.0667p[l + a ( T - 293)] /A
+0.006^/[p(l + a [T - 293]) /A]:
£ (T) = 0.5737i//j[1 + a ( T — 293)] T - 0.1769p [1 + a (T - 293)] T
MATERIAL TEMPERATURE,°C
TOTAL NORMAL EMISSIVITY
Polished aluminium 0 0.03Polished aluminium 100 0.05Anodised aluminium 100 0.55Polished gold 100 0.02Polished iron 40 0.21Oxidised iron 100 0.64Polished steel 100 0.07Oxidised steel (800°C) 100 0.79
Lampblack 20 0.95White paper 20 0.93Wood 20 0.90Polished glass 20 0.94Human skin 32 0.98Water 1 0.92Snow 0 0.80
These relations are, however, subject to certain limitations. For example, for tungsten, it is necessary to have A > 2/xm, for gold and silver A > 1.5 //m and for nickel and iron A < 5/im.
The state of the surface and the oxidation of the metal can change the emissivity quite considerably.
3.7.4 T he effect o f the angle o f incidence on em issivity
The emissivity of a material depends, on the one hand, on the angle of observation and, on the other, on the polarisation of the radiation being considered. This leads to Fresnel relations for the reflectance of the separation boundary between two media.
Emissivity Spectral Within wavelength interval AA. Total
In direction of incidence (p
dRiA. r )Hi
f dR(A,T) ^r . L ■* - dX
r dR(A.T) ^I -x - dXT = ---------- —-----------U ̂
ex,ip (in (X.t)cn d A
AA.<p dR (X. T j/ •£r-X - - •< * AA dA
Sp » dR (A.T)
In solid angle Q.
f dR(A.T) _J --- —r--- COS U> dSin
/ / dR(A--T3 cos <p dft dA _ ft AA dA
f f dRULn cos(p dQ dA ft 0 dA
A.ft dR (A.T)/ — -----cos ^ft dA
eAA.ft dR (A.T)J / — ----- cos u> dft dAft AA dA
‘'ft °° dR (A.T)f J ——~r----- cosipdftdAft 0 dA
In hemisphere
( dR(A.T) _ ^ J --- ------ cos ip dft2 71
J J dRU J2 cos tp dn dA . . 271 AA
/ / ^ hV' ‘ ̂ C0S ^ ^ ^2 71 0eA.h dR <A.T)
; ;o5 «, on 2-n
"AA.h dR (A.T)/ / — ----- cos tpdft dA2n AA
h • dR (A.T)/ / — ----- cos (pdftdA2" o
48 Infrared Thermography
HencedI w — = —a d x
and, on integrating,
or
log I = —a x + constant
I — K exp ( - a x )
The constant of integration K is obtained by putting I = /0 (incident intensity) when x = 0:
I = Io exp (—ocx)
dtA
- V1
0> 1
d X
Fig. 3.29 Absorption in a medium
The transmission factor of a slice of thickness x is given by
r = y - = exp (—a x)I o
If we suppose th a t there is no reflection at the interfaces between successive slices, the corresponding emissivity of a slice of thickness x becomes
e = l - r = l - e x p ( - a x )
Hence the basic emissivity of a slice dx is
de = a exp (—a x) dx
andd R = de Rx
where R x is the exitance of the black body a t the local tem perature Tx of the slice dx.
A thick object (large x) thus has a total exitance
[mR = R x a exp {-ocx)dx Jo
(3.34)
52 Infrared Thermography
Fig. 3.35 Biased np junction
Photodiodes based on these principles emit in a narrow optical band (A/AA = 10) a t A = 0.85 pm with time constants of the order of a few nanoseconds.
The gallium arsenide (AsGa) photodiode emits A = 0.85 /im at T = 77 K (the tem perature of liquid nitrogen). At T = 300 K (ambient tem perature), the wavelength is A = 0.95/im. The gallium phosphate (G aP) photodiode emits a t A = 0.620 /im at 300o/ f
3.9.4 Sources em ploying stim ulated em ission (lasers)
(a) Stimulated emission
Consider a volume v containing N atoms, each with two energy levels E \ and E 2. If the populations of levels E\> E 2 are n i , n 2, we have
N = m + r*2
Transitions between the two levels can release or absorb energy
hv — Ei — E\
Energy balance between the medium and radiation produces a spontaneous em iss ion term (proportional to the number of atoms in the upper level (random emission of a photon) and an absorption term (which depends on the number no of photons present in the volume and the number n i of atoms in the lower level).
The rate of change of energy in the volume v is given by
60 Infrared Thermography
Occupied states
Unbiased junction Junction bias voltage V
Fig. 3.47 Principle of the semiconductor laser
the valence band of the p-type region. The passage of a current through such a junction imposes a bias V which shifts the Fermi level by the am ount E = E f n — E pp = e V and thus gives rise to a population inversion a t the junction. Photons are released as the levels relax. The parallel polished faces of the semiconductor close the cavity. The materials used are AsGa (A = 0.9 pm ), AsAlGa (A = 0.82 //m) and InAs.
The ou tpu t power is of the order of a few milliwatts in the continuous mode, but can reach a few hundred watts in pulsed operation. The small dimensions of the cavity (lOOx 2 fi)m produce considerable beam divergence by diffraction (5° — 20°).
Fig. 3.48 Diode laser
(e) Triggered lasers
I t is possible to use ruby or YAG lasers to generate very short giant pulses. This is accomplished by blocking relaxation oscillations during the pumping process by, for example, electro-optic shutters in the cavity. The pulse is em itted when the higher energy level is almost saturated . These pulses carry a few hundred millijoules and their length is of the order of 20ns which corresponds to peak power th a t can reach about 10 MW. The lasers are used for illumination and telemetry. Very much higher power ou tpu ts can be achieved in special applications (welding lasers, controlled nuclear fusion and so on).
Polished parallel faces
Laser light
You have e ither reached a p a g e tha t is unava ilab le fo r v iew ing or reached your v iew ing lim it fo r thisbook.
You have e ither reached a p a g e tha t is unava ilab le fo r v iew ing or reached your v iew ing lim it fo r thisbook.
Path length 1.852 km Wavelength, f im Visiblity (A = 0.6 /im: Dv = 20 kmPrecipitation h = 17 mm Resolution: AA = 0.025a*0.050 fim
Fig. 4.2 Transmission of the atmosphere
68 Infrared Thermography
4.3 A TM O SPH ER IC T U R B U L E N C E
Solar radiation absorbed by the surface of the E arth produces a heatingeffect in the lower layers of the atmosphere. The gas density therefore takeson local values th a t depend on thermal conditions.
Density fluctuations create gas motion by convection and, hence, atm ospheric turbulence.
In the first approximation, the refractive index n of air is given by Gladstone’s law
n - \ = K p (4.2)
where fC is a constant.T he density of air a t constant pressure is inversely proportional to its
absolute tem perature T t and therefore
dn dp - d Tn - 1 ~ ~ J ~ T
so tha t
d n = - ^ d T
For example, for n = 1.0003, T = 300°/ f and d T = 10K , we have d n = - 10-6
More generally, fluctuations in the refractive index of the atmosphere depend on winds, therm al convection currents, the gravitational field, humidity and so on. These relations are difficult to determine and involve param eters th a t are themselves random functions of position and time.
The atmosphere is not normally homogeneous. A common simplifying assumption is th a t its random fluctuations are uniformly distributed in all directions, i.e., we have homogeneous and isotropic turbulence.
When optical radiation enters this type of medium, characterised by a refractive index n(r, t) th a t is a function of position and time, the propagation ceases to be rectilinear and light is deviated, partially or totally, depending on the inhomogeneities th a t behave like diffracting objects.
If <j> is the diameter of the beam and / is a typical linear dimension of an inhomogeneity, we observe the following.
(a) For <t> -C /, the light rays become curved. This is due to the s tra tification of the air into layers with different refractive indices in the lower atmosphere, and is responsible for mirages and errors in visual targeting.
(b) For <t> < I, the image fluctuates. The fluctuations are caused by the rapid displacement of the refractive-index inhomogeneities when the geometrical dimensions are large compared with the transverse dimension of the light beam. The fluctuations produce variable inclinations of the wavefront, which is perceived as a shift of point images on a plane.
72 Infrared Thermography
Fig. 4.13 Structure function
Exam ination of meteorological param eters reveals a variation of these average values. We will therefore use the structure function, defined in the following way. If n(ar) is a non-stationary random function, i.e., n (z ) is not constant, then the difference n(x + r) — n(x) is stationary and its value does not change when n(x) varies as a function of small fluctuations in r.
T he structu re function is the correlation function for this difference:
Fn (r) = [n(z) - n(x - r ) ] . [n(x + r) - n(z)]
The above difference is stationary, i.e.,
n(z) — n(x — r) — n(x -f r) — n(z)
andFn (r) = [n(x + r) - n ( i)]2 (4.4)
The structu re function can also be expressed in another way:
Fn (r) = [n(x + r)]2 + [n(ar)]2 - 2n(x) n(x -j- r)
wherer n (r) = n(x + r)n (x )
andr„ (o) = R i) F = [n(, + r)]2
so th a tFn (r) = 2 [r„ (0) - r„ (r)]
andFn (r) = 2 W W - r „ (r)] (4.5)
Tatarski has shown in his book Wave propagation in a turbulent medium (McGraw-Hill, New York, 1961) th a t the s tructure function for refractive indices can be written in the form
Fn (r) = C ^r i 10 < r < L0 (4.6)
76 Infrared Thermography
num ber of sharp lines whose envelope generally resembles a Lorentz curve. The absorption coefficient for a simple spectral line is given by
v _ /oK i°) ~
TT J(<7 - <70)2 + A(72
The full width a t half height, Atr, varies with tem perature and pressure as follows:
A A P [T*A ct = Aero — y —
In general, calculations of atmospheric transmission must take into account the structure param eters of absorption bands, including the band width, the number of spectral lines within each band, the relative line spacing, the number of lines with the same intensity, the full width a t the half height of a line, the line shape and the gas pressure and tem perature.
There are three main m ethods of modelling molecular absorption th a t employ these parameters.
4.4.1 T he ‘line-by-line’ m ethod
This m ethod is based on a procedure of combining a very large number of facts on the molecular s tructure of the atmosphere and on the position in the spectrum of all possible lines (U S A ir Force Cambridge Research Laboratories, Massachusetts, 1973). The transmission spectrum is obtained by averaging such d a ta over constant spectral intervals Act,-. However, the ‘line-by-line’ m ethod is very effective for a line spectrum, but does not apply to absorption in regions where the band structure is continuous.
Monochromatic lines
Spectral absorption
A 0,-
Fig. 4.19 The line-by-line method
4.4 .2 T he band m odel m ethod
Here the intensities and positions of absorption lines are assumed to have a distribution th a t can be described by a simple m athem atical model, and absorption is once again obtained by averaging over a given band.
Larmore (published in Atmospheric Transmission, Rand paper, p. 897, Santa Monica, California, The Rand Corporation, 11 July 1956). Even though the figures contained in these tables may have been obtained from a theoretical model, they are sometimes adjusted in order to fit experimental data. The Passman-Larmore tables given a t the end of this Section refer to a horizontal path a t sea level.
The spectral absorption bands become narrower with increasing altitude because of lower pressure. This has the effect of improving the transmission, especially since the water content decreases with altitude.
80 Infrared Thermography
Fig. 4.24 Relative humidity as a function of altitude
In practice, it is often interesting to know the total transmission within the atmospheric ‘windows*. The curves in Fig. 4.25 show the transmission within these windows as a function of the height of precipitable water.
(b) The determination of 7 c o a The concentration of gaseous carbon dioxide in the atmosphere is almost independent of pressure; at zero altitude, this param eter can be considered constant. The transm ittance of gaseous carbon dioxide, 7 c o 2> is therefore a function of only the distance travelled D. The Passman-Larmore tables refer to a horizontal path a t sea level.
(c) Example We shall evaluate the atmospheric transm ittance due to molecular absorption under the conditions corresponding to the experimental plots of Taylor and Yates (Fig. 4.25). The conditions are:
tem perature t = 40.5° F = (40.5 - 32)(5/9) = 4.72°C relative humidity R H = 0.48 distance travelled D = 16.2m wavelength A = 10/im
88 Infrared Thermography
CO2 gas continued
5.0 - 6.9 /im
Wavelength,/im0.1 0.2 0.5 i
Distance, km
2 5 10 20 50 100 200 500 1000
5.05.15.25.35.4
5.55.65.75.85.9
6.06.16.26.36.4
6.56.6 6.7 6 86.9
0.999 0.998 0 997 0 995 0.994 0.990 0.986 0.979 0.968 0 954 0 935 0.897 0.8550 999 0.999 0 998 0.998 0.996 0.994 0.992 0.988 0 984 0.976 0 961 0 946
0.986 0 980 0 968 0 955 0 936 0 899 0 857 0.799 0.687 0 569 0 420 0.203 0.0720.997 0.995 0 993 0 989 0.984 0.976 0 966 0.951 0 923 0 891 0 846 0.760 0 666
I I I I I I I I I I !
Carbon dioxide gas7.0 — 9.4 /im
Wavelength,/im 0.2 0.5
Distance, km
2 5 10 20 50 100 200
7.07.17.27.37.4
7.57.67.77.87.9
8.0 8 18.28.38 4
8.5 8 6 8.7 8 88.9
9.09.19.29.39.4
I I I I I I I I0 999 0.999 0.998 0.995 0.991 0.978 0 955 0.9140 999 0.998 0.995 0.991 0.982 0.955 0.913 0 834
0.999 0.997 0.995 0.990 0.975 0.951 0.904 0.776 0.605 0 3630.993 0.982 0.965 0.931 0.837 0.700 0.491 0.168 0.028 0.001
92 Infrared Thermography
Fig. 4.31 Rayleigh scattering as the cause of the blue colour of the sky
r /A = 1.25 and Ko « 3,
7 = ir txK qt2 = 3.14 x 200 x 3 x 25 x 10~8 = 4.71 x 10-4
which for a pa th length of D = 100 m gives
T, = e -7 D = exp ( -4 .7 1 x 10~4 x 104) = 0.009
For A = 40 / i m , r/A = 0.1, Ko « 0.05 and D = 100 m the result is
7, = exp ( -3 .1 4 x 200 x 0.05 x 25 x 10-8 x 104) = 0.92
It is interesting to note th a t for a path of only 100 m, the 4-/im radiation is almost completely absorbed, but suffers hardly any attenuation at wavelengths ten times larger (unfortunately, this spectral range corresponds to molecular absorption by water vapour). For A = 12/im, the transm ittance is 50% under comparable conditions.
Actually, this method of calculation is difficult to implement because we need to know the number and size of the particles in the atmosphere. These param eters can be determined by laboratory methods th a t yield curves similar to those reproduced above.
(c) Relation between Ts and visibility To approach the scattering problem from a more practical point of view, it is interesting to use the relation between the scattering coefficient 7 and the wavelength A. This takes the form
7 - A"a
96 Infrared Thermography
4 / im = 0.84 TH20 4 / i m = 0.90 7cOa 4 / i m = 0.98
10 /im = 0.95 TH2O 10/im = 0.30 T co2 10/im = 0.99
so th a t the total transmissions for D = 10 km are T4tim = 0.74 and Ti0 ̂ =0.28.
Finally, when an average rain shower is superimposed on this type of configuration, we obtain, for D = 10 km,
= 0.92 X 0.74 = 0.68
no/im = 0.80 x 0.74 = 0.59
and, for D = 1.852 km,
?4 ̂ m = 0.74 x 0.20 = 0.50
no/im = 0.28 X 0.20 = 0.06
100 Infrared Thermography
Atmospheric transmissionDistance, 2.00 km. Precipitation, 7.00 mm.Temp. 20 deg C. Rel. humidity, 70%. Visibility, 15 km.
Jim
Atmospheric transmissionDistance, 2.00 km. Precipitation, 42.00 mm.Temp. 20 deg C. Rel. humidity, 70%. Visibility, 15 km.
^im
Fig. 4.37 Effect of temperature
104 Infrared Thermography
e div E = p
1 T T ^ E ^curl H = £— + 7 E at
div H = 0
(5.2)
(5.3)
(5.4)
The field vectors may be resolved into components along the rectangular Cartesian axes O z ,O y , Oz. Thus, the components of E are E X} E yyE z and those of curl E are
(cu rlE )x =
(c u r lE ) , =
d E , d E y
d y d z
d E z d E zd z d x
6 E y d E xd x d y
MoreoverdivE = ^ +^ v + d E *
dx ' d y d z
The equation of propagation of an electromagnetic wave will now be derived on the assumption th a t there are no free charges in the medium (p = 0) so th a t div E = 0. I t will also be assumed th a t there is no current in the medium (dielectric), so tha t 7E = 0. Differentiating (5.3) with respect to the time, we obtain
, d H d 2E d Ecurl -dT = £- d ^ + ^
(differentiation with respect the tim e commutes with the curl which entails only space differentiations).
Equation (5.1) gives d H / d t , so th a t
1 d*E dEcurl curl E = - e / i - 7/i —
The ar-component of the vector curl curl E is given by
d dEy d E x d \ d E x B E ,d y d x dy . d z dz
»d x
d 2Ey d 2E 1d yd x d y 2
d 2E x + d 2E xd z 2 d z d x
Adding and sub tracting the term d 2Ex / d x 2 we obtain
(curl curl E )r = —d E x | d E y | d E z 1d x dy dz
’d 2E x + d ^ _ d 2Exd x 2 d y 2 d z 2
If propagation takes place in a dielectric, the conductivity is very low, so tha t 7 is practically zero:
n2 = e/tc2
K 2 * 0
In vacuume0ii0c2 ~ 1
where €o is the perm ittivity of empty space and //o = 4tt x 10“ 7(H m _1) is the magnetic permeability of em pty space. Hence
2n — ------
It is often the case tha t // is very little different from /io, which allows one to write
n K = ——2u>6q
so tha tn 2 = — (5.10)
€0In a conducting medium 7 is large, and in the infrared, where the wavelength A is long, the angular frequency = 2nc/X is low and the difference n 2 — K 2 remains constant whilst the product n K increases, so that
n 2 « * 2 = 7 ^2u>
In the general case, the amplitude of the electric field vector describing an optical phenomenon may be written
E x = aexp j u ( t - exp (5.11)
i.e., the refractive index n affects the phase of the wave propagating with speed v = c /n .
The index of extinction K affects the amplitude of the wave, attenuating it exponentially. The radiant intensity is given by
I = E XE* = a 2 exp = a 2 exp (—az)
where the absorption coefficient is
2!jJ I\ 4 7T° = — = t k < 5 1 2 >
108 Infrared Thermography
112 Infrared Thermography
F ig . 5 .6 T h in anti-reflective coating
At normal incidence, the phase difference between rays reflected by the first surface of the layer of index n' (1) and the surface of the medium of index n (2) is given by
2tr A d<p = T - = «
Rays (1) and (2) are in antiphase and interfere destructively, yielding a negligible amplitude for reflection by the system, especially if the two waves have the same amplitude, and n ' = y/n.
For germanium (n = 4) and A = 10 /im, for example, a zinc sulphide film with n' = 2.2 and thickness / such that
n'l = A/4,
we have/ A 10 I = — - = —- = 1.14 um
4n ' 8.8 pand the reflectance of the system is
* - ( s S ) ■
This may be compared with the untreated air-germanium interface for which R — 0.36.
The reflectance of the interface between two media can be reduced over a wide wavelength range by means of a multilayer coating. The materials commonly used in such antireflective films are cryolite (A I F 3 — NaF, n’ — 1.3), magnesium fluoride (M gF2, n ' = 1.38), silicon monoxide (SiO, n' = 1 .6 - 1.9), cerium oxide (CeC>2, n ' = 2.2) and zinc sulphide (ZnS, n' = 2.2)
and is called the coefficient of linear expansion. Materials may be classified, as far as their thermal expansion is concerned, in terms of this coefficient.
The melting and softening points are of great importance in defining the tem perature ranges in which a material may be used. Some typical values are listed in the table below.
116 Infrared Thermography
MATERIAL
LINEAREXPANSION
COEFFICIENT,106 (°C)->
MELTING POINT, °C DENSITY
Invar 0.8 1 495 8(Fe-Ni-C alloy)Platinum 9Steel -10Iron 12 1 490 7.78Copper 17 1 082 8.5Brass -18Silver 18 820Aluminium 23 660 2.71Silica gel 0.6 1 700 2.2Germanium 5.5 958 5.33Ordinary glass -9 500 variousFluorine 24 1 360 3.18Plastics 90 to 179 66 to 123
5 .3 .3 C o s t o f m a te r ia l s
The cost of optical materials ranges widely, no doubt because of difficulties
120 Infrared Thermography
• Fused silica Si(>2 (Corning, General Electric, Infrasil).
T>
0,5 1 2 3 4 5 6 7 8 9 10 11 12 15 20 25 30A,/jm
Refractive Index : n 0 5 = 1.46ni = 1.45 n2 = 1.438 n3>5 = 1.406
- Density: 2.2- Melting point: « 1700°C- Knoop hardness: 461:(200 g)
- Coefficient of expansion: 0.55 x 10“ 6(°C )_1- Insoluble in water.
• ( Calcium aluminate glass - BS - 37A - Barr and Stroud; IR - 10 - 11 - 12 - Bausch and Lomb)
0,5 1 2 3 4 5 6 7 8 9 101112 15 20 25 30A , / i m
Refractive index : no s = 1.67n i = 1.654 n 2 = 1.640 n 3 = 1.627 n 4 = 1.607
- Density: 2.9-3.4- Melting point: « 800°C- Knoop hardness: « 600- Coefficient of expansion: « 9 x 10_ 6(°C )” 1- Solubility: glasses sensitive to water.The 2 .8 /im absorption band can be eliminated by fabrication in vacuum.
• Arsenic trisulphide glass - AS2S3 (Barr and Stroud, American Optical
124 Infrared Thermography
- Density: 5.85- Index of dispersion: i / | = 154; i/g1 = 209- Knoop hardness: 45- Coefficient of expansion: 5.7 x 10~6(°C )“ 1- Insoluble in water.
♦ Amorphous selenium - Se
*6
A , / i m
Refractive index : no s = 3.1 n4 = 2.45n i = 2.55 nio = 2.45 n 2 = 2.5
- Density: 4.26- Softening point: ft; 40°C- Coefficient of expansion: 37 x 10~6(°C )“ 1- Insoluble in water.
♦ Chalcogenide glass, Selenium arsenic (Se-As), Kodak
T>
0,5 1 2 3 4 5 6 7 8 9 10 1112 15 20 25 30
A , / i mRefractive index : n i = 2.58 n 4 = 2.484 nio = 2.477
n 2 = 2.50 n 5 = 2.481 n 12 = 2.475n3 = 2.488 n8 = 2.478 n 14 = 2.474
- Density: equivalent to Se- Softening point: « 70°C- Coefficient of expansion: 34 x 10“ 6(°C )_1- Insoluble in water.The last two materials become relatively fluid as the tem perature rises; the second has a higher softening point because it contains arsenic. These are difficult materials to use.
128 Infrared Thermography
Lithium fluoride (LiF synthetic single crystal), 0 1 2 /im < A < 9//m
T>
Refractive index : n 0 5 = 1.394 n 3 = 1.367 n 5 = 1.327 n x = 1.387 n4 = 1.349 n 6 = 1.298n 2 = 1.379
- Density: 2.6- Melting point: 870°C- Hardness 110: (600 g)- Coefficient of expansion: 37 x 10” 6(°C )_1- Solubility in 100 g of water: 0.27 g.This m aterial has a relatively low solubility and may be used for windows and laboratory apparatus. Lithium fluoride absorbs slightly a t 2.8 /im, which may be eliminated by fabrication under vacuum. The dispersive characteristics of this material are such th a t it may be conveniently combined with arsenic trisulphide (AS2S3) to make achromatic elements for systems working near 4 /im.
• Caesium iodide (Csl, synthetic single crystal), 0.24/im < A < 70/im
i— 1 1 1 1 1 rn-----1— r, 2 3 4 5 6 7 8 9101112 15 20
Refractive index : no.5 = 1 804 n 2o = 1.727ni = 1.757 n 30 = 1.706n4 = 1.75 n 40 = 1.677nio = 1.739 n50 = 1.63
- Density: 4.5- Melting point: 621°C- Knoop hardness: not measurable- Coefficient of expansion: 50 x 10“ 6(°C )_1- Solubility in a 100 g of water: 44 g.Caesium iodide transm its over a wide spectral range (0.24 to 70 /im). It isoften used for the fabrication of prisms in the infrared.
132 Infrared Thermography
(b) Semiconductors
Silicon (Si; synthetic single crystal), 1.2/im < A < 15/im
75
0,5 1 2
Refractive index : 3.453.4323.4253.422
3.4183.418: 3.418 = 3.418
- Density: 2.33- Index of dispersion : = 236 and i/g1 = 3454- Knoop hardness: 1150- Melting point: 1420°C- Coefficient of expansion: 4 x 10“ 6(°C )“ 1- Insoluble in water.Silicon is a very hard material; the results of optical polishing are excellent. Because of its high refractive index, reflection losses are high (46% for two surfaces). Transmission may be improved by the deposition of ZnSn film. As all other semiconductors, absorption increases with tem perature.
• Germanium (Ge; synthetic single crystal), 1.8/im < A < 23 /im
73
0,5 .1 2 3 4 5 6 7 8 9 10 1112 15 20
Refractive index : n 2 = 4.101 ng = 4.005n 3 = 4.049 719 — 4.004n 4 = 4.024 n l0 = 4.003n 5 = 4.015 n i2 = 4.002
- Density: 5.33- Index of dispersion : = 88a n d ^ x = 1112- Knoop hardness: 700-850- Melting point: 958°C- Coefficient of expansion: 5.5 x 10_ 6(°C )“ 1- Insoluble in water.
136 Infrared Thermography
ical wavefront E arising from each point M of the object into a wavefront E ' with finite dimensions, which generates the image M ' of M . In a perfect system, the image wavefront is a portion of a sphere S ' centred on the image M ' of M . All rays tha t, by definition, are normal to the wavefronts, then converge on M ' and every point of the object has a point image. The surface S ' is called the reference sphere; the defects of the optical system, or aberrations , may be described in terms of the normal deviation A of the actual wavefront E ' from the sphere S ' of an ideal system. In practice, rays normal to E ' do not converge to a unique point, but give rise to a spreading of the image point, i.e., the so-called circle of least confusion, whose dimensions may be predicted from the normal deviation. The normal deviation A corresponding to an aberrated optical path nA in the medium of refractive index n is a function of the space coordinates on the wavefront.
Plane object
wavefront, EImage wavefront, E ' Reference sphere
Normal deviation
Optical system
Plane image
F ig . 6 .1 Image form ation
The aberrations usually encountered in optical systems are of two kinds: geometrical aberrations, associated with the aperture and the field in the system, and chromatic aberrations th a t arise from the dispersive properties of the different optical media. The materials must of course be transparent in the spectral range to be used.
6.2 A B E R R A T IO N S OF O PTICA L SYSTEM S
6.2.1 Chrom atic aberrations
Consider a thin lens of focal length / , or convergence C = 1 / / , and refractive index n. The radii of curvature of the two faces will be denoted R and R', respectively.
140 Infrared Thermogra.phy
The rays B O and O B ' will be called field rays and the angles 9 = (OA, O B ) and O' = (O A \ O B ') the object and image field angles, where
. A B ytan 0 = 777 = “ O A x
tan# ' = = —O A ' x '
The quantities 9, 9' or y ,y ' will also be referred to as the field param eters of the system. The ray B O B ' will be called a principal or field ray.
A ray originating a t A y crossing the optical system at a point M and finally arriving a t A' will be called on aperture ray. The aperture angles a = (A O , A M ) and a' = (A '0 ,A 'M ) are given by
O M ht a n a = = = —
AO' x
, O M 1 h tan a = . = —
OA' x '
where h = O M and the quantities a , a ' and h>h' are called aperture param eters of the system.
F ig . 6 .5 Definition of geom etrical param eters
Rays passing through the rim of the system at maximum aperture (/imax and a ^ ^ ) are called marginal rays.
W hen the system suffers from aberrations, rays from A and B will not all pass through A ’ and B \ respectively, bu t through points more or less
where the function <£(»') represents the deviation of the system from the sine condition.
It can be shown th a t the normal deviation in a plane of azimuth <f> in the case of coma is
A / * / • / \ cos (f>A c = ( n y s in a — n y sin a ) — —
A c = — (nhd -f n 'y1 sin a') — 5-— (6.10)
A c = (nysin a + n 'h ’O')n '
where the last two relations apply to an object at infinity and an image at infinity, respectively. The expression for the normal deviation, which yields zero when the sine condition is satisfied, can be written in the form
A c = y' $ ( a / ) sin a ' cos 0 (6.11)
In the third-order approximation, the first term of the series expansion of $(<*') is
$ ( a ' ) = 6sin2 a
where 6 is the coma coefficient.The normal deviation therefore takes the form
A c = b y' sin3 a ' cos <j> (6.12)
and the lateral spread is described by
dy' = y' <J>(ct') (2 -f cos 2<j>)
(6.13)dz ' = y' <£(<*') sin 2<j)
The blur spots obtained with an aperture a! and field size y' are shown in Fig. 6.16, constructed from a circle of radius
pc = b y‘ sin2 a ' = y' ^ ( c / ) (614)
and two straight lines tangential to this circle and intersecting a t the paraxial image B ' at an angle of 60°.
148 Infrared Thermography
F ig . 6 .1 6 Flared im age produced in com a
152 Infrared Thermography
Fig. 6.20 Astigmatism and field curvature
It can be shown th a t the curvatures of the focal surfaces are
C , = V - A
Ct = V - 3 A
where V is the sum of optical powers
and A is a measure of the astigm atism of the system. T he la tter is afunction of the positions of the object and pupil
C3 - C t s ' - t '
The field curvature is defined by
r _ Ct + C3 _ s' + t'2 y*
and the normal deviation due to astigmatism and field curvature is
c*/2 y'2A A tc — ------ — (C — A cos 2<j>)
156 Infrared Thermography
tI
B
F ig . 6 .2 5 Spherical m irror w ith a s top a t the cen tre of cu rva tu re
limited in aperture a t its centre of curvature, presents no astigmatism. The ray passing through the centre C is then the axis of revolution of the mirror. For the same reason, there is no coma and the image B ' of B is subject to spherical aberration alone.
(d) Distortion
We have seen th a t distortion is measured by terms involving y'3 and does not depend on the aperture h of the system. It is present even for h = 0, i.e., for the ray passing through the centre of the pupil (principal ray).
This ray comes from the object point B and cuts the plane of the paraxial image A 'B ' a t D ‘ which is different from B'. The difference between D' and B ‘ depends on the ray angle 0 or O’.
and the edge of the pupil and the principal ray crossing the center of the pupil and the edge of the field.
6.3.1 Path o f a marginal ray - imaging by an objective
Consider a spherical refracting surface centred at C, apex a t S, radius of curvature S C = R and refractive indices n and n' on the two sides. An object point A is situated in the medium of index n, a t a distance S A = x along the optical axis defined by S C .
A ray from A crosses the refracting surface a t I located a t a height h above the axis and, after refraction, cuts the optical axis at A \ such th a t S A ' = x ' . The angles of incidence and refraction at I are i and respectively. The problem is to determine the image angle a ' in terms of the object aperture angle o r , and then to calculate the position x' of the image A! of A.
160 Infrared Thermography
Let us erect the normals CI\ and C K ’ to A I and I A \ respectively. In the right-angled triangle A C K )
C K — AC sin a = (R — x) sin a
and, similarly, in the right-angled triangle IKC
CK = 1C sin i = R sin i
so th a tR - x .
sin z = — —— sin a R
(6 .21)
164 Infrared Thermography
W hen R is very large,
, _ sin 6 cos \ (S' + j ' ) ^ ^ s in # ' cos ^ (0 -f- j )
and when R = oo (plane surface),
. n' cos S' t a n aV = P-------- j =P~----- 7n cos 0 tan a '
Finally, when the stop is a t infinity (p = oo),
. . V SmJ = R
9' = j ' - j
6 .3 .3 P a r a x ia l r a y s - t h e G a u s s ia n a p p ro x im a t io n
(a) Small apertures
We now assume th a t the optical system has a m oderate aperture, i.e., small a , so th a t sin a can be taken as equal to a (in radians). The angles i, i' and a ' are then small and the sines of these angles can be taken as equal to the angles themselves. In th a t case, the above relationships nowbecome
R — x o , ii = — —— a a = a 4- 1 — i
R
n .1 = “7* n '
where the subscript 0 labels paraxial quantities. These equations enable us to eliminate the angles a , a 7, i and i \ so th a t we finally obtain
n ' n n' - n= - + - s - (6-31)
where, n 'xo R
Xr\ —n R + x o (n ' — n)
i.e., we have reached the Gaussian approximation, or the first-order approximation, since we have kept only the first terms in the series. This can be described as paraxial optics.
from which we obtain the lateral spread and the normal deviation:
dy1 = — dx1 tan oc‘
168 Infrared Thermography
A , = -dx' Ck
(6.36)
wheredx'
a =Q/2
is the spherical aberration coefficient.
F ig . 6 .3 5 Spherical ab erra tion
6 .3 .6 T h e c a se o f a p la n a t ic o p t ic s - A b b e ’s s in e c o n d i t io n
Let us again consider a refracting surface and a small object A B of height y whose image A 'B ' has a height y'.
The relationships obtained previously enable us to write
R s in i = (R — x ) sin a
R sin i' = (R — x ') sin a'
The ratio of these two equations and Snell’s law give
sin i R — x sin asin if n R — x ' sin a'
and since the transverse magnification of the system is
R - x ' R — x
(6.37)
172 Infrared Thermography
We now have
B'0S ' = S ' ------- 9 -0 cos e '
andf = B qT ' cos 9' = r cos O' - x 'q
s' = B qS ' cos = 5 ' cos 9' - Xq
It is thus possible to evaluate the following normal deviations:• astigmatism
/2A ^ = (s ' - *') cos 2<t>
• field curvaturea '2
• lateral spread in the plane of paraxial image
dy‘ — —a ' t' cos <f>
dz' — —a ' s ' sin<£
astigmatic difference
Pa = S" n (6-48)
(6.45)
(6.46)
s ' - <' (6.47)
• radius of the circle of least confusion and its position
6 .3 .9 D is to r t io n
The height of the final image produced by the system after the m -th surfaceis
y'm = K m - Pm) tan 6m (6-50)
where x'm is the abscissa of the paraxial image relative to the last surface, p'm the abscissa of the image of the stop corresponding to the field angle
factor if we replace the plane of observation with a sphere of radius R , centred on 0 .
In the neighbourhood of the optical axis, it is of course possible to identify these spheres with the corresponding tangential planes, so tha t, with the above assumptions, we have
A {yi'Zi) « J J Ap (ypt Z p ) exp dyp dzp
We now define the point P in terms of the angular coordinates (/?,7 ) (see F ig .6.39)
P R
176 Infrared Thermography
zp7 = R
so th a t
Ai{Vi,* i) « R e x p ( - j k R ) JJ A p ( p , y ) e x p \ j k (0 y( + y Zi)]d(3 d f
Finally, substitu ting 0 = A/i and 7 = \ u or
Up ZpLi — — — 1/ — — —^ A fl A R
(p and v have the dimensions of spatial frequencies), we obtain
Ai(yi>Zi) « X 2R e x p ( - j k R ) JJ A P (ji, j /)exp[2*r;'(/i y, + i/z,-)] d/zdi/
(6.52)This shows tha t, apart from a constant factor, the disturbance at M is the Fourier transform of the amplitude A p(p,i/) on the aperture.
The result of all this is th a t the image of a point of light produced by a perfect optical system with a circular aperture cannot be a point.
Indeed, a point object produces a spherical wavefront with a uniform am plitude distribution. After passing through the system, the wavefront is cut by the circular edge of the aperture, and the distribution A (p , i / ) is given by the circular function defined by
A p (n, v) = l f + S < P 2= 0 elsewhere
where p is the radius of the aperture p = h / \ R. The image spot is then given by the Fourier transform of the above function, i.e.,
A i ( y t , Z i ) = 2 * p 2 —
Z7T rp
184 Infraied Thermography
Object
Fig. 6.47 Imaging a test pattern
Lo(Y’z) objectD(y'.z')
/ \ Impulse
/(response
VXY/v ^
E(y\z’)
image
or tawIh<u
• *-l3£
F ig . 6 .4 8 Im age form ation
The spatial frequencies of the image pattern remain identical to those of the object pattern , and the function characterises the lowering ofthe contrast in the image.
Had the system contained aberrations, we would have had
E ( y ' , z ' ) = 1 + (M TF) cos2 7 T z 1
+ (PT F)
Degradations o f the optical transfer function
Generally, the presence of aberrations (A p complex) reduces the modulus of the optical transfer function. However, for small aberrations, which is
192 Infrared Thermography
Numerical example
If we letS \F \ = f \ = —180 mm
S \ F = d = 44 mm
we obtain
g2 = - L = 1.667 Jl
, _ 5 2 ( / i - d ) _ _______________
/2 = -----o---- :— = — zlUm m9 2 - 1
52F = (1 - 02)/2 = 140 mm
S 2S \ = (1 - g2) f i - d = 96 mm
2h2 = 2(1 - ^2) f 2j = 81.66mm
S = 7r (/i2 — /ij) = 1.88 x 104 m m 2
We note tha t the above evaluation of the diameter of the secondary mirror does not take the optical field into account. In order not to stop down the rays a t the edge of the field, especially in a wide field, we have to increase the radius h 2 of the small mirror by dh2l where
dh2 _ TTS~2 y ~ TTf
and y = f 9 is the height of the image a t the focal point. Hence
—»» Ray in max field
F ig . 6 .5 6 A p ertu re field
196 Infrared Thermographyy
F ig . 6 .5 9 H yperboloidal m irro r
Aplanatic telescope This is a telescope corrected on the axis, whilst accepting a large field by correcting for coma. This type of telescope is obtained by a modification of the Cassegrain whereby the mirrors are altered to have a meridional section satisfying the sine condition (Ritchey-Chretien solution).
6 .4 .6 C a ta d io p t r i c te le s c o p e s
The above systems use mirrors with aspherical surfaces th a t are sometimes difficult to make. For systems with a wider field, it is often preferable to use spherical mirrors and then correct the wavefront by means of a refracting system. A large number of correctors has been devised. We present the most popular solutions (Fig. 6.60).
All these solutions use mirrors th a t enable the systems to have an entrance pupil with a large area without an excessive increase in construction cost. The different solutions are compared in the Table below. The Schmidt telescope offers very good performance for an aperture of //2 and focal length of 100 mm. The other solutions show markedly poorer performance.
6 .4 .7 E v a lu a t io n o f im a g e - s p o t a b e r r a t i o n fo r d if fe re n t s im p le o p t ic a l sy s te m s
By considering the propagation of rays in different optical systems, we have obtained relationships for the rapid evaluation of the angular width dO of
200 Infrared Thermogra.phy
ddtotai = (2.5 + 0.95 + 0.78 4- 0.016) 10"3 = 4.25 x 10~3 rad
• Simple lens o f better shape (germanium, n = 4; v | = 88 and z/g1 = 1112) Spherical aberration
0.0087 0.0087_ = = 1 0 - 3 f a d
N 3 (2)
Coma
d0c = — ;— - rrr = ----------- — ----------« = 2.6 x 10 4 rad16 (n + 2) N 2 16 x (4 + 2) x (2)
Astigmatism
Chrom atic aberration (a t 3-5 /im)
= 77—77 = t;— i — - = 2.8 x 10~3 rad2 u N 2 x 8 8 x 2
Chrom atic aberration (a t 8-11/im )
= ——77 = - — — - — - = 2.2 x 10"4 rad 2 v N 2 x 1 1 2 2 x 2
^ to ta l (3 - 5 pm ) = 6.5 x 10_ 3 rad
6 .4 .8 R e f ra c t iv e o p t ic s
Refractive systems are now frequently used in the infrared. The relative abundance of materials and very efficient surface coating facilitate the fabrication of compact, high-grade optical systems.
The sensitivity of infrared detectors is linked to the aperture of the optics, and it is quite common to come across optical systems with apertures greater than / / I in this spectral range.
The resolving power of infrared optics is generally much lower than th a t of systems designed for the visible range.
We saw th a t a simple germanium lens with a focal length of 100 mm and an aperture of f / 2 produces a diffraction spot of the order of 5 mrad for a to ta l field of 12°.
A well-corrected refractive system, consisting of several elements, would produce an elementary field close to the angular limit set by diffraction,i.e., 0.5 mrad in the range 8-13/im . The size of the image spot is then 0.05 mm, and the corresponding resolution is 20 lines per mm. In the visible
We shall calculate the path of aperture rays through the two surfaces ofthe lens. For a half-field 0 = 6° and height of object y\ = 105.1mm, wehave:• O bject aperture angle c*i
h\ h\ 25 _ _. _tan ax = ===== = -------= —-— = 0.025
j4iSi xoi ( - 1000)
a ! = 0.0249947
sin a ! = 0.0249921
• P ath of a paraxial aperture ray First refracting surface:
m = 1; n[ — 4; zoi — —1000; R \ — 107.0088
/ Tlj *01 R\A « ■ ■ i - '
ni R i - xoi ( n i - n[)4 x (-1000 ) x 107.0088
“ 1 + 107.0088 - (-1000 ) (1 - 4)
= 147,95592 mm
P ath through the second refracting surface:
X02 = x'0l — e = 147.95592 — 5 = 142.95592 mm
n 2 = n[ = 4
ti2 — M3 — 1
R 2 = 160,5132
204 Infrared Thermography
Second surface; position of the paraxial image: S ^A '^ = x'02
n 2 xq2 R 2x 02 —
T I 2 R 2 - x 02 ( n 2 - n 2 )
1 x 142.95592 x 160.5132 “ 4 x 160.5132 - 142.95592 x (4 - 1)
= 107.63565 mm
P ath of marginal ray:
S \ A i = £01 = x \
First refracting surface:
sin i'i = R l ~ J l sin ft! = 107,° ^ ~ (~ 10° 0) x 0.0249921 = 0.2585439l07.U08O
208 Infrared Thermography
♦ Size of the exit pupil
h'2 = h\ x 70 = 25 x 0,977 = 24.429 mm
(d) Spherical aberration
• Longitudinal spherical aberration
dx 2 = x 2 — x'02 — 106.2555 — 107.63565 = —1.380 mm
• Transverse spherical aberration
dy'2 = —dx'2 tan a 2 — +1.380 x (—0.2337379) = —0.323 mm
• Normal deviation
<fx'2a '22 1.38 x (0.2296156)2= 0.018 mm
• Third-order spherical aberration coefficient
dx'2 1.38= -26.174
'22 (0.2296156)2
(e) Coma
• Difference due to the sine condition
go $ ( a ') =rii sin c*i 1 x 0.0249921n'2 sin a '2 90 1 x (0.2276032)
$ ( a ') = -
+0.1114 = 1.594 x 10-3
1.594 x 10" 3 1.594 x 10" 3= —0.01431
9 o
• Coma with spherical aberration
-0.1114
= -0.01431 +1.380
107.63565 + 1.2214634= -1 .6 3 x 10" 3
dy2 = y'2 - y'02 = -1 1 .7 1 1 4 5 + 11.7082 = -3 .2 5 x 10 -3 mm
212 Infrared Thermography
• Relative distortion
dr/2 -3 .2 5 x IQ- 3 4^ -11.7082
(h) Conclusion
The above calculations enable us to estimate the value of the spread for each aberration:
• spread due to spherical aberration: Da = 2dt/2 = 0.65 mm• spread due to the coma: D c = 3/? = 0.06 mm• spread due to astigmatism: D a — 2dy2 = 1 mm.In practice, by setting the image plane in the plane of the circle of least
spherical aberration, the latter is reduced to the quarter of its value. The required defocussing is
3 , 1.38 x 3-d x n — -------- ------= —1.035 mm4 1 4
and nD , = — = 0.16 mm
4D c = 0.06 mm
D a = - 2 a 2(t - = 2 x (—0.2296)(—2.216 + 1.035) - 0.54 mm
The values of dO = D / f are:
d03 = 1.6 x 10-3 rad
d0c = 6 x 10“ 4 rad
ddA = 5.4 x 10-3 rad
These are slightly greater than the results given in the example on spreading in an optical system. Here we are dealing with an optical system working at a finite distance, initially planned to give the image of objects located at infinity.
216 Infrared Thermography
F ig . 7 .3 R ad iom etry and sp a tia l analysis
E m (y ' , z ' )d S ' = d F ( y ' , z ' ) m ( y ' ,z ')
The field optics affects this flux through its transm ittance Tc, so th a t the detector finally receives
dFfet = TcdF (y ' , z ' ) m ( y , >z')
The grid can, for example, take the form of a grating of equally-spaced alternate opaque and transparent lines. The grating travels with uniform velocity v' in the Oy' direction in the plane of filtering. Its image, projected back on to the object space moves with the velocity v in the O y direction.
The filtering function in object space is then m ( t / , vt, z) and the flux element received by the detector at time t is
dFdet = TatTopTc^ L (y, z) m (y + vt, z) dydz
The resultant contribution of all the elements of the object is obtained by integration: