physical principles of remote sensing
TRANSCRIPT
Physical Principles of Remote Sensing
Manuscript of the Lecture Course, W7147, University of Bern, Autumn Semester 2008
Deutscher Titel
Physikalische Grundlagen der Fernerkundung
Skript zu Vorlesung, W7147, Universität Bern, Herbst-Semester 2008
Christian Mätzler
Institut für Angewandte Physik (IAP) Sidlerstrasse 5
3012 Bern, Switzerland
http://www.iap.unibe.ch
Downloads from
http://www.iapmw.unibe.ch/teaching/vorlesungen/remotesensing/
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Cover Picture: Global microwave radiometer data: AMSR-E 36 GHz, horizontally polarised brightness temperature of 31 Dec 2005. Values range from violet (115K), dark blue (160K), light blue (180K), green (225K), yellow (260K) orange (275K) to red (294K). Courtesy of the US National Snow and Ice Data Center, Boulder, CO.
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Physical Principles of Remote Sensing Contents 1 Introduction.........................................................................................................................1
1.1 Remote Sensing ............................................................................................................1
1.2 Key requirements, and a dilemma .................................................................................2
1.3 Active and passive methods ..........................................................................................3
1.4 Examples ......................................................................................................................3
1.5 The significance of system models as a motivation .......................................................8
1.6 Conclusions ..................................................................................................................9
1.7 Literature ......................................................................................................................9
2 Electromagnetic Waves .....................................................................................................11
2.1 From Maxwell's Equations to the Wave Equation .......................................................11
2.2 Plane EM waves .........................................................................................................12
2.3 Polarisation of EM waves ...........................................................................................14
2.4 Interaction between EM waves and homogenous media ..............................................15
2.5 Kramers-Kronig relations, and the Hilbert Transform .................................................16
2.6 The electromagnetic spectrum.....................................................................................17
2.7 Literature ....................................................................................................................19
3 Sensors for EM Waves ......................................................................................................20
3.1 Antenna ......................................................................................................................20
3.2 Radar ..........................................................................................................................25
3.3 Radiometer .................................................................................................................28
3.4 Literature ....................................................................................................................30
4 Effective Medium, and Dielectric Mixing Formulas ..........................................................31
4.1 Maxwell-Garnett Formula...........................................................................................31
4.2 Semi-empirical mixing formulas .................................................................................34
4.3 Literature ....................................................................................................................35
5 EM Waves and Boundaries................................................................................................36
5.1 Boundary conditions ...................................................................................................36
5.2 The Fresnel Equations and Snell's Law of Refraction..................................................36
5.3 Waves in layered media ..............................................................................................43
5.4 Lorenz-Mie scattering.................................................................................................47
5.5 Rayleigh scattering .....................................................................................................51
5.6 Literature ....................................................................................................................53
6 Microscopic View of Matter ..............................................................................................54
6.1 Electric dipole, and polarisation of dielectric media ....................................................54
6.2 Types of polarisability ................................................................................................55
6.3 Electronic polarisation ................................................................................................57
6.4 Resonance absorption .................................................................................................58
6.5 Polar molecules in a static field...................................................................................59
6.6 Debye relaxation in polar liquids.................................................................................61
6.7 Space-charge polarisation ...........................................................................................63
6.8 Summary ....................................................................................................................63
6.9 Literature ....................................................................................................................64
7 Spectra of Matter that Matter .............................................................................................65
7.1 Recapitulation and Introduction ..................................................................................65
7.2 The earth atmosphere..................................................................................................66
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7.3 Hydrosphere and cryosphere .......................................................................................72
7.4 Biosphere: Vegetation.................................................................................................83
7.5 Soils and rocks............................................................................................................86
7.6 Software and datasets..................................................................................................89
7.7 Literature ....................................................................................................................91
8 Radiation ...........................................................................................................................93
8.1 Radiance and related quantities ...................................................................................93
8.2 Radiation in thermal equilibrium.................................................................................95
8.3 Radiation in Local Thermodynamic Equilibrium: Kirchhoff's Law..............................96
9 The Radiative Transfer Equation .......................................................................................99
9.1 Radiative transfer without absorption and scattering ...................................................99
9.2 Absorbing medium ...................................................................................................100
9.3 Radiative transfer with absorption, emission and scattering.......................................101
9.4 Formal solution: integral form of the RTE.................................................................102
9.5 Plane-parallel medium ..............................................................................................103
9.6 Solutions without scattering ......................................................................................104
10 Radiative Transfer with Volume Scattering....................................................................107
10.1 Introduction ............................................................................................................107
10.2 The lamella pack as a simple model ........................................................................107
10.3 The transfer equation ..............................................................................................108
10.4 Results and discussion ............................................................................................110
10.5 MATLAB functions................................................................................................112
10.6 Literature for Chapters 8 to 10 ................................................................................112
11 Surface Scattering and Emission....................................................................................114
11.1 Introduction ............................................................................................................114
11.2 Bistatic scattering....................................................................................................115
11.3 Smoothness criteria and specular reflectivity...........................................................118
11.4 Bistatic surface scattering in geometrical optics ......................................................119
11.5 Shadowing effects by the relief ...............................................................................122
11.6 Final remarks ..........................................................................................................124
11.7 Literature ................................................................................................................124
Index ..................................................................................................................................125
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1 Introduction
1.1 Remote Sensing
Remote Sensing is understood as the collection of information relating to objects without
being in physical contact with them. Thus our eyes and ears are remote sensors, and the
same is true for cameras and microphones and for many instruments used for all kinds of
applications.
The term, Remote Sensing (Télédétection, in French, Fernerkundung, in German), arose
around the year 1900 when balloons (and later airplanes) became carriers of people to
altitudes well above the surface. These platforms allowed unprecedented views of the environment, and especially of the earth surface. The bird's eye view enabled an accelerated
progress in Earth Sciences.
The impact of elevated platforms was most pronounced in areas with flat horizons where
natural viewpoints are missing. Therefore masts on ships are remote-sensing platforms. An even less stable platform was used by Inuit hunters: They threw a man or a child up in the air
to search for seals. Thus "remote sensing" has been essential for survival.
Most dramatic was the appearance of the first images
of the earth received from satellites. The figure to the
right is the first picture obtained by the first weather satellite, TIROS - 1 in April 1960. "The TIROS
Program's first priority was the development of a
meteorological satellite information system. Weather
forecasting was deemed the most promising application of space-based observations".
Although not of high quality, these television images
already indicated the potential to obtain geophysical information over large regions, e.g. on the spatial
distribution of clouds and snow and ice cover because
their reflective properties were clear signatures above the darker terrestrial background.
Remote sensing originated from (1) human vision on special platforms, complemented by (2)
the recording of vision information, and (3) photogrammetry, the quantitative exploitation of image records. Later the methods were extended to spectral ranges beyond the human eye.
For this purpose special sensors and instruments had to be developed. This process is still
ongoing.
Remote Sensing is not a scientific discipline in the classical sense; it is rather a collection of
a large variety of diagnostic methods, mainly using electromagnetic waves covering the
spectrum from radio waves (wavelength > 1 m) to gamma rays ( < 10-12 m). In some
cases sound waves or other elastic waves are also in use, especially where electromagnetic methods fail. It is obvious that very different techniques and skills are required in the different
parts of remote sensing. Not only the techniques are multidisciplinary, the applications cover
a wide range of human disciplines, e.g. archaeology, botany, climatology, geology, hydrology, meteorology, security aspects, etc.
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1.2 Key requirements, and a dilemma
There are four requirements for any remote sensing method:
1) An instrument on a given platform is needed that can detect and measure the information-
carrying signal. The signal must be calibrated to defined standards to allow reproduction of
the observation under the same conditions. Errors (geometry, timing, radiometry, spectrum, polarisation) should be specified.
2) A signal from the object to the observer must be propagated in an unambiguous way and
without serious loss. Ideally, the propagation is along a straight line with constant velocity and without attenuation. In other words, the propagation medium should be transparent and
homogeneous, like free space for electromagnetic waves.
3) An interaction must exist between the sensing wave and the object to be measured. The interaction can be emission or scattering of radiation, or it can be a modulation (scintillation)
or delay of a propagating wave.
4) The signal must be unambiguous to allow extraction of the correct information. We say
that the signal must contain some sort of an object signature. The transformation from the calibrated sensor signal to the object information is called inversion. The link between the
object and the signal is a model that is able to simulate the signal from the object information.
Such forward models are based on the physics of the interaction between the object and the sensing waves. The development of forward models is a main task in remote sensing.
Models derived from physical principles are more general than empirical rules, but both types
exist. The forward models are also useful to assess the sensitivity of a signal at a special frequency and polarisation to a given perturbation of the environment. In this way the models
contribute to the optimisation of existing methods and development new methods.
All 4 requirements are fulfilled in successful methods. They may utilise different types of
waves and different types of physical interactions. However, a dilemma of remote sensing is posed by the opposite needs of Requirements 2 and 3: A transparent medium does not
interact with the wave. But an interaction is needed with the object to be sensed. Since the
object is usually embedded in the propagating medium, and often, the propagating medium is the object to be sensed, the dilemma is between the needs for and against the interaction.
Remote Sensing needs a balance between interaction and transparency. The problem is
relaxed if the two media are clearly separated, for instance for sensing of surface properties.
We should realise that remote sensing is not perfect, and that it cannot be used in all diagnostic tasks. Often there are ambiguities in the interpretation. We say, the inversion of
remote-sensing data is ill conditioned. The skill of remote sensing depends on additional
information available from various sources, such as
1) in-situ observations
2) previous remote-sensing observations
3) the quality of the forward model
4) the understanding of the processes describing the behaviour of the objects
5) maps and inventories, for parameters that can be regarded as constant
6) imposed limitations of the parameter range
and on the best combinations of them.
Due to measurement errors and model errors, the various inputs can lead to conflicting
results. This may happen if there are more data points than parameters to be determined.
Then the task is to find the best or optimum solution. These problems are studied in the field of optimal estimation (Rogers, 2000).
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1.3 Active and passive methods
1) Passive sensing methods make use of naturally available signals, such as thermal
radiation of terrestrial objects (Figure 1.1: a), sunlight (b), starlight, the cosmic background
radiation, but also lightning and thunder, the song of birds and whales. Sometimes, artificial
radiation of opportunity, such as man-made nightlight, or radio waves from radio stations, is used for passive remote sensing, too. But more often man-made signals are problematic,
impeding the application of remote sensing. Therefore the use of the radio-frequency
spectrum is regulated by the radio regulations of the International Telecommunication Union (ITU). Thanks to these rules, certain frequency bands are kept free for passive sensing.
2) Active methods (Figure 1.1: c) sense artificially produced waves after they interacted with the objects to be sensed. Examples are radar, sonar, lidar, GPS, but also a photo camera
with a flashlight. An active method is called monostatic if transmitter and receiver are
collocated, otherwise the method is called bi-static, or even multi-static if several receivers at
different locations are used.
Figure 1.1: Illustration of
examples with
passive (a, b) and active (c) methods
in remote sensing
(from Schanda
1986).
1.4 Examples
1) Panorama for geographic and topographic information (mapping and orientation)
Instrument / platform Propagation
(wavelength range)
Interaction Signatures
Human vision,
drawing, compass / mountain top, tower
Same for optical camera, or
triangulation
Through clear air
Limited to daylight and to suitable
illumination
Scattering of
sunlight
Contour, brightness
and colour of landscape and objects
versus direction and
elevation; objects in front of others are
closer
Fig. 1.2: Partial panorama (direction SE) from the Zimmerwald Observatory (Dec. 12, 2006).
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2) Underwater exploration, bathymetry
Instrument / platform Propagation
(wavelength range)
Interaction Signatures
Optical camera /
ship, diver
Clear water, shallow
depth ( 0.4-1 μm)
Scattering of sunlight Reflectance spectra,
contours.
Sonar / ship,
submarine
Deep penetration of
sound waves
Backscattering of
sound waves from
bottom surface, from animals, ships, etc.
Depends on object
Usually easy to
locate depth to ocean-bottom
Sound navigation and ranging (sonar), or echo-sounding, in underwater exploration works in
the same way as bats use sound for aerial navigation. There is a clear advantage of sonar over electromagnetic sensors because sound waves can propagate in water over very long
distance whereas electromagnetic waves are strongly damped in water, only visible light
allowing penetration into 10 to 100 m deep water. A problem with sonar in water is the
significant dependence of the sound speed* on salinity, temperature and density (or depth) of the water (Table 1.1). This means that sound may not propagate along straight lines, the
location of objects becoming ambiguous.
*Comment: The wave path is determined by the Principle of Fermat, which states that the
wave chooses the path with the fastest propagation.
Table 1.1: Speed of sound (m/s) in pure water and in sea water at P=0.1 MPa (sea surface) and at 100 MPa (10 km depth), from http://www.akin.ru/spravka_eng/s_i_svel_e.htm
Temperature (C) Pure water
(surface, 10 km)
Sea water S=3.5%
(surface, 10 km)
0 1402, 1578 1449, 1623
10 1447, 1618 1490, 1659
20 1483, 1650 1522, 1687
30 1511, 1677 1546, 1710
3) Detection of snowcover (regional, global)
Instrument / platform Propagation
(wavelength range)
Interaction Signatures
Optical camera /
tower, aircraft, satellite
Through clear air,
( 0.4-1 μm)
Scattering of sunlight High reflectance,
especially for fresh snow.
Thermal IR imager /
satellite
Clear air
( 8-14 μm)
Emission of thermal
radiation T 273.15 K
Microwave
radiometer / satellite
Clear air, clouds
( 3 mm – 10 cm)
Scattering of sky
radiation and
emission of thermal
radiation
Dry snow with
characteristic
reflectance,
wet snow with high emissivity
Microwave radar /
satellite
Clear air, clouds,
precipitation
( 3 cm – 10 cm)
Backscattering of
microwave radiation
Low backscatter,
especially for wet
snow
Gamma-ray detector
(scintillation counter)
/ low-flying aircraft
Short range < 300 m
in air ( < 10-10 m)
Natural radiation of
minerals (e.g. 40K, 238U, 208Tl) in the top 10 cm of the soil
Attenuation by snow
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Figure 1.3: Example of a
satellite true-colour image of
Scandinavia showing snow
on 15 March 2002 using the visible Bands 1,4,3
RGB of MODIS,
from Xavier Planella Robisco (2005).
http://modis.gsfc.nasa.gov/
The snow signature is quite
clear with respect to snow-
free land and sea. However ambiguities appear with
respect to clouds and sea
ice. Question: How could the
ambiguities be resolved?
4) Measurement of atmospheric water vapour H2O
The following examples derive the amount of vertically Integrated of Water Vapour (IWV),
also called total column water vapour (CWV).
Instrument / platform Propagation
(speed)
Interaction Signatures
GPS system with
network of fixed receivers / surface
and satellites
All weather
(very close to speed of light in
vacuum)
Delay of signal
speed by water vapour and dry air
Unique for total column
of H2O if surface air pressure is known
Microwave radiometer / surface
Air, smoke, clouds Emission line of H2O
frequency f 22 GHz
( 14 mm)
Distinction from cloud emission by 2nd
frequency
Microwave
radiometer / satellite
Air, smoke, clouds Emission line of H2O
frequency f 22 GHz
( 14 mm)
Distinction from surface
and clouds by additional channels (frequency,
polarisation)
Sun photometer /
surface
Clear air with
sunlight
Absorption band of
H2O ( 940 nm)
Distinction from aerosols
by additional channels
MERIS / satellite
(ENVISAT)
Clear air with
sunlight
Absorption band of
H2O ( 940 nm)
Distinction from surface
and aerosols by
additional channels
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Figure 1.4: Top: GPS Zenith Total
Delay (ZTD) and Zenith Hydrostatic
Delay (ZHD), both in m, versus time during 20 days at Gütsch
(Andermatt) in November 2000.
Bottom: Derived integrated water vapour (kg/m2).
The ZTD curve is the delay measured by GPS. The delay is the
difference in distance between a
wave travelled through vacuum and
through air. This distance is related to the refractive index. ZHD is
computed from the measured air
pressure at the surface, whereas the difference, ZTD-ZHD, is due to
changes of IWV.
It is obvious that water vapour changes more rapidly than
pressure.
Figure 1.5: Water-vapour column (g/cm2) above the alpine region as derived from MERIS on ENVISAT. Note the disturbances in areas where clouds occur.
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5) Detection and localisation of lightning
Instrument Propagation (speed) Interaction Signatures
Human observer Viewing lightning
(speed of light),
Hearing thunder (sound speed)
Light flash and
sound burst emitted
by lightning
Characteristic flash
in view direction,
delay of thunder proportional to
distance
Electromagnetic
lightning detector network
All-weather capability
(ground-wave close to speed of light)
Radio burst (sferic)
emitted by lightning
Propagation distance
from time-of-arrival measured at several
stations.
Figure 1.6: European Network of
lightning stations
http://www.sferics.physik.uni-
muenchen.de/
6) And many more
The presented examples were intended to illustrate the concepts of widely different methods
and applications. There exist many more. Add your own favourite examples...
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1.5 The significance of system models as a motivation
As mentioned in Section 1.2, the potential and success of remote sensing heavily depends
on the existence of complete information on the states and processes to be investigated. The tasks involved are, apart from the generation of forward models, advancements of models
that describe the processes to be studied. Such process models or system models should
represent the state-of-the art of the scientific understanding behind the observations. These models represent the laws, which govern the components of the earth, like the atmosphere,
the ocean, the cryosphere, the land, the solid earth, any subsystem, but also the interaction
between these components.
Earth System Models are developed through a complex and systematic process of comparison with observations, called biases, point to the incorrect representation of some
processes that must be improved in the models, or to systematic observational errors that
must be corrected (ESA, 2006). Once the biases are reduced to a minimum, the remaining random differences between the models and the observations can be exploited to further
improve the model formulation, or to create a set of model variables representing the reality
at a specific point in time. The model can then be used for predictions. This process is called assimilation. It forms the heart of Earth System Science. An example is shown in Figure 1.7.
Data assimilation opens the way for optimised state analysis, and subsequently more reliable
prediction of the state of the Earth, and for re-analysis. Re-analyses are long time series of
historical data obtained from state-of-the-art data assimilation, using all available observations. The European Centre for Medium-Range Weather Forecasts (ECMWF) has
led the meteorological community in the creation of re-analysis. The most recent of these
projects, ERA-40, has allowed the production of a 45-year (1957-2002) time series, using all available satellite and in-situ observations of the physics and dynamics of the atmosphere.
Figure 1.7: Principle of
variational retrieval with data-assimilation. The
background information is
given by a system model, here vertical profiles of
temperature and dew-point
temperature. The observations are radiometer
data at various frequencies.
An optimal solution is found
by minimising the overall error (from Hewison and
Gaffard, 2006).
Future re-analysis projects will encompass several components of the Earth System. In the
not-too-distant future, the Earth Science community will have the capacity to produce coupled re-analyses of the Earth System, including weather, atmospheric composition, state
of the ocean, amount of moisture in the continental soils, hydrology of large rivers, and the
state of the biosphere and cryosphere. This, in turn, will open the way to an objective verification of the predictional capacity of Earth System models. Special challenges of Earth-
System Science are listed in ESA (2006). For all these reasons it is important for remote
sensing to develop accurate and physical forward models that can be applied to the system models for required comparison with observations.
The above statements are the motivation for following the present lecture and, well beyond,
for studying the physics of remote sensing.
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1.6 Conclusions
This introduction gave an overview on the properties and principles of remote sensing. It was
stated that for any successful application, four requirements are to be fulfilled: (1) technical
and logistic feasibility (instrument and platform), (2) transparency of the propagating medium
from the object to be sensed to the observer, (3) interaction between the sensing wave and the object, and (4) a signature, to allow the retrieval of the requested information from the
observed signal. It was found that a balance between transparency and interaction is needed
to get optimum results.
Several examples were discussed to illustrate the methods and the meaning of the 4
requirements. For certain tasks, several methods were identified, each of which having
different properties. Whereas satellite observations are optimal to unveil large-scale features, surface-based observations are best to monitor dynamic processes.
The motivation for the physical approach is based on the required link with Earth-System
models allowing the application of data assimilation, leading to predictions and to a deeper
understanding of the Earth or its major components as complex systems.
In the further chapters we will concentrate on the physical properties of nature relevant to the
requirements for remote sensing, including an introduction to remote-sensing instruments.
More information can be found elsewhere, see e.g. the list of previous lectures:
http://www.iap.unibe.ch/content.php/teaching/. Scripts are also available at the ExWi library.
1.7 Literature
The following list is a small selection of monographs mainly from the inventory of the ExWi Library (BEWI):
Introductory books
A booklet written by
E. Schanda (1986), Physical Fundamentals of Remote Sensing, BEWI: XKA 118 etc. is still very useful today. Similar are:
W.G. Rees (2001), Physical Principles of Remote Sensing, BEWI: XKA 212, and
C. Elachi and J. van Zyl (2006), Introduction to the Physics and Techniques of Remote Sensing, 2nd Ed. BEWI: XKA 214.
G.W. Petty (2006), A First Course in Atmospheric radiation, 2nd Ed. Sundog Publishing, Madison, Wisconsin. Excellent introduction also for the physics of remote sensing, BEWI: XJX 205
In-depth studies
A Manual of Remote Sensing has been published by the American Society of Photogrammetry. The two volumes
of the second edition from 1983 are available at our library BEWI: XKA 116, 117.
Ulaby, Moore and Fung (1981, 1982, 1986), Microwave Remote Sensing, Vol. 1,2,3 (BEWI: XKA 129, 130).
Monographs on special topics
L. Tsang, J.A. Kong, and R.T Shin (1985), Theory of Microwave Remote Sensing, BEWI: XKA 136
G.L. Stephens (1994), Remote Sensing of the Lower Atmosphere, BEWI: XKF 203
M.A. Janssen (Ed), (1993), Atmospheric Remote Sensing by Microwave Radiometry. BEWI: XKF 201.
C. Mätzler (Ed), (2006), Thermal Microwave Radiation: Applications for Remote Sensing, IET Electromagnetic Waves Series 52, London UK, BEWI: XKJ 209.
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H. Sauvageot (1992), Radar Meteorology, BEWI: TEF 202. Lecture notes on this topic are also available, see
http://www.iapmw.unibe.ch/teaching/vorlesungen/radar_meteorologie/
C.L. Rodgers (2000), Inverse Methods for Atmospheric Sounding (BEWI: MAF 206).
Earth System Models
ESA (2006), The Changing Earth – New Scientific Challenges for ESA's Living Planet Programme, SP-1304
Journals
Many journals deal with remote sensing, the following ones being fully dedicated to the topic: 1) IEEE Transaction on Geoscience and Remote Sensing (at BEWI)
2) Remote Sensing of Environment (at BEWI)
3) International Journal of Remote Sensing (library of the Geographical Institute)
Additional reference
T. J. Hewison and C. Gaffard (2006), Combining data from ground-based microwave radiometers and other instruments in temperature and humidity profile retrievals, WMO Technical Conference on Meteorological and
Environmental Instruments and Methods of Observation, TECO-2006, Dec. 4-6, Geneva, Switzerland.
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2 Electromagnetic Waves
2.1 From Maxwell's Equations to the Wave Equation
The electromagnetic (EM) fields, i.e. the electric field E, the displacement field D, the
magnetic field H, and the magnetic induction B, are governed by Maxwell's Equations (Kong,
1986):
H =Dt
+ j; D = e (2.1)
E =Bt
; B = 0 (2.2)
where =dx,dy,dz
is the Nabla Operator (here applied as rotation, , and divergence,
), j is the electric current density and e the electric charge density. To understand how
EM waves propagate, we simplify the situation to homogeneous and isotropic media far
away from regions with sources (no transmitters, i.e. no isolated charges: e=0). We will
consider complex time-harmonic fields with the time dependence, exp( i t) = cos( t) -
i sin( t), where i is the imaginary unit, i = 1, t is time and the angular frequency.
Complex fields are chosen as usual convention for easier computation. The physical fields
are their real parts, e.g. Re e i t( ) = cos( t) . Based on Ohm's Law,
j= E (2.3)
a (complex) current density j will be excited in a conducting medium by the electric field E of
the wave, where is the conductivity of the medium. Now, Maxwell's Equations become
H = i D+ E; D = e = 0 (2.4)
E = i B ; B = 0 (2.5)
We eliminate B and D by the linear Constitutive Relations
D = 0E + P = ' 0E and B = μ0H +M = μμ0H (2.6)
where ' is the (relative) dielectric constant (also called relative electric permittivity),
0=8.854 10-12As/V/m the vacuum permittivity, μ the relative magnetic permeability, μ0=4 10-7
Vs/A/m the vacuum permeability, and P and M, respectively, are called electric and magnetic
polarisability. We can further simplify the right-hand side of the first equation in (2.4) i D+ E = ( i ' 0 + )E to i 0E , by defining a complex relative dielectric constant
= '+i " ; "=0
(2.7)
The real part is the original relative dielectric constant, and the imaginary part is related to
the conductivity by (2.7). The complex has a full physical meaning. Conductivity and
imaginary permittivity are different representations of the same effect. It turns out that also the magnetic permeability can be complex μ = μ'+iμ". The final form of Maxwell's Equations
for harmonic EM waves in homogeneous media then read as follows
H = i D = i 0E; E = D = 0 (2.8)
E = +i B = +i μμ0H ; B = H = 0 (2.9)
with the generalised constitutive relations (note the change of D which now includes E)
D = ( ) 0E and B = μ( )μ0H (2.10)
12
where and μ usually depend on angular frequency .
Remarks:
1) In chiral and in bi-isotropic media, also the electric and magnetic fields are linearly related (Kong, 1986; Sihvola, 1999).
2) In anisotropic media and μ are tensors. Plane waves propagate independently for
special eigenmodes (dichroism).
Elimination of H from (2.8) to (2.9) leads to
E = 2μμ0 0E . But E = ( E) E ,
and since source regions are avoided, E = 0. Then we get the Wave Equation for E in the
unbounded homogenous medium:
c 2 E =2E (2.11)
The same equation can be found for the fields, D, B, H, where =2
dx 2+
2
dy 2+
2
dz2
is the
Laplace Operator, and
c 2 0μμ0( )1=c0n
2
; c0 =1
0μ0=2.99793 10
8m/s (2.12)
It turns out, see Equation (2.32), that the real part of c is the phase velocity of the wave in the
medium, c0 is the speed of light in vacuum, and n is the refractive index of the medium:
n = μ ; for μ =1 n = . (2.13)
2.2 Plane EM waves
An important type of solution of (2.11) in a homogeneous medium is a plane wave
propagating in an arbitrary direction given by the wave vector k. For the electric field of the
wave we use the Ansatz
E(r, t) = E0 exp(ik r i t) (2.14)
where E0 is the amplitude and r is the position in space. The divergence equation in (2.8)
leads to E0 k = 0. This means that the electrical field is oriented perpendicular to the wave
direction, k, but otherwise is arbitrary. Furthermore, also the magnetic field is of the form
H(r, t) =H0 exp(ik r i t) (2.15)
H is perpendicular to both k and to E. This follows from H = ik H = 0 , together with
E = ik E = i μμ0H (2.16)
Inserting (2.14) in the wave equation yields with E = (ik ik)E = k 2E , ( k 2 k k )
k = ±c
= ±n
c0= nk0; k0 c0
(2.17)
and k0 is the vacuum wave number. The Equation for k is the dispersion relation of EM
waves in unbounded space. The ± signs mean that the wave can propagate forward or
backward in k direction. Since k is arbitrary, the dispersion relation holds for all directions, and it is independent of the direction of the electric field (polarisation).
Equation (2.16) allows to relate the amplitudes of the fields, defining the wave impedance
13
ZE0
H0
=μμ0k
= cμμ0 =μμ0
0μμ0=
μμ0
0
= Z0μ
; (2.18)
Z0 =μ0
0
377 ; (2.19)
and Z0 is the vacuum impedance.
The wave intensity is the power flux (W/m2) transported by the wave. This quantity follows from the Poynting Vector, defined by
S E H (2.20)
S is a vector pointing in direction k of wave propagation. It turns out that the wave intensity I is given by the time average of the physical part of S, which can be written as
S = 0.5Re E H*( ) (2.21)
where * means conjugate-complex value. Inserting (2.14-18) in (2.21) and taking the
magnitude, we find
I = S =E0
2
2Re
1
Z *
; if μ=1, then I =
E0
2
2Z0n' (2.22)
Note that Z may be complex. From (2.22) we learn that the wave intensity is proportional to
the square of the electric field.
Figure 2.1: One period of a plane EM wave with
linear polarisation
(direction of the E field is constant). The horizontal
axis is the phase kr t with the propagation path, r.
Problems
1) Proof Equation (2.21). Hint: Express E = (E'+iE")e i t , H = (H'+iH")e i t and note that
the real parts of these quantities are the physical fields. Express these parts, multiply them
and average over time to confirm (2.21). For further reading, see e.g. Schanda (1969), p. 30-
31.
2) Explain why the wave equation (2.11) does not apply for inhomogeneous media. Hint: Show what additional terms appear in the derivation from (2.8) and (2.9) if depends on the
location.
14
2.3 Polarisation of EM waves
Polarisation of EM waves refers to the direction of the electric field. If it is constant as in
Figure 2.1, then we call the polarisation linear. Let us consider (Fig. 2.2) an electromagnetic
wave propagating in the z direction with the E – field in the x-y plane given by
Ex = E1 exp(ikz i t), Ey = E2 exp(ikz i t) (2.23)
x
E(t=0) E1
z y
E2
x
E(t=0) E(t)
t
z y
E(t= /2 )
Fig. 2.2a: E field at z=0 for E1=E2. The two
components oscillate in phase. Polarisation is linear, but rotated by 45°. E is shown for t=0.
Fig. 2.2b: E field at z=0 for real E1, E2=iE1
at 3 different times. Polarisation (E) is rotating clockwise with time. This is called
circular polarisation.
Each component alone describes the E field of a linearly polarized wave. But together, the
situation is more complex. In Figure 2.2a, the two components oscillate in phase, and thus a
linear polarisation results again. In Figure 2.2b at t=0, only an x component exists, because
Ey is purely imaginary. With increasing time the physical y-component first increases as E1 sin t , while the x component decreases as E1 cos t . The two components oscillate with
a phase difference of 90°. This is the motion of a circle with constant radius E1.
In the general situation the polarisation is elliptical, that is, the tip of the E vector describes an ellipse. The actual behaviour depends on the relationship between the complex amplitudes,
E1 and E2. The rotation can be clockwise or anti-clockwise.
Stokes Parameters
Instead of dealing with phase angles and complex numbers, an easier way to describe the polarisation of a wave is by the Stokes Parameters I, Q, U, V . All parameters have the
dimension of an intensity, and they are defined by
I =1
2Z0Ex
2+ Ey
2
or I1 =
1
2Z0Ex
2 (2.24a)
Q =1
2Z0Ex
2Ey
2
or I2 =
1
2Z0Ey
2 (2.24b)
U =1
Z0Re ExEy
*[ ] (2.24c)
V =1
Z0Im ExEy
*[ ] (2.24d)
Here the brackets mean averaging over time (usually many periods). Alternative
conventions are used for the quantities of (2.24a, b), where I1 and I2 are called modified Stokes parameters. Note that Q and U depend on the coordinate system used, but the
degree of polarisation
15
p =Q2
+U 2+V 2
I (2.24e)
is independent of the coordinate system used. For unpolarised radiation p = 0 . Linear
polarisation is described by Q and U, whereas circular polarisation is described by V.
Problem: Express the Stokes Parameters for the 2 examples in Figure 2.2.
2.4 Interaction between EM waves and homogenous media
For homogeneous, unbounded media the interactions between EM waves and matter
describe wave absorption and wave velocity. The medium is characterised by (including )
and μ, and it turned out that the refractive index and the impedance are related parameters.
Note that for natural media n=n'+in" is complex, and essentially non magnetic (μ=1), the
right-most form of (2.13) applies for n, and the impedance follows directly from n, since
Z=Z0/n. Thus we are left with one independent, complex quantity, , consisting of a real and
an imaginary part. Then we have from (2.13): = '+i "= n2 = n'2 n"2 +2in'n" to find
'= n'2 n"2 ; "= 2n'n" (2.25)
n'='+ '2 + "2
2
', "<< '
" /2, '<< "
; n"=
"
2n'=
'2 + "2 '
2 (2.26)
The meanings of n' and n" become apparent from the properties of a plane wave propagating
parallel to an arbitrarily chosen r axis. Then
E = E0 exp(ikr i t) (2.27)
where k of (2.17) is complex
k = k'+ik"= ±nk0 ; k'= ±n'k0 , k"= ±n"k0 ; k0 c0 (2.28)
Inserting (2.28) into (2.27), we get for the + sign: E = E0 exp(ik'r i t)exp( k"r) ; The physical field (for real E0) is given by
Re(E) = E0 cos(k 'r t)exp( k"r) (2.29)
For the - sign we get
Re(E) = Re E0 exp( ik'r i t)exp(k"r)( )=E0 cos(k'r + t)exp(k"r) (2.30)
Whereas (2.29) is a wave propagating in the positive r direction, (2.30) applies for waves in
the negative direction, and both waves are exponentially damped along their path (Figure 2.3). The damping is due to Ohmic currents, which transform the wave energy into heat. This
is called wave absorption. The distance,
ds =1
k"=
c0n"
, (2.31a)
after which the field is reduced by a factor e 1, is the field-penetration depth (or skin depth).
Since the wave intensity is proportional to E2, its spatial variation is an exponential decay
exp( 2k"r) . The damping coefficient, 2k", of the intensity is called absorption coefficient
16
Figure 2.3: Damped EM
wave and its electrical field envelope; here c stands for
c0, and propagation is along
the z axis.
a , and it is related to the field-penetration depth
a = 2k"=2
ds=2n"
c0 (2.31b)
The phase velocity cph (speed of points with constant phase in the propagation direction) and
the group velocity cg (speed of a wave train or signal), respectively, of the wave are
determined by
cph =k'
=c0n'
and cg =d
dk (2.32)
The frequency is the number of periods per second: = /2 , and the wavelength is the
spatial period of the wave: = 2 /k ', in vacuum: 0 = 2 /k0 .
Problem: Determine the wavelength and skin depth of a medium with = 2 + 0.3i , μ=1, at the
frequency =2.4 GHz.
2.5 Kramers-Kronig relations, and the Hilbert Transform
The medium descriptors n, and μ depend on frequency (or wavelength), and on the
physical state and chemical composition, see von Hippel (1954). It turns out that the real and
imaginary parts of these functions are not completely independent. The assumption of causality requires that an effect cannot exist before its cause. Here the cause is the electric
field E, and the effect, D, is the excited displacement field. The formulation of causality
together with the linearity between E and D leads to the integral relations named after Kramers and Kronig (Kong, 1986):
'( ) =1PV
"( ')
'd '
+
= Hi "( ){ } (2.33)
"( ) =1PV
'( ')
'd '
+
= Hi '( ){ } (2.34)
where PV means the Cauchy Principal Value of the integral (due to the singularity at '= ).
Equation (2.33) is also known as Hilbert Transform, Hi{ }, and (2.34) as inverse Hilbert
Transform (which is equal to the negative Hi Transform). They mean that if either the
complete real or the complete imaginary spectrum is known, the other spectrum follows (apart from a constant high-frequency limit ) from the above relations. They are useful to
check the physical correctness of model functions and of experimental data. Similar
equations also hold for μ and n. Note that the Hilbert Transform of a constant is zero.
17
Therefore a constant 1 is subtracted from '. Some Hilbert Transforms are shown in
Table 2.1.
f (x) Hi f (x){ }
cos x sin x
sin x cos x
sin x
x
cos x 1
x
1
1+ x 2
x
1+ x 2
(x) 1
x
Table 2.1: Hilbert
Transforms of some
functions (from Bracewell,
1965). Note that
Hi Hi f (x){ }{ } = f (x)
The fact that sin and cos functions are Hilbert Transforms allows us to formulate the following
rule: If the real part of the dielectric constant can be expressed by a Fourier series of cos
functions, then the imaginary part is the respective series of sin functions. This is always the case because ( ) = * ( *) ; thus '( ) is a symmetric function of the real frequency axis
whereas ' '( ) is antisymmetric.
2.6 The electromagnetic spectrum
The frequency = /2 is the independent variable of waves, and the functional
dependence on frequency is called spectrum. The electromagnetic spectrum is extremely
wide, and therefore different units are in use as shown in Figure 2.4. The main unit of frequency is the cycle per second (s-1), which is also called Hertz. For vacuum (n=1),
Equation (2.32) uniquely relates the frequency with the vacuum wavelength, 0, according to
0 =c0 =
2
k0 (2.35)
Therefore the inverse wavelength, 1/ 0, is proportional to frequency and to k0, which is called
wave number. As a matter of confusion, the term wave number is also used by spectroscopists for the spatial frequency: 1/ 0 = k0 /2 .
The photon energy Ep = h h (where h is the Planck constant) is another measure of
frequency. If the photon energy is expressed in eV (electron Volt) and the frequency in Hz
(Hertz), then
Ep = 4.1383 10 15 (2.36)
The range of values for all types of units is shown in Figure 2.4.
18
Fig. 2.4: The electromagnetic spectrum in terms of wave number, wavelength (both in
vacuum), photon energy and frequency ( f ), and some of the nomenclature of spectral
bands used in engineering (from Kong, 1986).
19
Visible spectrum
Fig. 2.5: Visible spectrum on a wall
created by sunlight after crossing a glass
prism.
The wavelength ranges (in nm) of the visible colours are, according to Petty (2006):
violet 390 – 460
dark blue 460 – 490
cyan (light bl) 490 – 510 green 510 – 550
yellow-green 550 – 580
yellow 580 – 590 orange 590 – 620
red 620 – 760
Ultraviolet (UV)
Extreme UV 10 – 100
UV- C 100 – 280 absorbed in mesosphere (>50 km) by O2 UV- B 280 – 320 reduced by O3, responsible for sun burn
UV- A 320 – 390 99% of solar UV at sea level, not dangerous for living tissue
2.7 Literature
R. Bracewell (1965), The Fourier Transform and its Applications, New York, BEWI: GQE 119.
D. J. Griffiths, Introduction to Electrodynamics, (1999) 3rd Ed., BEWI: OGA 151. Good introductory book, includes
introduction to mathematical concepts (e.g. vector analysis.
A. von Hippel, Dielectrics and Waves, 1st Ed. (1954) BEWI: TEA 149, 2nd Ed. (1995) BEWI: VTZ 201.
J.A. Kong, Electromagnetic Wave Theory, New York 1st Ed. (1986) BEWI: TEA 150, 2nd Ed. (1990) ETH Library.
G.W. Petty (2006), A First Course in Atmospheric radiation, 2nd Ed. Sundog Publishing, Madison, Wisconsin.
Excellent introduction also for the physics of remote sensing, BEWI: XJX 205
In German:
G. Eder, Elektrodynamik, BI Hochschultaschenbuch 233 (1961), BEWI: OGA 130.
E. Schanda (Ed.), Theorie der elektromagnetischen Wellen, Birkhäuser Verlag, Basel (1969), BEWI: OGF 120, TEA 152, 156. Dieses Buch kann als kompakte Einführung in Vektoranalysis (Kapitel von H. Carnal), ins
Verständnis der Maxwellgleichungen (Elektrizitätslehre und die Maxwellsche Theorie von E. Schanda) und Wellen, Antennen (weitere Kapitel) empfohlen werden.
20
3 Sensors for EM Waves Remote sensing is based on information obtained from electromagnetic waves and radiation.
This chapter is limited to the concepts and serves as an introduction focussing on essential properties of sensors, which can be used for this task. As mentioned in Chapter 1, we
distinguish between active and passive remote sensing. Active sensors need signal
generators and transmitters, and both methods need receivers and detectors.
3.1 Antenna
Transmitters and receivers need elements that allow the transition between the propagating radiation in free space and the guided radiation in the sensor, and vice versa. The transition
is realised by the antenna. This term is derived from Latin for 'sail' in analogy to wind and
surface waves and is well known for radio- and microwaves. Although not often used in other domains, the term, antenna, is of relevance to other wavelength ranges, but may be hard to
realise. Special to the antenna is that a single, polarised wave mode is exited from a given
feed point, meaning that there is an unambiguous phase and field relationship between the feed point of the antenna and any point in space. Thus antennas radiate fully polarised
radiation. Incoherent radiators and detectors, on the other hand, cannot provide this property.
3.1.1 Antenna types
There exists a large variety, from nearly isotropic dipole radiators to highly focused reflector
and array antennas. Some examples are shown in Figure 3.1. All types can be characterised by the parameters to be described in the following section. Here we will focus on horn
antennas.
Figure 3.1: Antenna types,
from Ulaby et al. (1981).
21
3.1.2 Transmitting antenna
Most important is the antenna pattern, i.e. the directional distribution of the radiated power.
As the antenna is a point source, its directional pattern can be described in spherical
coordinates with the antenna in the centre.
Figure 3.2: Spherical
coordinates for antennas
(top), and transition to quasi-plane wave in the far field
(bottom), from Ulaby et al.
(1981).
Figure 3.3: Typical radiation
pattern of a horn antenna.
Shown is the normalised
radiation intensity p when expressed in dB (pdB),
defined as pdB = 10 10log(p).
The x axis corresponds to the angle 0 of Figure 3.2
where 0 marks the direction
of the peak intensity (usually 0 or 90°).
The directivity D( , ) describes the directional distribution of the radiated power P1. In the
far field (Figure 3.2), that is at distances r larger than
r >2d2
(3.1)
where d is the maximum diameter of the antenna, the radiation intensity S( , ) is
proportional to 1/r2. Therefore we can define a quantity that is independent of distance by
D( , ) = 4 r2S( , )
P1 (3.2)
22
Here D( , ) is the directivity in direction ( , ) . Integration of S( , ) over a sphere with
radius r around the antenna must give the total radiated power P1. Therefore the integral over
directivity must give
D( , )d = D( , )sin d d00
2
= 44
(3.3)
Also used is the antenna gain G( , ) . It is similar and proportional to D( , ) , with the
difference that in (3.2) the radiated power in the denominator is replaced by the total power
Pin fed into the antenna. Thus,
G( , ) = rD( , ) (3.4)
where r =P1 /Pin is the radiation efficiency of the antenna. Ideal antennas are lossless,
meaning that gain and directivity are the same.
The quantity shown in Figure 3.3 is the directivity normalised to the maximum
p=D( , ) /Dmax ( , ) and expressed in decibel (dB).
Special cases:
1) Isotropic antenna: D is independent of direction. Then, with the requirement in Equation
(3.3) we find that D=1. Small antennas in comparison to the wavelength are nearly isotropic.
Therefore their directivity is always on the order of 1.
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
0
200
400
600
800
1000
1200
D
teta
Figure 3.4: Directivity of a boxcar antenna with Dmax=1000. From
Equations (3.5) and (3.6) we get
max = 0.0632 , corresponding to a full
beam width of 7.2°.
2) Boxcar antenna: D=Dmax inside the antenna beam defined by a solid angle e, and D=0
elsewhere. Insertion into Equation (3.3) leads to
Dmax = 4 / e (3.5)
This expression relates the solid angle of the antenna beam with the directivity. Therefore
Dmax gets very large for narrow beams. For a circular-symmetric beam around the pole =0,
the effective solid angle becomes
e = 2 sin d = 2 (1 cos max ) max2
0
max
(3.6)
where the last equation is valid for narrow beams ( max <<1). Note that the beam diameter is
given by =2 max .
In analogy to Special Case 2, an effective solid angle e can be defined for any antenna by
23
=
4max
1Dd
De =
max
4
D ; =
e
dDb ),(4
1 (3.7)
The second equation defines the fraction b of the radiation transmitted in directions within
the beam e ; b is called beam efficiency. The boxcar antenna described above is ideal with
b=1. High beam efficiency is important in remote sensing to relate signals to a well-defined
direction and position.
3.1.3 Receiving antenna
The receiving antenna collects radiated power from the radiation field expressed by its intensity at the position of the antenna. This power is related to an effective area Aeff
collecting radiation. The power P2 collected by an antenna and guided to the output port Pout
is
Pout = S Aeff (3.8)
The principle of reciprocity (e.g. Kong, 1985) allows relating the characteristics of the
receiving antenna with those of the identical antenna, when used for transmission. Therefore the transmitting antenna can be used for reception as well and, Aeff is related to D( , ) . It can be shown that
Ae ( , ) =2G( , )
4= r
2D( , )
4; Pout = rP2 (3.9)
Remember that r describes the internal power loss. For sufficiently large antennas, the
maximum value of Aeff of a good antenna ( r 1) is often close to the geometrical antenna
cross section A. On the other hand, for small antennas Aeff, max cannot be much smaller than 2 because Dmax is at least 1; however, for antennas there is no lower size limit.
3.1.4 Directivity and diffraction
A simple model for a directional antenna is shown in Figure 3.5.
Figure 3.5: Simple model
for a horn antenna:
Geometry of an excited
wave in z direction diffracted at the
rectangular aperture
A = x y at z=0 in k
direction to point P.
A wave is radiated through the aperture of a horn. At the horn aperture the radiation is
diffracted. From scalar diffraction theory the directivity can be related to the electric field in the aperture plane:
D( , ) =42
E(x,y)exp(ikx x + iky y)dxdyA
2
E(x,y)2dxdy
A
(3.10)
where the nominator contains the diffraction integral over the aperture area A. This integral is
the two-dimensional Fourier Transform of the aperture field E(x,y,z=0) if E is set to 0 outside
the aperture. The denominator of Equation (3.10) represents the radiated power passing
24
through the aperture. Using spherical coordinates of Figure 3.2 we have for the wave vector
components kx and ky perpendicular to the z direction:
kx = k sin cos , ky = k sin sin , k = 2 / (3.11)
Example: Let E be the scalar electric field amplitude of a plane wave in z direction, and let
x = y = a , and A = a2 . Then E is a constant within the aperture, leading to
D( , ) =4
A 2 exp(ikx x + iky y)dxdyA
2
=4
A 2 exp(ikx x)dxa / 2
+a / 2 2
exp(iky y)dya / 2
+a / 2 2
, and thus to
D( , ) = Dmax
sinX
X
2sinY
Y
2
; X =akx2
, Y =aky2
, Dmax =4 A
2 (3.12)
This is a well-known result for the Fourier Transform of a boxcar function. Note that since
(sinX)/X 1 for X 0, the maximum directivity is simply given by Dmax, and according to
Equation (3.9) for a loss-less antenna ( r =1), the maximum of Aeff is indeed equal to A.
Example: The radiation pattern for Dmax =1000 is realised if the antenna size a = 8.9206 .
Inserting this value in Equation (3.12) for = 0 , we get X=28.025sin . The directivity is
shown in Figure 3.6.
Figure 3.6: Directivity in the plane = 0 of a square antenna with homogeneous aperture
field for Dmax=1000.
Problem: Compute and plot the directivity of this antenna for = 45°. Furthermore compute
e and b (limit integration to 0< <90°), and compare the results with the boxcar antenna of
Section 3.1.2.
Comments: The presented diffraction model is limited to radiation in the forward direction,
and we must assume that no radiation is transmitted in the backward hemisphere
(180> >90°). We must also assume that the far-field condition of Equation (3.1) applies. This
latter condition can be relaxed by quasi optics (see Lecture Microwave Physics and Quasi
Optics). Polarisation has been neglected here, but must be included when needed.
25
3.2 Radar
3.2.1 Radar – Principle
Radar originally was the abbreviation for "Radio Aircraft Detection And Ranging". Today it
stands more generally for "RAdio Detection And Ranging", meaning that a wave is transmitted, and its echo is used to extract information about remote objects, especially their
distance. Today radar is used for many more purposes in remote sensing (rain rate, wind
speed in the atmosphere, waves and oil spills on the ocean, se ice, ice caps, snowcover, vegetation, temporal changes of the relief at mm scales, etc.
Figure 3.7: Bistatic radar configuration with separate transmitter and receiver.
We distinguish the following configurations: • Monostatic Radar (collocated transmitter and receiver, most common type)
• Bistatic Radar (as in Figure 3.7)
• Multistatic Radar (using more than one receiver)
In order to locate a wave train it is necessary to modulate the transmitted wave, either by
creating sufficiently short pulses of duration in the μs (10-6s) or even in the ns (10-9s)
range, or by a (linear) frequency modulation. With such means the total travel time t = l /c (path length l, speed of light c) between transmitter and receiver can be measured to an
accuracy (error t ), which is limited by the uncertainty relation
t 1 (3.13)
where is the frequency bandwidth of the radar signal. The equal sign can be reached if a
so-called matched filter is used (Ulaby et al. 1982). From Equation (3.13) we get the path-
length error
l = c t c / (3.14)
In monostatic radar the path length is two times the range, r (distance between transmitter and scatterer); then
r = 0.5c t 0.5c / (3.15)
Furthermore radar types distinguish between coherent and incoherent systems, types with a single polarisation for both transmission and reception, with multiple polarisation, and fully
Transmitter
Receiver
scatterer
scattered wave
26
polarimetric radars that can transmit any state of polarisation (transmit Stokes Parameter)
and detect any polarisation upon reception (receive Stokes Parameter). A coherent radar
measures the phase relationship between transmit and receive signals (Figure 3.8) to
determine travel time t. For this purpose a very stable oscillator is required especially for long propagation paths. In non-coherent radars t is measured with a clock triggered by the
transmit pulse.
3.2.2 Coherent radar
Antenna
U1
It
U2
Qt
Figure 3.8: Principle of coherent radars.
In a coherent monostatic radar, the phase difference 2kr between a transmitted voltage
U1 =U0 cos( t) (3.16)
(with the circular frequency =2 ) and the received echo voltage U2 after travelling the
distance 2r is to be measured
U2 =L U0 cos( t 2kr) (3.17)
where L <<1 describes the signal loss during propagation. The detection is accomplished
with a so-called I-Q mixer. Its outputs consist of It =U1 U2 and Qt, which is the same
product, but after a phase delay in U1 of 90°. The low-frequency parts of these products are
registered. With the frequency control, the signal can be modulated, for instance a linear
sweep over a time T, (=TR/2 in Figure 3.9).
(t) = 0 + at , for 0 < t < T (3.18)
This sweep has a modulation bandwidth ( = fm in Figure 3.9) :
= aT /(2 ) (3.19)
It and Qt contain the difference frequency between transmission and reception (Figure 3.9):
f = 2ar /c (3.20)
3.2.3 The radar equation
The radar equation relates the transmitted power Pin, with the received echo power Pout, the
distance r to the scattering object and to its radar (or backscatter) cross section b
Pout = Pin2
(4 )3G1G2
r4 b=P1Aeff1Aeff 2
4 2r4 b (3.21)
where G1 and G2 are the transmitter and receiver antenna gains in the object direction and
Aeff1,2 are the corresponding effective antenna areas.
Oscillator
Frequency Control
90°
27
Figure 3.9: Frequency variation in a Frequency-Modulated Continuous-Wave
(FM-CW) radar with time of the transmitted
and received signals with linear frequency modulation.
3.2.4 Radars for remote sensing
Precipitation radar: surface based radar scanning 360° in azimuth in near horizontal direction
to monitor precipitation determined from backscatter from hydrometeors (rain drops, hail,
snow) in the atmosphere. See lecture notes on "Radar Meteorology" (2003) and appropriate literature.
Radar wind profiler (surface based): Measures Doppler velocity of the turbulent atmosphere
to determine horizontal wind velocity.
Altimeter: Radar operated from satellites and aircraft to measure the altitude above the
surface, but also to observe surface properties, especially over the ocean (wind, waves, currents).
LIDAR is the optical equivalent of radar. It is used to measure the surface shape with high spatial resolution, and for atmospheric sensing of water vapour, aerosols, clouds, wind, etc.
Synthetic Aperture Radar (SAR) operated from satellite and aircraft to image the backscatter
with high spatial resolution (a few m). See lecture notes on ENVISAT (Mätzler, 1998), SAR Basics from Gamma Remote Sensing (2008).
Figure 3.10: Example SAR image of the Bern area from
ENVISAT-ASAR on 15 Oct
2007 is shown here
28
3.3 Radiometer
3.3.1 Radiometer, Spectrometer and Polarimeter
Passive remote sensing requires instruments to receive and analyse radiation of
"opportunity", i.e. radiation that exists naturally. Most common is thermal radiation (to be described later in this lecture). The instrument to quantify the radiation intensity or radiance is
the radiometer. If all Stokes parameters (I,Q,U,V) are to be measured the radiometer is
called a polarimeter; and if the measurements resolve the spectrum of the radiation, the instrument is called a spectrometer. All types are basically radiometers, but with multiple
channels. We will concentrate on the single-channel radiometer.
3.3.2 Principle of the radiometer
The basic principle of an ideal (not necessarily realistic) radiometer is shown in Figure 3.11.
Figure 3.11: Basic idea of a
radiometer to measure the radiation at a given position
in a given direction, given
polarisation and given frequency band as defined
by the antenna and
frequency filter. All other
radiation is rejected by the filters. The selected
radiation is absorbed in the
red absorber inside a perfectly insulated box
whose temperature is to be
measured.
The radiometer collects thermal radiation from an object at temperature T. Part of this
radiation passes through the filter into the absorbing box. If we wait until thermal equilibrium has been reached, the measured temperature inside the box will be T. The radiance can
then be quantified by one half (one polarisation only) of the Planck function for temperature T
at the specified frequency. See Chapter 8. This type of radiometer is called a bolometer,
because the detection uses a thermal property of the radiation. Often the radiative heating power is too weak, requiring other detection methods.
3.3.2 Microwave radiometer
In the microwave range, the thermal radiation is very weak requiring strong amplification
before it can be detected. A typical block diagram of a microwave radiometer called after the inventor, Robert Henry Dicke, is shown in Figure 3.12. By rapid switching between the
signals from the antenna TA and from a known reference radiator at the temperature of the
Dicke load TD, the radiometer can eliminate the influence its own radiation. The output voltage U is proportional to the difference, TA-TD. Therefore the measured brightness
temperature can be expressed as
TA = TD +G U + (3.22)
where G is the radiometer gain factor and is a correction term.
29
TA U
TD
Antenna Dicke Switch Filter Isolator Amplifier synch. Power Det Integrator
Figure 3.12: Block diagram of a Dicke radiometer using synchronous switching and detecting
between the signals of the antenna (TA) and of a reference (Dicke load at temperature TD).
The following figure shows a set of Dicke radiometers constructed and operated at the IAP:
Figure 3.13: Mobile
radiometers at IAP,
University of Bern, with conical and
rectangular horn
antennas operating
at frequencies of 11 GHz (MORA, right),
21 and 35 GHz
(centre) and 94 GHz (left), here for
measuring
emissivities of surface materials.
More sophisticated IAP radiometers are operated on the ground for dedicated observations:
MIAWARA: spectrometer for water-vapour profiling of the middle atmosphere (altitude range:
20 to 70 km)
ASMUWARA: for all-sky scanning of the troposphere for clouds, water vapour and temperature profiling
TROWARA: for monitoring integrated water vapour and cloud liquid water of the atmosphere.
SPIRA: imaging polarimeter at a frequency of 91 GHz
Imaging radiometers have been operated from satellites for global earth observations for more than 30 years; consult literature and internet for SSMR, AMSU-A, AMSU-B, SSM/I,
AMSR, Windsat Radiometer, and others. An example is shown by the cover image.
3.3.3 Radiometers in the infrared and visible spectrum
Radiometers in the infrared and visible spectrum follow the same principles as in the microwave range, but in the technical realisation there are significant differences. See
literature for more information. As an example, see SEVIRI on Meteosat 2nd Generation:
http://www.esa.int/msg/pag5.html
30
3.4 Literature
Gamma Remote Sensing, SAR Basics, Documentation Theory, Version 1.4, Gümligen, Schweiz (2008).
Mätzler C. ENVISAT: der neue Europäische Fernerkundungssatellit für die Umweltforschung Universität Bern,
1998, BEWI SK1 PH 1998: 1.
Ulaby, Moore and Fung (1981, 1982, 1986), Microwave Remote Sensing, Vol. 1,2,3 (BEWI: XKA 129, 130).
Information about IAP radiometers: s. IAP Homepage, Microwave Physics: Research Projects and Publications.
31
4 Effective Medium, and Dielectric Mixing Formulas Particles that are small with respect to a sensing wavelength are invisible, or they may
appear as point-like scatterers without any structure. The medium appears like a
homogeneous medium, but with an effective dielectric constant, , which depends on the
dielectric constants of the components. Effective-medium theories use this property of
radiation to determine effective mean values, (r) and μ(r), for heterogeneous or granular
media, especially for structures much smaller than the wavelength. The effective-medium
properties are expressed by mixing formulas (Sihvola, 1999). The topic is important because
1) nature is very often heterogeneous 2) the theory allows significant simplifications without losing the physical basis.
Here we will present some of the most important mixing rules.
4.1 Maxwell-Garnett Formula
The effective dielectric constant of a heterogeneous medium is defined on the basis of
fields (D and E) averaged over a volume sufficiently large to smear out the heterogeneity, but still small with respect to the wavelength. The averaging is expressed by brackets < >. Let us
assume a host medium of dielectric constant 1 with embedded particles of dielectric
constant 2 and volume fraction f . Then the following equations can be set up for the mean
fields (Index 1 in host, Index 2 in particles):
< D >= 0 < E >
< D >= (1 f ) < D1 > + f < D2 >
< E >= (1 f ) < E1 > + f < E2 >
< Di >= i 0 < Ei >; i =1,2
(4.1)
In addition we need a relationship between the electrical field strengths in the particle and in the host medium. Let us assume proportionality:
< E2 >=K < E1 >; (4.2)
The set of equations then leads to an expression for the effective :
=(1 f ) 1 + f 2K
1 f + fK (4.3)
Generalisation to multi-component mixtures
Equation (4.1) can be generalised to multiple-component media, e.g. a host Component 1 as
above, but now with inclusion Components j, j=2 to N of volume fractions fj. Then the porosity
is f1 =1 f , where f = f jj= 2
N, and with the field ratios K j of the inclusions, Equation (4.3)
becomes
=(1 f ) 1 + f j jK jj= 2
N
1 f + f jK jj= 2
N (4.3a)
From (4.3) or (4.3a) it appears that the problem of the mixing formula is solved if we find the correct value of K or Kj. It turns out that the K factors mainly depend on particle shape. In
general K has to be determined either experimentally or theoretically. We will give
expressions for some examples.
32
Sphere
For a single sphere in an infinite host volume the electrical field E2 inside the sphere is
homogeneous and can be expressed by (4.2) where K is given by the electrostatic
expression (4.4).
K =3 1
2 + 2 1
(4.4)
Figure 4.1: Electric field in and around a dielectric sphere (from von Hippel, 1954).
Maxwell-Garnett formula for a 2 component mixture of spherical particles
Assuming that (4.4) is also applicable in a granular medium consisting of spherical particles,
we get the Maxwell-Garnett (1904) mixing formula by inserting (4.4) in (4.3):
= 1 1+ 3 f 2 1
2 + 2 1 f ( 2 1)
(4.5)
This formula is in wide use in diverse fields of application (Sihvola, 1999). Equation (4.3) can
be regarded as a generalised form of (4.5).
Ellipsoid
Electrostatic field expressions exist also for ellipsoids. For an ellipsoid in an infinite host volume with the electrical field parallel to a principal axis, the electrical field E2 inside the
spheroid is again homogeneous and can be expressed by (4.2), where K now depends on
the axis ( i = a,b,c ) along the E field, and is given by
Ki = 1
1 + Ai 2 1( ) (4.6)
and Ai is called the depolarisation factor of the ellipsoid along the i-axis. These factors can be computed from an elliptical integral. For i = a we have
Aa =abc
2
ds
s+ a2( ) (s+ a2)(s+ b2)(s+ c 2)0
(4.7)
and for i = b, c the integral follows by respective replacements. In the integral a, b, c are
the 3 semi axes of the ellipsoid. The sum of all factors equals one:
Aa
+
+ Ab + Ac =1 (4.8)
Problem: Show, by integrating (4.7) for the special case of a sphere ( a = b = c ), that A =1/3.
33
Spheroid
In the case of spheroids (ellipsoids with one axis of symmetry), the integral (4.7) can be
solved analytically. The results are given in Table 4.1, and the values for Ab = Ac are plotted
versus the axial ratio in Figure 4.2.
Tab. 4.1: Depolarisation factors of spheroids with eccentricity e and semi axes a, b = c .
Type of spheroid Aa Ab = Ac
Oblate b = c =a
1 e2 1
e21 e2
e3arcsine
1 Aa
2
Prolate b = c = a 1 e2 1 e2
e21
2eln1+ e
1 e1
1 Aa
2
Fig. 4.2: Depolarisation
factors perpendicular to the symmetry axis of
prolate (dashed) and
oblate (solid line)
spheroids versus x=minor/major axis.
For oblate spheroids x = a /b , for prolate ellipsoids x = b /a .
The effective dielectric constant of a medium, consisting of parallel-oriented ellipsoids or
spheroids with dielectric constant 2 in a host medium ( 1) can now be formulated in the
Maxwell-Garnett Approach, using Equation (4.6) for Ki and inserting this expression in (4.3):
i = 1
1 f +f 2
1 + Ai( 2 1)
1 f +f 1
1 + Ai( 2 1)
; i = a, b, c (4.9)
This quantity clearly depends on the direction of the electric field (parallel to main axis i). The dielectric constant is anisotropic, and - in a proper coordinate system - it can be expressed
by a diagonal tensor
=
a 0 0
0 b 0
0 0 c
(4.10)
The mixing formula becomes scalar if the particles are isotropically oriented. Then the
average K factor
34
K =1
3Ka + Kb + Kc( ) (4.11)
has to be inserted in Equation (4.3).
Numerical computations with MATLAB
function result = mixmaxgarnett(e1,e2,f,ax) % Maxwell Garnett dielectric mixing formula for % isotropic-oriented spheroids % Input: e1=host epsilon, e2=particle epsilon % f= volume fraction of particles % ax=a/b=(symmetry-axis length)/(cross-axis length) >=0 % Matzler, January 2007 >> mixmaxgarnett(1, 3+2i, 0.2, 6) ans = 1.3158 + 0.1948i
Improvements of the Maxwell-Garnett formula
The limitation of the Maxwell-Garnett mixing formula is due to the assumption that the ratio
between the mean fields inside and outside the particles is determined without any influence of neighbour particles. This means that the formula gets inaccurate with high volume fraction
of particles. Therefore a number of improvements have been proposed to take the field
interactions between particles into account (Sihvola, 1999; Sihvola, 2006, in Chapter 5 of
Mätzler, 2006). The most important improvement is the mixing formula by Polder and van Santen, also called Bruggeman formula. For isotropically oriented ellipsoids it reads:
= 1 +f
3( 2 1)
+ Ai( 2 )i= a,b,c
(4.12)
Note that this is an implicit formula, where the unknown value of appears on both sides of
the = sign. The solution of (4.12) for is often found by iteration. The idea behind the formula
is that a particle with a dielectric constant 2 is embedded in a host medium with the effective
mean value (instead of the actual 1). In this way the interaction with the neighbour particles
is taken into account.
An advantage of the physical mixing formulas is their applicability to complex dielectric
constants of both host and guest materials. That is, a single formula describes the behaviours of both the real and imaginary part of the dielectric constant.
4.2 Semi-empirical mixing formulas
Due to the lack of sufficiently accurate information on the shape of particles, there is a need for practical formulas for certain materials (e.g. soil, vegetation). Therefore a number of
empirical formulas have been found that fit certain measurements quite well. One of the most
important examples is the refractive mixing formula, meaning that the refractive index is
mixed in a linear fashion with the volume fractions fi of the N different components being mixed, and ni are their refractive indices:
n = finii=1
N
; where fi =1i=1
N
(4.13)
Now since for non-magnetic media, = n2 and i = ni2 for all i =1,... N , the refractive mixing
formula reads, in terms of the dielectric constant
35
= f i i
i=1
N
2
; where fi =1i=1
N
(4.14)
A physical motivation of the refractive formula is based on the fact that the real part n' of the
refractive index is proportional to the propagation time of waves travelling through the material. An average propagation time in a mixed material can be represented by an average
n' as expressed by (4.13), see Figure 4.3. The same reasoning applies to the imaginary part
n". The average absorption of the wave can be expressed by the average n".
s1 s2 s3
n1 n2 n3
Propagation time t over total path s = s1 + s2 + s3 :
t =1
c0n1's1 + n3 's3 + n3 's3( ) =
n's
c0.
Therefore we have
n'= n1' f1 + n2 ' f2 + n3 ' f3 ; where fi = si /s, i =1,2,3
This corresponds to Equation (4.13).
Figure 4.3: Illustration of refractive mixing for the propagation time t of a wave travelling from
left to right through the medium consisting of 3 different volumes.
Problem: Show that the same result is obtained for the imaginary part n", again assuming
that the wave propagates through the three volumes along path s from left to right.
The imagination of a wave travelling first through one particle followed by another particle
implies a model such as geometrical optics where the particles are much larger than the wavelength. Nevertheless, (4.14) is also used in situations with particles being smaller than
the wavelength. Note that the refractive model fails if the propagation direction is vertical in
the example of Figure 4.3. It only works in a serial way.
A generalisation
A generalisation of (4.14) is the following form
= f i i( )i=1
N
1/
; where fi =1i=1
N
(4.15)
The refractive formula is represented by =0.5, the linear formula by =1, and a physically-
based formula for spherical particles called after Looyenga (1965) by =1/3.
4.3 Literature
H. Looyenga, "Dielectric constants of heterogeneous mixtures", Physica 31, 401-406 (1965).
J.C. Maxwell Garnett, "Colors in metal glasses and metal films", Trans. Royal Soc. (London), CCIII, pp. 385-420 (1904).
A. Sihvola, Electromagnetic mixing formulas and applications, IEE Electromagnetic Waves Series 47, London UK
(1999), BEWI: OGE 201.
36
5 EM Waves and Boundaries In previous chapters we described EM waves and fields in effectively homogeneous media.
More complex is the situation of inhomogeneous media. We will limit ourselves to tractable situations of piecewise homogeneous media separated by boundaries. Then Maxwell's
Equations, in the forms of (2.8) and (2.9), are still valid in each homogeneous region. The
inhomogeneity is limited to the boundaries; and these can be treated by boundary conditions.
5.1 Boundary conditions
The simplest situation is a composite of two homogeneous media ( = 1, μ=μ1 in Medium 1,
= 2, μ=μ2 in Medium 2.) separated by a smooth boundary. If we know the solution of
Maxwell's Equations on one side, say in Medium 1, we can compute the transmission of
waves into Medium 2 from boundary conditions at the interface. In this way we are able to
solve Maxwell's Equations in the whole medium. This is the method to compute the reflection, absorption and transmission of EM waves at a
plane boundary between two half spaces (Fresnel Equations), the scattering and absorption
of EM waves by spheres (Lorenz-Mie Theory), and many other EM problems.
Maxwell's Equations lead to the following boundary conditions for the fields at the interface:
1) Tangential components of the E fields are the same on both sides of the interface:
E1 = E2 (5.1a)
2) Tangential components of the H fields are also the same:
H1 = H2 (5.1b)
3) Normal components of B are the same on both sides of the interface:
μ1H1 = μ2H2 (5.1c)
4) Normal components of D are also the same:
1E1 = 2E2 (5.1d)
Exceptions occur if one medium has infinite conductivity: then electrical surface charges (for
D) and surface currents (for H) must be taken into account (Kong, 1986).
5.2 The Fresnel Equations and Snell's Law of Refraction
The results of this section are very important in many aspects of remote sensing. However,
proper application of the Fresnel Equations in nature requires the knowledge of potential
disturbances. Therefore we will have to go beyond a simplistic discussion. Let us first consider in Figure 5.1 the classical situation for the Fresnel Formulas, namely two
half spaces separated by an infinite plane, here defined by z=0. Medium 1 of the upper half
space has values indexed by 1, and Medium 2 of the lower half space has values indexed by
2. The figure also shows an example of fields of EM waves: in Medium 1 an incident wave
(Index i) from a direction given by the incidence angle 1 and a reflected wave (Index r), and
in Medium 2 a transmitted wave (Index t). The H fields are in the plane of incidence, whereas the E fields are perpendicular to the plane of incidence, but parallel to the interface between
the two media. In remote sensing the interface is often the horizontal, terrestrial surface;
therefore this situation is called horizontal polarisation. It is also called transverse electric (TE) or perpendicular because the E field is transverse (perpendicular) to the plane of
37
incidence. The orthogonal situation to Figure 5.1 is called vertical polarisation, also called
transverse magnetic (TM) or parallel. It is obtained by exchanging the E and H fields, more
exactly by replacing E by H and H by –E. This procedure follows from the principle of duality
(Kong, 1986). A general EM wave can be decomposed into a fraction in vertical polarisation and another fraction in horizontal polarisation.
Figure 5.1: Geometry of
horizontally polarised incident, reflected and
transmitted waves at a
horizontal interface (z=0)
between two half spaces, consisting of
medium properties
indexed by 1 and 2 (from Ulaby et al. 1981).
Based on Figure 5.1 the electrical fields of the 3 waves only have y components, which can be written as
Ei = E0 exp(ik1x x ik1zz i t) (5.2a)
Er = RhE0 exp(ik1x x + ik1zz i t) (5.2b)
Et = ThE0 exp(ik2x x ik2zz i t) (5.2c)
where k1x = k1 sin 1; k1z = k1 cos 1; k2x = k2 sin 2; k2z = k2 cos 2 (5.3)
and Rh and Th are reflection and transmission coefficients at horizontal polarisation to be
determined. Note that the incident and reflected waves have the same value of k1x and
opposite values of k1z. This choice is necessary to get a solution. It means that the reflection
angle is equal to the incidence angle 1.
The total field on the upper side of the interface at z=0 consists of the sum Ei+Er, and on the
lower side of Et: The two are required to be equal according to (5.1a), giving
(1+ Rh )exp(ik1x sin 1) = Th exp(ik2x sin 2) (5.4)
This equation can only be true for all x if the following equations are both valid
1+ Rh = Th (5.5)
k1 sin 1 = k2 sin 2 (5.6)
38
and since ki =nic0
; i =1,2 , Equation (5.6) leads to Snell's Law of refraction (5.7). The law
determines the direction (angle 2) of the refracted wave, given the refractive indices and 1:
n1 sin 1 = n2 sin 2 (5.7)
Figure 5.2:
Graphical demonstrations of
Snell's Law of refraction (a), and
of the law of reflection (b) for a water surface. The wave is
incident from medium 1, here
called medium i, transmitted in medium 2, here called medium t.
From Petty (2006).
In remote sensing, Medium 1 is usually the atmosphere where n1 is very close to 1, and n2 is
the refractive index of the surface material. Note that if n2 is complex, Equation (5.7) tells that
the refracted angle 2 is complex, too.
To solve for Rh and Th, a second equation is needed, which is found from Equation (5.1b) for
the case of Figure 5.1 and from the relation (2.18) between the E and H fields:
H0
E0
Z=E0
Z0 μ. This leads to
1
μ1(1 Rh )cos 1 = 2
μ2Th cos 2 (5.8)
Solving Equations (5.5) and (5.8) for Rh gives the Fresnel Equation for h polarisation:
Rh =
1
μ1cos 1
2
μ2cos 2
1
μ1cos 1 + 2
μ2cos 2
=Z2 cos 1 Z1 cos 2
Z2 cos 1 + Z1 cos 2
; Th =1+ Rh ; (5.9)
39
In analogy to the derivation found above, the following results are obtained for vertical
polarisation:
Rv =
μ1
1
cos 1
μ2
2
cos 2
μ1
1
cos 1 +μ2
2
cos 2
=Z1 cos 1 Z2 cos 2
Z1 cos 1 + Z2 cos 2
; Tv =1 Rv ; (5.10)
For non-magnetic media, the reflection coefficients of Equations (5.9) and (5.10) can be
expressed by the more familiar refractive indices:
Rh =n1 cos 1 n2 cos 2
n1 cos 1 + n2 cos 2
; Rv =n2 cos 1 n1 cos 2
n2 cos 1 + n1 cos 2
(5.11)
Although the two formulas look very similar, the results are different. The Fresnel Formulas
can be expressed in various ways, e.g. by eliminating 2 by Snell's law. For cos 2 we can
write, using (5.7),
cos 2 = 1 sin2 2 = 1n12
n22 sin
21 (5.12)
Equation (5.12) also helps to understand the meaning of the complex angle. The fraction of reflected power (or intensity) is called reflectivity, or reflectance, denoted by rh
and rv. Furthermore, the fraction of transmitted power is called transmissivity, or
transmittance, denoted by th and tv. Since the intensity is related to the fields by (2.22):
I = 0.5 E0
2Re 1/Z *( ), the rh and rv are simply given by
rh = Rh
2; rv = Rv
2 (5.13)
The transmissivities have to include the change in Z from Medium 1 to Medium 2. But since energy is conserved, the transmissivities also follow from
th =1 rh ; tv =1 rv (5.14)
Of special interest is the Brewster Effect, a situation with completely vanishing reflection for real n1 and n2. By inserting (5.12) into (5.11) it can be confirmed that for nonmagnetic
materials, the Brewster Angle occurs at vertical polarisation, and it is given by
Brewster = arctan(n2 /n1) (5.15)
For a medium with an imaginary refractive index n2, the reflectivity is a minimum at the
Brewster angle defined by the real part of n2 in (5.15). Total reflection is another phenomenon to be mentioned. This occurs for upwelling waves in
Medium 2 at the interface if 2 is larger than a critical value 2,c with
sin 2,c =n1n2
(5.16)
Then there is no real solution for 1 according to Snell's law. The only solution is total
reflection back to Medium 2.
40
Numerical computations
We will use MATLAB to illustrate results. A general function for non-magnetic media is
fresnel.m; it computes all quantities defined above, and in addition a few more. Some more
functions are available as explained below.
function result = fresnel(ei, et, thetai) % Matlab function for the calculation of parameters related % to the transmission and reflection of electromagnetic waves % on a planar surface between two non-magnetic media % (Fresnel Formulas) % Input parameters: % ei: relative dielectric constant of first medium % et: relative dielectric constant of second medium % the constant can be complex with Imag >0 % thetai: angle [rad] of incident beam (first medium)% % Lorenz Martin, 2001-01-18, 2002-03-12, 2005-03-23 % adapted by Matzler 2007 result.ei = ei; % dielectric constant first medium result.et = et; % dielectric constant second medium result.ni = ni; % refractive index first medium result.nt = nt; % refractive index second medium result.thetai = thetai; % angle [rad] of incident beam (first medium) result.thetat = thetat; % angle [rad] of transmitted beam (second medium) result.brewster = brewster; % Brewster angle [rad] result.totalrefl = totalrefl; % angle of total reflection [rad] result.s = s; % field penetration depth [vacuum wavelength] % TE mode result.p0tTE = p0tTE; % imped.ratio, s. Kong 1986, p. 113 result.RTE = RTE; % reflection coefficient result.TTE = TTE; % transmission coefficient result.rTE = rTE; % reflectivity result.tTE = tTE; % transmissivity % TM mode result.p0tTM = p0tTM; % imped.ratio, s. Kong 1986, p. 116 result.RTM = RTM; % reflection coefficient result.TTM = TTM; % transmission coefficient result.rTM = rTM; % reflectivity result.tTM = tTM; % transmissivity function result = fresnelreflectivity(eps,mu,teta) % Computes Fresnel reflectivities and reflection coefficients % Input parameters: % teta: incidence angle(s) teta (rad), can be a scalar or a column vector % eps: relative dielectric constant of lower medium (upper=1) % mu: relative magnetic permeability of lower medium (upper=1) % values can be complex, Matzler, November 2006 result=[teta,rv,rh,Rv,Rh];
function result = fresnelreflectivityplot(eps,mu) % Plot of Fresnel reflectivities from 0 to 90 deg incidence angle % based on fresnelreflectivity
41
Examples
Figure 5.3a: Fresnel reflection coefficients versus incidence angle for
1 =1; 2 = 5; μ =1. Note that
Rv passes through 0 at the Brewster angle of 66°.
Figure 5.3b: Fresnel reflectivities versus incidence angle for 1 =1; 2 = 5; μ =1.
Note the vanishing reflectivity rv at the Brewster angle of
66°.
For small angles of incidence, the Fresnel reflectivities show small angular variation. The
behaviour changes towards large angles, always leading to rh =rv =1 at grazing incidence
( 2=90°). The difference between small and large dielectric constants can be seen in Figure
5.4 showing the reflectivity of water at 22 GHz and in the visible range. Note that in the
microwave range the reflectivity is nonzero at the Brewster minimum because of the large imaginary dielectric constant.
42
Figure 5.4: Fresnel
reflectivity of water versus incidence angle:
Black curves for the microwave range
(22 GHz, = 22 + 32i ) Red curves for the
visible range ( =1.8).
In remote sensing the incidence angle is often not a freely selectable parameter because it is either a fixed value (for conically scanning sensors), or else it changes with the scan angle
and thus with the position of the observation point. Therefore it may be useful to see what
type of information can be obtained from the measurement of the reflectivity at h and v
polarisation, by assuming a fixed value of 1.
Figure 5.5 shows the behaviour at the incidence angle of 57° (1 radian). Both Fresnel curves
(x and o) are very similar, indicating that the reflectivities are unable to give independent
information on the real and imaginary parts of . This property is useful as signature of
Fresnel surfaces. Rough surfaces, on the other hand, are characterised by very small
polarisation, i.e. by rv = rh, as represented by the dashed curve.
Figure 5.5: Fresnel rv
versus rh at 1=1 rad (57°)
(x) for real dielectric
constant with =1+ a,
where a refers to the values: 0.2, 0.4, 0.7, 1, 1.5, 2, 2.5, 3, 4, 5, 6, 7, 8, 9, 11, 14, 19, 24, 29, 39, 49, 69, 99, 199, 299,
and (o) for a lossy medium
with "= ' 1=a
2 .
Thus for both curves
12
= a2
The dashed line
represents rv = rh.
43
5.3 Waves in layered media
In nature, the boundary between two media is often not exactly as needed for the Fresnel
formulas to be applicable. Even if the surface is perfectly flat, deviations due to
inhomogeneities of the subsurface can have an influence on the reflectivity. Some media,
like snow, are strongly layered. Then several reflections from different layer interfaces will play a role. A similar problem occurs if the transition between the upper and lower medium is
not an abrupt one, but extends over a certain distance. Both cases can be handled by
general formulas for layered media. The solutions are derived from the superposition of Fresnel reflections and Fresnel transmissions with proper phase adjustments. The resulting
formulas are suitable for numerical computations. z layer m+1 n(m+1) d(m+1) incident beam ------------------------------------------------- layer m n(m) d(m) ------------------------------------------------- layer m-1 n(m-1) d(m-1) ------------------------------------------------- . . . ------------------------------------------------- layer 1 n(1) d(1) transmitted beam Figure 5.6: Geometry of the layered medium with n( j) = n j '+in j" the complex refractive
index of Layer j and d(j) its thickness.
Recursion formula
The reflection and transmission coefficients (R and T), and the reflectivity and transmissivity
of a multi-layer sandwich can be computed either by a matrix method (Dobrowolski, 1995,
Born and Wolf, 1975), or by a recursive method called invariant embedding (Adams and Denman, 1966). The recursion formula will be presented here. However, note that for m>>1
the matrix method is much faster.
Let us consider (Figure 5.6) a multi-layer medium with plane-parallel boundaries (planes with
constant z). The bottom medium has Number 1 with propagation angle 1 to the z axis,
relative dielectric constant 1 and relative magnetic permeability μ1. Furthermore we call F1
the Fresnel reflection coefficient and G1 the transmission coefficient at the boundary from
Medium 2 to Medium 1. Medium 2 is a layer of thickness d2 above Medium 1.
For Medium 2, we define the respective parameters as d2, 2, 2, μ2, F2 and G2, and so on for
all layers to the uppermost layer (No. m) of the sandwich with parameters, dm, m, m, μm, Fm
and Gm. An EM wave is incident from the halfspace above the layered medium with
parameters m+1, m+1, μm+1. Now the recursive formulas read for j=2,... m
R j =Fj + R j 1Qj
2
1+ FjR j 1Qj2 ; Tj =
GjTj 1Qj
1+ FjR j 1Qj2 ; j=2, 3, ..., m (5.17)
and for j=1, the values are simply given by
R1 = F1; T1 =G1 (5.18)
44
The factor Qj describes the complex phase change of the wave on its way through Layer j:
Qj = exp(ik jd j cos j ) (5.19)
We have to remember that Fj and Gj depend on polarisation. Then the final reflection and
transmission coefficients R and T at the top of the layer are the values of (5.17) for j=m, but now indicating also the polarisation (h, and v)
Rv,h = Rm; Tv,h = Tm (5.20)
Part of the wave intensity is reflected, another part is transmitted to the bottom of the
sandwich, and the remaining intensity is absorbed by the sandwich. The respective fractions
are the reflectivity r, transmissivity t, and absorptivity a at v and h polarisation, respectively,
of the sandwich, and they are determined from (assuming that the top and bottom media are
identical):
rv,h = Rv,h
2; tv,h = Tv,h
2; av,h =1 rv,h tv,h (5.21)
Matlab functions
function result = layers(freq, thetai, e, d) % General multilayer program using recursive formula of R.N.Adams, % & E.D.Denman, Wave Propagation and Turbulent Media, % American Elsevier, New York, 1966 % Input parameters: % freq: frequency [GHz] % thetai [rad]: Incidence angle of incident beam (top at j=m+1) % e: vector with complex dielectric constants % d: vector with thicknesses [m] of layers % d and e are both of length m+1 (dummy values for d at j=1 and j=m+1) % Lorenz Martin, 2005. function result = lamella(fGHz, thetad, epsilon, d, nphase) % Transmission, Reflexion und Absorption EM Wellen an ebener dielektrischer % Lamelle (Dicke d) in Vakuum. Kohaerente Rechnung bis max Phasendifferenz % darueber inkohaerent. % Literatur: Kong, J. A. (1986). Electromagnetic Wave Theory. % Adams, R.N. & Denman (1966): Wave Propagation and Turbulent Media % Input: % thetad: Incidence Angle [deg] % fGHz: Frequency [GHz] % epsilon: complex, relative epsilon of lamella % d: Thickness of Lamella [m] % nphase=1,2,3,.. number of half waves+1/4 with coherent computation function result = layermat(freq, thetai, e, d) % General multilayer program using the matrix method of Dobrowolski (1995), % s. also Born and Wolf (1975). % For large m this method is much faster than the recursive method of % invariant embedding (layers.m) % Input parameters: % freq: frequency [GHz] % thetai [rad]: Incidence angle of incident beam (top at j=m+1) % e: vector with complex dielectric constants % d: vector with thicknesses [m] of layers % d and e are both of length m+1 (dummy values for d at j=1 and j=m+1) % C. Matzler, Nov. 2008
45
Problem
Show that for n j"<< n j ' , the one-way phase k jd j cos j of Qj can be written as
k0n j 'd j cos j,eff +in j"
cos j,eff
where j ,eff is the real refracted angle resulting for a layer with
real refractive index n j ', i.e. for n j"=0. Use Snell's Law of refraction.
Note that k j = n jk0 where k0 = /c0 is the vacuum wave number, and c0 is the speed of light
in vacuum.
Thus the real part of the phase through the layer decreases with increasing incidence angle,
whereas the opposite is true for the imaginary part.
Examples
1) Reflectivity of a frozen lake at a frequency of 18 GHz: Ice layer ( 2=3.188 + 0.0017i) on top
of water ( 1=21.7 + 32.5i). The following example was computed for an ice thickness of 20
cm. With increasing , the phase factor between the ice and water surface decreases, a
strong interference phenomenon appears. It disappears at the Brewster angle of ice near 60°
(v pol). The method could be used to measure the lake-ice thickness.
Figure 5.7: Reflectivities
at 18 GHz, vertical (solid)
and horizontal (dashed)
polarisation versus incidence angle of a
frozen lake with a smooth
ice layer of 20 cm thickness.
2) If a surface has a gradual transition layer from air to soil, the dielectric constant changes
over a certain depth range from air ( =1) to soil. The reflectivity of such a transition can be
modelled by a number of layers with gradually changing . We want to find out under which
condition this effect is relevant. For simplicity we represent the transition region by a single
layer of thickness d, thus m=2, and we assume loss-less media. Then 1 is the real dielectric
constant of the bottom halfspace, 2 is the real dielectric constant of the transition layer and d
is its thickness. The upper halfspace has 3=1. The value of 2 is intermediate: 1< 2 < 1.
From (5.17-9) for m=2 we get directly the solution
r = R2
=F2 + F1e
i2
1+ F2F1ei2
2
; k2dcos 2 (5.22)
For loss-less (and approximately also for low-loss) media, the Fresnel reflection coefficients are real. Then we get, after some transformations
46
r = r01 2 sin2
1 r02 sin2
; 2 F1F2F1 + F2
(5.23)
where r0 is the Fresnel reflectivity without the transition layer (i.e. from 3 to 1), and is the
ratio of the geometrical to the arithmetic mean values of F1 and F2. If F1 = F2, =1, otherwise
<1. The reflectivity r changes with phase angle . For =1, r can be suppressed completely
for sin =1. This happens, for instance, if n2dcos 2 = (1/4) 0 .
The variation r /r0 with increasing layer thickness, expressed by , is shown in Figure 5.8 for
3 values of r0, all for =1. For <10° the reduction of r with respect to r0 is negligible.
However, for =30° ( d = /12) the reduction is already significant, especially if r0 < 0.4. The
reduction leads to zero reflection at 90°, that is when d is a quarter wavelength. This effect is used in optics to suppress reflections of glasses and lenses by anti-reflection coatings.
With still higher values of r0, the reflectivity in Figure 5.8 approaches the properties of a
Fabry-Perot resonator.
Figure 5.8: Relative
change r /r0 of the
reflectivity by a transition layer whose phase is Beta
(deg.) for different values
of the reflectivity r0 without a transition layer. For
F1=F2 ( =1).
The change is smaller for
other values of .
47
5.4 Lorenz-Mie scattering
For a more detailed description, see literature cited below or the lecture notes on Radiative
Transfer.
5.4.1 Introduction
Scattering of electromagnetic waves at homogeneous and layered spheres can be computed
in the analogous way as we did for plane surfaces and plane layers, i.e. by fitting the fields of the incident plane wave to the sum of a scattered wave and internal fields at the surface
boundaries of the sphere. The method is complicated by the fact that the boundary
conditions cannot easily be expressed when plane waves meet a sphere. The main task is to express a plane wave by a superposition of spherical waves. An excellent description can be
found in the textbook of Bohren and Huffman (1983), in short BH. The theory was developed
by Ludvig V. Lorenz in 1890 and by Gustav Mie (1867-1957) in 1908. Here the basic results are presented, making use of BH and MATLAB functions developed and described in IAP
reports and lecture notes Mätzler (2002-2004). For scattering by coated spheres, and for
more details on the physics and numerical problems, see the mentioned references.
A spherical wave is described by exp(ikr i t)
kr. BH use the convention of Equation (5.24)
for the scattered electric far field components, where the time factor is omitted, and where
parallel and perpendicular field components are defined with respect to the scattering plane
as shown in Figure 5.8.
E s
E s
=
S2 S3
S4 S1
E i
E i
exp(ikr)
ikr (5.24)
x2 Ei Es
Es
s
x3 , i scatterer
x1 , Ei
Figure 5.8: Scattering geometry with incident fields on the left scattered by a sphere in (x2, x3)
plane showing parallel and perpendicular components of the incident (Index i) and scattered
(Index s) electric fields, and scattering angle .
In general the scattering matrix is composed of 4 complex numbers Sj. In case of spheres,
for reasons of symmetry, only S1 and S2 are different from 0. They turn out to be given by
S1(cos ) =2n +1
n(n +1)(an n + bn n )
n=1
;
S2(cos ) =2n +1
n(n +1)(an n + bn n )
n=1
(5.25)
The functions n and n describe the angular scattering patterns of the spherical harmonics
used to describe S1 and S2 and follow from the recurrence relations
n =2n 1
n 1cos n 1
n
n 1 n 2; n = ncos n (n +1) n 1 (5.26)
starting with
0 = 0; 1 =1; 2 = 3cos ; 0 = 0; 1 = cos ; 2 = 3cos(2 ) (5.27)
48
The elements Si are scattering amplitudes; they describe the angular and polarisation
dependence of the scattered wave.
5.4.2 Mie Coefficients
The key parameters are the Mie Coefficients an and bn to compute the amplitudes of the
scattered field, and cn and dn for the internal field, respectively. The coefficients are determined by the boundary conditions of the fields at the sphere surface, and they are given
in BH (1983) on p.100. The coefficients of the scattered electrical field are:
)]'()[()]'()[(
)]'()[()]'()[(
)]'()[()]'()[(
)]'()[()]'()[(
)1()1(
1
1
)1(
1
)1(2
1
2
mxmxjxhxxhmxj
mxmxjxjxxjmxjb
mxmxjxhxxhmxjm
mxmxjxjxxjmxjma
nnnn
nnnnn
nnnn
nnnnn
μμ
μμ
μμ
μμ
=
=
(5.28)
where prime means derivative with respect to the argument. The Index n runs from 1 to ,
but the infinite series can be truncated at nmax; for this number BH proposed
243/1
max++= xxn (5.29)
and this value is used here as well. The size parameter x is defined by x ka, a is the radius
of the sphere, and k=2 / is the wave number, the wavelength in the ambient medium,
m=( 1μ1)1/2/( μ)1/2 is the refractive index with respect to the ambient medium, 1 and μ1 are the
permittivity and permeability of the sphere and and μ are the permittivity and permeability of
the ambient medium. The functions jn(z) and yn(z), and )()1(zh
n=jn(z)+iyn(z), are spherical
Bessel functions of order n of the arguments, z= x or mx, respectively. The derivatives follow
from the spherical Bessel functions themselves, namely
)()()]'([);()()]'([ )1()1(
1
)1(
1 znhzzhzzhznjzzjzzj nnnnnn == (5.30)
Relationships exist between Bessel and spherical Bessel functions:
)(2
)( 5.0 zJz
zj nn += (5.31)
)(2
)( 5.0 zYz
zy nn += (5.32)
Here, J and Y are Bessel functions of the First and Second Kind; for n=0 and 1 the
spherical Bessel functions are given (BH, p. 87) by
zzzzzyzzzy
zzzzzjzzzj
/sin/cos)(;/cos)(
/cos/sin)(;/sin)(
2
10
2
10
==
== (5.33)
and the recurrence formula can be used to obtain higher orders
)(12
)()( 11 zfz
nzfzf nnn
+=+ + (5.34)
where fn is any of the functions jn and yn. Power-series expansions for small arguments of jn
and yn are given on p. 130 of BH. The Spherical Hankel functions are linear combinations of
jn and yn. Here, the first type is required
)()()()1( ziyzjzh nnn += (5.35)
49
To describe the internal field we need additional coefficients as described in BH; see also the
lecture notes on Strahlungstransport (Radiative Transfer) with MATLAB functions and many
examples.
5.4.3 Cross sections and efficiencies
In contrast to the situation of the Fresnel Equations with planes of infinite extent, only a finite part of the plane wave actually interacts with the sphere of radius a . Therefore a new
concept has to be introduced: cross sections and efficiencies. The efficiencies Qi for the
interaction of radiation with a sphere are cross sections i normalised to the geometrical
particle cross section, g= a2, where i stands for extinction (i=e), absorption (i=a), scattering
(i=s), backscattering (i=b), and radiation pressure (i=pr), thus Qi i / g . It turns out that
Qs =2
x 2(2n +1)( an
2+ bn
2)
n=1
(5.36)
Qe =2
x 2(2n +1)Re(an + bn )
n=1
(5.37)
Qa =Qe Qs (5.38)
The infinite series are truncated after nmax terms. Furthermore, the asymmetry parameter
g= cos indicates the average cosine of the scattering angle with respect to power; it is
used in radiative transfer, and it is related to the efficiency Qpr of radiation pressure:
Qpr =Qe Qs cos (5.39)
Qs cos =4
x 2n(n + 2)
n +1Re(anan+1
*+
n=1
bnbn+1* ) +
2n +1
n(n +1)Re(anbn
* )n=1
(5.40)
Finally, the backscattering efficiency Qb, applicable to monostatic radar, is given by
2
12
)()1)(12(1
=
+=n
nn
n
b banx
Q (5.41)
5.4.4 Examples
MATLAB functions:
mie_ab(m, x) computes an and bn for n=1 to nmax
mie(m, x) computes Qext, Qsca, Qabs, Qb, g=<costeta>, for non-magnetic spheres
mie2(eps1, mu1, x) computes Qext, Qsca, Qabs, Qb, <costeta>, for magnetic spheres
mie_xscan(m, nsteps, dx) and Mie2_xscan(eps1, mu1, nsteps, dx) are used to compute and plot the efficiencies
versus size parameter x in a number (nsteps) of steps with increment dx from x=0 to x=nsteps dx.
Magnetic sphere with x=2, eps1=2+i, mu1=0.8+0.1i
The command line
>> eps1=2+1i; mu1=0.8+0.1i; x=2; mie2_ab(eps1,mu1,2)
leads to the Mie Coefficients [an; bn] for n=1 to nmax=9: 0.3745 - 0.1871i 0.1761 - 0.1301i 0.0178 - 0.0237i 0.0010 - 0.0016i
0.3751 + 0.0646i 0.0748 + 0.0294i 0.0068 + 0.0044i 0.0004 + 0.0003i
0.0000 - 0.0001i 0.0000 - 0.0000i 0.0000 - 0.0000i 0.0000 - 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i
0.0000 - 0.0000i
0.0000 + 0.0000i
50
whereas the command line >>mie2(eps1,mu1,2)
returns the Mie Efficiencies Qe, Qs, Qa, Qb, g=<costeta> and Qb/Qs
= 1.8443, 0.6195, 1.2248, 0.0525, 0.6445, 0.0847
Mie Efficiencies are plotted versus x (0 x 5) by Mie2_xscan(eps1, mu1, 501, 0.01) in Figure
5.9. To plot the angular dependence of the scattered power in the two polarisations, the
function Mie2_tetascan(eps1,mu1,x,201), for x=0.2, is used to provide Figure 5.10.
Figure 5.9: Mie
Efficiencies versus size
parameter for a
sphere with =
2+i, μ=0.8+0.1i.
MATLAB command:
mie2_xscan(eps1, mu1, 501, 0.01)
Figure 5.10: Mie angular pattern
for a sphere where S1 is
shown in the
upper and S2 in the lower half
circle, with x=0.2,
= 2+i,
μ=0.8+0.1i. Note
that here backscattering is
stronger than forward
scattering, in
agreement with negative values
of < cos > at x=0.2 in upper
figure.
MATLAB command: mie2_tetascan(eps
1,mu1,x,201), for
x=0.2
51
5.5 Rayleigh scattering
For small size parameters, Lorenz-Mie scattering reduces to the well-known and much
simpler Rayleigh scattering. The expressions for the efficiencies can be derived either from
the lowest order terms of the Taylor-series expansion of the Mie formulas, or from the
integral representation for scattering amplitude as directly derived from the Maxwell Equations. The latter were discussed by Ishimaru (1978), using a slightly different
convention. The following description should help to improve the insight in electromagnetic
scattering.
5.5.1 General scattering integral
Es(r) = f(s,i)Ei
exp(ikr)
r (5.42)
The vector f(s,i) is the scattering amplitude, i and s are unit vectors in the directions of the
incident and scattered wave, respectively, and the time factor is omitted. Note that f has the dimension of a length. According to Ishimaru (1978, p.10-17) f is given for a dielectric
scatterer with volume Vs in vacuum:
f(s,i) =k 2
4 Ei
s s E(r ')[ ]{ } (r ') 1{ }exp( ikr' s)dV 'Vs
(5.43)
The integral requires knowledge on the electric field E(r') inside the scatterer. Since the
incident and scattered intensities Ii and Is are proportional to the squared absolute value of the respective fields, we get
Is = Iif2
r2 (5.44)
The nominator f2= d is the differential scattering cross section d. And the bistatic scattering
cross section bi, is defined by
bi 4 d = 42
),( isf (5.45)
The total-scattered power Ps is the integral of Is over a closed surface A around the scatterer.
Ps = IsdAA
= Isr2d
4
= Ii d d4
= Ii s (5.46)
The last quantity s is the scattering cross section, which can also be expressed by
s =Ps
Ii=1
4 bi4
d (5.47)
In analogy, the absorption cross section a uses the absorbed power Pa:
a =Pa
Ii (5.48)
A formal expression for Pa, again requiring the internal E field, was given by Ishimaru (1978)
Pa = 0.5 0 "(r')E(r ')V
2dV ' (5.49)
This power is just the Ohmic loss due to the electrical current density j= E excited in the
particle.
52
5.5.2 Rayleigh scattering
The internal field is given by the electrostatic solution:
E(r') =3
+ 2Ei (5.50)
This is a homogeneous field and parallel to the incident electric field.
Scattering amplitude
Inserting (5.50) in Equation (5.43) for a point-like scatterer, and noting that the volume is
given by Vs = 4 a3 /3, the scattering amplitude becomes
f(s,i) = a3k 21
+ 2s s e i[ ]{ } (5.51)
where ei is the unit vector describing the direction of the incident electric field. Equation (5.51) can also be regarded as the scattering amplitude from an electric dipole with dipole
moment De = 0e i for a unit-amplitude incident field and polarisability = 4 a31
+ 2 of
the sphere:
f(s,i) =k 2
4 0
s s De[ ]{ } (5.52)
This is a more general expression, which is valid for all kinds of small particles.
Cross sections and efficiencies
If we denote the angle between incident electric field and the scattering direction by , we get
for the bistatic scattering cross section
bi = 4 f( )2= 4 a6k 4
1
+ 2
2
sin2 (5.53)
The scattering cross section follows from the integral of over all scattering directions
s = f( )2d
4
= 2a6k 41
+ 2
2
d sin2 sin d0
/ 2
0
2
=8
3a6k 4
1
+ 2
2
(5.54)
and the scattering efficiency becomes:
Qs = s
a2=8
3(ak)4
1
+ 2
2
=8
3x 4
1
+ 2
2
(5.55)
Due to the 4th power of x=ak, Rayleigh scattering rapidly diverges to unrealistically large
values when x approaches or exceeds 1. Finally, for the absorption efficiency we get an
expression linear in x:
Qa =12ak
+ 22 "=
12x "
+ 22 and a =
12x a2 "
+ 22 (5.56)
53
Figure 5.11: Dipole radiation pattern for Rayleigh scattering: solid lines for parallel, dashed
line for perpendicular field. The lengths of the vectors s are proportional to the scattered field strength.
Problems
1) Express S1 and S2 for Rayleigh scattering. Hint: Compare the definitions (5.24) and (5.42) for the scattered field and use the result (5.51) for the scattering amplitude.
2) Compare the results of Lorenz-Mie scattering with Rayleigh scattering by doing your own
computations for Qs and Qa, and/or see Section 4.9 of the lecture notes on Radiative
Transfer. 3) Show that Pa of (5.49) represents the power loss due to the induced Ohmic current in the
scatterer.
5.6 Literature
R.N. Adams, E.D. Denman (1966), Wave Propagation and Turbulent Media, American Elsevier, New York, BEWI: TEE 123.
C.F. Bohren and D.R. Huffman, Absorption and Scattering of Light by Small Particles, John Wiley, New York, NY
(1983). BEWI: TDD 122.
M. Born and E. Wolf, Principles of Optics, Pergamon Press (1975).
J.A. Dobrowolski, Optical properties of films and coatings, Ch. 42 in M. Bass et al. (Eds) Handbook of Optics, Vol. 1 (1995)
A. von Hippel, Dielectrics and Waves, 1st Ed. (1954) BEWI: TEA 149, 2nd Ed. (1995), BEWI: VTZ 201.
A. Ishimaru, Wave propagation and scattering in random media, Vol. 1, Academic Press, Orlando (1978), BEWI:
TDD 111.
C. Mätzler, “MATLAB Functions for Mie Scattering and Absorption”, IAP Res. Rep. No. 02-08, Institute of Applied Physics, University of Bern, June (2002).
C. Mätzler, “MATLAB Functions for Mie Scattering and Absorption, Version 2”, IAP Res. Rep. No. 02-11, Institute of Applied Physics, University of Bern, August (2002).
C. Mätzler and L. Martin, “Advanced Model of Extinction by Rain and Measurements at 38 and 94 GHz and in the
Visible Range”, IAP Res. Rep. No. 2003-1, February (2003).
C. Mätzler, “Mie Scattering With and Without Diffraction”, IAP Res. Rep. 2004-02, April (2004).
F.T. Ulaby, Moore and Fung (1981), Microwave Remote Sensing, Active and Passive, Vol. 1. BEWI: XKA 124, 125, 205, 207.
54
6 Microscopic View of Matter
6.1 Electric dipole, and polarisation of dielectric media
So far we have used the material constants and μ as parameters to describe
electromagnetic waves and their interaction with matter. Their spectral behaviour strongly
influences the properties of remote-sensing signatures. Therefore we are interested to
understand the physical nature of these parameters.
The term polarisation of media refers to the displacement of charges in atoms and molecules
due to the action of the electromagnetic fields, and is not to be mixed up with the polarisation
of EM waves. The displacement of positive and negative charges results in electric dipoles.
Figure 6.1: Model of a water molecule as a dipole aligned in an electric field (left), charges
and field of an electric dipole (middle), and schematics of the effect of polarisation of matter by electric dipoles in a dielectric medium of a capacitor (right). Charges on its plates are due
to an applied voltage. Bound charges result from the neutralisation of dipoles, thus
enhancing the charge-storing capacity (from Mike Schwank, and A. von Hippel).
We will focus on because most media are essentially non magnetic (a similar consideration
would apply to μ for magnetic media). The ability of matter to become polarised by an electric
field is the basis of the dielectric behaviour. If matter gets polarised positive and negative charges of atoms and molecules are separated, forming electric dipoles with the electric
dipole moment De = qd where q is the positive electric charge and d the displacement vector
from the negative (–q) to the positive (+q) charge centre. The electric polarisation vector P is
the density of electric dipole moments in a given volume element V and is related with the
dielectric displacement vector by
D = 0E + P = 0 E; and P 0( 1)E =DeV
V= N De (6.1)
where N is the number density of dipoles, brackets mean spatial averaging, and
e = 1 is called electric susceptibility. Insight in this quantity means understanding the
distribution of electric dipoles in matter. We distinguish two cases:
1) In dilute media such as gases, where 1, the electric field acting on the molecules is the
externally applied field E.
2) In dense media, like liquids and solids, the molecular fields of the microscopic dipoles
interact, leading to modifications of E around the dipoles. Therefore we denote the field acting on a given dipole by E'. For sufficiently small fields the average dipole moment has a
proportional relationship with E', and the factor of proportionality is called polarisability .
De = E' (6.2)
55
According to a Model by Mosotti, E' can be approximated by (von Hippel, 1954)
E' E +P3 0
=E3( + 2) (6.3)
For 1, Equation (6.3) confirms Case 1 withE'= E. There exist improvements to the
Mosotti field (e.g. by Onsager, see von Hippel, 1954, Part II, Section 23).
Accepting (6.3) allows to eliminate E' and P by (6.1) to find a relationship between the dipole
number density, polarisability and
N = 3 0
1
+ 2 (6.4)
which gives for
=3 0 + 2N
3 0 N (6.5)
Note that if N approaches 3 0, diverges to infinity. This is called the Mosotti Catastrophe. It
implies a ferro-electric medium with spontaneous polarisation. At the low end, if N <<3 0, we
get the simpler result =1+ N / 0 .
6.2 Types of polarisability
There exist different types of electric dipoles as shown in Figure 6.2, and they can be
distinguished by their temporal response or frequency variation as shown in Figure 6.3.
Figure 6.2: Electric
dipoles in matter without and with an
applied E field:
a) induced polar. in
free atoms (positive
nucleus is displaced
versus negative electron cloud)
b) in free molecules (atmosphere)
c) orientation of permanent molecular
dipoles and dipole
chains in liquids (also
in gases and solids)
d) space-charge
polarisation of mobile charges in crystals
(conductors and semi
conductors)
56
Figure 6.3:
Polarisability versus
frequency of a complex medium
(from Schanda,
1986).
The fastest response is due to the electron cloud in an atom or in a molecule. The electronic
polarisability e reacts to the wide frequency range from the static up to the ultraviolet (UV)
range where resonances occur due to electronic transitions. Atomic spectra are governed by quantum theory.
A slower response is due to the deformation of single molecules where atomic or ionic
motions are involved (molecular vibrations and rotations). The corresponding polarisability is
denoted by i. Since atoms are heavier than electrons (typically by a factor 104), the
resulting resonances occur at much lower frequencies, namely in the infrared (IR) and mm
wavelength range. Quantum theory and observations show that vibrations of polar molecules
result in absorption bands and lines in the wavelength range from about 1 to 100 μm,
whereas rotations result in absorption lines at still longer wavelengths extending to about 1
cm.
Electronic transitions also occur in condensed matter. However, individual vibrations and rotations of molecules are strongly hindered in liquids and solids. The absorption features
lack sharp resonances. In ordered media, such as crystals, the electric field can interact with
elastic deformations (transverse and longitudinal modes). Insight is provided by solid-state
physics. In polar liquids, like water or alcohols, the electric polarisability d is due to the orientational
force of the electric field on the dipoles of the polar molecules. Thermal motion and friction
forces counteract the electric field. The resulting response is a relaxation phenomenon called after Debye, who studied the process in detail (Debye, 1929). Debye relaxation dominates
the interaction in the microwave and radio-frequency range.
At the low-frequency end, freely moving charges (ions or electrons) are the dominant contributors s to . They are displaced if electric field is applied. The limitations are given
by geometric boundaries of the conductor. Space charges of the limited conductors result in
dipoles. They are called Maxwell-Wagner effect. The build up of space charges is often slow
due to the properties of capacity and resistance. The low-frequency response in molecular crystals (ice) is due to Bjerrum Defects, i.e.
imperfect crystals due to variable orientations of the molecules. The resulting response is a
very slow Debye relaxation. The total polarisability of the medium is the sum of all contributions, thus
= e + i + d + s (6.6)
If there are no freely moving charges, s = 0, and if there are no permanent dipoles, d = 0.
Since the terrestrial atmosphere mostly consists of molecules, the first two terms are usually
non zero.
To get a physical idea on we will discuss some models based on classical physics. More
details can be found in the book of von Hippel (1954).
57
6.3 Electronic polarisation
A classical model of an atom consists of a positively charged nucleus with charge Ze (where
Z is the atomic charge number, and e=1.602 10-19 Coulomb) surrounded by an electron cloud
with a constant charge density within a sphere of radius r0, and no charges beyond r0.
Without an external field (E=0) the charges are concentric, and De=0. If E 0 the nucleus is
displaced by the vector d (with d = d ) in direction E with respect to the centre of the electron
cloud. Note that even for the extremely high field, 105V/m, d=10-17m, which is much smaller
than the atomic radius of about 10-10m. Thus d in Figure 6.4 is highly exaggerated.
Figure 6.4: Displacement d of a nucleus (charge +Ze) with respect to the electron
cloud (charge -Ze) by an externally applied E
field (from von Hippel 1954).
To balance the externally applied force F = ZeE on the electrons, a restoring Coulomb force Fc will be set up in the atom by the separation of the charges. To compute this force,
the contributing charge only consists of the electron cloud inside a sphere with radius d
(Figure 6.4). This charge is Qd = Ze(d /r0)3, and the restoring force is
Fc =ZeQdd4 0d
3 = (Ze)2d
4 0r03 = F = ZeE (6.7)
The linearity with d indicates an elastic behaviour. The excited dipole moment is
De = Zed = eE; (E=E' in gases) (6.8)
From (6.7-8) we get a simple expression for the electronic polarisability
e = 4 0r03
= 3 0V0 (6.9)
where V0 is the volume of the electron cloud. Inserting (6.9) in the low-density form of (6.5)
we get for the relative dielectric constant of an atomic gas
=1+ 3NV0 =1+ 3 f (6.10)
where N is the number density of atoms, and thus NV0 is the fraction of the volume occupied
by atoms. In dielectric mixing models this is called volume fraction f. Equation (6.10) means
that the electric susceptibility, -1, is just 3 times the volume fraction of the atoms in the gas.
Comment: The same equation follows from the Maxwell-Garnett mixing formula (4.5), if the
particle dielectric constant 2 , like for a perfect conductor.
For hydrogen atoms we approximate r0 by the Bohr Radius of 0.053nm. Under standard
conditions (T= 0C, P= 1000 hPa), N = NL= 2.687 1025/m3 (Loschmidt Number), and we get
=1.00005. The actual value would be higher (1.0002) because the actual electron cloud
extends beyond the Bohr radius.
58
6.4 Resonance absorption
The elastic force (6.7), acting on the electron cloud in atoms and molecules, and also on
ions, leads to resonances based on the classical equation of motion under an acting electric
field assumed to be in the z-direction
md2z
dt 2+ A
dz
dt+ Bz = eE ' (6.11)
In (6.11) the first term is the inertial force due to acceleration, the second term is an assumed friction force, the third term is the elastic restoring force, and E' is the local field acting on the
particle with charge e and mass m.
We can rewrite (6.11) as a differential equation for the polarisation P by assuming that the
medium is composed of N oscillators per unit volume with electric dipole moments De = ez ,
and P = NDe = Nez , and assuming the Mosotti internal field E'= E + P /(3 0) we get for the
field components in the z direction
md2P
dt 2+ A
dP
dt+ B
Ne2
3 0
P = Ne2E (6.12)
For a time-harmonic electric field E = E0ei t
, also P is harmonic, P = P0ei t
, and we get
1=P
0E=
p2
r02
p2 /3 2 ia
=p2
r2 2 i2 a
(6.13)
where p =Ne2
m 0
is the angular plasma frequency, r0 =B
m is the angular resonant
frequency of the diluted medium ( p << r0), p =p
2, =
2 are the corresponding
frequencies, and aA
m is the damping rate due to friction. Note that (6.13) is a complex
susceptibility. The effective resonant frequency
r = r
2=1
2 r02
p2 /3 = r0
2p2 /3 (6.14)
is lowered by the influence of the internal polarisation interactions in the medium. An
example of the resonance phenomenon is shown in Figure 6.5. For sufficiently low frequencies the susceptibility is real and approaches the value
1=p, j
2
r, j2
j
(6.15)
which corresponds to Equation (6.10) found in the previous section. Below the resonant frequency, the real part of 1 (and thus of n 1) is positive and increases with frequency.
This is called normal dispersion. It is observed for most transparent media in the optical
wavelength range because the electronic resonances are in the ultraviolet (UV). Through the
resonance region and n strongly decrease (anomalous dispersion), whereas the imaginary
parts show strong positive peaks. This means strong absorption near r. Therefore the
resonance is also called an absorption line. In the more general case of various oscillator types without mutual coupling, (6.13) may be
generalised to a sum of terms over all oscillator types identified by Index j
59
1=p, j
2
r, j2 2 ia jj
(6.16)
This is a system of many absorption lines. The theoretical determination of the resonant frequencies is the task of quantum theory. Results will be shown later.
Figure 6.5:
Resonance of 1 according
to Equation
(6.13) for
p / r = 0.126
a / r = 0.159
6.5 Polar molecules in a static field
Polar molecules have a permanent electric dipole moment. Exposed to an electric field they
suffer a torque force, which tries to align the dipole parallel to internal field E'. But due to
thermal motions the alignment is by far not perfect. The following discussion gives an idea on the necessary field strength for appreciable alignment. The potential energy of a dipole
pointing at an angle to the electric field is U = De E'= DeE 'cos . The probability of the
alignment is given by the Boltzmann Distribution Aexp U /(kbT)( ) , where kB = 1.38 10-23 J/K
is the Boltzmann Constant. To the effective polarisation field P each dipole only contributes
its component De,E = De cos parallel to E'. The sum of these components can be expressed
as an integral of the Boltzmann Distribution over all directions. The result is (von Hippel,
1954)
De,E
De
=ex + e x
ex e x
1
xL(x) ; where x
DeE '
kBT (6.17)
and L(x) is a function called after Langevin who first derived it for magnetic dipoles in
paramagnetic media.
60
Figure 6.6:
The Langevin Function L(x) . For x<<1, L(x) = x /3.
In a typical polar molecule, the distance d between the charges is about 10-10 m, and the
charges correspond to ±1 electron charge. Then De = 1.6 10-29 Coulomb m. For an extreme
electric field of 1 Mio V/m and a temperature of 300 K we get x = 3.8 10-3. This means that
the thermal motion dominates the orientation of the dipoles, and that L(x) = x /3. This
linearity also means that does not depend on E'. The average moment due to orientation
becomes xDe/3, thus
De,E =De
2
3kbTE ' (6.18)
Now we can summarise the static polarisability due to electrons (Eq. 6.9), bound ions and dipolar-orientation (6.18)
= e + i +De
2
3kbT (6.19)
The constant terms and the last term can be determined experimentally from the temperature
dependence as shown in Figure 6.7.
Figure 6.7: Molar polarisability versus 1/T for
polar gases (H2O, BrCl, HBr) and for a non-polar
gas (C2H2). Note that is defined by N0
3 0
where N0 = 6.023 1023 is Avogadro's Number. It
gives the number of particles per Mol.
(From von Hippel, 1954)
61
6.6 Debye relaxation in polar liquids
The microwave spectrum of the dielectric constant of water (Figure 6.8) is characterised by a
decrease of the real part, whereas the imaginary part undergoes a broad peak with a
maximum at a frequency of about 10 GHz (T=0C). The functional dependence on frequency
follows a Debye Relaxation spectrum (Debye, 1929). It is of the form
= + s
1 i / 0
(6.20)
where s is the static dielectric constant, is the value at very high frequencies and 0 is
the relaxation frequency. Other polar liquids show the same behaviour, and the same is even
true for many solids, such as ice.
The Debye Formula (6.20) can be understood from the differential Equation (6.12) under neglect of the inertial force (Term 1), by limitation to frequencies well below the resonances:
AdP
dt+ B
Ne2
3 0
P = Ne2E (6.21)
For time-harmonic fields E = E0ei t
, P = P0ei t
, we get
1=P
0E=
Ne2 / 0
BNe2
3 0
iA (6.22)
Equation (6.22) corresponds to (6.20) by proper assignment of the constants.
The reason for relaxation is the dynamic behaviour of the permanent dipoles, which feel the rotational torque by the electric field. At low frequencies the dipole motion follows the field
variation as in the static case. With increasing frequency the friction force more and more
hinders the rotational motion, leading to a reduced dielectric response, and to friction losses
expressed by the imaginary part. The friction corresponds to the dielectric loss as expressed earlier by the conductivity = " 0 = 2 " 0 " / c , where c =17.975 GHz if is
expressed in MKS Units ( -1m-1). Figure 6.8 shows an example of the Debye Function. It
corresponds to the values of water at T=0C, where the relaxation frequency is about 10 GHz.
Figure 6.8: Debye Relaxation
spectrum of = '+i " including conductivity
= " / c , example of water
at T=0C, where
s = 88, = 4.9, 0 = 9GHz.
62
There are similarities between the relaxation spectrum and the resonance curve of Figure
6.5. Both show a symmetrical peak of the imaginary part " in the semi-logarithmic
presentation and the real part ' decreases from the low- to the high- frequency end.
However, in Figure 6.8, ' nowhere increases with increasing frequency, and the transition
from high to low values as well as the peak of the imaginary part " are very broad (note the
4 frequency decades here, but only 2 decades in Figure 6.5).
Figure 6.9 shows " versus ' for similar parameters and over the same frequency range as in
Figure 6.8. The curve is a semicircle with its centre on the real axis.
Figure 6.9: Cole-Cole Plot
of a Debye Relaxation for
s = 80, = 5. Zero
frequency is at the right-
hand end of the curve,
the relaxation frequency
at the maximum of ", and
the highest frequency at
the left-hand end.
Debye (1929) gave a classical interpretation of the relaxation time =1/ 2 0 of a spherical
molecule rotating in a viscous liquid:
=3 V
kbT (6.23)
where V is the volume of the spherical molecule and is the viscosity (friction parameter) of
the liquid. For more details, see von Hippel (1954). Since strongly decreases with
increasing temperature, strongly decreases with T (Figure 6.10).
0.1 1.0 10.0 100.0 10000
20
40
60
80
100
Real
Imaginary
0C30C
Epsilon
Frequency in GHz
Figure 6.10:
Comparison
of dielectric constant
spectra on
log-
frequency scale of pure
water at 0C
(blue) and 30C (red),
showing the
strong temperature
dependence.
63
Although the Debye Equation (6.20) for the relaxation phenomenon and Equation (6.23) for
give good descriptions of the general behaviour, the absolute values found for polar liquids
are more complex. Other theories have been developed, but they are very complex, and are
still under development. Today, results from empirical measurements are used in models for remote sensing. Therefore we will also concentrate on such data.
6.7 Space-charge polarisation
As already indicated by the curve of Figure 6.3 the effect of freely moving charges can also
lead to relaxation phenomena. Examples are dielectric mixtures of conducting particles
hosted in non-conducting media. The resulting dielectric losses are called Maxwell-Wagner losses. Nature produces many media of this type (e.g. vegetation, soil). The related
relaxation frequencies are mostly very low (MHz range and lower).
If the medium is conductive even at zero frequency, " diverges for 0 according to
Equation (2.7). This applies to metals and other conductors.
6.8 Summary
It was shown that microscopic effects of dielectric media lead to three different types of polarisability and thus of dielectric spectra: (1) a very fast response due to the adjustment of
the electron cloud together with similar adjustments with slower response with a frequency-
independent polarisability below a certain limit, (2) resonance phenomena, and (3) relaxation processes. They were attribute to electronic, ionic, orientation and space-charge polarisation.
Figure 6.11 combines the phenomena for assumed parameters. The combined spectrum has
the form
=p2
r2 2 i2 a
+s p
2 / r2
1 i / 0
(6.24)
which is obtained from (6.20) by replacing the dummy parameter by the resonance (6.13)
in the first place and by its low-frequency limit in the second term. The equation is defined by
the 5 parameters given in the title of the figure (f standing for ).
Figure 6.11: Complex dielectric spectrum with one relaxation and one resonance according
to Equation (6.24) with the parameters given in the title.
64
Not discussed before, but of interest is the behaviour of the real and imaginary parts, ' and
" of (6.24):
'=p2
r2 2( )
r2 2( )
2+ 2 a( )
2+
s p2 / r
2( )1+ / 0( )
2 (6.25a)
"=2 a p
2
r2 2( )
2+ 2 a( )
2+
/ 0 s p2 / r
2( )1+ / 0( )
2 (6.25b)
and their low-frequency behaviour ( 0):
'= s ; "=2 a p
2
r4 +
s p2 / r
2( )0
(6.25c)
Whereas the real part is the constant static value, the imaginary part increases proportionally
with frequency in both terms. This will be important to understand why the low-frequency
absorption of many media increases with increasing frequency in proportion to 2.
Problem
Note that ' decreases when the frequency increases through a resonance, or through a relaxation phenomenon. For the relaxation process the decrease ' corresponds to twice
the maximum loss '= 2 "max , which occurs at the relaxation frequency. Show that a similar
rule exists for the resonance effect:
'="maxQ
, where Q = r is the quality factor of the resonance, where the maximum
of " occurs, and = 2 a is the full width at half maximum of ".
6.9 Literature
Debye, P. "Polar Molecules", first published by Chemical Catalog Company, New York (1929), reprinted by Dover Publications, New York (1945), German Edition: "Polare Molekeln", Leipzig (1929).
A. von Hippel, Dielectrics and Waves, 1st Ed. (1954) BEWI: TEA 149, 2nd Ed. (1995) BEWI: VTZ 201.