Illumination path   Observation path

Figure 6.1: Schematic illustration of the interaction between radiation and mat- ter for the purpose of object visualization. The relation between the emitted radi- ation towards the camera and the object feature can be disturbed by scattering, absorption, and refraction of the incident and the emitted ray.

 

If we want to measure certain object features, however, such as density, temperature, orientation of the surface, or the concentration of a chem- ical species, we need to know the exact relation between the selected feature and the emitted radiation. A simple example is the detection of an object by its color, i. e., the spectral dependency of the refl ection coeffi cient.

In most applications, however, the relationship between the parame- ters of interest and the emitted radiation is much less evident. In satellite images, for example, it is easy to recognize urban areas, forests, rivers, lakes, and agricultural regions. But by which features do we recognize them? And, an even more important question, why do they appear the way they do in the images?

Likewise, in medical research one very general question of image- based diagnosis is to detect pathological aberrations. A reliable decision requires a good understanding of the relation between the biological parameters that defi ne the pathological aberration and their appearance in the images.

In summary, essentially two questions must be answered for a suc- cessful setup of an imaging system:

1. How does the object radiance (emitted radiative energy fl ux per solid angle) depend on the object parameters of interest and illumination conditions?

6.2 Waves and Particles                                                               147

 

2. How does the irradiance at the image plane (radiative energy fl ux den- sity) captured by the optical system depend on the object radiance?

This chapter deals with the fi rst of these questions, the second question is addressed in Section 7.5.

 

Waves and Particles

Three principal types of radiation can be distinguished: electromagnetic radiation, particulate radiation with atomic or subatomic particles, and acoustic waves. Although these three forms of radiation appear at fi rst glance quite diff erent, they have many properties in common with re- spect to imaging. First, objects can be imaged by any type of radiation emitted by them and collected by a suitable imaging system.

Second, all three forms of radiation show a wave-like character in- cluding particulate radiation. The wavelength λ is the distance of one cycle of the oscillation in the propagation direction. The wavelength also governs the ultimate resolution of an imaging system. As a rule of thumb only structures larger than the wavelength of the radiation can be resolved.

Given the diff erent types of radiation, it is obvious that quite diff erent properties of objects can be imaged. For a proper setup of an image system, it is therefore necessary to know some basic properties of the diff erent forms of radiation. This is the purpose of this section.

 

Electromagnetic Waves

≈ ×

Electromagnetic radiation consists of alternating electric and magnetic fi elds. In an electromagnetic wave, these fi elds are directed perpendicu- lar to each other and to the direction of propagation. They are classifi ed by the frequency ν and wavelength λ. In free space, all electromagnetic waves travel with the speed of light, c 3 108 ms− 1. The propagation speed establishes the relation between wavelength λ and frequency ν of an electromagnetic wave as

λ ν = c.                                                       (6.1)

The frequency is measured in cycles per second (Hz or s− 1) and the wave- length in meters (m).

As illustrated in Fig. 6.2, electromagnetic waves span an enormous frequency and wavelength range of 24 decades. Only a tiny fraction from about 400–700 nm, about one octave, falls in the visible region, the part to which the human eye is sensitive. The classifi cation usually used for electromagnetic waves (Fig. 6.2) is somewhat artifi cial and has mainly historical reasons given by the way these waves are generated or detected.

148                                                                             6 Quantitative Visualization

 

 

Frequency [Hz]

Wave length [m]

Photon energy [eV]

 

15

1024          _

10

 

1fm

 

109

 

 

1GeV

cosmic rays

 

0.94 GeV rest energy of

proton, neutron

 

 

12

1021          _

10

 

_9

1018

 

 

1pm

 

 

 

 

106

 

 

gamma rays             

 

 

1MeV

      

 

ultraviolet (UV)

8 MeV binding energy/nucleon 1 MeV e-e+ pair production

0.5 MeV rest energy of electron compton scattering

 

diameter of atoms

grid constants of solids

 

photoelectric effect, electronic transitions of inner electrons electronic transitions of

outer electrons

1015

        visible (light) UV/Vis spectroscopy

1µm  1

10 6

_

IR spec- troscopy

infrared (IR)

molecular vibration, thermal radiation at environmental temperatures (300 K)

 

_3

1012                                                               

 

Band

 

11 EHF

3K cosmic background radiation

10

 

1 GHz 109

1

 

1 MHz 106

1mm  10

 

6

1m

_

10

 

 

 

6 MF

radio waves

 

molecular rotation electron-spin resonance

 

8 VHF

9 UHF

10 SHF

nuclear magnetic

7 HF

resonance

 

 

103

 

102

103      1km    _

 

9

12

10

10

6                        _

10

 

4 VLF

20 kHz

 

2 ELF

3 VF

sound frequencies 50 Hz

 

Figure 6.2: Classifi cation of the electromagnetic spectrum with wavelength, fre- quency, and photon energy scales.

6.2 Waves and Particles                                                               149

 

In matter, the electric and magnetic fi elds of the electromagnetic wave interact with the electric charges, electric currents, electric fi elds, and magnetic fi elds in the medium. Nonetheless, the basic nature of electro- magnetic waves remains the same, only the propagation of the wave is slowed down and the wave is attenuated.

=

= +

The simplest case is given when the medium reacts in a linear way to the disturbance of the electric and magnetic fi elds caused by the electro- magnetic wave and when the medium is isotropic. Then the infl uence of the medium is expressed in the complex index of refraction, η n iχ. The real part, n, or ordinary index of refraction, is the ration of the speed of light, c, to the propagation velocity u in the medium, n c/u. The imaginary component of η, χ, is related to the attenuation of the wave amplitude.

Generally, the index of refraction depends on the frequency or wave- length of the electromagnetic wave. Therefore, the propagation speed of a wave is no longer independent of the wavelength. This eff ect is called dispersion and the wave is called a dispersive wave.

The index of refraction and the attenuation coeffi cient are the two primary parameters characterizing the optical properties of a medium. In the context of imaging they can be used to identify a chemical species or any other physical parameter infl uencing it.

Electromagnetic waves are generally a linear phenomenon. This means that we can decompose any complex wave pattern into basic ones such as plane harmonic waves. Or, conversely, we can superimpose any two or more electromagnetic waves and be sure that they are still electro- magnetic waves.

This superposition principle only breaks down for waves with very high fi eld strengths. Then, the material no longer acts in a linear way on the electromagnetic wave but gives rise to nonlinear optical phenom- ena. These phenomena have become obvious only quite recently with the availability of very intense light sources such as lasers. A prominent nonlinear phenomenon is the frequency doubling of light. This eff ect is now widely used in lasers to produce output beams of double the frequency (half the wavelength). From the perspective of quantitative visualization, these nonlinear eff ects open an exciting new world for vi- sualizing specifi c phenomena and material properties.

 

Polarization

The superposition principle can be used to explain the polarization of electromagnetic waves. Polarization is defi ned by the orientation of the electric fi eld vector E. If this vector is confi ned to a plane, as in the previous examples of a plane harmonic wave, the radiation is called plane polarized or linearly polarized. In general, electromagnetic waves are not polarized. To discuss the general case, we consider two waves traveling

150                                                                             6 Quantitative Visualization

 

=

in the z direction, one with the electric fi eld component in the x direction and the other with the electric fi eld component in the y direction. The amplitudes E₁ and E₂ are constant and φ is the phase diff erence between the two waves. If φ 0, the electromagnetic fi eld vector is confi ned to a plane. The angle φ of this plane with respect to the x axis is given by

=

φ   arctan E 2 .                                               (6.2)

E₁

=

±

= ±

Another special case arises if the phase diff erence φ 90° and E₁ E₂; then the wave is called circularly polarized. In this case, the electric fi eld vector rotates around the propagation direction with one turn per period of the wave. The general case where both the phase diff erence is not 90° and the amplitudes of both components are not equal is called elliptically polarized. In this case, the E vector rotates in an ellipse, i. e., with changing amplitude around the propagation direction. It is important to note that any type of polarization can also be composed of a right and a left circular polarized beam. A left circular and a right circular beam of the same amplitude, for instance, combine to form a linear polarized beam. The direction of the polarization plane depends on the phase shift between the two circularly polarized beams.

 

Coherence

An important property of some electromagnetic waves is their coherence. Two beams of radiation are said to be coherent if a systematic relation- ship between the phases of the electromagnetic fi eld vectors exists. If this relationship is random, the radiation is incoherent. It is obvious that incoherent radiation superposes in a diff erent way than coherent radia- tion. In case of coherent radiation, destructive inference is possible in the sense that waves quench each other in certain places were the phase shift is 180°.

Normal light sources are incoherent. They do not send out one con- tinuous planar wave but rather wave packages of short duration and with no particular phase relationship. In contrast, a laser is a coherent light source.

 

Photons

Electromagnetic radiation has particle-like properties in addition to those characterized by wave motion. Electromagnetic energy is quantized in that for a given frequency its energy can only occur in multiples of the quantity hν in which h is Planck’s constant, the action quantum:

E = hν.                                                      (6.3)

The quantum of electromagnetic energy is called the photon.

6.2 Waves and Particles                                                               151

 

In any interaction of radiation with matter, be it absorption of radia- tion or emission of radiation, energy can only be exchanged in multiples of these quanta. The energy of the photon is often given in the energy unit electron volts (eV), which is the kinetic energy an electron would acquire in being accelerated through a potential diff erence of one volt. A photon of yellow light, for example, has an energy of approximately 2 eV. Figure 6.2 includes a photon energy scale in eV. The higher the frequency of electromagnetic radiation, the more its particulate nature becomes apparent, because its energy quanta get larger. The energy of a photon can be larger than the energy associated with the rest mass of an elementary particle. In this case it is possible for electromagnetic energy to be spontaneously converted into mass in the form of a pair of particles. Although a photon has no rest mass, a momentum is as- sociated with it, since it moves with the speed of light and has a fi nite energy. The momentum, p, is given by

p = h/λ.                                                      (6.4)

The quantization of the energy of electromagnetic waves is important for imaging since sensitive radiation detectors can measure the absorp- tion of a single photon. Such detectors are called photon counters. Thus, the lowest energy amount that can be detected is hν. The random na- ture of arrival of photons at the detector gives rise to an uncertainty (“noise”) in the measurement of radiation energy. The number of pho- tons counted per unit time is a random variable with a Poisson distribu- tion as discussed in Section 3.4.1. If N is the average number of counted

photons in a giv√ en time interval, the Poisson distribution has a standard

deviation σ _N =   N. The measurement of a radiative fl ux with a relative

standard deviation of 1 % thus requires the counting of 10 000 photons.

 

Particle Radiation

Unlike electromagnetic waves, most particulate radiation moves at a speed less than the speed of light because the particles have a non-zero rest mass. With respect to imaging, the most important type of partic- ulate radiation is due to electrons, also known as beta radiation when emitted by radioactive elements. Other types of important particulate radiation are due to the positively charged nucleus of the hydrogen atom or the proton, the nucleus of the helium atom or alpha radiation which has a double positive charge, and the neutron.

Particulate radiation also shows a wave-like character. The wave- length λ and the frequency ω are directly related to the energy and momentum of the particle:

ν =  E/h  Bohr frequency condition,

λ =  h/p  de Broglie wavelength relation.

 

(6.5)

152                                                                             6 Quantitative Visualization

 

These are the same relations as for the photon, Eqs. (6.3) and (6.4). Their signifi cance for imaging purposes lies in the fact that particles typ- ically have much shorter wavelength radiation. Electrons, for instance, with an energy of 20 keV have a wavelength of about 10− 11 m or 10 pm, less than the diameter of atoms (Fig. 6.2) and about 50 000 times less than the wavelength of light. As the resolving power of any imaging sys- tem — with the exception of nearfi eld systems — is limited to scales in the order of a wavelength of the radiation, imaging systems based on electrons such as the electron microscope, have a much higher potential resolving power than any light microscope.

 

Acoustic Waves

In contrast to electromagnetic waves, acoustic or elastic waves need a carrier. Acoustic waves propagate elastic deformations. So-called lon- gitudinal acoustic waves are generated by isotropic pressure, causing a uniform compression and thus a deformation in the direction of propa- gation. The local density ρ, the local pressure p and the local velocity v are governed by the same wave equation

0

∂ 2ρ       2               ∂ 2p      2                                                                 1

∂ t2  = u  ∆ ρ,        ∂ t2  = u  ∆ p,           with  u =, ρ   β ad ,              (6.6)

where u is the velocity of sound, ρ ₀ is the static density and β _ad the adiabatic compressibility. The adiabatic compressibility is given as the relative volume change caused by a uniform pressure (force/unit area) under the condition that no heat exchange takes place:

1 dV

β _ad = − V dP.                                                   (6.7)

Thus the speed of sound is related in a universal way to the elastic prop- erties of the medium. The lower the density and the compressibility, the higher is the speed of sound. Acoustic waves travel much slower than electromagnetic waves. Their speed in air, water, and iron at 20°C is 344 m/s, 1485 m/s, and 5100 m/s, respectively. An audible acoustic wave with a frequency of 3 kHz has a wavelength in air of about 10 cm. However, acoustic waves with a much higher frequency, known as ul- trasound, can have wavelengths down in the micrometer range. Using suitable acoustic lenses, ultrasonic microscopy is possible.

If sound or ultrasound is used for imaging, it is important to point out that propagation of sound is much more complex in solids. First, solids are generally not isotropic, and the elasticity of a solid can not be described by a scalar compressibility. Instead, a tensor is required to describe the elasticity properties. Second, shear forces in contrast to pressure forces give rise also to transversal acoustic waves, where

6.3 Radiometry, Photometry, Spectroscopy, and Color                   153

 

the deformation is perpendicular to the direction of propagation as with electromagnetic waves. Thus, sound waves of diff erent modes travel with diff erent velocities in a solid.

Despite all these complexities, the velocity of sound depends only on the density and the elastic properties of the medium. Therefore, acoustic waves show no dispersion, i. e., waves of diff erent frequencies travel with the same speed. This is an important basic fact for acoustic imaging techniques.

 

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