1. Introduction
Waves have specific properties, which do not depend on their physical substance. It means they are common for all kinds of waves as electromagnetic, mechanical, etc. It deals mainly with such typical demonstrations as interference, diffraction, and wave displaying connected with them.
2. Wave function
The wave function is a function of space coordinates and time. It describes the wave quantity in a proper place and time, for example,
The general form of the wave function can be expressed as
where the sign “−” stands for a wave propagating in
2.1 Wave polarisation
The wave quantities are vector quantities, which have their direction relating to the direction of the wave propagation. If this direction is defined, we speak about a polarised wave. For example, an acoustic wave generated by a converter or EM wave generated by an antenna is polarised. One of the polarizations is a
There are also elliptically polarised waves in a plane parallel to the propagation direction. It can be considered a combination of the transversely polarised and longitudinally polarised waves. One example is the surface wave on the water. If we observe the movement of a tiny body floating on the surface of the water with the surface wave, we can see that the body oscillates in a vertical direction, but it also performs an oscillating motion in the longitudinal direction. It follows an ellipse.
If the wave is generated by many individual uncoordinated sources of polarised waves, the polarisation of the resulting wave cannot be unambiguously determined, and such a wave is called a non-polarised one. Bulk acoustic waves in gases and liquids are always longitudinally polarised because in these media the transversally polarised waves cannot propagate due to the fluidity of the medium (this does not apply to the surface waves in the liquid). If the EM wave is generated by a temperature source (hot fibre, infrared radiation of the body surface, starlight, etc.), the individual sources are single atoms of the substance that emit their waves accidentally. Each elementary wave is transversally polarised, and all polarisation directions are present equally. The resulting wave is therefore non-polarised. Typical sources of such non-polarised light are bulbs, gas discharge lamps, LEDs, etc.
Some applications require the use of polarised radiation. In this case, a source of polarised radiation, for example, LASER, can be used. Another possibility is to use a polarizer (polarising filter), a tool that suppresses (filters) one polarisation of non-polarised radiation. Polarizers use anisotropic crystals, lattice structures or reflective elements. Periodic structures (dense optical gratings) are most often used as light polarizers. Polarizers are used on glasses, cameras, or microscope lenses, polarising filters are part of LCDs (digital watches, laptops, mobile phone screens, etc.). The polarisation of light also occurs when reflected from shiny surfaces.
Reflection and transmission of light at the plane interface of two media express Fresnel’s relations
where index r denotes a wave polarised in the plane of incidence, index k a wave polarised perpendicularly to the plane of incidence (parallel to the plane of the interface), index d the incident wave, index “o” the reflected wave and index “p” the passing wave. The angle
As we can see from the relation (2), for
where
A wave polarised parallel to the plane of incidence is not reflected.
If the
The incident wave with r-polarisation is reflected with the same phase as incident one when
Light affects charged particles in biological tissue. Apart from the thermal effect, EM waves also have a direct effect on cellular structures, it can support or suppress some cellular processes. The beneficial effects of light utilise
2.2 Waveform coherence
In an ideal harmonic wave, the phase is defined at each place and time by the wave function
In the case of light sources, the coherence length is limited by the fact that light is generated by individual atoms of a substance. Their photon emission is more or less coordinated. In the case of temperature sources (filament lamps, gas discharge lamps, LEDs), coordination restricts to a very small space, and thus the length of coherence is very small. The large coherence length is achieved with sources that use optical resonator—lasers. Approximate values of length of coherence are filament lamp ∼ 1 μm, mercury lamp ∼ 10 μm, LED ∼ 100 μm, semiconductor laser about 1 m, gas He-Ne laser up to 100 m, fibre laser up to 100 km.
Apart from the coherence of the waves, we also define the coherence of wave sources. The wave sources are coherent in the case of their constant phase relationship. They are then the sources of coherent waves. At the same time, there is a constant phase difference between the waves generated by such sources. For example, two loudspeakers connected to a common AC generator, or two antennas connected to a common transmitter are such coherent sources. Another example of coherent sources is an ultrasonic transducer used in ultrasonography. It is assembled from many segments powered by the same electric generator. Waves generated by two independent sources cannot fulfil the conditions of coherence. The slight difference in the frequency will earn a considerable phase mismatch. To ensure the coherence of two wave beams, they must be coupled to each other, so that the phase difference of the source signals is constant. Coherent sources can be created by letting a harmonic wave fall on two or more slits in a shielding grating. Single slits act as coherent sources for the space behind the grating (e.g., an optical grid). A similar effect is also achieved when the harmonic wave hits a suitable mirror assembly (e.g., a segmented reflector).
2.3 Plane, cylindrical, and spherical waves in lossless medium
For simplicity, consider waves in a lossless, homogeneous, isotropic, and linear medium. In the free space (outside the source), the wave equation has a form
where
The solution is generally complicated and usually obtained by numerical methods. There are several software tools for modelling wave fields. The concrete solution of the equation depends on initial and boundary conditions. The initial conditions are determined by the time dependence of the source quantity, the boundary conditions result from the geometrical arrangement of both the source and the objects influencing the propagation of the wave.
In the following section, we will show some simple cases of wave propagation from sources with significant symmetry.
2.3.1 Plane wave
If the source is a sufficiently large and planar one that generates a wave with the same amplitude and phase over its entire surface, the wave quantities depend only on the coordinate
The solution of the equation has a shape of the wave function
as shown for mechanical or electromagnetic waves. A wave depending on only one spatial variable is called a
In the case of the plane wave in a lossless medium, the shape of the wave propagating along the
A plane wave is only an idealised model of a real wave. In practical cases, the wave has a more complex spatial distribution, since the source surface is never infinitely large, which is a prerequisite for the formation of a plane wave.
2.3.2 Cylindrical wave
Simple cases of spatial two- and three-dimensional waves include cylindrical and spherical waves.
The cylindrical wave is generated by a very long rectilinear line source that generates a wave of equal amplitude and phase along its entire length. In this case, the wave quantities do not depend on the longitudinal coordinate
The solution of this equation depends on the specific time dependence of the wave function given by the time dependence of the source quantity. Due to the harmonic time dependence of the excitation, the solution is the Bessel functions, which for longer distances converge towards the shape
The direction of the vector
As you can see, the displacement
A cylindrical electromagnetic wave occurs, for example, around the line antenna or in the transverse direction within the coaxial line or other cylindrically symmetrical structures.
2.3.3 Spherical wave
We consider a point (spherical) isotropic source that generates waves to the surrounding homogeneous and isotropic medium. Due to the source point symmetry, the generated field has also the same point symmetry. Preferably, the spherical (spherical) coordinates
The solution to this equation has the form
where
In the case of a harmonic wave, the solution is the wave function
where
Note: The correctness of solution (14) or (15) can be approved by direct substitution into the Eq. (13).
The wave intensity is proportional to the square of the wave quantity and therefore decreases with the square of the distance from the source
From a long-distance
In the case of an anisotropic source, which emits different intensities in different directions (e.g., acoustic loudspeaker, directional radio transmitter), the dependence on distance 1/
If we observe the wave at a long-distance
Example 1 Solar constant
The Sun emits an essential part of its energy in the form of EM radiation with a wide spectrum of frequencies. The total irradiated power gives us Stefan-Boltzmann’s law for the radiation of a black body
where
Numerical results are
There is radiation intensity at the distance of the Earth’s orbit around the Sun
where
After substitution, we get
For each square meter perpendicular to the direction of radiation, this power (partially reduced by the atmosphere), which heats the Earth’s surface, evaporates water and causes rain and water to supply rivers, drives atmospheric currents and thus wind, provides energy to living organisms, etc. The energy of these renewable sources, as well as direct energy, is used to generate electricity for human needs.
2.3.4 Waves propagation in waveguides
The wave may propagate in an open space or a structured transmission medium. A special case is represented by waveguides; spatial boundaries that guide the transmission of waves, and hence signal and energy of waves. The simplest case represents a homogeneous waveguide having the same properties along the entire direction of wave propagation. A classic case is a coaxial power line (coaxial cable), which is commonly used to transmit mainly a high-frequency signal. The coaxial cable has an inner conductor and an outer coaxial conductor, the space between the conductors being filled with a dielectric. Between the conductors, there is a radial electric field with an intensity
However, the waveguide does not need an inner conductor and the form of a tube with a rectangular or circular cross-section is sufficient. A waveguide performs its function if the wave is reflected from its walls without loss and remains within. It uses a full reflection of the waves at the interface of the internal and external media. The complete reflection of the EM wave occurs on a perfectly conductive wall, in which case it is the metal waveguide of the EM wave. Total reflection also occurs at the interface of dielectric media if the propagation speed
Example 2 Conductive waveguide EM wave
A simple idea can be obtained from the EM waveguide according to Figure 1.
We consider two parallel planar conductive walls with a mutual distance
The condition of spreading of the
(if
The phase velocity of the waveform
The group velocity of the wave progression in the
The velocity
If the waveguide cross-section is rectangular (
where
The condition for the transmitted wave frequency and the cut-off frequency is
The higher the mode numbers, the greater the geometric dispersion of the waves (as opposed to the material dispersion); it means the dependence of phase velocity on the frequency and thus distortion of the transmitted signal. Modes with low mode numbers TE10, TE01, TE11 are therefore used (both numbers
Analogously, in the waveguide can be excited a wave that has a magnetic intensity
Waveguide modes also propagate in waveguides with other cross-sections, most often circular. Similarly, like for rectangular, there exist TE
The same principle is used by optical fibres—waveguides for optical waves. Instead of a perfectly reflecting conductive wall, a total wave reflection from the dielectric interface with different refractive indices (fibre core
If the cylindrical metal waveguide has an inner conductor (coaxial line), the EM wave has only a transverse character, that is, a TEM mode, which is characterised by signal propagation without distortion due to geometric dispersion. However, in the coaxial line dielectric, there are heat losses due to dielectric imperfection, and therefore the coaxial line is not used to transmit very high EM wave power when the dielectric becomes overheated. Coaxial lines are particularly advantageous for signal transmission.
2.4 Transmission of information utilising waves
The harmonic wave propagating in a medium does not transmit the information itself, but it is only an information carrier. To transmit information, the wave must be modulated by an appropriate information signal, Wyatt [1].
As an example, consider transmitting a data pulse. It can be decomposed into harmonic components using the Fourier integral. For a rectangular pulse with time length
is a complex Fourier image of the pulse function
This is a sin
The propagation of individual harmonic components can be expressed by the wave function
If the pulse is modulated upon a base wave with an angular frequency
The carrier wave with angular frequency
The modulation envelope is represented by the parenthesis, and thus the modulated signal propagates as a wave with parameters Δ
which characterises the rate of signal transmission in a medium.
If we have
2.5 Wave modulation
In the previous paragraph, we showed that waves can transmit information. For example, the sound of our voice is a superposition of single waves with frequencies from tens of Hz to several kHz. It is similar in case of the tones of musical instruments or EM waves generated by lightning during a storm.
Sometimes we need to utilise proper conditions for the propagation of waves in some range of frequency or to transmit several parallel information channels by the wave. In such a case, we use the so-called carrier wave of the required frequency
Suppose the signal given by its time dependence
where
2.5.1 Amplitude modulation: AM
The wave with amplitude modulation (AM) represents the relationship
where it applies to phasors
If two independent information channels are to be transmitted, their carrier frequencies
The wave amplitude changes with modulation. Since the frequency component with the carrier frequency
AM is used to transmit radio broadcasts on long, medium, and short waves. The officially agreed bandwidth of the radio station is 9 kHz so that the maximum frequency of the modulation signal spectrum is 4.5 kHz. This is enough for clear speech transmission but not enough for high-quality music transmission.
A special case is pulse amplitude modulation. It is used mainly for digital signal transmission. Pulsed AM is used, for example, when sampling an analogue signal. The bandwidth of the EM wave required to transmit pulses with a length
The advantage of amplitude modulation is the relative simplicity of modulators and demodulators. The main disadvantage is the liability to interferences and a higher signal-to-noise ratio.
2.5.2 Frequency modulation: FM
In VHF and UHF radio and TV channels with a carrier frequency of about 100 MHz, there is enough space for a wider band (for radio 40 kHz, for TV 7–8 MHz) that provides high-quality audio and video transmission. The fast transfer is also required for the transmission of large data files in telemedicine (e.g., CT and MRI images, online video transfer of the medical operation, etc.).
Concerning transmission better security (suppression of signal error and interference) and reduction of the signal to noise ratio, frequency modulation of the waves is used. Frequency modulation consists of the control of the frequency of the wave by the modulation signal. The amplitude and thus the power remains constant (as opposed to AM when the power fluctuates and is thus more susceptible to interference).
The frequency modulated wave can be described by function
which can be expressed as a series of Bessel functions, where the spectral components of the wave are the same as in the case of AM.
Pulse FM used in data transmission, consists of a change of frequency at the time pulse duration. Since this change is limited in time, the same condition as AM applies to bandwidth Δ
2.6 Material dispersion of the wave
The wave velocity depends on the parameters of the medium, depending on the frequency of the wave. This phenomenon is called material dispersion. As shown in the previous paragraph,
The material dispersion appears significantly at a large relative bandwidth Δ
The material dispersion in the case of the transmission of (pulse) data information in the optical fibres is reduced by selecting a frequency domain with minimal material dispersion, for example, in glass fibres around
3. Ray optics
For the wave at a great distance from the source, that is, at the distance much greater than the dimensions of the source (so-called far Fraunhofer region), the ray representation of the wave can be used.
A ray is a line along which a wave passes. In addition to the rays, there are defined wavefronts. They represent the surfaces of the constant phase, in the case of a harmonic wave, and the surfaces that the wave reaches over some time, in the case of pulse or other waves. The wavefronts and rays are orthogonal to each other. At any point, the beams are perpendicular to the wavefronts.
Wave propagation obeys two basic principles, the Fermat principle, and the Huygens principle, which follow from the basic wave equations.
3.1 Fermat’s principle
Fermat’s principle says that the wave propagates from point A to point B along a spatial curve at which the time
where the symbol δ represents a variation of the respective functional (expression in parentheses).
For EM waves, phase velocity
where
Fermat’s principle then says that EM waves propagate from point A to point B along the shortest optical path.
Example 3 Reflection and refraction of waves on the plane boundary of two homogeneous media
It follows from the Fermat’s principle that in a homogeneous medium, where
At Figure 2 there is a ray going out from point A and reflected to point A′ and refracted to point B.
The ray OA′ has the same length as its mirror image OA′′. As the ray travels in the same medium, the shortest wave path is equal to the shortest geometric path whose length corresponds to the length of the abscissa between points A and A′′. The point of reflection O lies on the line AA′′so that the angles
Consider now the transition to the second environment from point A to point B. We denote the distances x and d, as shown. We are looking for a minimum of expression
We express the relation for
We obtain the minimum
from where we get Snell’s refraction law
Example 4 Total reflection
We express the relation for the angles of incidence and refraction in the form
From this inequality, we get the condition of the refraction of the beam
For
If
where the refraction of the ray occurs when the condition
For the angles of incidence
The total reflection phenomenon is used, for example, in optical fibres, or generally in waveguides. If the refractive index
In biomedical applications, optical fibres are used in laser lithotripsy, laser scalpel, endoscopy, and the like.
Example 5 Optical fibre
Many applications use waves led in optical fibre. These are mainly signal transmission (e.g., optical computer networks), illumination of inaccessible areas (e.g., an internal organ in the body using an endoscope), the transmission of radiation power (e.g., in laser lithotripsy). The essence of the optical fibre transmission is that the wave is completely reflected from the fibre walls and cannot escape from the fibre. This creates an optical waveguide. The basic condition is that the wave must strike the fibre wall at an angle of incidence
The beam of parallel rays is centred by the lens S on the inlet surface of the cylindrical filament, the angle of incidence on the filament front being
If
The wave proceeds along the fibre axis at a phase velocity
The optical cable consists of many fibres. In endoscopes, an optical cable is used to transfer the image so that the image produced by the lens of the objective is projected onto a bundle of optical fibres, each transmitting 1 pixel of the image. The cable consisting of 100,000 fibres with a diameter of 5 μm (for a light wavelength of about 500 nm) has a diameter of approximately 1.5 mm. After leaving the cable, the light strikes the detector (CCD chip).
3.2 Huygens and Huygens-Fresnel principle
The propagation of waves in space is described by the
In this way, we can construct one wavefront after another and gradually depict the whole wave field. The procedure is shown in Figure 4. In the figure, three wavefronts correspond to times
In the figure, the elementary wavefronts do not have the same radius, which relates to different velocities of wave propagation in different places of the non-homogeneous medium. The rays are orthogonal to the wavefronts. We can see that in a non-homogeneous medium, the rays decline towards parts of the space with the lower velocity of propagation, lower right part in the figure.
Fresnel extended Huygens’ idea to add a quantitative dimension. The wave quantity at a given point P of the following wave is a superposition of individual waves of elementary sources in the points M of the previous wavefront S, for illustration, see Figure 5.
where
Example 6 Plane wave and the Huygens-Fresnel (H-F) principle
As an illustration of the application of the H-F principle, let us give an example whose result we know. Consider a plane wavefront of a plane wave as a source (points M) and examine the further propagation of the wave—to the point P at a distance
The H-F integral then has an expression
Since
If
and we adjust the H-F integral to this shape
which is a complex function of the plane shifted by
From the initial planar wavefront, we get the following parallel planar wavefront, which corresponds to the propagation of the plane wave in space.
This example has shown that (34) correctly describes the wave propagation and the construction of the next wavefront, which is more than
The Huygens-Fresnel principle will be used in the next section to explain the diffraction phenomenon.
4. Wave interference
If several waves propagate in a linear medium, they compose—
4.1 Constructive and destructive interference
Let us consider two plane harmonic waves with the same polarisation, and the same frequency, which propagates in the same direction and the phase difference φ between them. The resulting wave is
We set the relationship to shape
The resulting wave has the wave properties of the original waves, the amplitude
and phase shift for the wave
If the waves have the same amplitude
and
Under constructive interference
The resulting wave has twice the amplitude and the same phase as the original waves.
Under destructive interference
In the first case, the wave is amplified to double the amplitude, in the second one the waves are mutually suppressed.
Example 7 Reflection of waves from thin film
A typical interference phenomenon is the reflection of the wave from a thin layer of substance. Consider the perpendicular impact of the wave from the medium with propagation velocity
The incident wave is reflected from the first interface backward with the opposite phase, that is, Δ
The interference is constructive when Δ
If the layer has a thickness
As we can see from the result, constructive reflection depends on the wavelength. If white light falls on the layer, only a certain colour is reflected. If the layer thickness changes, the colour of the reflected light changes as well. This is used, for example, for measuring the thickness of thin films. If there is a thin oil layer on a surface of the water, which does not have the same thickness everywhere, different colour patterns are formed because of interference reflection. We can see it on water puddles. The effect is most noticeable when
Destructive interference occurs when Δ
A layer of a thickness dd reflects minimally, and thus maximally transmits. This is used to form anti-reflective layers, which we use to get the maximum of the incident light. They are used, for example, for glasses to increase the brightness of the observed objects, for telescope lenses, etc.
As we can see, the layer is anti-reflective only for certain wavelengths, but for other wavelengths, it should be reflective. The thickness of the layer can be set to wavelength according to our demand to support or suppress the light transmission. Since the wavelength of yellow light is about twice the wavelength of ultraviolet one, the layer can be reflective for the UV light and at the same time transparent for a yellow one.
4.2 Wave beats
In another case, consider two harmonic waves with the same direction of polarisation and the same amplitude, which propagate in the same direction and whose angular frequencies differ by a small difference Δ
We adjust the resulting wave function to shape
The composition of waves results in a wave that represents a carrier wave with a mean angular frequency of
Wave intensity is
The wave intensity
Example 8 Stereo sound reproduction
The preferred way of listening to recorded music is stereophonic reproduction. The record is scanned by two microphones located at an appropriate distance. The recording is then played back from two parallel speakers whose distance is
The listener P moves along a line parallel to the speaker apertures at a distance
Suppose that the electric current through the loudspeakers has the same frequency and phase. Both speakers thus generate coherent waves with the same amplitude and phase. The waves from both speakers are composed at the listener position P
where cos
The sound intensity in the point P of the listener is
where
The dependence of the relative sound intensity in front of a pair of speakers is shown in the graph at Figure 7. For the clearness of the illustration, the plots for two different frequencies are drawn. We can see that in some places the sound of some frequency is not heard. Thus, the spectral composition of the music you listen to depends significantly on the position P of the listener. It results in the change of tone colour, and thus the whole harmony of the music. The only place where we do not hear this distortion is on the axis of the system
Suppression of these negative phenomena in two-source stereo is solved by a set of more speakers, for example, a quadraphonic system.
4.3 Standing waves and resonators
Consider two harmonic plane waves with the same polarisation and frequency that propagate in opposite directions. The resulting wave is the superposition of both waves
The term can be broken down into two parts
The resulting wave has two different components. The first one is a moving wave that propagates in the
If the amplitudes of the two waves propagating against each other are the same, a pure standing wave occurs
Points where
Conversely, points where
Standing wave or partially standing wave arises when the wave is reflected from the interface of two media by the composition of both direct and reflected waves.
The relation (22) shows that the maximum displacement is
The quantity
defines the standing wave ratio—
The principle of standing wave formation is used in resonators. One example of a resonator is the reflection of the wave from the thin film described above. The resonator may be used as a wave amplifier. Multiple wave reflections, inside the resonator, can cause its amplification, like resonance in a serial RLC electrical circuit, wherein the resonance state the voltage at the capacitor is Q times a voltage of the source. Q is the quality factor (in the low-loss circuit Q ≫ 1). Similarly, in the wave resonator, the intensity of the wave in the resonator is many times greater than the intensity of the coming wave, resp. of the wave generated near the node. A typical example is creating a sound on a piano string—the hammer gently strikes the string near the knot (point of string fixation). The string sounds only with tones corresponding to the resonance condition (basic and higher harmonic frequencies). It is like the air column of the wind musical instrument, where the tongue vibrates the air near the knot, and the entire column sounds with multiple intensities.
Example 9 Standing wave on the string—vocal cords
The vibration of vocal cords whose frequency determines the tone of the voice can be modelled with a simple model of standing wave on the string. The string (the musical instrument) is fixed at the endpoints, which cannot move. They represent, therefore, standing wave nodes. For the standing wave then applies
On the string, there arise standing waves with discrete frequency spectrum
The vocal cords are the muscle bundles between which there is a gap. As the air flows through the gap, the vocal cord muscles tremble, the fundamental frequency of oscillations being dependent on the thickness of the vocal cords, and the force that strains the vocal cords. Children and women have vocal cords thinner, so they have a higher voice frequency. In men in adolescence, the vocal cords coarsen (mutation), and the voice becomes deeper. You can control the pitch of a tone by changing the vocal cord tension, so you can intonate when singing (intonation—pitch control).
Example 10 Standing wave in a tube—ear canal
The outer ear canal is an acoustic resonator that increases the sensitivity of hearing at the middle frequency of audible sound (about 3 kHz). The incident longitudinally polarised wave is reflected on the eardrum and interferes with the wave passing directly. This creates a standing wave and resonantly amplifies the sound.
A simple model idea is provided by the description of the standing wave in the air column in the canal. If the tube is closed at the end, there is a standing wave node at the end, there is an anti-node in the open mouth. The length of the ear canal thus represents a quarter of the wavelength.
Substituting the values of
Example 11 Ultrasonic transducer
To generate ultrasound, for example, in sonography or lithotripsy, piezoelectric transducers are used. The ultrasonic transducer is a plate of an anisotropic piezoelectric material with suitable orientation and electrodes on its surface. When the AC voltage is applied, the mechanical stress in the plate changes alternately, and thus the deformation (plate thickness slightly alternately increases and decreases) generates oscillations that represent a standing wave in the plate. If the wave impedance of the outer medium is less than the impedance of the plate, there occurs reflection of the wave on the surfaces of the plate and the standing wave arises in the plate. Thus, resonance occurs when the plate thickness is approximately equal to half the wavelength. For values used in sonography, for example,
Example 12 Infrasound in the room
The danger for a human is represented by vibrations with frequencies below the audible limit of
For a room with a length of
5. Diffraction
5.1 Radiation of the planar source
Planar sources are often used to generate waves, for example, piezoelectric plates for generating ultrasound, reflectors of sources of EM radiation, or funnel antennas of microwaves. The planar source can also be an aperture in a screen, which is hit by a plane wave—the aperture becomes a planar source for additional space.
5.1.1 Long rectangular strip source
Thin strip transducers, especially ultrasonic ones, are often used. Such a source may be also a slot that is hit by a plane wave, for example, the light.
Using the Huygens-Fresnel principle, the strip of width
The interference of cylindrical waves of elementary sources is expressed as
Consider a wavefield at a distance
The wave function argument is expressed
and after calculation of the integral, we obtain
The wave propagates from the elementary source as a cylindrical one in all directions, but the wave is not isotropic and has a distinct directional pattern depending on the angle φ under which we observe the radiation.
The wave intensity is
where
In the graphs in Figure 9, we see the diffraction function F(
The maxima of the function F(
The value
In Figure 10, the radiation characteristic of a strip with
The width Δ
It results in the numerical solution
The width of the main lobe is then
For the characteristic at Figure 10 (
For a wide strip with
If we want to achieve a narrow characteristic, for example, in sonographic imaging, we must choose a small ratio
The above-described diffraction occurs at a distance
Only from a distance
from where
For
5.1.2 Rectangular and circular planar source
To illustrate the diffraction effect, there are shown, in Figure 12, diffraction patterns formed on the projection screen after irradiation of a rectangular aperture (left image) and a circular aperture (right image) with a plane wave of the laser beam. The series of bright points on the horizontal axis (left) corresponds to the diffraction maxima according to Figure 9. The rectangular source can be taken as the combination of two mutually perpendicular strips. Each side induces diffraction in a direction perpendicular to the respective side with parameter
In the case of a circular source, the Huygens-Fresnel integral has a form
where
Considering
and the surface element of the source
By solving this integral we get
where J0(
where J1(
where
In Figure 13 (right) we see the irradiation characteristics. At 50% of the maximum, the width of the main lobe is Δ
The central bright circle is called the Airi disk. It is surrounded by a circle of zero intensity F(
For the values in Figure 13 is
The results indicate that the angular diameter of the Airi disk and thus the scattering angle of the radiated wave decreases as the diameter of the source increases.
Again, the result is valid in the far Fraunhofer region
5.2 Structured plane sources
5.2.1 System of parallel strip sources with the same phase
Consider a simple transducer structure consisting of a grid of N parallel equal strips (according to). As in the previous paragraph, we observe the angular dependence of the radiation intensity. The distance between the centres of the strips is
If we express the sum of the geometric progression, we get
and radiation intensity
where
The function F(
For the width of this maximum, we have Δ
For small values of angular width, we get approximate relation
The relation (68) shows that the original beam of incident waves is divided into individual diffraction beams deviated from the original direction by angles
Example 13 Diffraction grating spectroscope
A diffraction grating is used to separate the wave components of different wavelengths from one another. If white light falls on the grating, it is broken into individual colour components. The first diffraction maximum is used for the decomposition of light (
For grating with
The diffraction grating can distinguish wavelengths with the difference δ
from where
For the given values δ
In the light spectrum of the sodium lamp that emits yellow light, there is an emission double line (doublet) with wavelengths
Example 14 Diffraction grating monochromator
In some applications, monochromatic radiation (single wavelength) is required. In such a case, the light with the desired wavelength can be obtained using its selection from intense white light, for example, of halogen lamp, by a diffraction grating monochromator.
The principle is the same as in the previous example. The white light is dispersed by the diffraction grating into individual colour components, and the desired wavelength is selected using a slit-shaped aperture. In addition to the transmission grating, a reflective one is often used as well. It consists of parallel strips with a reflective surface that is illuminated by white light. The individual strips thus represent, in reflected light, a set of parallel wave sources, which interfere with each other as in the case of a transmission grating.
The arrangement of the monochromator is in Figure 14. The white light source and the aperture are fixed. The reflective grating is deflected so that light with the desired wavelength
In spectrometers, the sample to be examined is placed behind the aperture. Its absorbance is determined from the signal of the detector versus wavelength (deflection of the grating). The result is recorded in a spectrogram, a picture on the right, from which it is possible to determine the presence of certain substances in the sample, for example, the haemoglobin spectrum or the fat content in the blood.
From the results (69), resp. (70), transducers with multiple periodic structures have a significantly narrower radiation pattern than a single strip source. If the aim is to concentrate the wave power to a small scattering angle Δ
Example 15 Sonographic probe
A structured transducer consisting of more parallel piezoelectric strips is used as the ultrasonographic probe. In the case of
Such a structured transducer has a sufficiently high angular resolution.
5.2.2 Electronic deviation of the radiated wave
Many applications require changing the direction of radiation, for example, radars, or ultrasonography. This can be achieved by a mechanical deviation of the antenna, for example, swinging, or rotating radar antennas, or USG probes with a mechanical deviation of the ultrasonic transducer.
The controlled deviation of the radiation characteristic can also be achieved electronically. In the previous case of structured multistrip transducer, all strips generated their waves with the same phase on their surface. If, however, the individual strips are excited, so that the phase shift
where γ =
The main maxima correspond to
If
Due to varying the phase shift
5.2.3 Electronic focusing of the radiated wave
To achieve the necessary resolution of the ultrasonographic image of the tissue at the desired depth, it is necessary to focus the wave to that depth. The elder method used an acoustic lens on the surface of the ultrasonic transducer. The current method consists of focusing the wave electronically. It is achieved by a suitable phase shift of the individual waves radiated by the strips of the transducer, Figure 15 on the right.
The individual waves from the strips are concentrated in the focus P when they encounter in the same phase, and constructive interference occurs.
The path of the
If the electrical exciting signals of the individual strips are programmed so that their phase decreases gradually according to (73), the resulting wave concentrates in the focus P at a depth
Different generated beam profiling can be achieved through digital processing of the exciting signal. The digital control of phase shifts of the individual elements of the transducer using a computer is considerably simpler than the analogue solution.
6. Wave imaging
Waves, both mechanical or electromagnetic, occur in a wide range of phenomena and applications. One of the technical and natural phenomena is the transmission of information. We can see objects by vision or perceive sound through hearing. The basis of human perception of the waves is processing many signals provided to the brain by biological sensors, which are light-sensitive cells on the retina of the eye or cells of the cochlea sensitive to vibrations in the inner ear. Paired sensor organs (a pair of eyes, a pair of ears) provide the brain with information to create a stereo effect. In space, we orient ourselves by sight and hearing. The essence of sensory perception is the detection of the spatial–temporal distribution of the respective wave, which is created by its source and carries information about the properties of the source. By detecting this field, we infer the configuration of the waveform and create such an image of the source (optical, acoustic, etc.). The observed image is formed in our brain. The projection of an object into the structure of brain cells represents a certain transformation by which the object is transmitted to the brain cell’s structure using neural signals. From the created image in the brain, we infer the position, arrangement, and other observed properties of the object. But the image in the brain only shows the perceived waves field. Between the object and the human sensors, the perceived field may be modified by many obstacles. It means, the image created in the brain need not match the object faithfully, for example, if we look into a crooked mirror the image created in the brain does not match the real object (we do not believe our eyes). In the perception of vision, the waves emanating from the observed object are modified before their impact on the eye retina cells by the eye lens and cornea. On the retina, the incident wave creates a two-dimensional image of a three-dimensional object. For example, kilometres large object is displayed as a centimetre image on the retina, and then it is sensed by sensory cells.
The brain has a memory, and it stores received images. Human memory is subjective so that various methods of wave-field recording are used for objective image preservation. If not only the amplitude but also the phase information is recorded, we get a complete record—a hologram. But two-dimensional recording only amplitude information is much simpler and widely used, for example, an ordinary photo. A suitable combination of amplitude records can create a spatial impression, for example, an acoustic stereo effect using headphones with a separate signal for the right and left ear, or optical by creating separate two-dimensional images for the right and left eye (e.g., using stereo glasses). However, this stereo effect is incomplete because it provides a spatial view from only one direction. Full digital image recording, tomography, is allowed by computer memory which can store a series of two-dimensional images of thin slices (
Wave imaging follows three basic goals. The first is to make the observed object visible or audible to the human senses, for example, observation of infrared, X-ray, ultrasound, or infrasound and ultrasound, or other waves “invisible” to humans. The second one is to allow observation of very distant, very large or very small objects, observation of very fast or very slow events, etc. The third goal is to record images for both analysis and archiving purposes. In addition to artistic recordings (paintings, sculptures), there are recordings on storage media, which may be chemical emulsions (classical photography), mechanical, magnetic or optical sound recordings (turntables, magnetic tape, magnetic disc, optical disc) or digital recording in computer memory. Records can be temporary (short term) or permanent (long term).
Short-term recording (computer display, microscope or telescope image, computer RAM recording, etc.) serves mainly to analyse the current structure and properties of the observed object, for example, observation of tissue by a microscope, observation of internal organs by an endoscope, etc. The long-term records are mainly used for archiving, for example, photo albums, X-rays images, etc. A modern and very economical way of archiving is archiving in digital form on various storage media. Digital archiving saves considerable space in healthcare facilities instead of storing printed reports or photographic images. Also, archived digital records allow rapid search and operative communication using communication networks.
The basis of imaging is to create a readable image, either for the observer’s eye or for the respective recording medium. Wave imaging, except for volumetric holography, is two-dimensional (eye retina, film, camera CMOS or CCD chip, tissue tomography slice). The wave imaging aims to display the object as accurately as possible on the surface of the recording medium. For this purpose, serve the individual elements of the display systems, such as mirrors and lenses, or different complex systems, for example, optical, and acoustic microscopes, telescopes, photographic objectives.
6.1 Elements of display systems
6.1.1 Mirrors
One group of display system elements are mirrors. The mirror may be planar or curved. In Figure 16 are several types of used mirrors.
(a) The planar mirror creates an image virtual, straight, and right-left inverted (that is why persons know themselves differently from a mirror and a photograph). When properly positioned, a planar mirror allows looking around the corner (known application is a periscope). (b) The 3D corner reflector consists of three mutually perpendicular planar mirrors forming a “corner”. At the present 2D figure, only a pair of mirrors is shown. The ray of wave incident at any angle on one mirror is reflected from the other one exactly in the opposite direction. The principle of the 3D corner reflector is the same. It is used, for example, in roadside reflecting pieces, or in the safety reflecting elements, where the light of a vehicle headlamp is always reflected towards the driver. (c) Curved concave mirror (usually spherical or parabolic in more demanding applications). The beam of rays parallel to the optical axis incidents in the mirror and is reflected the focal point F located at the centre of the distance between the top of the mirror and the curvature centre. It is used in medicine, for example, as an ORL mirror that the doctor puts on the head. It concentrates the light from the lamp on the examined place, usually an ear. The doctor looks at the place through the aperture A at the top of the mirror. In advance, the mirror shields the disturbing direct light falling into the doctor’s eye, thus increasing the contrast of the object under investigation. If a detector is placed in the focus of the concave mirror, the energy of radiation striking the entire surface of the mirror (parabolic antennas) is concentrated there. The larger the mirror radius, the stronger the signal is detected. This principle is used, for example, at antennas for satellite reception or microwave radio transmission of a digital signal. Similarly, the principle also uses a parabolic antenna to receive mechanical waves of distant acoustic sources. (d) If a point source of the wave is placed in focal point F, the wave, when reflected from the concave mirror (reflector), forms a parallel beam of rays. It is used, for example, when illuminating the operating field during surgery or dental procedures. By changing the position of the source on the axis, a slightly diverging or converging beam of radiation is achieved. (e) The curved concave mirror is also used to display the object. If the object P is placed between the focus F and the top V of the mirror, a direct and enlarged virtual image is produced, with a greater distance from the mirror. This is used, for example, as dental mirrors for the diagnosis of teeth or cosmetic mirrors. These principles are common to any wave, regardless of its physical nature, whether mechanical (sound, ultrasound) or electromagnetic (light, infrared radiation, radio waves, etc.).
6.1.2 Lenses
Unlike mirrors, which use reflection, lenses use a transition through the material of the lens with a refractive index different from the refractive index of their surroundings. The basic types of lenses are shown in Figure 17 with the indicated passage of the light rays through the lenses.
Lenses in the figure have refractive index
(a) The converging (positive) lens concentrates a beam of rays parallel to the axis into the image focus F′ and forms a diverging beam behind the focus like a point source located in the focus. The intensity of the wave in the focus is greater, the larger the area of the lens from which it gathers incident rays. Like an optical lens, an acoustic lens works by concentrating an incident ultrasonic or sound wave into the focal plane. Acoustic lenses have been used in ultrasonography to focus the ultrasonic beam. Today they are replaced by digital focusing. (b) If we place a point source in the object focus F of the lens, for example, halogen bulb, the lens forms a parallel beam of rays similar to a spherical mirror (while in the case of a mirror the source shields part of the reflected beam, this does not occur when the lens is used). It is used in projection devices as a condenser. (c) The figure shows the display of points lying outside the lens axis. If the object is located at a distance
Lenses (glasses, contact lenses) are used to correct long-sightedness (positive lenses) or short-sightedness (negative lenses) if the eye lens is not capable of the necessary accommodation.
6.2 Imaging systems
Various imaging systems are assembled from the individual optical elements to display the object for observing it by a human eye with sufficient resolution or an image recording by a proper device (photograph, file on a memory disk). Imaging systems allow observing very distant objects (binoculars—telescopes) or very small objects (microscopes). The displaying system can be supplemented, for example, by diffraction grating for spectral analysis of the incident radiation. See also Splinter [2].
6.2.1 Photographic camera and projector
The simplest optical system is a photo camera, projector, or the human eye. The principle of the camera is in Figure 17 on the left. The device objective can be a single lens or a set of lenses. Due to the dispersion of light in the lens material and correction of this undesirable phenomenon, complex objectives composed of several lenses (converging and diverging) of different materials are created for cameras to correct these defects. The camera objective at all is a converging system that concentrates the rays from the object on a recording medium (photographic film or plate or detecting sensor—CCD or CMOS chip) to create an image. As shown in the section on lenses, a distant object creates an image in the focal plane of the lens. The focal length
To display distant objects a telephoto lens with a focal length of up to 400 mm is used, which has a small viewing angle but displays very distant objects on the recording medium. Optical devices also use lenses with variable focal lengths, zooms, which allow using both a wide-angle and a telephoto lens when retuning. The zoom can be optical (optical zoom) when the focal length of the imaging system changes, or electronic (digital zoom), when the selection of image points on the recording medium is electronically changed. By selecting the material of the elementary parts of the optical system and the detecting medium, it is possible to realise cameras for displaying visible light (VL)—photo cameras, or cameras for displaying infrared radiation (IR)—thermographic cameras. The IR camera works on the same principle as a VL camera, only the elements of the optical system must be transparent for infrared radiation and opaque for visible light (e.g., monocrystalline germanium), and the detector must be sensitive to IR radiation. CMOS detectors have the best features (sensitivity, noise) for it. IR cameras are used in thermographic diagnostics, whether technical or medical.
Projector, Figure 18 on the right, is used to project film slides or digital recordings on a screen. The current modern tool is a digital data projector. The projector contains an intense light source (usually a halogen lamp). The diverging beam of the point source is changed to a collinear beam using a condenser (positive lens). The light hits a film or digital recording and projects it onto the screen using an objective. Projectors are used to present movies, images, or digital recordings from a variety of storage media.
6.2.2 GRIN lens
In an optically or acoustically inhomogeneous medium, the rays of the waves are curved. It is used in various technical applications. The optical inhomogeneity of the medium is used, for example, in optical fibres or GRIN lenses.
Example 16 Curvature of the acoustic beam in the surface layer of water in the sea
In the sea, the chemical composition and pressure of the saltwater change with depth. This affects the velocity of sound propagation in water. At steady conditions, the velocity of sound in saltwater decreases approximately linearly to a depth of about
We show that the sound beams are curved and have the shape of a circular arc, Figure 19. The figure on the right shows the elementary section of the beam with the radius of curvature
The length of the elementary section of the ray can be expressed by
The beam is, therefore, a curve with a constant curvature, that is, circular arc. If an acoustic wave is emitted from a source at point A, it propagates along a circular ray to point B at the same depth
then it continues along a next circular arc to point C at the depth
Elements of optical systems with a modulated refractive index, are mainly used in micro-optical applications. As mentioned, the basic element with a transversally modulated refractive index is an optical fibre.
Another element, especially of endoscopic imaging systems, is the GRIN lens (Gradient Refraction Index). The principle of focusing is indicated in Figure 20, like Example 16. In the middle is a sample of a GRIN lens with its typical dimensions. On the right is a sample of the endoscope tip.
GRIN lenses are particularly advantageous in that they have very small dimensions and do not require any specially shaped surface like conventional lenses. By suitable profiling of the refractive index, a special shaping of the optical beam can be achieved. Figure 20 indicates the use of GRIN technology to implement a converging lens. This application is mainly used as an end optical element of endoscopes.
6.2.3 Endoscopy
The endoscope is an optical imaging device that allows investigating internal organs inaccessible to direct observation. The endoscope is a cable that is inserted into the examined area. It contains illumination of the investigated area and a camera for its displaying. Typical applications are gastroscopy (endoscope inserted into the oesophagus), colonoscope (endoscope applied through the rectum to the large intestine), and the like. In the endoscope, the GRIN lens has two different functions. They are used as collimators to illuminate the field of view. A laser beam propagates through the optical fibre, and the GRIN lens extends it into a diverging light beam. The second function is in getting the image, where it has the same function as the camera objective.
The GRIN lens displays the subject in the image plane, from which it is captured. Two techniques are used. Most endoscopes are fiberscopes that introduce images into the optical fibre bundle (about 100,000 fibres with a diameter of several μm) in the image plane, and the fibres guide the light of the individual pixels of the image to a detection device at the outer end of the endoscope. The second option is to use a CCD sensor that transforms the image in the image plane into an electrical signal. Fiberscopes have a smaller head and a higher signal-to-noise ratio. They are mainly used in laparoscopic operations and endoscopy of fine structures (in angiography, neurology, etc.). The CCD chip has larger dimensions and a lower quality signal, but it allows an electric connection with an output. It is used, for example, in endoscopic capsules, Figure 21. The capsule with a diameter of about 10 mm contains an optical system with a GRIN lens and LED illumination around the circumference (see the front view of the capsule). The image is captured using a CCD, and the electric signal is transmitted via Bluetooth to an external sensor on the body surface. The capsule has an internal power supply. It is used to investigate the small intestine, which cannot be reached by a standard endoscopic probe, gastroscopy of the stomach and duodenum, or colonoscopy of the large intestine. After swallowing, the probe moves through the digestive tract and transmits images of the intestinal wall.
6.2.4 Optical system of the telescope
A telescope is an optical system that is mainly used to image distant objects or to expand or compress a wave beam.
The basic set of the telescope consists of two lenses, where the image focus F1′ of the first lens (objective) is identical with the object focus F2 of the second lens (eyepiece). Lens telescope has an objective converging lens. The first Galileo telescope used an eyepiece the diverging lens, Figure 22a. The focal length
Then the angle
The disadvantage of lens binoculars is the loss of part of the power of the incident radiation by reflection at the lens-air interfaces. This is solved by mirror telescopes, in which the objective lens is replaced by a hollow mirror (Newton’s or Cassegrain’s telescope). The intensity of the image depends on the aperture of the lens
The intensity of the compressed beam
For
Binoculars are also used together with a recording device mounted behind the eyepiece instead of the eye, or with a Doppler analyser, which allows determining the speed of movement of the observed object. Such a device uses, for example, police to measure speed and record road vehicles. The camera (photo or video) with a telephoto objective also performs a similar function.
6.2.5 Optical system of the microscope
Unlike the telescope, the microscope is used to image very small and near objects. We will explain the principle of the optical microscope, but the same principle uses infrared or acoustic one.
The microscope, Figure 23, consists of two converging lenses S1 and S2. There is an optical interval
Conventional microscopes are used to image only thin planar (2D) samples. The image of the volume (3D) sample is blurred. Figure 24 is the construction of images of objects that lie in different places and object planes r1 and r2. Let us first consider two objects A, B in one object plane r1 the microscope is focused on. If we observe the course of the characteristic rays, objective S1 creates real images A1, B1 in the focal plane of the eyepiece. Behind the eyepiece, beams of parallel rays at different angles concerning the axis correspond to each of the objects. There is another imaging lens behind the eyepiece, either the lens of the eye in the case of visual observation or the camera lens if the image is recorded on a recording medium. In modern digital microscopes, the recording medium is a CCD chip. We can see that the lens S3 focuses the rays sharply to the points A2, B2 in the only image plane r3.
Even in this simple figure, we see that the distance A2B2 is greater than that of AB. The image is then projected on the LCD screen, or the digital recording is transferred to a computer. The advantage of a digital microscope is that it allows the image to be saved to disk, archived, or further processed.
Consider now point C, which is in another plane r2 of the 3D object. If we observe the course of the characteristic rays, we see that the objective S1 creates a real image C1, which is displayed by the eyepiece to a point C1′ at the intersection of the converging rays. The lens S3 then creates an image C2 behind the detection plane. The beam pointing to point C2 covers the trace indicated by a little ellipse in the detection plane. Object C thus does not create a sharp point image in the detection plane, but a wider track. Points that are in an object plane other than r1 thus cover the resulting image in the detection plane with such surface tracks and blur the image.
Thus, a conventional microscope is not suitable for imaging 3D objects. However, it is suitable for imaging thin planar slices or liquid samples placed between two microscope slides, for example, in the investigations of various histological findings, blood particles, movement of sperm in the semen, the occurrence of chromosomes in cells in genetic tests, etc. Conventional optical microscopes are used in precision surgery, ophthalmology, neurology, dentistry, and the like.
6.2.6 Confocal microscope
A digital confocal microscope (CLSM—Confocal Laser Scanning Microscopy) is used to observe 3D objects. The scheme of the microscope is in Figure 25a. There is displayed only point A of the object plane. The overall image of a given sample plane is obtained by scanning the sample in two x, y directions perpendicular to the system axis. The confocal microscope displays only the points of the object plane on which the microscope is focused. Points from other planes of the object are suppressed and do not blur the resulting image (like a conventional microscope). In Figure 25 are, for comparison, images of a defect on the skin surface: (b) from a conventional microscope and (c) from a confocal microscope focused on different planes of the object. It can be seen from the images that the confocal microscope images are significantly sharper than the image (b). By moving the sample in the z-direction, we get images of different planes of the sample layer by layer. By combining these images in a computer, a 3D image of the object can be created. It is an optical tomography. An example of a 3D image of cancer cells is shown in Figure 25d. A confocal microscope is standard equipment in the pathology department and the biochemical laboratory.
The principle of imaging is in Figure 25a. The laser beam of parallel beams is focused by the lens L1 on a small aperture of the screen S1, thus creating a point source of coherent light. The diverging beam is reflected by the semi-transparent mirror SM onto the lens L2. It focuses the rays to point A in the object plane. At point A, the light is reflected or scattered, and the rays going back from this point of the object incident on the lens L2 are returned to the semi-transparent mirror SM. Some of the light reflects towards the laser, but now we are interested in the light passing through the semi-transparent mirror.
At the same distance from the SM as the screen S1, there is a screen S2 with a small hole in the straight direction. Since the beams from point A are focused on the centre of its aperture, only the light from point A passes through the aperture to the lens L3 of the recording detector. The magnitude of the detector signal is directly proportional to the reflectivity of point A of the object. By moving the sample (scanning), the detector signal is changed, which is digitised and recorded.
The laser light incident on the sample illuminates also points outside point A. The light emanating from point A′ at another depth also hits the lens L2 but forms a converging beam (dashed in the figure) that is not focused on the small aperture S2. It does not, therefore, pass through the aperture. The light emanating from a depth other than the depth of point A thus does not affect the detector signal and, therefore, cannot blur the image of the O plane. Similarly, the points in the plane O sideways from point A form an emanating beam, which is deflected by direction and thus also does not pass through the aperture S2. The detector signal thus corresponds only to point A of the displayed sample and its very close surroundings. The created digital image is, therefore, very sharp.
6.2.7 Microscope with phase contrast
The subjects of observation are often samples, especially in biochemistry and biology, which contain transparent objects like cells, small microorganisms, etc. A standard microscope evaluates the intensity (amplitude) of light, and this is not affected by transparent objects. Such objects are very faintly visible in the sample image, Figure 26a. However, the investigated objects have a different refractive index than the surrounding medium. As light passes through the sample, a different phase shift of the light beam occurs. If such a beam can interfere with a suitably arranged reference beam, the light may be increased or reduced depending on whether the interference is constructive or destructive. Thus, the phase shift caused by the different refractive index of the object is converted into amplitude information visible in the resulting image. In this way, originally invisible objects become visible—phase contrast is achieved, Figure 26b. The phase-contrast microscopes belong to the fundamental equipment of laboratories working with biological samples (pathology, biochemistry, etc.)
6.2.8 Fluorescence microscope
The fluorescence microscope is used to image organic and inorganic components of samples that are characterised by fluorescence. It uses the excitation of the fluorescent components of the object by ultraviolet radiation and detects the visible light that the substances emit when relaxed. The construction of the fluorescence microscope is like that of a conventional one. It is only extended by a source of UV radiation, Figure 27. The light from the UV lamp passes through the filter F1. UV light with the selected wavelength proceeds to the semipermeable mirror SM and is reflected towards the sample. It induces fluorescence in the sample. The emitted visible light enters the microscope objective and passes through the SM to the optical filter F2, the eyepiece, and the detector. Filter F2 suppresses reflected UV radiation. The image is observed directly with eyes or captured with a digital camera. The parts of the sample containing a fluorescent substance are displayed markedly in the resulting image. Since different molecules or cellular components have different excitation energy, it is possible to change the display of the individual fluorescent components by retuning the F1 filter.
In Figure 27 right is an apparatus for direct observation or with digital output and display of the observed sample on the LCD. The figure also shows an image of a cell with structures that emit light of different wavelengths (colours). In some cases, the primary fluorescence of certain molecules (e.g., some proteins—amino acids) is used. More often objects are “coloured” with fluorescent dyes, which penetrate biological structures and make them visible. In this way, the processes taking place in them can also be observed (for example, nucleic acids—DNA, RNA—are visualised). Fluorescence microscopes are mainly used in biological research laboratories.
6.3 Resolution of the wave imaging
If we use waves to image a certain structure, we are interested in what details can be observed, that is, resolution. Suppose we observe a small circular detail with radius
For the first minimum, the relation (30) applies. If the lens is at a focal length
The main information of the zero-order lobe is in its width at the level of −3 dB (50%). The contours of the image of the circular detail will no longer be sharp but can still be distinguished. For
It follows this analysis that wave imaging allows observing objects with minimum dimensions of about one-half of the wavelength of the used wave. With an optical microscope, details with dimensions of up to about 200 nm can be observed in blue light.
If we observe a sample with naked eyes, the radius of the pupil is approximately
The resolution increases if the object can be observed from a closer distance so that the if angle
In edge cases, blue light is preferable to red one. X-rays provide greater resolution than light, but the device is technically more complicated. A significant rise in resolution provides electron beam imaging (electron microscopy), in which the electron wavelength
where
Bacteria, cells, larger viruses, etc., can be observed with an optical microscope. The electron microscope can image such small objects as macromolecules (e.g., DNA), proteins, small viruses, etc. The mentioned resolution rule also applies to ultrasound. USG imaging uses ultrasound with frequencies of 2–10 MHz. For fine structures, for example, eyes or nerves, higher frequencies > 10 MHz are used.
6.4 Focusing a wave beam
If the source is a strip, the wavefield described by the relation (26) arises. For a circular source (e.g., a circular hole or a laser output hole), the relevant relation (29) for the spatial distribution of the wavefield is similar, with Bessel functions instead of harmonic ones. We can see from the shape of the resulting relation for the strip source that the relation (25) for the angular dependence of the radiation characteristic corresponds to a Fourier image of a rectangular pulse with length
A lens with a circumference diameter
Using Eq. (27) we get a condition
and after adjustment
The 2
This fact must be respected, for example, in gentle operations with a laser scalpel in neurology, eye surgery, etc. We get a smaller focused track using blue light, in confront with a red light or IR radiation. The condition of the best focus applies to coherent waves, for example, laser. When using incoherent radiation, the possibility of focusing the beam is always worse (e.g., sunlight or LED).
7. Non-linear phenomena
7.1 Non-linear wave interactions
In a linear medium, wave interference occurs, which means the waves add at the place of their common occurrence, but the waves themselves do not affect each other. However, if the medium is non-linear, the wave is deformed. It leads to arising higher harmonic components, and due to a non-linear interaction to different combination frequencies. These combination components may propagate in different directions relating to the direction of the original waves. These interactions are most often described by the quantum model. For example, the acoustic wave modulates the optical refractive index of the medium, and the electromagnetic wave thus passes through the medium with a periodic change of the propagation parameter, like through an optical grating. It causes a diffraction pattern, that is, the main lobe and lateral ones. Due to the movement of the grating at the speed of the acoustic wave, there arises also a shift of the EM wave frequency by the frequency of the acoustic wave.
Example 17 Acoustic modulator of the laser beam
In some information transmission by light, for example, optical fibre, it is necessary to modulate the light beam frequency. One way is an acoustic-optical modulator, which uses the nonlinearity of the modulation crystal, Figure 30. The carrier of the signal is a frequency-modulated acoustic wave, which causes a temporal change of the refractive index of the crystal corresponding to a transient change of the signal. A laser beam enters this crystal perpendicularly to the direction of propagation of the acoustic wave and is diffracted. The modulated first-order diffraction beam is used, which is angularly separated from the 0-order beam.
The principle of the deflector (modulator) is easily explained using the quantum concept of waves. EM waves represent photons with energy
In the interaction, where the photon captures the phonon, the law of conservation of energy and momentum is fulfilled.
If we consider only a very small change in frequency
and
The deflected beam is frequency modulated by an acoustic signal and angularly separated from the unmodulated 0-th order beam.
Example 18 Two-photon excitation microscopy
Another example of the nonlinear interaction of a laser beam with a substance is fluorescence microscopy used in biochemistry, which uses the principle of two-photon excitation of atoms. Conventional fluorescence microscopy—see Section 6.2.8, utilises excitation by high energy photons of UV radiation, which have a shorter depth of penetration into the sample due to the higher frequency and which produce a relatively strong white background of the image.
Two-photon excitation uses the excitation of molecules by a pair of low-energy photons. If we observe the fluorescence of the most common fluorophores (molecules exhibiting fluorescence) in the optical range of 400–500 nm, IR laser radiation of 800–1000 nm is used for excitation. If an electron captures a photon in the ground state, it is excited to a higher energy level. The electrons from this excitation level relax back to the ground state, emitting photons in the IR band, that is, optically invisible.
However, if it is excited again to the next higher energy level before relaxation, due to the relaxation from this second level photons of the visible light are re-emitted—fluorescence occurs. Each fluorescent molecule has its characteristic excitation energy, so that specific fluorophore can be discovered. The probability of double capture of excitation photons is very small and increases with an increasing energy density of radiation. The conditions for double excitation are only in the focus of the focused laser beam. In such a way, the location of the fluorophore can be precisely addressed. The method of scanning individual layers of the sample is used for the display. It permits creating a 3D image of the structure.
7.2 Photo-acoustic imaging
Other modern diagnostic tools use a nonlinear photoacoustic (PA) effect, for example, Xu [3], Beard [4]. As mentioned in Chapter 3, the ultrasound can be excited thermally (by thermal pulse) using a powerful pulse laser. The laser focuses at a location on or near the surface of the object and emits a very short (nanosecond) pulse of radiation. The absorbed energy causes a local increase in temperature, and thus a mechanical expansion of the material. This thermal deformation causes a mechanical shock wave to propagate from this point to all sides. If a series of pulses is applied, a series of shock waves is generated which has a harmonic ultrasonic component with a frequency equal to the repetition rate of the pulses. Since the pulse length is significantly shorter than the repetition period, it is sufficient to return the temperature to the original site temperature before the next pulse arrives. The amplitude of the generated mechanical wave depends on the amount of energy absorbed, and this is directly proportional to the absorption coefficient (absorbance) of the substance at a given location. The generated mechanical (ultrasonic) waves are detected by ultrasonic sensors. In this way, the absorbance distribution of the sample, which is related to the tissue structure, can be mapped using a photoacoustic phenomenon. For example, the blood has a large absorbance for a red light, and therefore the bloodstream and small blood capillaries are displayed very well. The method has a very good resolution due to the possibility of focusing the light beam on an area with dimensions of units of μm. Another advantage is that, unlike light, the attenuation of mechanical waves is relatively small and the generated wave shocks propagate well through an investigated sample. Some specific imaging devices work on the principle of the photoacoustic phenomenon. Some of them are listed below.
7.2.1 Photoacoustic microscopy
The first method that uses the principle of the photoacoustic phenomenon is photoacoustic microscopy (PAM), for example, Wang [5]. The laser beam and the detection acoustic transducer are focused on the examined point of the sample. The energy of the laser beam is thus concentrated to one point and a shock wave with a repetition frequency of 500 MHz, is detected by an acoustic transducer focused at the same point. PAM is a scanning method. The sample is shifted in the transverse directions
PAM allows displaying a depth up 3–4 mm. By composing images from different depths, a 3D image can be created. Figure 31 is the basic principle of PAM. The box with an optical lens and a focused acoustic transducer contains water as a transmission medium for the acoustic wave. The box is acoustically bound to the displayed object using a binding gel. Point A is the common focus and displayed point. The picture shows four scans from different depths and a 3D reconstruction.
7.2.2 Photoacoustic computing tomography
The second principle of information processing of a photo-excited sample is photoacoustic computed tomography (PACT). Unlike PAM, the light pulse of the power laser illuminates the entire examined surface of the sample. As a result, all points with higher absorbance in the sample become a source of acoustic waves. The sample is surrounded by a system, for example, 512 small acoustic detectors, arranged on a hemisphere to obtain a 3D image, or on a circular ring around the sample to create a 2D image. The signals of all detectors are scanned and fed to a computer. They are converted by a complex integral transformation into a spatial or planar image of the distribution of points with different absorbances. In this way, different tissues in the surface layer of the sample are displayed. PACT can display the structure up to a depth of 50 mm, in the case of using a set of detectors arranged on a hemispherical surface, with a resolution of 0.5 mm at a pulse repetition frequency of 5 MHz.
In Figure 32 on the left is a 3D image of the tissue perfusion of the top layer of the skin. Different levels of the signal are distinguished in colour, which corresponds to different absorbances of the cells. Such an image allows, for example, detection of skin tumour (melanoma)—the picture on the right.
PAM and PACT provide similar resolution as ultrasonography (USG). Against USG, they have a much smaller depth of view, which is limited by the attenuation of light in the tissue but allows seeing the information that USG does not provide. This is due to another mechanism of ultrasound generation. While different acoustic impedances of tissues are applied in USG, different absorbance of tissues is decisive for PA methods. Photoacoustic imaging methods are not yet widely used but are the subject of intensive research.
7.2.3 Photoacoustic spectroscopy
Another method that uses the photoacoustic phenomenon is photoacoustic spectroscopy (PAS), for example, West [6]. It uses different spectral compositions of absorbance of different substances in the tissue.
Figure 33 is the spectrum of various substances in the blood: melanin, Hb-haemoglobin, Mb-myoglobin, and others. The difference in the absorbance of HbO2 and HbR (oxidised and reduced haemoglobin) at a light wavelength of about 680 nm can be used to distinguish oxygenated blood from deoxygenated blood. In this way, it is possible to determine the degree of oxygenation of the blood—
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