Progressive wave

1. Progessive wave

  • A progressive wave is an oscillation or vibration that transfers energy and information.
  • The particles oscillate about their fixed positions but do not move to a different place.

Waves are represented in two different ways:

  • A wavefront joins points on a wave that are at the same point of the cycle as their neighbors.
  • A ray is an imaginary line showing the direction the wave travels in. It joins the position of the wave source to the wavefronts.

Figure 1  The energy spreads out from the centre and Water spreading out from a ripple.

2. Particle movement in wave

  • particles to vibrate perpendicular to the direction energy is transferred
  • The amplitude is the maximum displacement from the particle’s undisturbed position and the larger the amplitude is, the more energy is transferred.
  • The distance between wave peaks is the wavelength (represented by λ and unit is meter)
  • λ is the distance between two equivalent points in successive cycles.

Figure 2  Wave terminology

  • The frequency of a wave, is the number of cycles or vibrations per second.
    • Unit is hertz
    • Formula f=1/T
    • Frequency Represented by f
    • T is represented by the period
  • Period is the time for one complete cycle, measured in seconds.
    Since 1Hz is a very low frequency we also measure hertz in kHz (103Hz), MHz (106Hz) and GHz (109Hz).

Figure 3 The period for one oscillation is 0.8 seconds.

3. The wave equation:

  • Different waves travel at very different speeds, but in each case the speed is calculated the same way using the equation
  • [math] \text{Speed} = \frac{ \text{distance}}{\text{time}} \qquad (1) [/math]
  • For a wave, the wavelength, λ, is the distance traveled in one cycle and the time to complete one cycle is the period, T. This means that:
  • [math] \text{wave Speed} = \frac{\text{wavelength}} {\text{period}} \qquad (2) \\
    \text{speed of light(c)} = \frac{λ}{T} \qquad  (3) \\
    \text{speed of light(c)} = λ \frac{1}{T} \qquad   (4) [/math]
  • The period is the time (in seconds) for one complete cycle
  • The frequency,
  • [math] f = \frac{1}{T} [/math]
  • the equation4 becomes
  • [math] \text{Speed of light (c)} = \text{Wavelegth} (λ) * \text{Frequency (f)} [/math]
  • Now by units
  • [math]  C( m.s^{-1}) = λ(m) . f(Hz) \qquad (5) [/math]

This is known as the wave equation

4. Phase Difference

  • Phase difference is measured as a fraction of the wave cycle between two points along a wave, separated by a distance x.

  • One complete rotation involves turning through 2π radians (2π= 360°)

 Figure 4 The phase difference between two sine waves

  • Particles along a wave that moves in phase move in the same direction with the same speed.
    • The particles have the same displacement from their mean position.
  • Particles along a wave that moves out of phase are at different points in their cycle at a particular time.
    • Do not move together in the same direction and with the same speed.
  • Particles in parts of a wave that moves in opposite directions and at the same speed are moving in antiphase, or completely out of phase.
    • The particles have opposite displacements from their mean position.
    • Particles moving in antiphase are separated by a distance of a whole number, n, of wavelengths plus an extra half wavelength ( nλ + λ/2)

Figure 5 Three different phase differences

 The Math of phase difference

  • One complete rotation involves turning through 2π radians, so (2π= 360° )
  • The motion of the wave in Figure 4 is sinusoidal with a period of T.
  • the value of the angle in the sine function must be 2π radians.

This happens when t = T for the angle, then

  • We can describe the vertical displacement of the particles in the wave using an equation of the form

So equation (6) becomes

     

  • where y is the vertical displacement at a time t
  • A is the amplitude of the wave.
  • f is the frequency of the oscillations.
  • The oscillations will be out of phase. This phase difference is a fraction of the angle 2π
  • Points that are half a wavelength apart have a phase difference of π.
  • Points three-quarters of a wavelength apart have a phase difference of 3π/2
  • Two points separated by a distance x on a wave have a phase difference of

    • where Ø is the phase difference

    ⇒Phase change on reflection:

  • When a wave is reflected off the surface of a denser medium, it undergoes a phase change of 180°.

  • Figure 6 Red line shows the incident line while the blue line determines reflection

    • 5. Longitudinal waves and transverse waves

    • Mechanical waves
      • Mechanical waves need a medium to travel through and can travel as transverse or longitudinal waves.
      • Examples are seismic waves, sound waves, etc.
      • cannot travel through a vacuum.
    • Longitudinal waves
      • When the motion of the particles in a mechanical wave is back and forth along the direction of propagation, then create longitudinal waves.

      • Figure 7 Propagation of longitudinal wave

      • A compression is a region where the particles are closest together.
      • A rarefaction is a region where the particles are furthest apart.
      • When sound travels through a solid, energy is transferred through intermolecular or inter-atomic bonds.
      • Sound travels quickly through solids because the bonds are stiff and the atoms are close together
      • In gases the energy is transferred by molecules colliding
      • the speed of the sound depends on the speed of the molecules
      • Sound waves travel fastest in solids (5100m-1 in aluminium)
      • less quickly in liquids (1500ms−1 in water)
      • even slower in gases (340ms−1 in air).
    • Transverse waves
      • Waves in which particles of the medium oscillate perpendicular to the direction of propagation of the wave is known as transverse waves.

    Figure 8 Propagation of transverse waves

      • Crest is a point on the surface of the wave where the displacement of the medium is at maximum
      • A trough is a location on the wave’s surface where the medium’s displacement is the least.
      • Examples are seismic waves, electromagnetic waves (light waves, radio waves, etc.)
    • Electromagnetic waves
      • Electromagnetic waves do not require any medium for their propagation of waves
      • Those waves travel with the speed of light (3*108m/s) in a vacuum.
      • The electromagnetic wave’s electric and magnetic fields vibrate in opposition to one another and the direction of transmission.
      • The shortest path between two locations where the electric or magnetic fields are in phase is the wavelength of an electromagnetic wave.

    Figure 9 The energy in electromagnetic waves is carried by oscillating electric and magnetic fields

    • The properties of the waves vary considerably with their wavelength so we normally consider the spectrum as seven main groups

    Figure 10 The electromagnetic spectrum

    ⇒Effect of electromagnetic radiation on living cells

    • Electromagnetic radiation (EMR) can have both positive and negative effects on living cells, depending on the frequency, intensity, and duration of exposure.
      • Positive effects:
        • Visible light (a form of EMR) is necessary for photosynthesis, which supports the food chain and oxygen production.
        • UV radiation triggers vitamin D synthesis in skin cells, crucial for bone health and immune function.
        • Exposure to natural light-dark cycles helps regulate the body’s internal clock.
        • EMR is used in diagnosis (e.g., X-rays, MRI) and treatment (e.g., radiation therapy for cancer).
      • Negative effects:
        • High-frequency EMR (e.g., gamma rays, X-rays) can cause genetic mutations and cancer.
        • High-intensity EMR can alter cell membrane structure and function.
        • EMR can generate reactive oxygen species, leading to cell damage and inflammation.
        • High-intensity EMR can cause tissue heating, leading to burns or other damage.
        • EMR can interfere with cellular communication, potentially leading to changes in cell growth and behavior.

    ⇒Communication uses of electromagnetic radiation

    • Electromagnetic radiation (EMR) plays a vital role in various communication systems, enabling wireless transmission of information over long distances.
    • AM and FM radio broadcasting use EMR to transmit audio signals through the airwaves
    • Cell phones use radiofrequency EMR (RF-EMR) to connect calls, send texts, and access the internet.
    • These technologies use RF-EMR to enable wireless internet connectivity and device-to-device communication.
    • Satellites use EMR to transmit data, voice, and video signals across the globe.
    • Radar systems employ EMR to detect and locate objects, commonly used in aviation, weather forecasting, and military applications.
    • While not primarily a communication technology, microwave ovens use EMR to heat food.
    • While not a direct use of EMR, optical fiber relies on light (a form of EMR) to transmit data as light pulses through fiber-optic cables.
    • TV broadcasting uses EMR to transmit video and audio signals through the airwaves or via cable.
    • The Global Positioning System (GPS) uses EMR from satellites to provide location information and navigation.
    • EMR is used in wireless sensor networks for applications like environmental monitoring, industrial automation, and healthcare.
    • Radio Frequency Identification (RFID) tags use EMR to transmit data, commonly used in inventory tracking, supply chain management, and access control.
    • Services like SiriusXM use EMR to broadcast digital radio signals to subscribers.

    6. Polarization:

    • Polarization is a fundamental property of electromagnetic radiation, including light and other forms of electromagnetic waves.
    • The electromagnetic waves that are transmitted only have their electric fields oscillating vertically (and their magnetic fields oscillating horizontally). When the fields of an electromagnetic wave only oscillate in one direction, the wave is said to be a polarized wave.

    Figure 11 demonstrate polarization using laboratory equipment

    • In the diagram you can see a radio aerial transmitting 30cm radio waves, which are received by another aerial placed a meter away.
    • A high-frequency (1GHz) signal is applied to the transmitting aerial, which causes electrons to vibrate up and down.
    • In this case we say the wave is vertically polarized because the electric fields are confined to the vertical plane.
    • The polarization of the waves by rotating the receiving aerial: radio waves are detected when the receiving aerial is positioned vertically, as shown in the diagram because they cause electrons in the vertical aerial to oscillate up and down; no signal is detected when the aerial is orientated horizontally, because the electric field is not directed in a way that would cause the electrons to travel along the aerial.
      • Unpolarized light
        • In unpolarized light, the electric field vectors vibrate in multiple directions, which results in a random and chaotic pattern.
      • Polarized Light
        • In polarized light, the electric field vectors are restricted to vibrate in a specific plane, creating a well-organized pattern.
        • ⇒Polarization effect
        • Polarization effects are phenomena that occur when electromagnetic waves interact with matter or other fields, resulting in changes to the wave’s polarization state.
        • Light is polarized when it passes through a polarizing filter.
        • The filter only allows electric field oscillations in one plane because the filter absorbs energy from oscillations in all other planes.
        • The metal grille is a good model for a polarizing light filter.
        • Long molecules of quinine iodosulphate are lined up, and electrons in these molecules affect the light in the same way as the microwaves are affected by the grille.
        • Polarized light is less intense than unpolarized light because only half the energy is transmitted through the filter.
        • If a second polarizing filter is held at right angles to the original filter, all oscillations are blocked and no light is transmitted. This is called crossing the polarizers.

    Figure 12 (a) A polarized filter (b) reflected light is sometimes polarized

    • ⇒Polarization applications
      • Polarizing sunglasses (reducing glare)
      • Polarization filters (improving contrast)
      • Microscopy (enhancing image quality)
      • Polarization-division multiplexing (increasing data transmission rates)
      • Polarization-sensitive optical filters (enhancing signal quality)
      • Polarized light microscopy (studying material properties)
      • Circular dichroism spectroscopy (analyzing biomolecules)
      • Polarization-dependent optical properties (studying material behavior)
      • Polarized light scattering (characterizing material structure)
      • Polarized light scattering (diagnosing diseases like cancer)
      • Optical coherence tomography (imaging tissue structure)
      • Polarized light scattering (studying atmospheric and oceanic properties)
      • Polarimetric radar (monitoring land surface and vegetation)

    7. Refraction

    • When light travels at any angle other than along the normal, the change in speed causes a change in direction. This is called refraction
    • The frequency of a wave does not change when it travels from one medium to another but its wavelength does.
    • When light travels from air into a medium such as glass or water, the wave slows down and the wavelength gets shorter.
    • The light changes direction least when it enters water, and the most when it enters diamond.
    • The light travels fastest in water, slower in glass and, of the three mediums, the slowest in diamond.
    • ⇒Angle of incident
      • Angle of incidence is the angle between the incident ray and the normal.
      • The normal is an imaginary line at right angles to the boundary between two materials.

      ⇒Angle of refraction

      • Angle of refraction is the angle between the refracted ray and the normal.
      • Light refracts more when it enters diamond from air than it does when it enters glass from air.
      • The angle of refraction is least in diamond.
      • Diamond is said to have a higher refractive index than glass.

      Figure 13 Angle of refraction for different materials

    • ⇒Refractive index
      • Refractive index is the ratio of a wave’s speed between two materials.
      • It is normally quoted for light travelling from a vacuum into a material.
      • Refractive index is a ratio it has no units.
      • A refractive index is calculated using.
      • The speed of light in a vacuum is almost exactly the same as the speed of light in air.
      • The refractive index of air ≈ 1
      • the refractive index of a material
      • The refractive index of window glass is about 1.5.
      • The speed of light in air is about 1.5 times faster than it is in glass.
      • The refractive index of water is 1.33, so light travels 1.33 times faster in air than it does in water.
      • When light travels from one material to another (other than air) we can define the relative refractive index between them.
      • Figure 14 Light refracted between two materials
      • where 1n2 = refractive index between materials 1 and 2C1 is the speed of light in material 1C2 is the speed of light in material 2.
      • The advantage of this formula is that we only need to know one refractive index for a material, and we can calculate a new refractive index when light passes between any pairs of materials other than air.
    • ⇒ Law of refraction
      • hundreds of years scientists have studied refraction and have been able to predict the angles of refraction inside transparent materials.
      • Willebrord Snellius was the first person to realise that the following ratio is always constant for all materials.
      • where θ is the angle of incidence and  is the angle of refraction. This is now known as Snell’s law
      • More generally, Snell’s law of refraction is stated as:

      • 1n2 = refractive index between materials 1 and 2

        θ1= angle of incidence in material 1

        θ2= angle of refraction in material 2

        or, since

    • ⇒Total internal reflection
      • Total internal reflection of light is the complete reflection of light at a boundary within a material that has a higher refractive index than its surroundings.
      • Critical angle is the angle of refraction for which the angle of incidence is
      • When the angle of refraction reaches 90° all light is refracted internally. At this point the refracted angle is 90° and sin θ = 1
      • When light is in a material with refractive index n1 and is incident on a boundary with medium which has refractive index n2 , we can use Snell’s law to predict the critical angle as follows:
      • Now we can measure the critical angle using Equation 16
      • Figure 15 Total internal reflection
    • ⇒ Optical fibres:
      • Optical fibers are thin strands of glass or plastic that transmit data as light signals.
      • They consist of a core, cladding, and coating, and come in single-mode and multimode varieties.
      • Optical fibers offer high speed, long-distance transmission, security, reliability, and capacity, making them ideal for telecommunications, data centers, local area networks, sensors, medical imaging, lighting, and sensing applications.
      • These waves travel through the glass but are trapped inside by repeated total internal reflection.
      • The critical angle depends on the ratio between the refractive index of the optical fibre and its cladding (coating).
      • Step index optical fibre is an optical fibre with a uniform refractive index in the core and a smaller uniform refractive index for the cladding.
      • e the cladding has a different, smaller, refractive index.
      • By choosing a suitable material for the core and cladding, only certain wavelengths of light or infrared radiation can travel through the fibre by total internal reflection.
    • ⇒Material and modal dispersion
      • Dispersion, in the context of optics and photonics, refers to the spreading of light as it travels through a medium, resulting in different wavelengths (colors) propagating at different speeds.
      • Figure 16 Dispersion

      • Two types of dispersion occur in step index optical fibres:

        1. Material Dispersion
        2. Modal Dispersion
    • ⇒Material Dispersion
      • Material dispersion is the spreading of a signal caused by the variation of refractive index with wavelength.
      • Pulse broadening in material dispersion refers to the spreading of optical pulses as they travel through a material or medium, resulting in an increase in pulse duration. This occurs due to the material’s dispersion properties, which cause different wavelengths (colors) to propagate at different speeds.
    • Figure 17 Pulse broadening in material dispersion
    • ⇒Modal Dispersion
      • Modal dispersion is the spreading of a signal caused by rays taking slightly different paths in the fibre.
      • Figure 18 Modal dispersion
      • Rays taking longer paths take longer to travel through the fibre, so the duration of the pulse increases and the pulse broadens.
      • Modal dispersion is significant in multimode fibres, because these fibres are broad enough to allow rays to take different paths.
      • Modal dispersion is significant in multimode fibres, because these fibres are broad enough to allow rays to take different paths.
      • For communications, monomode fibres are used. These have a very narrow core, so that light is very nearly confined to one single path along the axis of the cable.
      • Figure 19 Modal dispersion: rays can take more than one path in the fibre.
    • ⇒Absorption
      • Absorption occurs when energy from a signal is absorbed by the optical fibre in which it travels.
      • The wavelengths commonly used are 650nm, 850nm and 1300nm.
      • It may also be necessary to amplify the signal if it travels long distances through the optical fibre.
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