Wave nature of light (Topic 5C)
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Diffraction occurs when waves pass through a narrow gap (aperture) or around an obstacle, causing them to spread out as they pass through. Diffraction is a property of all types of waves, including sound, light, and water waves.
Waves are usually represented in diagrams as wavefronts, where each front shows a line of constant phase.
Diffraction is most pronounced when the gap size is comparable to or smaller than the wave’s wavelength. If the gap is much larger than the wavelength, the wave continues with minimal diffraction.
Wavelength, frequency, and wave speed remain unchanged during diffraction.
Huygens’ construction models how a wavefront evolves. It states that every point on a wavefront can be treated as a source of secondary spherical wavelets that spread at the wave’s own speed.
The next wavefront is the envelope, the surface tangent to all these wavelets, which connects the peaks of these secondary wavelets.
At a slit, only the wavelets passing through contribute. Edge wavelets have no neighbours on the far side to interfere with, so the wavefront curves outward and the wave spreads. Spreading is greatest when the slit width is close to the wavelength. The same reasoning explains bending around obstacles.
Diffraction of a wave leads to a diffraction pattern, which consists of maxima and minima intensity points.
For light waves, this corresponds to a series of dark and light fringes.
A diffraction pattern is visible when a laser is directed at a narrow slit where the slit width is larger than, but comparable to, the wavelength of the laser light. For laser light, a bright central fringe appears, surrounded by smaller fringes of decreasing brightness on either side.
If polychromatic white light is directed at a single slit (which is larger than visible light wavelengths), the central maximum will be white, and each fringe beyond it will show a spectrum of colours:
- Violet and blue light (shorter wavelengths) appear closest to the central maximum since they diffract the least.
- Red light (longer wavelengths) appears further out since it diffracts the most.
Dispersion refers to the process of splitting visible white light into its constituent colours, forming a spectrum. This can be achieved using either a glass prism or a diffraction grating.
A transmission diffraction grating is a tool often used in spectrometers for high-resolution separation of light by wavelength. It consists of a glass or plastic slide with many closely spaced, parallel slits or lines. When light passes through these slits, it diffracts and spreads out, allowing for detailed analysis of the light’s components.
For example, a continuous spectrum from a halogen light may be passed through a sample and then through a diffraction grating to produce an absorption spectrum. This absorption spectrum can then be used to identify the sample’s chemical composition.
Diffraction gratings are particularly useful for analysing light from stars. By separating the light into its individual wavelengths, scientists can determine the composition of stars based on their emission or absorption spectra.
It is important to note that:
- Diffraction is the bending or spreading out of waves as they pass through a gap or move around an obstacle. The extent of diffraction depends on the wavelength of the wave and the size of the gap or obstacle. The effect is most pronounced when the gap size is comparable to the wavelength.
- Interference occurs when two or more waves overlap and combine, resulting in a new wave pattern. There are two types of interference:
- Constructive interference: When the crests of two waves align, their amplitudes add together, resulting in a wave with greater amplitude.
- Destructive interference: When the crest of one wave aligns with the trough of another, their amplitudes subtract, resulting in a reduction in amplitude or cancellation.
- Phase difference is the difference in phase between two points on a wave or between two waves. It measures how ‘in sync’ or ‘out of sync’ two waves are. It is typically measured in degrees or radians, where 360∘ (or 2π radians) corresponds to one full cycle.
Advantages of diffraction gratings
- Greater angular dispersion: Compared to optical prisms, diffraction gratings provide a much higher angular dispersion, meaning the colours are separated more distinctly.
- Sharper fringes: the fringes (or bands of light) produced by diffraction gratings are sharper than those created by a double-slit experiment, making them ideal for precise spectral analysis.
- Accuracy: Because diffraction gratings rely on the interference of light, they provide more accurate wavelength measurements compared to prisms, which are subject to imperfections in glass and material dispersion.
- Efficiency: Transmission diffraction gratings can transmit more light than a prism, making them more efficient for capturing fainter light sources, such as distant stars.
- Customization: Gratings can be manufactured with a variety of line spacings (grating density), allowing them to be optimised for specific wavelength ranges or applications. This flexibility makes them versatile for different types of spectroscopy.
- Compact Design: Diffraction gratings are often smaller and lighter than prisms, making them easier to integrate into compact and portable instruments, such as modern spectrometers.
Diffraction grating works by diffracting light through many slits, causing the light waves to interfere. Different wavelengths constructively interfere at different angles, producing a separated spectrum of colours.
The angles at which maxima of intensity (constructive interference) occur can be found using the diffraction grating equation:
Where:
- is the spacing between slits, typically in metres or millimetres,
- is the diffraction angle in radians or degrees depending on calculator settings,
- is the order of the maxima (first, second, etc.) and
- is the wavelength of the light, typically in metres or millimetres.
This equation may use and in either metres or millimetres as long as the same unit is used for both, as the units cancel each other out.
In exam questions, you may be given the number of lines per metre (or millimetre, nanometre, etc.) on the grating, called This can be used to calculate the spacing between slits, using the following equation:
This equation converts the number of lines per unit length, into the distance between adjacent slits,
To calculate the angular separation of each maxima, you can rearrange the diffraction grating equation to solve for
In this equation, is the angle measured from the centre (zero order) to the maxima.
Higher-order maxima (larger values of ) will occur at greater angles from the centre.
The angular separation between two maxima is simply the difference between their angles. For example, the separation between the first-order maxima, and second-order maxima is calculated as:
The highest order of maxima is observed when a beam of light is incident at a right angle to the diffraction grating.
This happens when the angle reaches 90°:
In this case, the highest order of maxima is found by rearranging the grating equation:
Remember,
However, must always be an integer. If the calculated value of is not an integer, you must round down to the nearest whole number.
For example, if the highest visible order is
Why must the order of maxima be an integer?
In diffraction, constructive interference occurs when the path difference between light waves from adjacent slits is an exact multiple of the wavelength This ensures that the light waves arrive in phase, reinforcing each other to create a bright spot at the order
Since the path difference must be a whole number of wavelengths for constructive interference, can only take integer values (first order, second order, etc.).
For non-integer values of the light waves are not perfectly in phase, resulting in partial interference that does not produce a distinct bright fringe.
Why must non-integer values of the order of maxima be rounded down?
When you calculate the maximum possible value of you sometimes get a non-integer value. Since must be an integer, and only integer orders produce visible maxima, you must round down to the nearest whole number.
For example, if rounding down gives meaning the highest visible order is the second order.
Rounding down ensures that you do not count an order that cannot physically exist, as an order like would not satisfy the condition for constructive interference if is less than three. This guarantees you stay within the physical limits of the diffraction grating, only counting the orders that produce visible maxima.
Question walkthrough
Diffraction Grating Spacing and Maximum Order
Calculates the line spacing of a diffraction grating from lines per mm, then uses the grating equation to find the highest observable order of maxima.
In 1924, de Broglie proposed that matter exhibits both wave-like and particle-like properties.
de Broglie’s hypothesis is known as wave–particle duality: particles can behave as waves and waves can behave as particles under different circumstances. Wave-particle duality applies to all matter.
The wavelength of a particle is known as the de Broglie wavelength.
Electromagnetic waves exhibit wave–particle duality.
The photoelectric effect provides evidence for light behaving as a particle. When light of high enough frequency is incident on a metal’s surface, electrons are emitted. This can only be explained by the light behaving as a stream of particles (photons).
Photons can also demonstrate wave properties. Light will diffract around obstacles and slits; it can also interfere with itself constructively and destructively, producing interference patterns.
Electrons also exhibit wave–particle duality, showing both particle-like and wave-like behaviour in different circumstances.
When behaving as particles, electrons can be accelerated by electric and magnetic fields due to their charges, and their motion can be described using classical mechanics.
Electrons can also display wave properties. When a beam of electrons is fired through very narrow openings, they can diffract and produce interference patterns.
To observe the wave nature of particles, the particles must exhibit wave behaviour, specifically producing an interference pattern. Such patterns are caused by diffraction and interference, both of which are properties of waves.
Diffraction effects are most pronounced when the de Broglie wavelength of the particles is of the same order of magnitude as the openings they pass through.
Graphite consists of layers of carbon atoms with a separation of This structure acts as an extremely fine diffraction grating, allowing the electrons to pass through and diffract.
The following experimental setup demonstrates electron diffraction.
- Electrons are fired from an electron gun in the direction of a thin piece of graphite and accelerated by an electric field so that their de Broglie wavelength is the same order of magnitude as the carbon atom spacing.
- The electrons diffract as they pass through the graphite in the same way as waves diffract when passing through a diffraction grating.
- A fluorescent screen is placed behind the piece of graphite to detect the electrons that emerge from it.
- An interference pattern consisting of concentric rings is observed due to the interference of the electrons.
Unlike a normal diffraction grating, where all openings are aligned, graphite has openings in random orientations, so overlapping diffraction from all the openings produces rings instead of discrete spots.
The setup is contained within a vacuum tube, ensuring that the electrons are not obstructed by air particles.
The wavelength of a particle is related to its momentum by the de Broglie equation, which states that:
where:
- is the momentum of the particle in
- is the Planck constant
- is the wavelength of the particle in
The de Broglie equation shows that the momentum of a particle is inversely proportional to its wavelength:
For particles travelling at the same speed, a greater mass results in a shorter wavelength.
The equation for the momentum of a particle in is:
Where:
- is the mass of the particle in and
- is the speed of the particle in
The de Broglie wavelength of a particle is inversely proportional to its momentum, and this can be observed in the electron diffraction experiment.
Slower electron acceleration by the electric field leads to lower momentum upon reaching the graphite grating. Consequently, the electrons exhibit a larger de Broglie wavelength, which is a closer match to the atomic spacing in the graphite. This results in increased diffraction and a broader interference pattern.
A greater momentum, which corresponds to faster electrons, leads to a smaller de Broglie wavelength. Consequently, the electrons diffract less, producing a narrower interference pattern.
Macroscopic objects, such as a football, have a large mass and therefore a large momentum. Since the de Broglie wavelength is inversely proportional to momentum, macroscopic objects have a negligible de Broglie wavelength:
- Diffraction is observed when the wavelength of the wave passing through an aperture is similar to the aperture width.
- Wave-like properties are not observed for macroscopic objects due to their small de Broglie wavelength.
An example of small de Broglie wavelengths is a tennis ball of mass travelling at a speed of , which has a momentum of:
Therefore, it has a de Broglie wavelength of:
This means it is physically impossible to observe the diffraction of a tennis ball as it is unable to fit through the opening that would cause diffraction.
Electrons are ideal for demonstrating the wave nature of particles because they can be accelerated to have de Broglie wavelengths comparable to atomic spacings, allowing for the observation of diffraction and interference patterns.
Refraction occurs when a wave passes from one medium to another, changing its speed. Refraction is best understood by considering light passing from one medium to another, such as from air to glass.
At the boundary between the two media, the light rays experience a change in direction, caused by a change in the light’s speed.
To understand how light bends, use the normal. An imaginary line perpendicular to the surface where the two media meet.
The angle of incidence (the angle of incoming light) and the angle of refraction (the angle of refracted light) are measured relative to this line, which is typically represented by a dotted line in diagrams.
When light passes from a medium with a lower refractive index to one with a higher refractive index, it slows down and bends toward the normal. This happens because wave fronts hitting the denser material slow down first on the side closest to the boundary.
Conversely, when light moves from a region of higher refractive index to one of lower refractive index, it speeds up and bends away from the normal. This is because the wave fronts speed up, beginning with the part farthest from the boundary.
If light travels along the normal, all wave fronts reach the new medium at the same time and maintain their straight path.
When light refracts, its speed and wavelength change to match the properties of the new medium, but its frequency remains constant. This is a consequence of the conservation of energy.
A straw in a glass of water illustrates that frequency remains constant when it passes between two media. The appearance of the shape changes (the straw looks bent), but the colour of the straw remains the same in both air and water.
The frequency of light determines its colour (for example, red light has a wavelength of . Therefore, if the colour is the same, the frequency is the same.
When light waves encounter a surface, they bounce off, creating a reflection. The law of reflection applies to waves reflected at a boundary – the angle of incidence is always equal to the angle of reflection :
- The angle of incidence is the angle between the incoming wave and the normal to the surface, and
- The angle of reflection is defined as the angle between the reflected wave and the normal.
Light reflecting off a mirror demonstrates the law of reflection:
A pulse-echo system emits a short pulse of waves that partially reflect wherever it meets a boundary between two media. The reflected pulse (the echo) then returns to a detector. The distance to any target can then be calculated as:
Where:
- is the distance to the reflecting surface,
- is the wave speed of the propogating medium, and
- is the time delay between transmission of the wave pulse and reception of the echo.
The factor of 2 in the denominator of the equation accounts for the return trip distance traveled by the signal.
Ships can use ultrasound beams for various imaging purposes. This is known as sonar (sound navigation and ranging). For example, ships may use ultrasound to detect fish, determine seabed depth, or locate submarines.
In the diagram above, an ultrasound beam is transmitted from the ship, and the waves is reflected by the seabed. The longer the time difference between the transmitted and reflected signals, the greater the depth of the seabed. The depth of a seabed can be determined using the expression:
Where:
- is the speed of sound in water in ,
- is the depth of the seabed in and
- is the time difference between transmitted and received signals in .
The pulse-echo technique is commonly used in medicine to visualise the inside of the body.
- Ultrasound waves are directed towards the target area in a narrow beam. When these waves encounter different surfaces within the body, various amounts of the sound are reflected back or transmitted through the tissues.
- The reflections, or echoes, are detected at different times, allowing a computer to construct an image of the area. This method is advantageous because it is non-invasive, enabling safe imaging without the need for surgical procedures.
One of the most notable applications of ultrasound is in imaging the fetus during pregnancy, which provides valuable information about the baby’s development and health.
Two features of a transmitted wave limit the information that the pulse-echo technique can reveal:
- Wavelength sets the smallest object that can be resolved: features much smaller than diffract the wave rather than reflecting it cleanly. Shorter wavelengths give finer detail. Hence, medical ultrasound uses frequencies to image millimetre-scale tissue.
- Pulse duration limits how close two reflectors can be while still giving separable echoes. If a pulse is too long, echoes from neighbouring boundaries overlap and merge. Shorter pulses give sharper timing and better resolution.