Advance organiser: photons modelled with spinning arrows, exploring contributions
Start with illuminating phenomena to promote noticing and discussion, to motivate the idea of
exploring space: the arrangements of the source and what goes in the surrounding space determine the brightness at different locations – so what turns up at the detector. Discuss the space into which the photons are emitted – develop a description from the point of view of the photons, for the photons of what's out there – that enables you to predict where it's bright, and where it's not. This will be the basis of an account of lighting, of illumination, of optics, based on photons.
First do some careful noticing, starting with seeing.
Seeing can be seeing things as brighter or seeing things as different colours. If ideas from describing sounds are available, then perhaps choose to re-activate these via the box of hearing.
The box of hearing
If not, then suggest directly that the box of seeing needs two axes – how bright, and what colour.
The box of seeing
The best description of light that we have is that it arrives in small chunks called photons – our eyes are not quite sensitive enough to notice each one arrive(5-8 photons enable a human to
a flash of light), but nocturnal insects and some animals can do much better, down at the single photon level.
So there are two independent quantities: the brightness and the colour. You can see something brightly lit or only dimly lit in many colours, or only in red; brightly lit or only dimly lit, or only in yellow; something brightly lit or only dimly lit, and green. And there may be other colours you cannot see, and perhaps some things which are too bright or too dim to see(humans are not all that special, and physics tries to be universal, so build that in from the start).
The (simple) character of the photon sets the colour: choose a simple spinning arrow to describe a simple thing. The number of rotations a second is the frequency, which we may see as a particular colour (if the number of rotation second-1
is within the range of human seeing). So to represent different colours just spin the arrows faster or slower. That's how you can start to think about photons fruitfully: to find out why spinning arrows are so fruitful, you need to think more about brightness.
Photons and the box of seeing
Engineered spaces around the source
Start with blue light: you can illuminate different things and make them appear bright or dim. Move the thing further away and it gets dimmer. Put a barrier in the way, casting a shadow over the thing, and it gets much dimmer. The brightness is set by how you arrange things in the space around the source of the photons. This space gets explored and in some places there get to be many photons, so it's bright, in others, fewer photons, so it's dimmer.
Vary what's between source and detector
To make it dimmer, just ensure some photons don't make it – put something in their way. Just absorb all or some photons, just using a darkened slide.
How to make the illumination dimmer somewhere, but also brighter somewhere else? Redirect some or all photons – use a mirror, a prism, a lens. These all ensure the photons end up somewhere else.
More photons arriving: it's brighter. Fewer photons arriving: it's dimmer. But can you predict how putting things in the space around the source affects where the photons end up?
Let's set up a way to figure out how the photons get to explore their space and from that predict where they'll show up – so where it is bright or dim. (Although people say that
photons explore all paths
, that's simply a way of describing the calculations to be done: no-one knows what the photons really get up to between source and detector.) Then show how this description predicts where it will be light or dark, still representing photons according the best available model.
A model to predict where photons will turn up
The spinning arrow of each photon explores the space, at the speed of light (light only knows this universal speed). See how to explore some of the space by choosing paths between source and detector. A single waypoint defines a simple path. The length of the path sets the trip duration and this fixes the rotation of the arrow.
A single path
Every path contributes an arrow, and the sum of these contributions predicts the brightness.
The photons are not
along particular trajectories, so there are no preferred paths. Photons explore all paths: but that's an impossibly time-consuming calculation to do for mere humans, so make a smart guess as to the important paths and calculate for these. Later, justify your guess by showing that calculating for additional paths makes not much difference and that your prediction from the few paths that you did try corresponds to what happens in the lived-in world.
Saying each photon
all paths is attributing an essence to the photon – a character – that is a translation of the mathematics that you do to calculate where the photon is likely to end up. If you don't like this essential approach , you could just say you're manipulating a mathematical model, but that may demand a metaphysics that is too austere for younger learners.
The contributions from different paths may be more or less in step. Start with two paths. The more the contributions are in step, the greater the resultant. A greater resultant predicts a greater brightness. (To be precise, the brightness is proportional to the resultant squared).
a pair in and out of step, depending on paths
Since it is only whether the arrows are in or out of step, you can choose to freeze the spinning, making the diagram simpler.
To explore all space you'd have to try all waypoints – and the photon does. But you can get a good idea what to expect by selecting fewer paths, and matching your selection to the particulars of the changes made to the space you're investigating. The selection of a few waypoints requires insight, experience and cunning – increasing the number should not affect the predictions about where will be bright by much. That way we can have a model which is fit for purpose – accurate and precise enough, but not too cumbersome (Calculating the contributions for all paths would take a very long time).
The result of all explorations of the many paths sets the brightness. See how this is done by varying the paths and seeing how the contributions set the resultant.
Four contributions in and out of step, depending on paths
Summary so far.
varying illumination is a consequence of altering the geometry
arrange for particular lighting by manipulating the space between source and the detectors
predict the illumination at a detector by calculating the contributions of significant paths between source and the detector
There is a big question – is this really how the photons are? To which the answer, however unsatisfying, is
. And perhaps they never could know: physics is concerned with phenomenological metaphysics – that is with building fruitful, plausible and intelligible descriptions of how quantifiable aspects of the lived-in world appear to us.
A general tool
The model is useable, but could be refined: you're looking for significant sets of paths, defined by source, waypoint and detector, where the contributions add to a significant resultant, so predicting a bright point. For other sets of paths, perhaps terminating in a different detector, the contributions don't add to much, predicting a lack of illumination at that detector. And in principle you've to sum these contributions over all possible paths.
There is a very neat way of working with significant groups of paths: working with three paths, as a triplet, is fruitful. More than three adds complexity without adding insight. Where the paths contribute significantly to the resultant, the contributing arrows line up, where there is a very small contribution, they curl up.
Curling up and Lining up
In any situation, picking out what's distinctive about the geometry let's you see how to explore the space with a triplet.
Start with a mirror, to see how this lining up and curling up plays out.
A triplet for the mirror
If you took out the paths near the end of the mirror, it wouldn't make so much difference, as the arrows contributed by the triplets of waypoints from those regions tend to curl up. But ignoring the contributions from nearer the centre does make a difference, so a rough and ready rule is that you'll only need to consider paths where
the angle of incidence = the angle of reflection
, which is comfortingly familiar.
Arrange a different change in what happens between the source and the detector, this time by arranging it so that the photons must change medium as they explore the space between the source and a detector: to explore refraction. Here you might guess that the significant paths will be those which have waypoints on the surface between the media. You'll want to vary the waypoints for a variety of source and detector positions to get a complete picture of how the illumination varies.
A triplet for refraction
Again you're looking for triplets of paths where the contributions line up, giving a large resultant, and so predicting a brightness, or the opposite, where the triplets curl up, predicting darkness.
Careful exploration will recover the empirical laws of refraction, that only specific relative locations for the source, interface between the two media, and the detector result in an illuminated detector.(An elaboration of this model can predict
which is the quantitative empirical law of refraction. Physicists often get excited when they can account for an empirical law with a theoretical model, so now might be a good time to
take a moment
As an extension you might look deeper, inot the variation in trip time between paths, as its the trip time that sets the arrow rotation. Where the difference in trip time between adjacent paths is small, the contributions from those paths will line up.
Exploring trip times for refraction
Photons don't even know to
in straight lines. However if there is nothing between the source and detector, light appears to travel in straight lines. So next you might try looking at lining-up and curling-up for a source and detector with nothing in the space separating them. Again you have to guess – but an intelligent guess – as to the most significant paths and then model that with the triplets.
Here only the contributions from the central area add much to the resultant – contributions from paths with waypoints far from the centre line curl up. Again you have recovered an empirical law: that light travels in straight lines. But once again you now have a theoretical reason why this is the case – it's no longer just to be accepted as fact.
But there is a wrinkle, which you can explore. It does look as if you can ignore all paths apart from those defined by a hole around the centreline. A commonly fruitful way of thinking in physics is to take things to extremes – so now try making the hole smaller by adding a barrier. In two dimensions on screen that will be a slit.
Restricted paths for propagation
Here is a situation where the paths are artificially restricted by two barriers. So the waypoints are restricted and you can investigate what effect this restriction has on the resultant of the contributions at different locations for the detector.
Use this array of resultants to predict the variation in brightness as you move the detector.
Here you have managed to predict diffraction using the same mechanism as you as accounted for reflection, refraction and propagation and so you should certainly take a moment to celebrate a bit of theoretical unifying but also demonstrate diffraction. The theory has shown you might expect to find something which you should then go out into the world and look for(an opportunity to exemplify a facet of reasoning in physics).
Restricted paths for reflection
The mirror was another situation where you've seen that the paths a long way from the obvious path did not contribute much. That is the meandering paths – those indirect paths explored by photons – did not result in contributions which made much difference to the overall brightness. Perhaps revisiting the mirror and restricting the paths might yield interesting results in the same way that restricting the paths did for propagation. Simply put, the ends of the mirror did not seem to make much difference, so let's start there.
Here are a triplet of paths with contributions that seem to curl up. Now restrict them in an interesting way, by removing the possibility of the central path. This results in an increase in the brightness as the contributions from the paths now tend to line-up rather than curl-up.
Delete a path on a mirror
Less can mean more: fewer contributions result in more illumination. As stated at the beginning, it all depends on what's in the space explored between source and detector and how that stuff is arranged.
Systematically removing the possibility of every other path is done by creating lines where reflection is not possible – in other words a grating of lines. The model has now predicted the existence of diffraction gratings – so again take a moment to celebrate this once you have shown a diffraction grating in action.
Just two slits
A further suggestion for restricting paths is to move from one to two slits. That is to have two gaps rather than a single gap, further modifying the model for propagation.(You might think, following on from restricted paths on the mirror, that a line of holes, or even an array of holes would be interesting. It would, and as an easy extension from what you're going to investigate here.)
Two slits, two waypoints, two contributions
The waypoints are again fixed by the slits, so use the model to explore the prediction about illumination at different locations for the detector.
You might prefer to work with triplets, rather than just the pair, in which case use:
Two pairs of triplets
Once you've seen that bands of brightness and darkness alternate, get some apparatus out and show this(it's the phenomenon of interference, central to the quantum world). Then celebrate a further unification.
The critical point here is that this alternating pattern of brightness and darkness still builds up even when the source is so dim that there is only a single photon in transit at once. Photons are just not like us – they really do behave as if they're exploring both slits at once (or if you have an array of slits, all of the slits at once). If the calculating we do is a reliable guide to how the photons are between source and detector then this is how they are. Not a stream of bullets. Not in any one place. The best we have is that they leave the source and we can predict how many will turn up at each detector by doing the calculations – here represented by exploring paths, defined by waypoints. Then summing the contributions to find the resultant, which predicts the relative brightness.
Engineering a lens
Lining up and curling up are core insights in representing the calculations. Looking for arrangements where all of the contributions line up predicts great brightness. That we can make reliable predictions enables us to engineers situations where the brightness ends up where we want it, such as a lens.
The photons appear to travel more slowly in the glass – they rotate at the same rate (because the colour remains constant) – but there are more rotations in a millimetre of glass than in a millimetre of air. So inserting extra glass effectively inserts more rotations.
Separate the central waypoints so as to insert more glass. This extra glass compensates for the shorter distance. So the contributions from the ends of this new object can line up with the contributions from the centre of the new object. The new object is of course a focusing lens.
Engineering a lens
It's a different account of the lens, but it is one that uses the same theoretical framework as we have use throughout. You really are showing that the idea of photons can account for a wide range of phenomena and be useful to build things.
Engineering a mirror
A curved mirror can also focus. Displaced the waypoints at the end of this mirror so that the contributions from all the waypoints all line up. Then you will have predicted or even designed a focusing mirror.
Engineering a mirror
The photon, a recap
The photon is perhaps the simplest example of a quantum object. It's therefore a good introduction to the quantum world – every aspect of the discussion here applies also to particles such as electrons, protons and neutrons. The world appears as a quantum place and this way of calculating what will happen is universal. It's the best model of the universe at a very small scale. It's certainly counterintuitive, but that's the way the world is. Physics has often progressed by encouraging us to think that things are not as they first seem and that our commonsense may not be as reliable as we imagined(think Aristotle, think Newton, think Copernicus, think Einstein, amongst others).
Why bother, and what about...
A century and a bit on from the introduction of the photon, and an increasingly quantum-technology driven world, together with a reconition of the cultural value of physics seems to make a strong case for teaching about light starting and ending with photons. This sketch of the core of a schema, once allied to essential tangible lived-in world experiences, shows that how this could be possible.
Later work might make use of wavefront descriptions and ray optics where these provide less fundamental but perhaps more useful insights for particular tasks. For an education in physics for all, I suggest starting out on optics with the photon.
That's it for an initial tour of optics, which I think could achieve a more coherent insight than the current collation of practices. But
keep on giving: the ideas can be exploited to give good accounts of power in pathways and of the interactions of radiations with matter, both accessible to pre-16 year old children and their teachers.
Photons, power, compensation
Photons are quantum entites, shifting energy to or from stores in discrete amounts. So simply counting the number of photons second-1
sets the power in the beam, for a particular frequency of photon.
The activity (which is the number of photons second-1
) at the source and detector set the power absorbed and emitted.
Emission and detection
Varying the frequency of the photon sets the energy shifted by each photon: the activity and the frequency together characterise what is
Seeing with photons
So varying the activity and the frequency might be expected to determine the power.
vary activity and frequency
And the activity and frequency are compensated quantities – a useful and repreating pattern to think of whenever considering power or energy.
compnesation and photons
Photons, devices, power, fractional absorbtion
Power in a pathways can be switched to another pathway by a device, here the source, absorber and detector are all devices, making the link with teaching about energy and power.
Devices, pathways and photons
By varying the materials between source and detector you can model macroscopic behaviours (Again, engineering the space between source and detector).
Varying what's between source and detector
Get an interesting (exponential) pattern from iserting successive slices of absorber between source and detector. There is an exponential relationship between acticities, the powers absorbed, and the powers transmitted.
Slices and steps
Photons and selective absorption
If the key fits the lock, then there is a result. Some colours trigger an effect: others do not. Photosynthesis, the colour of flowers, walls and bicycles are all determined by the interaction of photons with chemicals within the materials. As these are colour-dependent (and colour is related to frequency, and frequency to energy), photons can provide a neat account of the selective interaction of light and matter. A simplified band-gap approach.
Lock and Key
But keys may still operate locks, even if the fit is not precise, so allow a looser fit. Materials are of varying complexity, and sometimes only sharply defined frequencies are selected, sometimes the fit can be more relaxed. Roughly, it depends on the complexity of interactions within the material.
Resonance as fitting
For many processes the photon just needs to be more than a threshold value to permit the process to occur: any energy not directly enabling the process can be shifted to other stores. For example in photo-emission, energy above the threshold is shifted to the kinetic store of the emitted particle.
Gifts from photons
Introducing photons unifies and simplifies. Why wouldn't you?