Explanation of diffraction of a single light ray by Huygens' principle

Question

Huygens' principle is a great way to explain diffraction when light rays are incident on a slit, but how would one explain the diffraction of a single light ray crossing a slit?

Answer 1

I planned a comment, but after reading the existing ones, I think a direct answer addressing the main conceptual issue could be more appropriate.

There is nothing like a straight-line, one-dimensional ray. The only attempt at producing it would destroy such a ray.

It is a well-known misconception to think of the rays of geometric optics as trajectories of light particles. They aren't. Rays in geometric optics are just a set of infinite lines perpendicular to the surfaces of a constant phase of the wave.

Amazingly, a question on a single ray is embedded in the context of Huygens' principle, which should be enough to exclude it. Indeed, if a wavefront was reduced to a point, Huygens' principle says that this point would be the source of a system of spherical waves. Said in another way, if we try to make such a single ray by a screen with a hole, the smaller the hole, the more spherical the fraction of the wavefront outcoming the hole.

A final comment about photons. Photons cannot restore the concept of a "single ray." There is nothing like the trajectory of a photon, according to the basic principles of Quantum Mechanics.

Answer 2

Huygens principle works when light is treated as a wave. Here, the diffraction patterns arise due to the interference of different points of the wavefront. However, when you talk about a single ray, this is in the geometrical optics limit of light, and the Huygens principle does not hold anymore.

So you might ask, what do we expect if light is really taken as a single ray? But how do we even create a single ray (when even lasers have a Gaussian profile)? A good starting point would be to reduce the intensity of light, so we shoot only a few photons at a time. But then, a quantum mechanical description of light would need to take over Huygens's principle.

However, for good reasons, one avoids talking about trajectories of particles in QM. Instead, we talk about what can be observed. So again, we cannot talk about a ray. But that won't bother us if only a single photon arrives at the slit in a given interval of time. For such a case (partly quoting from Wikipedia) - In QM, during the light propagation through a slit, each photon is described by its wave function. The latter determines the distribution for the photon on the screen, which is a probabilistic choice of one of many possible paths in the electromagnetic field. These probable paths form the pattern: in dark areas, no photons are landing, and in bright areas, many photons are landing.

So if your light is of low intensity such that only a few photons arrive at the slit in a given interval of time, QM predicts that an interference pattern should be detected on the screen. The conclusion is akin to Huygens principle but now in the low-intensity limit.

Answer 3

A single photon doesn't have a trajectory. It is not a classical corpusle.When a photon takes part in a diffraction experiment it has nothing to do with Hygen's principle, because it is not a classical wave.'Hygen's principle' or 'wave theory of light' only works if we take an ensemble of photons. That's what we do in usual diffraction experiments. Hygen's principle can be used to predict a statistical outcome that emerges from the probabilistic nature of individual photons,only if we are working with a large number of identical photons. Though the visual interference pattern is an emergent property, the 'probability wave' is not a property of ensemble but associated with each individual photon.In the double slit interference experiment If you use an extremely weak source of light and keep the source on for a brief period of time you will see a few random dots here and there on the photographic film, and not a faint Interference pattern as Hygen's principle will tell you. If you don't swich off the light source and let the experiment run on, much more such dots will appear and at a point you will identify the interference pattern expected from the wave theory. More dots will appear where wave theory predicts constructive interference, less dot will appear where it predicts destructive interference.You usually don't see discrete dots because in ordinary diffraction experiments the dots merge with each other and the usual bright and dark fringes are noticed. Photons are getting absorbed one by one but the probability of it getting absorbed at a particular place on the screen is determined by the law of the addition of probability amplitude, it is a complex number which can add/interfere like a wave. Here take "the photon getting absorbed at a particular place on the detector screen" as the 'event'. In double slit experiment it can happen two alternatives ways, through slit A and through slit B. Now read the following:

/1. The probability of an event in an ideal experiment is given by the square of the absolute value of a complex number Φ which is called the probability amplitude:

P = probability,

Φ =probability amplitude

P = |Φ|^2

2.When an event can occur in several alternative ways, the probability amplitude for the event is the sum of the probability amplitudes for each way considered separately. There is interference:

Φ = Φ1 + Φ2

P= । Φ1 + Φ2 ।^2

3.If an experiment is performed which is capable of determining whether one or another alternative is actually taken, the probability of the event is the sum of the probabilities for each alternative. The interference is lost:

P=P(1)+P(2) /

: Feynman lectures of physics (Vol:3)

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A similar question was asked about the explanation of rectilinear motion of light considering it as a single photon. That answer may also help to understand the current question about the diffraction of a single photon.

..Light(photon) has an probability amplitude assigned to each path/trajectory possible and has a particular 'phase' association with each path/trajectory.

We know Fermat's principle of least time which roughly states that out of all possible paths that it might take to go from one point to another, light takes the path which requires the shortest time. So when light travels within a particular medium it takes the straight line, because if light doesn't change the medium it's velocity remains constant and the straight line is the path which requires shortest time.

But how light can choose a particular path? What actually happens is that: the path which represents the least time is also the path where the time for the nearby paths is nearly the same, so the 'phase differences' are really really small which allows the probability amplitudes to aligne in nearly the same direction and add up to a substantial magnitude where the probability amplitudes corresponding to numerous other paths cancels each other. That is why we can say light takes the path where time is least i.e it goes in straight line in vaccum or in a particular homogeneous medium. It doesn't break uncertainty principle which originates from the fact that microscopic particles can't have a trajectory (In single slit diffraction experiments we can see light isn't going in a straight line cause we have restricted the possible paths)

Feynman has discussed this in his lectures on QED .

There can be a misunderstanding. "...That is why we can say light takes the path where time is least i.e it goes in straight line in a particular homogeneous medium" doesn't mean it really does.

What it means:

In vaccum or Inside a particular homogeneous medium, without any kind of restriction on possible paths (what happens in diffraction experiments where we reduce possible paths significantly) if we consider light is going in a straight line and make predictions out of it we will find no significant error,We will get extremely fine, accurate results in any experiment whatsoever

Answer 4

You are looking at two different approximations to light.

Ray optics treats light at traveling in straight lines. At the surface of a lens or mirror, it changes direction.

Ray tracing is good for lens design. It was used long before there were computers. There has been a lot of effort to find ways to use it with an absolute minimum of hand calculation, and ways to calculate and minimize lens aberrations.

In this approximation, the slit casts a geometric shadow. Diffraction has been approximated away.

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As far back as Newton, there was debate whether light was a particle or a wave. Newton favored particles.

An early effort by Huygens to understand it as a wave can be illustrated by throwing a rock in a pond. A circular wave spreads out in all directions. You can understand rays as lines in the direction waves travel. This is perpendicular to the wavefront.

If you throw a long stick in the pond, you get a line of waves. You get the same thing if you first chop the stick up, so that it is a series of small pieces side by side. Each piece creates a circle. If you add up the waves from each circle, you get the line. To do this, you have to understand the phase of waves and interference. You do not have to understand what kind of wave you have.

This idea works in $3$ dimensions too. Light from a point source is an expanding spherical wave. An extended source is a series of point sources.

You can take this idea a step further. Suppose an extended source generates a plane wave that propagates to plane A and continues to plane B. If you put an extended source at A, you get the same wave at B. It is as if a plane wave is a series of point sources that continually generate expanding spherical waves as it propagates. This gives correct description of the propagation of a wave.

This idea makes sense in a mechanical wave, where a medium is disturbed, and each disturbed point is a source of further disturbance. If you think light travels through aether, this evidence supports your thinking.

If a plane wave travels through a slit, some of the point sources are cut off. If you add up the waves of those that remain, you get the correct pattern for single slit diffraction. So diffraction is a consequence of the wave nature of light. It can be described as the difference between ray optics and wave optics. You can understand it without knowing anything about light other than light is a wave that interferes as dictated by its phase.

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By and by, Maxwell came up with his equations. They showed that electric and magnetic fields can form waves. The predicted speed of those waves happened to be just the speed that light traveled at. A light bulb appeared over Maxwell's head.

Now we understand more about the waves that light is made of. Light is oscillating transverse electric and magnetic fields. Maxwell's equations are more powerful tools to predict its behavior. We can solve a wave equation with boundary conditions to predict how the wave propagates.

Diffraction comes out the same. Light is still a wave with a phase.

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Quantum mechanics changed things again. Light is photons. Photons are sort of like waves, but also sort of like particles. If you shine a very faint light on a fluorescent screen, a single atom lights up. Some time later, another atom lights up. Photons are like particles in that they can hit a single atom and miss all the others.

If you pass the faint light though a slit or a pair of slits, you still get single atoms lighting up. But some atoms are hit more often than others. If you turn up the brightness, you see the usual diffraction pattern. So photons are unlike particles in that they can go through two slits.

It turns out that photons are unlike anything classical. They are something like classical particles and classical waves. Thinking about them this way leads to some correct answers and provides some feeling that you understand them. But it also can lead you to expect classical behavior in cases where reality is different.

Whatever a photon is, it is described by a wave. The Schrodinger equation is a wave equation that predicts how the wave will propagate. Photons have a phase. The phase and interference govern how the wave propagates. This wave determines where the photon is likely to be. It determines which atoms the photon is likely to hit. Diffraction still is predicted.

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Responding to comments

The question of what happens to a single ray of light that passes through a slit needs clarification.

There is no such thing as a single needle like ray of light in isolation. The particle like aspects of a photon do not govern how it propagates. The wave like aspects do. Waves do not follow a single line. They are spread out.

If you have a plane wave arriving at the slit and are asking what happens to a single ray, the Huygens principle says it acts like a point source and adds to all the other point sources. It doesn't have a diffraction pattern in isolation.

You might have a very faint plane wave arriving. It can be so faint that only a single photon passes through the slit at a time. The photon passes through all of the slit, not just a point. Diffraction occurs because the photon has a wavelike nature and interferes with itself.

You might have light focused to a point inside the slit. This might be spherical waves headed toward the focus. An important real world example is a laser being focused to the smallest, most intense spot possible. Using ray tracing, it is possible to design perfect lens that focuses all rays to a perfect point. But light is a wave. Diffraction limits the minimum spot size.

A good quality laser typically has a Gaussian beam. The beam is approximately cylindrical. But the intensity falls off exponentially as you get away from the beam axis. And rays do not quite follow straight lines. The wave front is a slightly converging spherical wave that flatten into a plane wave as it passes a beam waist. Then it curves into a slightly diverging spherical wave front. The rays perpendicular to these wave fronts follow hyperbolic paths. Far from the laser the beam is approximately a cone with a very small divergence angle, typically a few milliradians.

A Beam Expander is two lenses just the right distance apart. Using ray tracing, one lens focuses the beam to a point. The other takes light expanding from the point and turns it back to a collimated beam. But what you really get from the first lens is a Gaussian beam that is a sharply converging cone. It passes through a very small waist and expands into a sharply diverging cone.

To produce a useful small spot, you might just use the first lens.

If you focus the laser to a spot that is smaller than the slit, the electromagnetic field does not fill the slit. Or equivalently, the wave describing the state of the photons does not fill the slit. The diffraction of the beam is the same with or without the slit.

Using Huygens' principle, you add up waves from point sources only where you have a wave. The diffraction pattern of the beam is similar to that of a pinhole the size of the beam. Smaller slits/pinholes produce wider diffraction patterns.

Answer 5

Huygens' principle can be phrased as all the points of the wavefront in the existing perturbation are the sources of secondary waves. Just from its formulation, there is a strong suggestion to not treat impending light as a bundle of rays (geometric optics). Hence, your image is conceptually wrong: on the left side from the obstacle you symbolically depicted, from what I understand, the ray you are talking about, while on the left side you drew surfaces of equal phase. Please, consider the image from Wikipedia, that is correct:

With that being said, if the wave (!) is incident at an angle, there is going to be a relative phase delay as it touches the surface, that will account for effects, related to incidence angle. You can examine the refraction image on that same Wikipedia page. A similar diagram can be constructed for the case of diffraction on an opening.

Answer 6

If we define a single photon (not use the word ray) as a somewhat confined disturbance/wave in the EM field then that is a good place to start, and let's for arguments sake give that energy disturbance a direction (I.e. towards a slit). "Current" thinking (Feynam 1960s, Dirac 1940s) says each photon acts on its own and the EM field guides everything. The slit material has its own EM field and it is fluctuating all the time, the screen (or the rest of your apparatus) is also involved with the EM field and fluctuations. There are 2 governing forces: 1) the EM field of the slit and 2)per "Feynman path integral" the fact that photons have a higher probability traveling paths an integer number of the wavelength (it appears the EM field likes to be resonant). Using these guidelines it would be believable that the random fluctuation in the EM fields ... the fact that a photon would never be perfectly aligned to the slit .... that the emission would have a partially random nature.

The Feynman part explains why we see "interference" patterns ... even with single photons .... when the energy enters the slit/apparatus the apparatus EM field resonates with the incoming energy .... more resonance is where the energy will go.

The Huygen's principle is 1700s! it is still taught today because it works mathematically .... especially for water.