# Rayleigh criterion when light phase is known

In summary, the conversation discusses the concept of measurement resolution in relation to light and electromagnetic waves. It touches on the idea of a theoretical measuring device that can measure light at a frequency of a quadrillion hertz and whether it would show a patch with oscillating intensity. The conversation also explores the Rayleigh criterion and whether it can be overcome with knowledge of the phase of two light sources. It mentions aperture synthesis and coherent detection, as well as the importance of signal to noise ratio in achieving better resolution.
Hi everyone,
this is sort of a soft question which I need to ask to make sure my understanding is correct, it relates to a little project I'm doing on measurement resolution. The first question is to clear up a general concept, the second is based on the first and is the actual question.

First, when light is directed on a detector, what is seen is a "patch" with a certain shape and a fixed intensity. However, being an electromagnetic wave, the magnitude of the field oscillates between 0 and a certain amount at a really fast rate. So if we had a theoretical measuring device capable of measuring the light hitting it at a frequency of, say, a quadrillion hertz, and we direct visible light at it, will the detector show a "patch" with oscillating intensity? If so, is the fixed intensity seen normally just our puny mortal eyes seeing the average of the intensity due to a lack of precision?

Now, assuming the answer for the above is somewhat positive, it's time for my real question regarding the Rayleigh criterion. The criterion (when relating to optics) says that if 2 given light sources are too close, or to be more exact, their 2 projections are too close, it will be impossible to tell whether said projection is a result of a single light source or 2 close light sources. My question is whether that would be the case if we knew the phase of the 2 light sources at all times.

For example, say we project 2 light sources with the same frequency ##f## and amplitude through a slit so each source creates a nice interference pattern, but since the sources are closer than the Rayleigh criterion, the peaks of the 2 patterns merge into what seems like a single peak. But say we have our super-accurate measuring device, and for convenience let's also say that the phase difference between the 2 sources is exactly ##\frac \pi 2##. If the assumption of the first question is true, then when the first signal is at its peak intensity, the second will be at its minimum (which is perhaps 0), this means that the peak of the first signal will be visible and much more prominent than the second peak. ##\frac 1 {4f}## seconds later, the opposite happens, so the second peak will be visible and the first will not. Clearly in this situation one will be able to tell whether the projection is a result of 2 sources or just 1, depending whether or not the main peak changes location every ##\frac 1 {4f}## seconds. This can also work when the phase difference is pretty much anything other than 0, although maybe to a lesser extent.

So, when the phase of 2 light sources is known and the difference between the phases is not 0, is it possible to overcome the Rayleigh criterion?

Can we assume the 2 light sources are at different points in space? If that's the case then, also assuming the emitted light waves are spherical, to overcome the Rayleigh criterion you would need to take a series of phase measurements at different distances from the sources, or at different angles, wouldn't you?

So, when the phase of 2 light sources is known and the difference between the phases is not 0, is it possible to overcome the Rayleigh criterion?
I believe this is the basis of aperture synthesis and suggest you track that down. Let me know the result!

<snip>
So, when the phase of 2 light sources is known and the difference between the phases is not 0, is it possible to overcome the Rayleigh criterion?

Most of your questions are easily answered in terms of millimeter-wave imaging (or radio astronomy), because phase-sensitive detectors are in existence. These detectors are primarily 'point detectors', meaning there is only a single pixel, but there are research efforts to construct array detectors.

What you are asking about is known as 'coherent detection', and the Rayleigh criterion is indeed affected by the relative phase between two separated mutually coherent point sources. Note, different stars are mutually incoherent sources.

sophiecentaur and hutchphd
it's time for my real question regarding the Rayleigh criterion
Before going any further with this, we need to know what the Rayleigh Criterion is, exactly. The RC was devised as a rule of thumb (pretty arbitrary but convenient) to apply to two independent (incoherent) light sources (e.g. equal brightness stars) to decide whether you would say they could be resolved (visually) as two sources and not a single source. Each source will produce a diffraction pattern when it passes through an aperture (originally an astronomical Telescope, I think). The (brightness) pattern for the two stars will be a smooth dip between one maximum and the condition for the Rayleigh Criterion is when the minimum of brightness at the mid point falls to half power. If the optics has been well charcterised and the signal to noise ratio is good (good 'seeing' with low light pollution) you can do much better if your imaging array is good and high res enough. That 'saddle' curve doesn't need to dip to half power if the stars are bright enough; you can resolve with a much shallower dip as long as the 'noise' doesn't fill it in. Stacking multiple images can suppress the noise. There is a lower limit of star visibility below which you cannot do better than the Rayleigh criterion - in fact you will do worse! It's the same with all imaging / measurements; signal to noise ratio is what really counts.

## 1. What is the Rayleigh criterion?

The Rayleigh criterion is a principle in optics that states that two point sources of light will be just barely distinguishable as separate when the center of one source falls on the first minimum of the diffraction pattern of the other source.

## 2. How does the Rayleigh criterion relate to light phase?

The Rayleigh criterion takes into account the phase of light when determining the distinguishability of two point sources. This means that not only the intensity, but also the phase of the light waves must be considered when applying the criterion.

## 3. Can you explain the mathematical formula for the Rayleigh criterion?

The mathematical formula for the Rayleigh criterion is θ = 1.22 λ / D, where θ is the angular resolution, λ is the wavelength of light, and D is the diameter of the aperture through which the light is passing. This formula takes into account the diffraction pattern of light and the size of the aperture to determine the minimum angle at which two point sources can be distinguished.

## 4. How is the Rayleigh criterion used in practical applications?

The Rayleigh criterion is commonly used in optical engineering to determine the resolution of imaging systems such as telescopes and microscopes. It is also used in the design of optical instruments to ensure that the distance between two point sources is large enough to be distinguishable.

## 5. Are there any limitations to the Rayleigh criterion?

Yes, the Rayleigh criterion is only applicable to point sources of light and does not take into account other factors such as the quality of the optics or the presence of atmospheric turbulence. In some cases, the criterion may also be too conservative and may not accurately predict the resolution of an imaging system.

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