Incoherence of S1 & S2: Explained w/ Interference Pattern

[Philip Wood]

Please consider this set up… A point source, P₁, (with a finite coherence time) lies on the perpendicular bisector of the line joining two slits, S₁ and S₂. Therefore the slits are (secondary) in-phase sources of waves from P₁,

A similar point source, P₂, sending out waves of the same frequency and amplitude as P₁, lies near P₁ but a little way off the perpendicular bisector, such that S₁ and S₂ act as (secondary) antiphase-sources of waves from P₂,

On the other side of the slits, no interference pattern will be formed, because wherever there is constructive interference due to waves from P₁, there is destructive interference due to waves from P₂, and vice versa.

My questions are: (1) Are S₁ and S₂ incoherent (secondary) sources? (2) Can the lack of an interference pattern in this case also be explained in term of coherence or lack of it?

Many thanks

 

[Chandra Prayaga]

Are P₁ and P₂ coherent with each other?

 

[Philip Wood]

Thanks for the response. No, they are not. I'm thinking of something like two diode lasers
.

 

[Andy Resnick]

Please consider this set up… A point source, P₁, (with a finite coherence time) lies on the perpendicular bisector of the line joining two slits, S₁ and S₂. Therefore the slits are (secondary) in-phase sources of waves from P₁,

I think you are mixing concepts- from what you describe, the relevant metric is the coherence area, not the coherence time. The two slits sample the wavefront at equal times but different positions. and if P1 and P2 are truly point sources, their coherence area is infinite. However, if P1 and P2 are independent, they will not be mutually coherent and there is no interference between diffraction patterns produced by P1+ S1/S2 and diffraction patterns produced by P2+ S1/S2

 

[Charles Link]

I think you @Philip Wood have the correct interpretation that you will have two separate two-slit interference patterns essentially superimposed on each other that will basically produce a pattern with little or no fringes. The one pattern (from P1) is an ordinary two-slit pattern and the other pattern (from P2) is a two slit pattern with each of the slits $\pi$ out of phase. (The slits are Huygen's sources). P1 and P2 are assumed mutually incoherent. If you compute the energy distribution, (if my quick calculations are correct), one pattern is of the form $I_1 (\theta)=I_0 \cos^2(\phi/2)$ ($\phi=(2 \pi /\lambda) d \sin(\theta)$ ) and the other pattern of the form $I_2 (\theta)=I_0 \cos^2(\phi/2+\pi/2)=I_0 \sin^2(\phi/2)$ so that $I_T (\theta)=I_1 (\theta)+I_2 (\theta)=I_0$ independent of angle. It does appear that in the diode configuration that you came up with, the interference energy fringe pattern of a single diode (through the two narrow slits) would be filled in by the energy pattern of the other diode. ... editing...One additional item is I think laser diodes are normally finite in extent so that in order to get this configuration to give optimal results, it might be necessary to make each diode into a pinhole type source.

 

[Philip Wood]

Thank you, AR and CL. I think I agree with both your responses, though I'm not sure that I have actually mixed concepts: my reference to coherence time was just to indicate that the sources were 'ordinary' ones. I'm afraid I'm still not sure about the answers to my questions. A little background to the question might help…

Jenkins and White (that worthy optics textbook of yesteryear) states that "If in Young's experiment the source slit […] is made too wide […} the double slit no longer represents two coherent sources and the interference fringes disappear." So the disappearance of the fringes is attributed to incoherence.

In a "more detailed" discussion later in the book, J and W attribute the disappearance of the fringes to overlapping of sets of fringes arising from light emitted from different points on the single slit (an extended source). This explanation is wholly convincing. What I did in my original post was to simplify the extended source to just two point sources (with a randomly changing phase relationship), and to consider a case when the interference patterns due to each point source were complementary.

While the point sources (P1 and P2 in my original post) have a random phase relationship (and, I presume, can be called 'mutually incoherent'), I'm not convinced that the same can be said of the slits S1 and S2, as they both give out waves from P1, and they both give out waves from P2. If $\phi_1$ and $\phi_2$ are randomly varying phase 'constants', then perhaps we can model the outputs from S1 and S2 by

$$y_{S1} = cos (\omega t + \phi_1)\ +\cos (\omega t + \phi_2)$$
$$y_{S2} = cos (\omega t + \phi_1)\ -\cos (\omega t + \phi_2)$$

There is some commonality, as it were. While it is true (see above) that "the double slit no longer represents two coherent sources" the double slits don't seem to me to be incoherent in the same way (or to the same extent) that P1 and P2 are (by hypothesis) incoherent.

It therefore seems to me that attribution of absence of fringes when a a double slit is illuminated by an extended source to the double slits "no longer representing two coherent sources" is potentially confusing. As you can see, it's confused me!

 

[Charles Link]

It therefore seems to me that attribution of absence of fringes when a a double slit is illuminated by an extended source to the double slits "no longer representing two coherent sources" is potentially confusing. As you can see, it's confused me!

When the two slits are illuminated by an extended monochromatic source (such as a monochromatic laser diode of finite extent), you will get a phase relationship between S1 and S2 that varies in a rather continuous manner as you move across the source P1. This will have a strong tendency to cause the interference pattern to wash out. That's why I suggested pinhole sources for the two diodes. Meanwhile, an additional detail: The location of a single pinhole source is not very important. You get a two slit pattern for any location of the single pinhole. It is the relative position of P1 and P2 that will cause the intensity patterns to wash out (having uniform intensity). Having sources from a continuum of locations can also cause the patterns to wash out. Jenkins and White is an ok textbook for optics, but I found Hecht and Zajac to be easier reading with better explanations.

 

[Andy Resnick]

It therefore seems to me that attribution of absence of fringes when a a double slit is illuminated by an extended source to the double slits "no longer representing two coherent sources" is potentially confusing. As you can see, it's confused me!

Your post shows pretty good understanding of the situation, it's hard for me to figure out exactly *what* you are confused about. Does it help to note that your random phases Φ = Φ(t)? Then, if you expand cos (ωt+Φ(t)) and take time averages of the intensities (don't forget to square!), you can determine the conditions that drive many of the terms to 1/2 (which is zero fringe visibility).

 

[Philip Wood]

I'm very grateful for the time and trouble you guys have taken to discuss this. I very much like the idea of the time-averaging. I've also begun to doubt the usefulness of my model of an extended source as two separated point sources. [An analogy: we may get an estimate of coherence time when a source is not pure monochrome as the time between interference minima for two slightly separated frequencies. But we know that what we get (beats) lacks some important characteristics that we're trying to model; for example it's possible with beats to predict the phase at any time in the future, as there is a clearcut phase change of pi between one beat-package and the next!]

Yes, I too prefer H&Z to J&W. The maths is neater and the explanations more succinct. But I retain a certain fondness for J&W, as was my first optics textbook.