Resolving Powers

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Once upon a time (I don't know now; much has changed after the Restructuring of the Curriculum), there was what was called the 4th Year Lab in the Physics Depatment at IIT KGP. Somehow or the other from the day I joined in 1965 till I retired 40 years later, I was always a fixture there, possibly because most of the Experiments there were in Electronics and I had a passing familiarity with the subject. And so I was 'allotted' Electronics Experiments for 'grading' purposes. And later, 'Spacetime Software' and 'Path Integral Software' for reasons I shall talk in a later post.

But it also had a Michelson Interferometer and a Fabry-Perot Interferometer sharing the same Sodium Lamp on opposite sides of it. Physical Optics always fascinated me and I believe, next to Thermodynamics, it is the toughest subject, mostly due to paucity of good books.

The two instruments were ok for a beginner: Enough Resolution to carry out the basic measurements and verify what is taught in the Theory. But, since I was not grading these experiments, students largely escaped my questions in the Lab Vivas except in the End-Term Viva, where I was always kind to a fault. But, before allotting the next Experiment, if I happen to be there, I used to insist that the Michelson students show me their 'white light fringes' and the Fabry-Perot show me the well-resolved Sodium Doublet 'ring systems'. By and large they did.

In the Grand Lab Viva I used to ask for the formula for the Chromatic Resolving Power (CRP) of the FP thing which they use to reel out in terms of Finesse or Reflection Coefficient. So far so good.

I then used to ask for the formula of the CRP of the Michelson thing. They would stare at me as if I was kidding them, because there is no such thing in any book. I used to leave it at that. They used to get the doublet separation by moving one of the mirrors (Fourier Transform Spectroscopy), but that doesn't mean any CRP.

For the last 5 years towards the end of my stay there, I was teaching the Jumbo First Year Theory Course in which I used to teach these two instruments rather intensively. In the end-sem Question Papers year after year, I used to sneak in this question of deriving the CRP of the Michelson Interferometer by giving enough hints leading them step by step. After all, it also has a formula for the Intensity of the fringe pattern in terms of 'theta'.

I used to ask them to derive [(delta theta)separation] and [(delta theta)width] precisely along the lines of the FP thing, equate the two for bare resolution to get the CRP. Doing so step by step as hinted, many students used to get the 'correct' result CRP = 2m where 'm' is the usual order of the ring. And then I used to ask them to justify the result by comparing it with the Grating CRP. In a flourish they would write that it is the same as Grating thing for N = 2, where N is the number of slits, or what is the same thing, the number of interfering beams. They used to get full marks till then, because Michelson is indeed a 2-beam Interferometer.

The next bit of the Question was that the answer is wrong, and that the CRP of the Michelson thing is precisely ZERO. And ask them to justify my statement and find the flaw in their logic. All, including quite a few tutors, apparently railed at this last bit and said gps is talking through his hat (no wonder), but the report reached only RSS and then KK, but never me directly; and as a rule, I never took steps against hearsay, rumor and calumny because there is no end to such things.

But, KK, being KK who suffered me in my Lectures and Labs for two long semesters each two decades ago, pinned me down. I told him that in any 2-beam interference like the double-slit, the dark fringes are as wide as the bright fringes and a blind application of the Rayleigh Criterion gives misleading results, because when the maximum of the second line of the doublet falls on the next minimum of the first line, the total Intensity, instead of showing the Rayleigh dip of 17%, just becomes uniform and the fringe pattern washes out as is well-known.

No double-slit can resolve any doublet, in any order, period. No 2-beam Interferometer can 'resolve' any doublet and so has to use the Fourier Transform method of moving one of the mirrors (Indeed Jenkins nd White mention that Michelson deduced that a line was a 'triplet' by plotting its Visibility over Distance and getting the complicated but beautiful Fourier Plots).

Then KK asked me what could be the minimum number of slits N of the grating that starts showing Resolution as given by the famous Grating CRP formula.

I told him I was just waiting for him to answer this question. This needs plotting the patterns of 3 slits, 4 slits, 5 slits and so on till the Rayleigh dip first appears. The formulae of the Intensity are easily well-known, but the rest of the calculation involves plotting graphs and superposing them, which is too laborious unless one writes a Computer Program to get the superposed Plots.

This was just KK's cup of Tea.

He did it and showed me his extensive Plots and got the answer to my question: "The minimum number N of slits after which the Grating CRP Formula CRP = mN starts making sense".

The answer is with KK. I forgot the number, maybe 5 or 6 or so.

Anyone interested can contact him or do it themselves.

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Sorry for this Heavy Blog. But I thought I should make amends somewhat for the usual levity once in a while.

Tomorrow, gossip again!


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