Interferometer Considerations: Arm Lengths and Vibration Noise

[WWCY]

Homework Statement

Hi all, I need some help regarding the following 2 problems.

1.
i) Why must we ensure that interferometer arm lengths $L_1, L_2$ are approximately equal?
ii) Is it also necessary for the distance from the beam-splitter to the laser, or the distance from the beam-splitter to the detector, to be equal to $L_1$ and $L_2$?

2. In what way is the quadrature Michelson interferometer more robust against vibration noise?

Assistance is greatly appreciated!

Attached are illustrations of the different setups.

Homework Equations

The Attempt at a Solution

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I have tried to do some reading (am not an Optics student) and I managed to come up with the following answers so far.

1i) When $L_1$ and $L_2$ are exactly equal and the mirrors perfectly perpendicular, the fringes produced are at their widest. As we start to vary $L_1$ and $L_2$ to make whatever measurements, having well-defined, wide fringes allows us (or our counting tool) to get more accurate counts. From (page 6): https://www.sheffield.ac.uk/polopoly_fs/1.14272!/file/L9.pdf

1ii) The beams reflecting from the mirror combine at the beam splitter, which is the key to obtaining fringes. This has little to do with the distance from the laser to BS and BS to detector (I think) and thus there's no need for these distances to be equal to $L_1$ and $L_2$.

2) I can't really tell how it makes the measurements more "noise-resistant" but I'm guessing it has something to do with the fact that the counts are done with a quadrature counter. I managed to get some information from TeachSpin (http://www.teachspin.com/modern-interferometry.html) that says:

"The counting electronics is arranged to make reversible (up-down) counting of fringes possible. Up-down counting can bring amazing vibration immunity to an interferometer. It is tell-tale that the apparent noise in the X(t) and Y(t) signals shown on the left, does not cause the dot in the (X,Y) display to wander about in the plane. Rather, the dot is confined to the locus shown. Hence, we can infer that the apparent noise is in fact signal, a measure of the instantaneous optical phase of the interferometer. And since the up-down counting system keeps track of the 'winding number' of that dot around the locus, it is easy to count thousands of fringes, even in the face of vibration."

However, I'm not really able to make sense of the main points.

 

[phoenix95]

1ii) The beams reflecting from the mirror combine at the beam splitter, which is the key to obtaining fringes. This has little to do with the distance from the laser to BS and BS to detector (I think) and thus there's no need for these distances to be equal to L1L1L_1 and L2L2L_2.

This part is correct.

For first question think about laser noise. By laser noise I mean all the noise that can originate from laser(example power noise). What is the effect of this at the output? Or more importantly how is this(noise appearance at detector 1) related to arm lengths? Use phasor diagrams if necessary.

For part 2, the method you listed is interesting but could be unnecessary extra machinery depending on what you use the interferometer for. I need some extra information here: what are you using it for? More specifically, are trying to measure phase or change of phase?

 

[WWCY]

Apologies for the delayed response, and thank you for your reply

For first question think about laser noise. By laser noise I mean all the noise that can originate from laser(example power noise). What is the effect of this at the output? Or more importantly how is this(noise appearance at detector 1) related to arm lengths? Use phasor diagrams if necessary.

Could you point me towards some readings for power noise? I have tried to do some reading about it but have found the concept confusing. I will attempt to answer it afterwards!

For part 2, the method you listed is interesting but could be unnecessary extra machinery depending on what you use the interferometer for. I need some extra information here: what are you using it for? More specifically, are trying to measure phase or change of phase?

The thing I'm specifically trying to measure is the number of interference fringes passing a certain point. I'm supposed to hook a quadrature counting device onto the oscilloscope producing the quadrature fringe signal. The counting device gives the number of fringes moving past.

Thank you for your assistance!