Relationship between interference and arm length in MZI

[boxfullofvacuumtubes]

A question about the Mach-Zehnder interferometer.

My understanding is: If its two arms are equally long, there is destructive interference in one output and constructive interference in the other output. So, the intensity of light detected at the first output should be 0% and at the second output 100% of the light that entered the interferometer. If one arm is longer, there is a phase shift proportional to the wavelength of the light and to the difference between the length of the first arm and the length of the second arm. The intensity of light detected at one of the outputs is proportional to 1+cos of this phase shift, while the intensity of light at the other output is proportional to 1-cos of the phase shift. For example, it can be 90% vs. 10%.

But this relationship cannot be true for any arbitrary difference in arm lengths, or can it? How much longer should one of the arms be for the interference pattern to totally disappear and be replaced with two gaussian blurs at the outputs, with an equal split of the original light intensity? Is there a formula for it? I haven’t found anything in my optics books.

 

[Charles Link]

The answer involves the coherence length of the source. A student recently posted something quite similar involving a Michelson interferomter: https://www.physicsforums.com/threa...michelson-interferometer.933638/#post-5902650 $\\$ You might also find an Insights article on interferometry of interest that I authored about a year ago: https://www.physicsforums.com/insights/fabry-perot-michelson-interferometry-fundamental-approach/ These interferometers all work with the same principles, as the article explains. Hopefully this is helpful. (And note: The concept of the interference resulting from two (coherent) beams, coming from opposite directions, incident on a single interface, is often not spelled out in the optics textbooks, but is very important in explaining the workings of the interferometer, as described in the Insights article). $\\$ Editing: One additional item: The line width $\Delta \lambda$ is related to the coherence length $L_c$: $L_c=\frac{\lambda^2}{\Delta \lambda}$. This is because $\lambda=c/f$, so that $| \Delta \lambda|=(c/f^2) |\Delta f|$. Meanwhile, coherence time $t_c=1/(\Delta f)$, and coherence length $L_c=c \, t_c$. You can check my algebra, but I think I got it right.

 

[boxfullofvacuumtubes]

Thank you, Charles! Coherence length is exactly what I was looking for. I just wish Δλ were easier to estimate.

 

[Charles Link]

Thank you, Charles! Coherence length is exactly what I was looking for. I just wish Δλ were easier to estimate.

Also, you might find it of interest=perhaps you know this already=if you observe what happens if you block one of the two beams of the interferometer when the beams are being recombined by the beamsplitter. You simply get a 50-50 energy split by the beamsplitter on the single beam. When you allow both beams to be incident simultaneously, instead of both beams now getting split 50-50, the energy gets selectively channeled in one direction or the other (depending on the relative phases of the beams). The Insights article describes this in detail, and the Fresnel coefficients $\rho=\pm \frac{1}{\sqrt{2}}$ and $\tau=\frac{1}{\sqrt{2}}$ still apply with linear principles, but not the energy $R=1/2$ and $T=1/2$ coefficients. This is the subject of the Insights article=perhaps you already know the details, but otherwise, you might find it good reading. :) $\\$ (The Maxwell equations are linear in the electric field, so the electric field must obey linear principles, but the same is not true of the energy equations which are to the second power of the electric field. (Intensity $I=nE^2$). Thereby the energy is not required to obey linear principles. Energy is conserved, but it doesn't get the 50-50 split with two beams that linear results would yield).

 

[Charles Link]

@boxfullofvacuumtubes And a follow-on: The fundamentals of interferometry, as presented in the article https://www.physicsforums.com/insights/fabry-perot-michelson-interferometry-fundamental-approach/ , is something that was not presented in the Optics books of my generation (my college days were 1974-1980). They did teach the Fresnel coefficients, but I never saw a textbook that presented the calculation of the interference of two beams incident on a single interface from opposite directions. $\\$ The Fabry-Perot interference was always presented as a dielectric slab with multiple reflections, and the Michelson interferometer was presented as two separate virtual images/sources (from the reflections) that were interfering with each other. In hindsight, it is quite simple, but it wasn't until 2009, that I figured out that both the Fabry-Perot interference and Michelson type interference are a result of the interference from two beams incident on a single interface from opposite directions. Anyway, I welcome your feedback on the article. $\\$ Meanwhile, for spectral line widths, the coherence length that results with an interferometer is probably one of the better ways to measure a narrow spectral line width. Even a high resolution diffraction grating spectrometer is not going to be able to measure the spectral line width of most atomic transitions.