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One spot on earth receives a constant stream of stellar radiation: this is the North Pole where the Pole star radiation flows constantly. Given that a point source of light is considered to emit a spatially coherent wavefront, what dimensions can we assign to the Polestar wave front as it falls on the North Pole, and at what frequencies? Issues to consider: given the curvature of the earth, as one moves away from the N pole while walking on the surface of the earth (snow, ice, water, etc), the distance from the Polestar will increase relative to the distance to the Polestar at the North Pole proper. Thus, we can expect that movement away from the N Pole will insert an out-of-phase relationship of radiation when measured simultaneously at the N Pole and at that new distance from the N Pole. At what radius distance would the phase relationship be considered "non-coherent" spatially? Also, given penetration of water by certain wavelengths, what is the additional considerations of maintenance of spatial coherence. E.g., to what degree do gamma rays maintain spatial coherence down to, say 500 feet, at a larger radius than, say blue light? What is the relationship between the radius of spatial coherence to electromagnetic wavelength at the N Pole regarding radiation from the Pole Star?

Thanks for the clarifying Q of the first responder. Yes, coherence refers to maintenance of the same phase relationship across the wave plane. Moving perpendicular to the plane even a few millimeters is equivalent to moving away from the North Pole, and thus introducing the issue of temporal coherence. So...how many feet, meters, miles does one move away from the N. Pole to introduce enough of that perpendicular movement such that temporal coherence reduces then ends? I expect that different wavelengths will have difference outcomes regarding this question. Is there a graph of "decoherence" one could produce such at at increasing wavelengths, one could move further away from the N Pole (Ie. move further away from the wavefront in that perpendicular direction) while still maintaining some semblance of phase consistency between point A (N pole) and B (some point away from the N pole)? Thanks for invitation to clarify.

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  • $\begingroup$ @JTS--hope I answered your Q enough. Quantitative answer is the goal. I'm writing sci fi novel about twins born at the N. Pole (50 feet below surface). The spatial coherence Q helps clarify how they could be psychically linked, given that their first breaths were simultaneous. (One lived off the umbilical cord while the other was coming out.) Plus they get linked to Polaris. (Radiation "sees" no time passing between emission and reception.) What is the radius of spatial coherence of various frequencies at the N. Pole. Especially those that can penetrate 50 feet of water and steel hull of a sub $\endgroup$ – John Sorflaten Apr 20 '19 at 22:38
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I answer the question on coherence without doing any calculations.

Coherence is not "having the same phase" but "having a fixed phase relationship" (more precisely, being statistically similar, but let us keep here in mind the simpler concept of having a fixed phase relationship). The radiation from the star is at the Earth to a very, very good approximation a plane wave. If you consider two different points on one wavefront of this plane wave, you will find always the same phase difference between the oscillations: zero. If you, though, move in the direction perpendicular to the wavefront (at a fixed time) will find that the field oscillates; for small distances (of a few thousands of a mm!!!), it will keep a phase that is very well correlated to the phase at the first plane. If you go a little bit farther away, the nice correlation will be lost. This happens because of the temporal coherence of the light emitted by the star, which is not affected by the distance at which you observe the star.

Probably to get clearer ideas on what happens in this particular case you could clarify your ideas on the difference between spatial and temporal coherence.

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  • $\begingroup$ Yes, coherence refers to maintenance of the same phase relationship across the wave plane. Moving perpendicular to the plane even a few millimeters is equivalent to moving away from the North Pole, and thus introducing the issue of temporal coherence. So...how many feet, meters, miles does one move away from the N. Pole to introduce enough of that perpendicular movement such that temporal coherence reduces then ends? I expect that different wavelengths will have difference outcomes regarding this question. $\endgroup$ – John Sorflaten Apr 21 '19 at 16:46
  • $\begingroup$ I think you can calculate it in this way: trace a plane perpendicular to the direction "star-North Pole"; move away from the North Pole in a given direction and calculate the distance between the position you are at and the plane you traced (if the star is in the direction of Earth's axis this measn tracing a plane tangent to the Earth's sphere at the North Pole location). I'll get back to your other question later (possibly tomorrow or the day after), and as well comment on the issue of wavelength. $\endgroup$ – JTS Apr 25 '19 at 10:17
  • $\begingroup$ I do not know anymore what I had in mind when I wrote "I'll get back to your other question later", so I write what I have in mind now :-). The coherence length of the starlight (this is the aspect temporal coherence takes if you do measurement along the propagation of light, at the same time) is a few micron. There is a theorem (Wiener-Khintchin) that implies that coherence time and spectral width are inversely proportional to each other. $\endgroup$ – JTS Apr 27 '19 at 22:54
  • $\begingroup$ From this idea of inverse proportionality you can figure out the wavelength dependence of coherence (hint #1: you have to have in mind how you measure coherence; hint #2 how would you measure the coherence for "a given wavelength"? hint #3 there is no dependence). let me know if what I am writing makes sense for you, and I will try to edit and improve my answer. $\endgroup$ – JTS Apr 27 '19 at 22:56
  • $\begingroup$ Finally the numbers. Approximating sinus and cosinus for small angles, you get that the distance from the sphere to its tangent plane is equal to $\frac{1}{2}\frac{d^2}{R}$ where $R$ is the radius of the sphere and $d$ is the distance from the point of tangency. Substituting numerical values, we get that for $d = 30 \; \mathrm{m}$ the sphere is about $10 \; \mathrm{\mu m}$ distant from the tangent plane, so farther than that and the coherence is lost (of course the decrease is gradual). $\endgroup$ – JTS Apr 27 '19 at 23:14

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